Kévin

The Advisory Council for Aeronautics Research in Europe (ACARE) has identified new incremental regulations required for the aviation industry for 2020, 2030 and 2050 to reduce CO2 emissions per passenger kilometer by 75% and NOx by 90%, compared to aircraft emissions in 2000. As the world' s air traffic is also expected to grow at an average rate of about 5% per year over the next 20 years, environmental sustainability issues (reduction of fuel burn and pollutant emissions) must be seriously considered to ensure the long-term survival of the aviation industry sector Je tiens à remercier les tous les membres de de mon jury, mes rapporteurs, Bénédicte CUENOT et Marc BELLENOUE pour la relecture minutieuse de ce manuscrit, les examinateurs Fabrice FOUCHET, Bruno RENOU, Stéfan ZURBACH et Stéphane RICHARD pour les échanges éclairés qui ont suivi ma défense, et enfin mes directeurs de thèse Frédéric GRISCH et Gilles CABOT, pour leurs encadrement tout le long de ce projet.

Je remercie l'ANR et Safran Tech pour le financement de cette thèse. Et tout particulièrement Stéfano PUGGELLI pour le suivi et l'accompagnement de ces travaux.

Je tiens également à remercier chaleureusement Vincent, Ghislain et Pierre, pour la chance qu'ils m'ont offerte de travailler sur Yales2 et leurs suivi riche d'enseignement, m'ouvrant la voie vers le monde de la simulation numérique.

Bien évidemment je remercie tous les membres de l'atelier du CORIA pour le travail titanesque qu'ils ont abattu durant ce projet pour la préparation des bancs, Benjamin, Phillipe, Thierry, Romain & Romain, et bien sûr Bruno soudeur hors pair et senseï des raccords.

L'équipe métrologie a également joué un rôle extrêmement important pour ces travaux. Je remercie donc Alexis, Carole et Frank, et bien entendu Gilles, avec qui j'ai passé des séances de réglage laser mémorables.

Enfin, je remercie chaleureusement Marcos, pour sa bienveillance infaillible, sa compétence, et sa patience exceptionnelle lorsqu'il pouvait m'arriver d'en manquer.

Cette thèse se serait déroulée sous un jour bien différent sans toutes les personnes géniales que j'ai pu rencontrer, et il me tient donc à coeur de tenter une énumération homérique. Les doctorants tout d'abord, et en premier lieux la team Grisch : Sylvain et son laser démoniaque, Maxime dit le paisible (ou le crabe violoniste selon l'étage), Aurélien, patron des barbecues, Eliot et son lance flamme, et Andreï, dit le jeune.

It is within this context that the PERCEVAL Industrial Chair was created, with the objective of providing state-of-the-art knowledge on the performances of innovative combustor and fuel injection systems, and developing pioneering optical diagnostics to extend the knowledge on the many interconnected physical processes involved. As such, the scientific focus of the current study aims at optimizing combustion efficiency and pollutant reduction in lean and staged combustors. To this end, an extensive experimental database was built around three fuel injection systems that were selected to measure the impact of specific technological modifications on their performances.

The full characterization of those injection systems were carried out in high pressure reactive conditions with Jet A1 in the HERON test rig. The aerodynamical field has been studied with high speed Particle Image Velocimetry. Then, the liquid phase was characterized with high speed tomography and Phase Dopler Anemotry, which has been thoroughtly detailed to ensure the use of an adequate methodology in such conditions. The gaseous fuel phase was studied with kerosene Planar Laser Induced Fluorescence (PLIF), which has been coupled with OH-PLIF and quantitative NO-PLIF so that the mixing process, flame topology and pollutant emission could be observed simultaneously. Finally, pollutant emissions were measured with a gas exhaust analyser.

Résumé

Le Conseil consultatif pour la recherche aéronautique en Europe (ACARE) a identifié les nouvelles réglementations progressives requises pour l'industrie de l'aviation pour 2020, 2030 et 2050 afin de réduire les émissions de CO2 par passager-kilomètre de 75% et les NOx de 90%, par rapport aux émissions des avions en 2000. Comme le trafic aérien mondial devrait également croître à un taux moyen d'environ 5% par an au cours des 20 prochaines années, les questions de durabilité environnementale, telles que la réduction de la consommation de carburant et des émissions polluantes, doivent être sérieusement prises en compte pour assurer la survie à long terme du secteur de l'aviation.

C'est dans ce contexte que la Chaire industrielle PERCEVAL a été créée, avec pour objectif de fournir des connaissances de pointe sur les performances des systèmes innovants de combustion et d'injection de carburant, et de développer des diagnostics optiques pionniers pour étendre les connaissances sur les nombreux processus physiques interconnectés impliqués. Ainsi, l'objectif scientifique de la présente étude est d'optimiser l'efficacité de la combustion et la réduction des polluants dans les chambres de combustion pauvres et étagées. A cette fin, une base de données expérimentale étendue a été construite autour de trois systèmes d'injection de carburant qui ont été sélectionnés pour mesurer l'impact de modifications technologiques spécifiques sur leurs performances.

La caractérisation complète de ces systèmes d'injection a été réalisée dans des conditions réactives à haute pression avec du Jet A1 dans le banc d'essai HERON. Le champ aérodynamique a été étudié par vélocimétrie par image de particules à haute cadence. Ensuite, la phase liquide a été caractérisée par tomographie à haute cadence et anémométrie de phase Dopler, qui a été minutieusement détaillée pour assurer l'utilisation d'une méthodologie adéquate dans de telles conditions. La phase gazeuse du combustible a été étudiée à l'aide de la fluorescence planaire induite par laser (PLIF) du kérosène, qui a été couplée à la OH-PLIF et à la NO-PLIF quantitative, de sorte que le processus de mélange, la topologie de la flamme et l'émission de polluants puissent être observés simultanément. Enfin, les émissions de polluants ont été mesurées avec un analyseur de gaz d'échappement.

Vient le tour maintenant des colloc', avec qui j'ai été heureux de partager mon toit et mes tacos ! Maxime bien entendu, fidèle pratiquant du Light Weight, du wakeboard, et bien sûr du 6 / 4 / +2 / 6 / 1. Edouard et Sofia, le jardinier et le diam's, et Julien, dit la légende. A ces premiers compagnons viennent s'ajouter Elodie et Chloé, pour ainsi former cette exceptionnelle colloc' républicaine, que je ne saurais oublier. Je remercie également chaleureusement la deuxième génération, qui m'a connu en rédaction, et donc bien plus bougon : Martin, dit le compotier, Quentin, dévoreur de papier, et Inès, la jeune Padawan. Enfin une pensée pour mes collocs pallois, adeptes du twerk, et de la patisserie ! Je remercie aussi mes aventuriers de Golarion, pour tous ces moments grotesques et hilarants ! Lucas, dit Lion, Thibaud, le dénommé Servant, Max a.k.a. Kraq, et Thomas mieux connu sous le nom d'Edmoïne. Puisse un jour revenir ces histoires dont l'absurdité de certains chapitres pouvaient rivaliser avec ceux de la grande tortue cosmique.

Mon épitrochlée droite aurait protesté si elle avait pu, mais je remercie maintenant les membres du RAC, Valentin et Hugo pour ces séances d'entrainements titanesques ponctuées d'Ajax et autre Kiai douteux. Puisse un jour le marteau arriver dans nos mains ! Il me reste à présent à remercier Félix, Yann, et Louise, dont l'importance dans cette aventure ne saurait se décrire en quelques lignes. Je m'efforcerais donc de n'utiliser qu'UN terme : systématiquement débile, toujours inattendu, et à jamais dans mes mémoires.

« There He aimed and stabbed, he gouged him down his glistening flesh and wrenched the spear back out. The brazen god of war let loose a shriek, roaring, thundering loud as nine, ten thousand combat soldiers »

To the stories that stayed with you That meant something Even if you were too small to understand why C2 -Confidential In August 2021, a major report was published by the Intergovernmental Panel on Climate Change (IPCC), drawing important conclusions about climate change and its future impact. According to this report, scientists are observing changes in the Earth's climate in all regions and across the climate system. Many of the observed changes in climate are unprecedented in thousands or even hundreds of thousands of years, and some of the changes already underway, such as continued sea level rise, are irreversible over hundreds or thousands of years.

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Table of contents

Global warming is defined by the IPCC as a combined increase in air and sea surface temperatures averaged over the globe over a 30-year period. The warming is expressed relative to the period 1850-1900, which is used as a rough approximation of pre-industrial temperatures. Human-induced warming reached about 1°C above pre-industrial levels in 2017, increasing by 0.2°C per decade. Since 2000, the estimated level of human-induced warming is equal to the observed level of warming, with a likely range of ±20%, considering the uncertainty in the C2 -Confidential contributions of solar and volcanic activity over the historical period. Warming above the global average has already been observed in many regions and seasons, with higher average warming over land than over the oceans. Most land areas are warming above the global average, while most ocean areas are warming at a slower rate. Depending on the temperature dataset considered, 20-40% of the world's human population lives in regions that have already warmed more than 1.5°C above pre-industrial levels for at least one season during the decade 2006-2015.

The IPCC has proposed a number of global warming scenarios with their realization conditions and consequences at a multitude of scales. The 1.5°C emission scenarios have been defined by the IPCC as those that, based on a current knowledge of climate response, provide a 33-50% probability of warming remaining below 1.5°C or returning to 1.5°C by 2100 following an overshoot. All 1.5°C trajectories involve limiting cumulative emissions of long-lived greenhouse gases, notably carbon dioxide and nitrous oxide, as well as substantial reductions in other climate forcing factors. Limiting cumulative emissions requires either a reduction in net global emissions of long-lived greenhouse gases to zero before the cumulative limit is reached, or negative net global emissions after the limit is exceeded. The real impact of each 1.5°C trajectory depends on its trajectory and in particular on the overshoot. In any case, the global transformation that would be required to limit warming to 1.5°C needs to consider a wide range of physical and other phenomena: geophysical, environmental-ecological, technological, economic, socio-cultural and institutional. In other words, to study the impact of the humanity on the earth system as a whole.

Assuming emissions are in accordance with current commitments under the Paris Agreement, global warming is projected to exceed 1.5°C above pre-industrial levels, even if these commitments are accompanied by a very ambitious increase in the scale and ambition of mitigation action after 2030. This voluntary action should achieve net zero CO2 emissions in less than 15 years. Even if that is achieved, temperatures should only be maintained below the 1.5°C threshold if the actual geophysical response falls within the lower part of the currently estimated uncertainty range.

Trajectories consistent with a warming of 1.5°C above pre-industrial levels can be identified under a range of assumptions about economic growth, technological developments and lifestyles. However, the lack of global cooperation, the absence of governance of the required energy and land transformation, and the increase in resource-intensive consumption are key impediments to reaching the 1.5°C trajectories. Assuming emissions are in line with the current commitments under the Paris Agreement, global warming is projected to exceed 1.5°C above pre-industrial levels, even if these commitments are complemented by a very aggressive increase in the scale and ambition of mitigation actions after 2030.

Limiting warming to 1.5°C depends on greenhouse gas (GHG) emissions over the coming decades. Indeed, lower GHG emissions in 2030 increase the probability of limiting the warming peak to 1.5°C. The available directions that aim for no or limited excess of 1.5°C limit GHG emissions in 2030 to 25-30 GtCO2/year in 2030. This contrasts with median estimates of the current unconditional national contributions, which are 52-58 GtCO2/year in 2030. Trajectories to limit warming to 1.5°C by 2100 after a temporary temperature overshoot rely on large-scale deployment of carbon dioxide removal measures, which are uncertain and bring obvious risks. In modelled trajectories with no or limited 1.5°C overshoot, global net anthropogenic CO2 C2 -Confidential emissions decline by about 45% from 2010 levels by 2030, reaching net zero by 2050. To limit global warming to below 2°C with at least a 66% probability, CO2 emissions are projected to decline by about 25% by 2030 in most pathways and reach net zero around 2070.

These mitigation pathways are defined by reductions in energy demand, decarbonization of electricity and other fuels, electrification of energy end-use, deep reductions in agricultural emissions, and some type of carbon storage on land or sequestration in geological reservoirs. Low energy demand and low demand for GHG-intensive consumer goods facilitate limiting warming to as close as possible to 1.5°C. Compared to a 2°C limit, the transformations required to limit warming to 1.5°C, are qualitatively similar, but are more pronounced and faster over the next few decades. Limiting to 1.5°C requires very ambitious and cooperative policy environments at the international level that transform both supply and demand. Specifically, the average annual incremental energy-related investments for the period 2016 to 2050 in trajectories limiting warming to 1.5°C compared to trajectories with no new climate policies beyond those in place today are projected to be around USD 830 billion2010 (range USD 150 billion to USD 1700 billion2010 over six models).

With a global warming of 1.5°C, heat waves, longer warm seasons and shorter cold seasons will become more frequent. With a warming of 2°C, heat extremes will reach critical tolerance thresholds for agriculture and health more frequently. But it is not only about temperature. Climate change is driving multiple changes in many regions, which will intensify as the temperature rises. These include modifications in humidity and dryness, winds, snow and ice, coastal areas and oceans. Examples include:

• Climate change is intensifying the water cycle. This brings more intense rainfall and associated flooding, as well as more severe drought in many areas.

• Climate change is affecting the precipitation pattern. In high latitudes, precipitation is expected to increase, while it is expected to decrease in large parts of the subtropics. Changes in monsoon precipitation are expected, which will vary by region.

• Coastal areas will face continued sea-level rise throughout the 21st century, contributing to more frequent and severe coastal flooding in low-lying areas and coastal erosion. Extreme sea level events that used to occur once a century could occur every year by the end of the century.

• Further warming will amplify the permafrost thaw, as well as the loss of seasonal snow cover, the melting of glaciers and ice caps, and the loss of Arctic sea ice in summer.

• Changes in the ocean, including warming, more frequent marine heat waves, ocean acidification and reduced oxygen levels have been clearly associated with human intervention. These changes affect both ocean ecosystems and, therefore, humanity, which depend on them, and will continue for at least the rest of this century.

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• For cities, it is possible that some aspects of climate change will be amplified, including heat (since urban areas are generally warmer than their surroundings), flooding from heavy precipitation, and sea level rise in coastal cities.

NOx regulations

While the world is attempting to put in place GHG regulations to reduce global warming, the significance of reducing nitrogen oxides was initiated several decades ago and has been successful. In 1985, Farman et al [START_REF] Farman | Large losses of total ozone in Antartica reveal seasonal CLOx/NOx interaction[END_REF] warned about the degradation of the ozone layer over Antarctica, showing that a large amount of ozone was gradually disappearing. More specifically, they also showed that chloride oxides and nitrous oxides play an important role in the ozone depletion. The loss of the ozone layer would be dramatic, as it absorbs ultraviolet radiation between 200 and 320nm emitted by the sun, which is harmful to most living things, causing sunburns and skin cancers. As a result, several chemical products were banned, such as chlorofluorocarbons, and regulations on NOx production were put in place, such as the Clean Air Act Amendments in 1990. These regulations put intense pressure on industries to develop solutions to reduce the production of ozone-depleting substances.

But while the ozone depletion has been stopped, nitrogen oxides have shown other harmful effects. The first regulation on NOx emissions was in fact set in 1967 in California on the automotive sector. Because of its geological layout and special climate, California suffered from smog clouds as early as 1943. This was first attributed to heavy industry, which was moved out of the cities, but the smog problems continued, causing eye and lung burns and nausea. It took almost two decades for scientists to establish a connection between smog and nitrogen oxides in cities and to show that they were mainly produced by the internal combustion engines. This led to the Clean Air Act, which regulates car emissions.

Clean combustion for aircrafts: the PERCEVAL Chair

The Advisory Council for Aeronautics Research in Europe (ACARE) has identified new incremental regulations required for the aviation industry for 2020, 2030 and 2050 to reduce CO2 emissions per passenger kilometer by 75% and NOx by 90%, compared to aircraft emissions in 2000. As the world' s air traffic is also expected to grow at an average rate of about 5% per year over the next 20 years, environmental sustainability issues (reduction of fuel burn and pollutant emissions) must be seriously considered to ensure the long-term survival of the aviation industry sector. Fuel consumption reduction can be achieved by various methods, such as increasing the overhaul pressure ratio, increasing the high-pressure inlet temperature, increasing the bypass ratio, and improving the efficiency of the engine components. Achieving the ACARE objectives will only be achieved by pushing these modern combustion technologies to their limits in terms of their ability to minimize pollutant emissions and maintain engine operability (combustion efficiency in part-power operation, ignition and light around, high altitude restart and pull-away capability, combustion dynamics in terms of noise and combustion instabilities). It is within this context that the PERCEVAL Industrial Chair was created, with the objective of providing state-of-the-art knowledge on the performances of innovative combustor and fuel injection systems, and developing pioneering optical diagnostics to extend the knowledge on the many interconnected physical processes involved. As such, the scientific focus C2 -Confidential of the current study aims at optimizing combustion efficiency and pollutant reduction in lean and staged combustors. To this end, an extensive experimental database was built around three fuel injection systems that were selected to measure the impact of specific technological modifications on their performances.

As NOx emissions depend on many variables, the following section is devoted to the description of their production pathways. Examples of combustor architectures that have already been developed with the aim of reducing NOx production are then presented. Finally, the fuel injection systems used in this study will be detailed.

NOx production mechanisms

The combustor is the main engine component required to convert the chemical energy contained in the liquid kerosene fuel into thermal energy by combustion with air. It is also the component with the greatest potential to significantly reduce the pollutants emitted by the aircraft. To achieve this, the combustion must produce the lowest emissions of environmentally undesirable chemicals, but without generating a large pressure drop. The initial effort to develop lean-burn technologies was initiated in the 1980s when NASA's High Speed Civil Transport (HSCT) research program was confronted with the problem of ozone depletion by chemical processes involving NOx [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF]. This led to political demands on engine manufacturers to develop low-emission combustors that could reduce pollutant emissions, especially NOx. Today, increasingly stringent emission reduction targets for commercial aviation are leading to increasing efforts, particularly in the European Union, to develop lean burn technologies for aero-engines [START_REF][END_REF]4]. The development of clean and efficient combustion technology is a key motivation as emissions are mainly driven by combustion.

Historically, stoichiometric combustion was aimed at providing maximum power to the engine. Fuel and oxidant were injected separately into the combustion chamber to avoid flashback. Thus, combustion was supposed to occur at the fuel/oxidant interface within a diffusion flame. However, the resulting excessive emissions of CO, unburnt hydrocarbons and nitrogen oxides (NOx) led to the design of low-temperature lean-burn engines. Lean combustion reduces NOx emissions through lower flame temperatures and CO2 emissions through improved thermal efficiency [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF]. As can be seen in Fig. 1, according to Seiffert and Walzer [5], when there is an excess of air, the area where NOx formation rates increase can be avoided, altogether with carbon hydroxide suppression. On the other hand, an equivalence ratio that is too low leads to an increase in carbon monoxide production and torque fluctuations.

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The control of combustion-related NOx requires an understanding of the chemical mechanisms involved in its formation. As NO is the main component of NOx emitted during combustion, this section focuses solely on the NO chemical mechanisms. NO is produced via various pathways, including fuel, thermal, prompt and nitrous oxide mechanisms. The fuel-NO mechanism is associated with nitrogen-bound fuels. It can be assumed that light distillates such as aviation fuels, with a nitrogen content of up to 0.06%, give negligible levels of fuel NO in a gas turbine engine. While fuel-NO can be largely neglected for aviation gas turbines, the thermal, prompt and nitrous oxide mechanisms are important in NO formation. The extent to which the different mechanisms contribute to the overall formation of NO can be estimated from a table presented by Bowman [START_REF] Bowman | Control of combustion-generated nitrogen oxide emissions: Technology driven by regulation[END_REF] which summarizes the NO values obtained from various references. This table helps to show the conditions that favor a particular mechanism, for example, the thermal mechanism dominating at conditions close to stoichiometry (ϕ = 1), the prompt mechanism occurring under fuel-rich conditions, the N2O mechanism occurring under fuel-rich conditions. The following sections discuss these NO mechanisms in more detail.

Thermal Mechanism

The thermal-NO mechanism becomes predominant in chemical reactions that attain temperatures around and above 1800 K [7]. Thermal-NO contributes to a large portion of the overall NO levels in most conventional gas turbine combustors, where temperatures greater than 1800K are likely to occur. The following reactions comprise the thermal-NO mechanism of Zel'dovich [START_REF] Zel | The Oxidation of Nitrogen in Combustion Explosions[END_REF] :

N 2 + O ↔ NO + N (1) N + O 2 ↔ NO + O (2)
with an additional reaction for the extended Zel'dovich mechanism:

N + OH ↔ NO + H

(3) Fig. 1 : Pollutant formation, specific fuel consumption and torque fluctuations in function of equivalence ratio. From Seiffert and Walzer [5] C2 -Confidential which becomes important in fuel-rich mixtures when O and O2 concentrations are low. The first reaction in the extended Zel'dovich mechanism (Eq. 1) is the rate-limiting step. A high activation energy is then required to break the triple bond in the N2 species. The rate of formation of NO via the Zel'dovich mechanism can be obtained using chemical kinetics, combined with steady-state assumptions for the N-atom concentration and a partial equilibrium assumption for the O-atom concentration. The resultant equation for the rate of NO formation shows an exponential dependence on the combustion gas temperature, as well as a weak dependence on the ambient pressure and on the concentration of O2 in the reacting flow [START_REF] Bowman | Control of combustion-generated nitrogen oxide emissions: Technology driven by regulation[END_REF]. For conventional gas turbine combustors operating with a primary zone with an equivalence ratio near 0.8, the thermal mechanism dominates NO formation. The corresponding plot of the NO concentration formed under the same conditions shows that the lower NO concentrations occur at fuel-lean (ϕ<≈ 0.6) and fuel-rich (typically ϕ>≈ 1.7, but can occurs for ϕ > 1.2), see Fig. 1. Note that the peak temperature occurs near the stoichiometric ϕ = 1 condition while the peak NO concentration is shifted left of the stoichiometric condition, toward the fuel-lean side. Note also that the presence of super-equilibrium O atom and OH concentrations in flames at lean and near-stoichiometric conditions leads to the increased levels of NO formation. Fig. 1 also suggests that the combustor should be operated at very fuel-lean and/or very fuel-rich conditions to avoid high thermal NO production rates. Note that the combustion of fuel-air mixtures at these conditions do not necessarily produce lower NO levels, as any spatial or temporal unmixedness leading to the formation of stoichiometric fuel packets increases NO production. Hence, the goal of thermal NO control heavily depends on ensuring a fuel-air mixture that is entirely uniform in fuel-rich and/or fuel-lean composition, which leads to the two main strategies of combustor designs: the RQL (Rich-burn Quick-mix Lean-burn) and Lean combustors respectively.

Prompt Mechanism

The prompt mechanism typically refers to the Fenimore [START_REF] Fenimore | Formation of nitric oxide in premixed hydrocarbon flames[END_REF] mechanism, and denotes NO formed at rates that are much faster than that achieved for the thermal mechanism. According to the classical prompt mechanism of Fenimore, NO is formed during the initial stages of combustion when hydrocarbon radicals or fuel fragments attack nitrogen molecules in the atmosphere. The following set of reactions describe the Fenimore prompt mechanism:

N 2 + CH ↔ HCN + N (4) N 2 + 𝐶 2 ↔ 2CN (5)
The nitrogen atom that is liberated in reaction (4) participates in NO formation via reactions [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF] and (3) from the Zel'dovich thermal mechanism. In addition, NO is also produced by the reaction of O and O2 with the CN molecule formed in reaction (5). Although the Fenimore prompt mechanism primarily dominates NO production under fuel-rich conditions, this mechanism cannot be neglected in lean-premixed combustion [START_REF] Nicol | The importance of the nitrous oxide pathway to NOx in lean-premixed combustion[END_REF].

Nitrous Oxide Pathway

The contribution of the thermal NO mechanism to the overall NO production diminishes as lower temperatures are attained in lean-premixed combustion. At the same time, the decrease in reaction temperature, coupled with a low overall NO formation rate, increases the importance of NO formation via the N2O pathway in lean-premixed systems [START_REF] Nicol | The importance of the nitrous oxide pathway to NOx in lean-premixed combustion[END_REF]. For reaction temperatures C2 -Confidential less than 1500 K, Malte and Pratt [START_REF] Malte | Combustion Science and Technology The Role of Energy-Releasing Kinetics in NO x Formation : Fuel-Lean , Jet-Stirred CO-Air Combustion The Role of Energy-Releasing Kinetics in NO : c Formation[END_REF] proposed that the nitrous oxide (N2O) pathway primarily contributes to the production of NO. N2O that is formed by the reaction :

N 2 + O + M ↔ N 2 O + M (6)
subsequently participates in the following set of reactions that, in part, produces NO:

N 2 O + O ↔ NO + NO (7) N 2 O + O ↔ N 2 + O 2 (8) N 2 O + H ↔ N 2 + OH (9)
Nicol et al. [START_REF] Nicol | The importance of the nitrous oxide pathway to NOx in lean-premixed combustion[END_REF] compared the contribution of the thermal, prompt, and nitrous oxide pathways to the production of NOx in lean-premixed combustion in gas turbine conditions. The study showed that each pathway could not be rejected when assessing their contribution to the overall NOx production levels. The nitrous oxide pathway, in particular, contributed to as much as 40 -45% of the overall NOx emissions at a pressure of 30 bar, and between 20 -35% of the NOx levels at a pressure of 10 bar. The nitrous oxide pathway was also responsible for producing up to 100% of the NOx formed for concentration levels below 10 ppmv (parts per million by volume, on a dry, 15% O2 basis). In relation to the fluid mechanics of the lean premixed combustion process, Nicol et al. recommended that the combustion flame zone should be minimized in order to decrease the exposure to the free radical concentration pools that lead to NOx production.

Pressure effect on NOx formation

Effect of pressure on NOx formation is complex to determine. Unfortunately, experimental studies carried out up to now are not clarifying, because of the difficulty of performing such experimental tests in high-pressure conditions. Some authors observed a pressure dependence of NOx production, following 𝑁𝑂 𝑥 ~ 𝑃 𝛼 with α = 0.4-0.8 [START_REF] Fenimore | Formation of nitric oxide in premixed hydrocarbon flames[END_REF][START_REF] Rink | Influence of fuel composition and spray characteristics on nitric oxide formation[END_REF] while others concluded that NOx is not dependent on pressure [START_REF] Heberling | Prompt NO measurements in high pressure[END_REF]. Later, some authors [START_REF] Maughan | A dry low NOx combustor for the MS3002 regenerative gas turbine[END_REF][START_REF] Sanjay | A Review of NO x Formation Under Gas-Turbine Combustion Conditions[END_REF] concluded that the lowest NOx levels result from the nitrous oxide and prompt mechanisms, that are dominant at low temperature and which do not depend on pressure. Concerning the higher NOx levels, they would be associated with the higher combustion temperature and are due to the thermal NOx mechanism, which depends on the square root of pressure.

Effect of atomization on NOx formation

Rink and Lefebvre [START_REF] Rink | Influence of fuel composition and spray characteristics on nitric oxide formation[END_REF] studied the effect of inlet temperature, combustion pressure conditions and spray characteristics in the NOx formation. Among other parameters, they tested three different mean droplet sizes, from 30 µm to 110 µm. They demonstrated that NOx emissions decreased with a better atomization process. This effect was attributed to the fact that, firstly, with a better atomization, the evaporation rates were higher, improving fuel-air mixing before burning. And secondly to the fact that with a finer atomization, the proportion of droplets interacting with the flame front is reduced, promoting a full premixed combustion rather than a partial premix with local diffusion flames.

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The authors pointed out that the manner and extent to which NOx are impacted by the sizes of the fuel droplets into the spray, is very dependent on equivalence ratio. The results presented in Fig. 2 were obtained using a flow combustor in which the mean drop size could be varied and controlled independently of other operating variables. It shows that NO emissions increase with a growth in mean drop size, but also that this effect is especially important at low equivalence ratios. At the high-pressure conditions used in the current study with preheated air (P3=12.7 bar and T3=573 K), evaporation rates are so fast that even for the larger droplets, the time required for their evaporation is small in comparison with their total residence time into the combustion zone. Yet, an increase in SMD means that a larger proportion of the total number of fuel droplets into the spray is capable of supporting "envelope" flames. These envelope flames, which surround the larger drops, burn in a diffusion mode at near-stoichiometric fuel/air ratios, giving rise to many local regions of high temperature in which NOx is formed in high quantities, as it was presented in Fig. 1. Reduction in mean drop size impedes the formation of envelope flames, so that a larger proportion of the total combustion process occurs in what is essentially a premixed mode, thereby generating less NOx. Although, in theory, combustion within these localized regions can take place at any equivalence ratio within the flammability limits, it tends to occur preferentially at the stoichiometric value, i.e., at the maximum flame temperature, thereby producing high-levels of NOx. This hypothesis explains why NOx emissions increase with SMD for lean mixtures. However, as the overall equivalence ratio increases toward stoichiometry [START_REF] Fenimore | Formation of nitric oxide in premixed hydrocarbon flames[END_REF], the local fuel/air ratio adjacent to the fuel droplets approaches the premixed value. According to this hypothesis, mean drop size should have no effect on NOx emissions for stoichiometric mixtures, and this is generally confirmed by the results shown in Fig. 2. This figure is important because it demonstrates that even at low equivalence ratios, where the average combustion temperature is so low that only negligible amounts of NO should, in theory, be formed, the presence of fuel droplets in the combustion zone gives rise to conditions in which combustion can and does proceed at near-stoichiometric equivalence ratios, regardless of the average equivalence ratio in the combustion zone. This is of great importance for lean-premixed combustion, as it will greatly impact the NOx production regardless of the global equivalence ratio.

Fig. 2 : From [START_REF] Rink | Influence of fuel composition and spray characteristics on nitric oxide formation[END_REF], EINOx in function of equivalence ratio for difference SMD. In lean conditions, the impact of the mean droplet size is much more significant than near the stoechiometry.

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Injection systems for a cleaner combustion 1.3.1 Atomizers for aircraft combustors

Before going any further in the history of injection system development, a handful of information will be given beforehand on the three main atomizers that were, and are still used in the architecture of an injection system. The term atomizers will be used to describe an atomization mean, such as the ones described in this small section, while injection system will denote an arrangement of atomizers and air swirlers. Three atomizers are commonly used in aeronautical injection systems: the pressure swirl, the film airblast and the jet in crossflow. The pressure swirl injection is based on the pressure drop between the injection line and the combustion chamber to create a fine spray. The main advantage is that it can work without any airflow and is thus usually used for ignition or idle regimes. The main drawback is the need of a hydraulic pump to pressurize the fuel line. Few designs are presented in Fig. .a), from Lefebvre's book [START_REF] Lefebvre | Atomization and Sprays[END_REF]. All the designs share the same principle: putting the liquid fuel in a rotating motion within a small chamber, and injecting it through a small orifice. The swirling motion creates a depression at the center of the spray that can enter the injection hole, creating an air cone inside the injection orifice, and thus an aerodynamical blockage. The effective injection surface found itself reduced which accelerates even further the liquid, helping its atomization. Airblast atomizers, as their names suggest, use high velocity air flow to explode a liquid sheet or jet. Within this work, the term film airblast will be used to describe an airblast atomization of a liquid film, whether it was created by the projection of liquid onto a wall or an annular canal injecting the fuel into an airflow. The term jet in crossflow will be used to denote the injection of fuel with circular or conics holes into an airflow. Two examples of airblast atomizers are proposed in Fig. 4, again from Lefebvre's book. In both cases, a primary air swirling flow starts the atomization process and pushes the liquid outwards, the secondary swirling flow, usually directed toward the centerline, completes the atomization process. A schematic view [START_REF] Lefebvre | Atomization and Sprays[END_REF] and a LES simulation from [START_REF] Leparoux | Primary atomization simulation applied to a jet in crossflow aeronautical injector with dynamic mesh adaptation Primary atomization simulation applied to a jet in crossflow aeronautical injector with dynamic mesh adaptation[END_REF] of jet in crossflows are shown in Fig. 5. The smallest droplets are located near the injection wall, while the largest penetrate the deepest. The penetration length is a key parameter to control fuel position and thus the flame positioning within the combustion chamber. It has been shown to depend on multiple parameters, but notably on the ratio of momentum between air and injected fuel, and the hole shape and size. C2 -Confidential

Rich combustor: the RQL strategy

From the conclusions drawn on NOx production mechanisms, two key effects arise to minimize NOx production: reducing the temperature of both the reactive zone and the burned gases, and controlling the fuel-air mixing to avoid localized stoichiometric combustion. Two main strategies have been commonly investigated: the RQL and the lean combustion.

The RQL technology exploits a clear rich primary zone, that has the advantage of being stable and resistant to extinction, followed by a strong dilution to rapidly bring down the equivalence ratio in the lean combustion region. Two main challenges arise from this design. Firstly, the control of the dilution is of paramount importance, as the quenching process has to be as fast as possible to avoid any reaction zone in the stoichiometric region. Otherwise, an important part of the fuel can be burned at the stoichiometric ratio, and create a large amount of NOx.

Secondly, the volume of the primary zone should be minimized, as well as the diameter of the fuel droplets, as a rich mixture is prone to the formation of carbon hydroxide, carbon monoxide and soot. Yet, the primary zone must be large enough to ensure proper mixing prior to the dilution. As an example of an RQL combustor, a cross section of the Rolls Royce phase 5 combustor is shown in Fig. 6.a)adapted from McGuirk and al. [START_REF] Mcguirk | The aerodynamic challenges of aeroengine gas-turbine combustion systems[END_REF]. The primary zone uses only 30% of the mass air flow to keep an equivalence ratio way above the stoichiometry. The majority of the air flow is used to cool down the rich mixture and is injected through the primary and dilution holes (see the blue arrows entering the combustion chamber). The rest of the combustion process occurs in the lean regime. Very similar combustors have been developed and used by both General Electric and Pratt & Whitney [START_REF] Mongia | GE aviation low emissions combustion technology evolution[END_REF][START_REF] Mckinney | The Pratt & Whitney TALON X low emissions combustor: Revolutionary results with evolutionary technology[END_REF]. The Fig. 6.b) shows a cross section of the GE "LEC" RQL combustor. The design is very close to the Rolls Royce Phase 5 RQL combustor, but this cross section gives an insight of the injection system used. It consists of a pressure injector composed of multiple injection points, with a surrounding swirled air flow. The fuel is projected onto a bowl to create a film, that is blasted by a second swirled air flow. A close injector design has been used by Pratt & Whiney for their TALON RQL combustor, and is reported on Fig. 6.c). The major difference in the injector design being the length of the bowl that goes up to the combustion chamber, so that there is no mixing of the fuel with the second swirled air flow within the injector itself.

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On those three RQL engines, the primary and dilution holes are separated, the primary ones being the closest to the injector. As shown previously, the idea behind an RQL design is to quench as fast as possible the rich mixture to avoid the stoichiometric region, so coincident dilution and primary holes would be preferable. Yet, those combustors used separated dilution holes. This is because the quenching can have a negative effect on the combustor relight capability, as it can light off the flame kernel that tries to propagate through the combustion chamber. To counter this effect, one could increase the volume of the primary zone, but doing so would induce an increase of soot formation. The separated primary and dilution holes were thus used, diminishing the efficiency of those RQL combustors in terms of NOx production, as it is more likely to have some combustion occurring near the stoichiometry. Those opposing constraints between operability and pollutant reduction efficiency lead to large effort on the other main concept for low NOx emissions: the lean combustion.

Lean Injection systems

The first motivation for operating aero-engine combustors under lean condition consists to decrease the flame temperature for which substantial NO formation can be minimized. Furthermore, these conditions prevent the oxidization of unburned hydrocarbons that allows attaining low CO and soot emissions, and high combustion efficiency. As a result, lean combustion provides benefits over the perspective of the whole emission reduction. To achieve the desired equivalence ratio in the reaction zone, up to 70 % of the total combustor air flow is used in the primary zone, leaving only 30% for air cooling and the control of the temperature profile of the gases entering the turbine. Thus, the advent of lean combustion requires complementary efforts in rapid and full fuel/air mixing strategies prior to combustion. Furthermore, lean combustion imposes drastic constraints in the design of combustor in relation to safety requirements. It has been shown that lean and fully premixed combustion increases the operational risk of flame blowout at partial load regimes and the generation of undesirable dynamic combustion instabilities generated from acoustic/flame interactions [START_REF] Cazalens | Combustion instability problems analysis for high-pressure jet engine cores[END_REF]. Premixing generates also the risk of the upstream flame propagation toward the fuel injection systems (flashback), leading to the failure of the components of the combustor. The mentioned safety concerns are generally less and less easily addressed with increasing overall pressure ratio.

Other trade-offs must be made concerning premixing that often relies on the production of high levels of turbulence and high velocities and hence increases the pressure drop, reducing the overall efficiency of the combustor. Until now, several strategies for achieving ultra-low emissions performance for aero-engines have been developed by manufacturers. With the shift away from traditional "diffusion" type combustion systems, various technologies that attempt to lower the peak combustion temperature have been proposed. These concepts are depending of the size and the overall pressure ratio of aero-engines. The most promising concepts are classified in the following subtypes [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF][START_REF][END_REF][4].

 Lean Premixed Prevaporized (LPP)

The Lean Premixed Prevaporized (LPP) concept is to supply in the injector upstream duct a lean homogenous mixture of vapor fuel and fresh air before its injection into the combustion chamber. Combustion is then performed at an equivalence ratio close to the lean blowout limit. In the premix duct, physical processes, such as fuel injection, fuel drop evaporation, and fuel/air mixing have to be optimized because a small degree of unmixedness could generate NOx. Disadvantages of this concept come from its narrow stability limits and their propensity to produce auto-ignition of the fuel/air mixture inside the duct under high-pressure and hightemperature operating conditions. Flashback is also another major risk encountered with this technology. The LPP combustion concept presents then operational risks for aero-engine applications. Nevertheless, the LPP injection system is still a promising concept for small engines presenting low OPR. Recent results in ducted-flame studies (EU NEWAC program) demonstrated the ability of this technology to reduce the NOx levels by 57% relative to CAEP2 regulations [START_REF] Arttic | NEWAC Publishable Final Activity Report[END_REF].  Partial Evaporation and Rapid Mixing (PERM) A variant of the preceding injection system is the so-called PERM (Partially Evaporation & Rapid Mixing) concept, also referred to as LP (Lean Premixed), which relies on premixing primary air with the liquid fuel, whereas fuel evaporation remains incomplete at the exit of the injection device.

Lean combustion is then generated by mixing more than 60% air through the nozzle with fuel in the aim to reduce NOx levels and to assure good operability for all required C2 -Confidential conditions. This method mitigates the issues of flashback and combustion instabilities, since only a portion of the fuel is pre-vaporized within the injection system. Benefit of this technology allows a reduction of NOx by 42% compared to CAPE2 regulations [START_REF] Arttic | NEWAC Publishable Final Activity Report[END_REF].  Lean Direct Injection (LDI) Lean direct injection consists in a direct injection of fuel and air into the combustion chamber without any premixing. As the name implies, the whole air flow enters the front end whereas fuel is injected directly into the combustion zone. Fuel/air mixture is lean so that the flame temperature and NOx are low. To eliminate hot spots that generate high levels of NOx, the LDI concept relies on a fast fuel/air mixing before combustion occurs. This fast mixing is obtained by adapting the distance over which the fuel and air flows are mixed. By breaking up a few large fuel injectors into many small ones, a homogeneous spray distribution and therefore a better homogenization of the fuel/air mixing are obtained, prohibiting auto-ignition or flashback [START_REF] Tacina | Experimental investigation of a multiplex fuel injector module for low emission combustors[END_REF]. Benefit of this technology allows a reduction of NOx levels by 70% relative to CAEP2 regulations [START_REF] Arttic | NEWAC Publishable Final Activity Report[END_REF], but it comes at the cost of the overall combustor increased weight and cost.

Lean Staged Injection Systems

Conventional combustors must operate over the entire duty cycle and thus represent a compromise design that can operate efficiently at low-and high-power conditions. In general, lean injection devices mentioned above, have narrow stability limits and are not able to operate over the entire engine cycle. To overcome this limitation, staged combustion devices have been introduced, allowing a safe variation of the power output by adapting the fuel flow ratio between the different stages or by deactivating the burner partially. This so-called fuel staging is achieved in various ways: the first mode is to divide the entire annular combustor into two rings (Double Annular Combustor). This technology already applied on CFM56 (-5B and -7B) and GE 90 combustors, and showed good results in terms of NOx emissions, but limitations in terms of the added weight and complexity of the architecture as well as difficulty of cooling such a large combustion chamber. The second mode uses SAC designs (Single Annular Combustor) in which fuel staging is achieved using internally staged injection systems.

Dual annular combustor

The "Pilot" combustor is designed to operate lightly loaded and provide all the temperature rise needed at startup, altitude relight, and engine idle conditions. At idle, the equivalence ratio of the combustion zone is selected to minimize the emissions of CO and UHC. The other annular combustor is specifically designed to optimize the combustion process at high-power settings. It features a small, highly loaded combustion zone of short residence time and low equivalence ratio to minimize the formation of NOx and smoke. The main advantage of radial staging is that it allows all the combustion performance goals to be achieved, including low emissions, within roughly the same overall length as a conventional combustor. This short-length feature is attractive from the standpoints of low engine weight and reduced rotor dynamics problems. Several drawbacks arise from radial staging. Firstly, the reduction of pollutants is achieved at the expense of increased design complexity and a marked increase in the number of fuel injectors. The larger liner wall surface area demands additional cooling air, which has an adverse effect on homogeneity of the temperature at the outlet of the combustion chamber. Furthermore, the peaks of the radial temperature profile could shift in radial position as a result of fuel staging, further increasing the inhomogeneity of temperature at the turbine inlet. Another challenge C2 -Confidential with radial staging is that of achieving the desired performance goals at intermediate power settings where both zones are operating well away from their optimum design points. The radially staged combustor shown in Fig. 7 was designed by the General Electric company [START_REF] Mongia | Engineering aspects of complex gas turbine combustion mixers part IV: Swirl cup[END_REF]. It achieved around 35% reduction in CO and UHC, and 45% reduction in NOx, in comparison with the corresponding single annular combustor. As the GE "LEC" RQL combustor, it relies on swirl cup air blast injectors, where liquid fuel is partially projected onto a cup, creating a film that is then atomized by a second swirled air flow.

Axial staging presents advantages over radial staging. Since the main stage is downstream of the pilot stage, ignition of the main stage directly from the pilot is both rapid and reliable. Also, the hot gas flow from the pilot injection into the main combustion zone ensures high combustion efficiency from the main stage, even at low equivalence ratios. On another hand, axial staging induces a longer combustion chamber in comparison with SAC or Radial DAC, which directly impact the shaft length and its dynamics. Also, the wall surface to be cooled off is still increased in comparison with a SAC, though to a lesser extent than with a Radial DAC. Furthermore, the fuel injection systems for the two combustion stages are placed at different axial positions and thus require separate feed arms, which involve two separate penetrations of the combustor casings and a further increase in cost and combustor weight. Fig. 8 shows a cross-sectional view of an axially staged combustor developed by Pratt and Whitney and equipping the P&W V2500-AS engine [START_REF] Koff | Aircraft gas turbine emissions challenge[END_REF]. The pilot combustion zone is specifically designed to provide wide stability limits and high combustion efficiency (low carbon monoxide and unburned hydrocarbons) while the main stage, located downstream is designed for high power regimes. In [START_REF] Koff | Aircraft gas turbine emissions challenge[END_REF], B. L. Koff also points out that the radially inward position of the pilot C2 -Confidential combustion zone is less impacted by heavy rain, as the water is centrifugated radially outward, which further enhance its stability. The injection system is not detailed, but the cross-section shown in Figure 5 seems to indicate the use of an LDI injector, with a pressurized fuel injector partially impacting the upper edge of the injector, and a single swirling air stream to facilitate atomization.

Single Annular Combustor

The operability of the lean injection systems has been addressed by the dual annular combustors at the expanse of the combustor weight, complexity and cost. Including the pilot and main function within a single injection system would help to resolve those drawbacks, and, with a single reaction zone, the temperature profile at the combustion chamber exit would be likely to be more favorable to turbine blade durability. The radial and axial combustors used two different strategies. With the radial DAC, the pilot and main combustion zones are fully separated: the main combustion zone is autonomous and activated only for the high-power regimes. With the Axial design, the pilot combustion zone is mixed with the main one, so that the combustion efficiency can be ensured by the pilot stage. For staged injection systems, two similar approaches have been imagined: including the pilot and main function without interaction, or having the pilot stage interacting with the main and ensuring its operability.

Because of the various advantages offered, staged lean injection systems were developed by numerous engine manufacturers and partners, including General Electric, Rolls-Royce and SAFRAN. As explained previously, developing such a system requires to understand and master a large range of physical processes from the atomization to the pollutant formation, while including all the constraints of aeronautical engine certification such as in-flight relight in cold conditions, heavy rain intake and a strong blow-off resistance. As such, a myriad of injection systems has been developed in the past, but designing one that validates all the aspects Fig. 8 : Pratt and Whitney Axial Dual Annular combustor cross section. From B. L. Koff [START_REF] Koff | Aircraft gas turbine emissions challenge[END_REF] C2 -Confidential enounced above remain an open challenge. This section will describe a handful of concepts to give an insight of the possibilities that arise while designing a lean staged injection system. Particular emphasis will be placed on the development of the TAPS injection system as this is an exceptionally large study, which has given rise to a significant number of patents, and is perfect material to illustrate the variety of technological developments that can, and should, be explored to achieve satisfactory staged and poor injection systems. Fig. 9 presents two lean staged injection system developed by CFD Research under a NASA contract, detailed by Nickolaus et al. in [START_REF] Nickolaus | Development of a Lean Direct Fuel Injector for low emission aero gas turbines[END_REF] and the object of the patent US 6,272,840 B1 [START_REF] Crocker | Piloted Airblast lean direct fuel injector[END_REF].

As the authors proposed various configurations, two variants of the injection system are presented, though their functioning principles remain the same.

In this concept, the main fuel is injected though a small canal, see the arrow "20", and partially atomized by a swirled air flow coming from the swirler "22". A part of the liquid is filmed onto a lip and blasted the air coming from another air swirler denoted "24". The main flame is stabilized at the injection system exit in a V shape. This dual atomization mechanism ensures a fine atomization but requires a fine tuning of the aerodynamic of the air flow to obtain the desired amount of fuel filming. The pilot function can be either created from an airblast atomized or a pressure swirl. In either case, the fuel is partially projected onto a lip, denoted "44" (and highlighted with a red square on Fig. 9), which creates a liquid film that is then blasted by a swirled airflow. This airblast can be ensured by the same airflow than the main, or a separate one (Fig. 9.a) and b) shows a dual or a single main air swirler). In both pilot configuration, the pilot spray is ejected axially so that the pilot flame does not interact with the main one. Because of this, there is no inner recirculation zone and the flame stabilization is controlled aerodynamically with the air flow velocity and the recirculation zones in between the main and pilot flame. As a consequence, the authors pointed out that the lean blow off (LBO) limit was very sensitive to the air flow repartition. In particular, they pointed out that the pressure swirl injector coupled with a bifurcated pilot air flow showed a great improvement of the LBO. The patent 6,389,815 [28] from General Electric has also been deposed in 2002 and present a lean staged injection system that uses interacting pilot/main flames, denoted the TAPS (Twin Annular Premixing Swirler), second generation. The actual cross section as well as a schematic are presented in Fig. 10.The pilot function is created with a pressure swirl injector, with a partial projection of the liquid fuel on a lip to create a prefilming air blast. The lip is opened radially outward to preserve the swirl number of the pilot air flow so that the pilot flame can have a V shape and be opened large enough to attain the main flame. The liquid fuel film from the pilot injection is blasted in between two swirled air flow. The main injection is created with jet in crossflows in the main air flow. A small part of the main air is injected by a film at the injector's exit to atomize the portion of the liquid fuel that didn't mix with the air and was projected onto the inner wall. The performance of this injection system in NOx reduction were presented by Mongia H. [START_REF] Mongia | TAPS -A 4th Generation Propulsion Combustor Technology for Low Emissions[END_REF]. A decrease of 46% in NOx has been measured over the LTO (Landing Take off) phase in comparison with the CFM56-7B, that used RQL technology, while the other key performances (operability, emissions, stability …) were satisfactory. Following those results, the authors denoted the importance of shifting from the RQL technologies that have been matured over decades but showed much less potential for NOx reduction.

A third generation of TAPS injection system was developed with the goal of reducing the NOx emission of the Gen-II TAPS injector. A cross section, extracted from the patent US 6,418,726 [START_REF] Foust | METHOD AND APPARATUS FOR CONTROLLING COMBUSTOR EMISSIONS[END_REF], is shown in Fig. 11. The main difference with the TAPS-II is the addition of a third fuel injection, used for mid-power and Cruise conditions. The fuel is injected as a jet in cross flow in a swirled air flow in between the pilot and main flow. The main air swirler uses two close swirlers to create a counter rotating shear layer to enhance the main jet ion cross flow atomization and the mixing with the air from the mid.

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The fourth generation of TAPS injection system was developed with a strong focus of the main air/fuel mixing. The main idea was to further develop the counter rotating shear layer in the inner part of the main air flow to enhance the performances of the jet in cross flow, that is, its capacity to create a fine atomization and a homogenous mixture of air/fuel. Numerous air swirler were depicted, a cross section of the injector and three different main swirlers are presented in Fig. 12, from the US Patent 7,581,396 [START_REF] Hsieh | MIXER ASSEMBLY FOR COMBUSTOR OF A GASTURBINE ENGINE HAVING A PLURALITY OF COUNTER-ROTATING SWRLERS[END_REF]. The first swirler is purely radial, with three channels, the one in the middle being counter rotating with the two others. The two channels closest to the inner wall, that is, the furthest from the combustion chamber, have a small height so that the very beginning of the jet in crossflow is subject to the counter rotating shear layer. The second main air swirler showed in Fig. 12 is a combination of two axial and two radial swirlers. With such a complex arrangement, predicting the flow field only from the scheme becomes a difficult task, and would require a well-resolved CFD computation. Though, one could imagine that the idea behind the addition of those two axial swirlers is to emphasize the shear layers mixing effect by putting them the closest as possible to the fuel injection points. Indeed, with the first swirler configuration, the shear layers can start to interact with one another and loose some of their fuel mixing potential by the time they reach the jet in crossflows. The third swirler is even more complex with a radial opening of the largest swirler, and three closely stacked little ones. It is likely that the idea behind this is again to create the shear layers as close as possible of the fuel injection points. Fig. 13.a) shows another variant of the fourth generation TAPS, with a modified pilot injector, from the patent US 8,171,735 B2 [START_REF] Mancini | MIXER ASSEMBLY FOR GAS TURBINE ENGINE COMBUSTOR[END_REF] published in 2012. The pressure swirled injector has been replaced with a film airblast, a swirled airflow being generated around the centerline of the injection system, exploding the fuel film ejected preferentially inward.

Fig. 11 : General Electric third generation TAPS injection system, from the patent US 6,418,726. The "mid" injection is a jet in crossflow used for mid power and cruise operating conditions.

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Finally, a fifth generation of TAPS injectors has been imagined, with a pilot injector assisted by plasma. The goal was to deal with the inherent combustion instability coming from lean regimes with an energy deposition right before the pilot fuel injection. It was designed originally for natural gas and presented in the patent US 6,453,660 [START_REF] Johnson | Combustor Mixer Having Plasma Generating Nozzle[END_REF]. Though it has been associated with the Gen-V TAPS, the patent is older than the Gen-IV ones and its application seem to have been limited, though the idea behind this patent is interesting and shows the extend of the possibilities to deal with the lean combustion challenges.

Overall, the work on the TAPS injection system led to a decrease of 87.5% of the NOx production compared to the best RQL technologies available in the late 90's, and 94% in comparison with the technologies used prior to any NOx regulations [START_REF] Mongia | Engineering aspects of complex gas turbine combustion mixers part V: 40 OPR[END_REF]. As this collection of patents propose a myriad of variants, the exact design used to obtain this final result is uncertain. Yet, two technological choices stand out. Firstly, most of the work has been made on the fuel air mixing, on both the pilot and main stages, and so by drastically increasing the injection system complexity. Secondly, interacting pilot/main flame have been used, so that the pilot burned gas can support and stabilized the main flame that is likely to function at a very lean equivalence ratio.

Fig. 12 : General Electric fourth generation TAPS injection system, from the patent US 7,581,396. Numerous main air swirler were described. Three of them are presented here, their goal is to create counter swirling shear layer to enhance jet in cross flow atomization

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From a research perspective, a massive amount of work has been carried out under the TLC EU program and the subsequent research initiatives, such as the TECC-AE, TIMECOP and LEMCOTEC UE programs. A number of publications resulting of those projects will be used in the next chapters, but covering the whole extend of the studies achieved under those programs is beyond the scope of this work. To continue the discussion on the key factors to conceive and operate an effective lean-staged injection system, a handful of results are thereby quoted. In 2009, Orain et al. [START_REF] Orain | Simultaneous measurements of equivalence ratio and flame structure in multipoint injectors using PLIF[END_REF] carried out simultaneous kerosene-and OH-PLIF measurements on a lean-staged injection system presented in Fig. 14a), in high-pressure reactive conditions. Notably the authors proposed a better understanding of the flame topology, in particular on the pilot/main flame interaction. The injection system was composed of a pressure swirl pilot injector creating a partial impact of liquid fuel on a cup, coupled with a double air swirler, and a jet in crossflow as a main atomizer. In 2012, Meier et al. [START_REF] Meier | Spray and flame structure of a generic injector at aeroengine conditions[END_REF] used Phase Doppler Anemometry and kerosene-PLIF to characterize the liquid and gaseous fuel phase from a double swirl film airblast injector under realistic conditions. More detailed will be shown in the PDA section of this manuscript. In 2014, Bhagwan et al. [START_REF] Bhagwan | Experimental investigation on lean blow out of a piloted aero engine burner[END_REF] worked on the understanding of the physical processes that controlled the pilot flame shape and its lean blow-out resistance. The injection system used a converging air/pilot separator towards the centerline to force the pilot flame to follow an axial development rather than a V opening, with the goal to enhance its resistance to extinction, leading to an arrangement of the reaction zones comparable to the one showed in Fig. 9.b). The same year, Berrino et al. [START_REF] Berrino | Experimental characterization of the flow field downstream of an innovative ultra low NOx injection system[END_REF], in collaboration with GE-AVIO, proposed a variant of the TIMECOP injector with an increased main swirl number. Few detailed are given, but one remarkable technological choice is the removal of the inner cup often used to create a film airblast in the pilot part of the system. The idea behind this modification was to evaluate the capacity of a Precessing Vortex Core (PVC) to enhance the fuel/air mixing. Particle Image Velocimetry (PIV) was used to study the aerodynamic of the flow and the topology of the PVC, found at ~1500Hz. The authors C2 -Confidential concluded that this use of an aerodynamical structure instead of a technological modification could be a promising solution for lean injection systems. They further detailed those results year after in [START_REF] Berrino | INVESTIGATION OF THE DYNAMICS OF AN ULTRA LOW NO X INJECTION SYSTEM BY POD DATA POST[END_REF]. In 2015, with a numerical study on another variant of a lean staged injector, Baurheim et al. [START_REF] Bauerheim | MULTIPHASE FLOW LES STUDY OF THE FUEL SPLIT[END_REF] described the PVC to be the main feature controlling the flame dynamic, and put in light the different modes that could arise within a realistic combustion chamber from this particular aerodynamical structure. The injection system used a jet in crossflow as a main injector, and a pressure swirl with converging lip as a pilot one. Finally, in 2018, Mazzei et al. [START_REF] Mazzei | Numerical and Experimental Investigation on an Effusion-Cooled Lean Burn Aeronautical Combustor: Aerothermal Field and Emissions[END_REF] showed with numerical simulations on an injection system of the LEMCOTEC UE program the different flame regimes depending on the engine operating conditions, and demonstrated the impact and importance of the diffusion regime on the CO and NOx production, and the necessity of functioning in the premixing regime.

The Multi Staged Fuel Injector (MSFI) concept

The MSFI concept studied within the current study is a lean-staged injection system, see Fig. 15. The pilot stage is constituted of a pressure swirl atomizer coupled with two radial air swirlers and a prefilming cup. The main stage consists of an air swirler and a jet in crossflow atomization. Dilution holes are present near the burner's exit, on both the main and the pilot stages. Three variants have been studied, one called the Lemcotec, which corresponds to one of the numerous variants developed within the LEMCOTEC framework. The MSFI-V0, an additive manufactured version of the Lemcotec injection system, and the MSFI-V2, a modified design which increases by 40-60% the width of the pilot/main separator, and between 10-30% the nozzle blade angle of the pilot's swirlers. The MSFI-V2 was also additive manufactured.

The different notes presented here on the NOx production pathways, the technological key features of the lean staged injection systems, the control of the flame topology and the impact of the PVC on mixing and operability are the basis of the current work, and looked upon in detailed in the results section. The last part of the current work to be introduced is the experimental tools used to study and further understand the physical processes detailed previously.

Experimental tools for aircraft combustion

Investigations to date have tended to focus to one or several specific combustion-related phenomena but lack sufficient depth to fully understand the physical and chemical mechanisms responsible for them. For instance, experimental studies have measured mean and fluctuating flame properties but they have been largely unable to track turbulent flow phenomena as they occur. The complexity of turbulent flame dynamics within aero-engines has equally led to reliance on simplifying assumptions and modelling of various parameters in numerical simulations. Based on these observations, there is a notable lack of comprehensive studies involving closely-coupled experimental and numerical investigations of lean combustion in combustor geometries of technical relevance especially under high-pressure conditions.

As highlighted earlier, aeronautical turbulent reacting flows involve a wide range of scales and complexities caused by the specific shapes of injection systems and the associated combustion regimes. A better understanding of the previously mentioned aspects of aero-engines combustion has been improved by the use of non-intrusive optical measuring techniques [START_REF] Höinghaus | Applied Combustion Diagnostics[END_REF][START_REF] Eckbreth | Laser Diagnostics for combustion temperature and species[END_REF]. From an experimental point of view, optical methods, in particular those that are laser-based are now recognized as the best candidates to resolve processes at small time and length scales.

MSFI-V0 MSFI-V2

C2 -Confidential Additionally, high-speed laser-based techniques can now address many of the needs such as temperature, species concentration and velocity distributions. Among their main advantages:  the ability to perform instantaneous non-intrusive measurements of one, 2D or 3D distributions of various scalar quantities  the capacity to image short-lived combustion radicals and heat release rates  the aptitude to measure various scalar parameters simultaneously and to capture, when possible, the dynamics of the reactive flow fields. For instance, dynamics of swirl flames was investigated using the combination of PIV and PLIF [START_REF] Boxx | Temporally resolved planar measurements of transient phenomena in a partially pre-mixed swirl flame in a gas turbine model combustor[END_REF][START_REF] Heeger | AT HIGH REPETITION RATES : ACQUISITION AND Benjamin B ohm[END_REF]. Flame-vortex interaction was investigated by applying OH-PLIF and PIV in a counterflow flame [START_REF] Meyer | Simultaneous PLIF/PIV investigation of vortex-induced annular extinction in H2-air counterflow diffusion flames[END_REF]. PLIF was also applied to investigate the mechanisms of flame stabilization [START_REF] Stöhr | Dynamics of lean blowout of a swirl-stabilized flame in a gas turbine model combustor[END_REF]. The role of equivalence ratio fluctuations and effects of unmixedness in lean premixed flames were investigated using chemiluminescence, infrared absorption measurements and laser Raman scattering [START_REF] Schuermans | A Detailed Analysis of Thermoacoustic Interaction Mechanisms in a turbulent premixed flame[END_REF][START_REF] Meier | Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame[END_REF]. Flame-vortex interaction inside GT combustors was also studied using PIV and OH-PLIF [START_REF] Sadanandan | Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor[END_REF]. Further examples of studies on LPP and LP injectors can be found in the paper of Dhanuka et al. [START_REF] Dhanuka | Unsteady aspects of lean premixed prevaporized gas turbine combustors: Flame-flame interactions[END_REF]. In most of these studies, the data measurements were usually performed at atmospheric pressure for reducing the complexity of experiments. At higher pressure, rare studies using optical techniques were reported until now. Among these studies, Griebel et al. measured the effect of pressure and turbulence on the flame speed of premixed turbulent flames [START_REF] Griebel | Flame characteristics of turbulent lean premixed methane/air flames at high pressure: Turbulent flame speed and flame brush thickness[END_REF]. Lückerath et al. [START_REF] Lückerath | Experimental Investigations of Flame Stabilization of a gas turbine combustor[END_REF] studied flame stabilization in a staged gas turbine model combustor, Strakey et al. [START_REF] Strakey | OH-planar fluorescence measurements of pressurized, hydrogen premixed flames in the SimVal combustor[END_REF] investigated the impact of hydrogen in gas turbine combustors. The impact of coherent structures on the establishment of the flame as well as mixing and reaction progress were also investigated [START_REF] Janus | Experiments on swirl stabilized non-premixed natural gas flames in a model gasturbine combustor[END_REF][START_REF] Ax | Experimental analysis of the combustion behavior of a gas turbine burner by laser measurement techniques[END_REF]. Lean burner concepts were studied using coherent anti-Stokes Raman spectroscopy by Thariyan et al. [START_REF] Thariyan | Dual-pump coherent anti-Stokes Raman scattering system for temperature and species measurements in an optically accessible high-pressure gas turbine combustor facility[END_REF] Recent efforts on the characterization of multi-point injection devices were also done by simultaneously applying OH-PLIF and kerosene-PLIF to study the interplay between kerosene/air mixing and flame structure and the CARS technique for measuring the temperature distribution under high-pressure operating conditions [START_REF] Orain | Fluorescence spectroscopy of kerosene vapour at high temperatures and pressures: Potential for gas turbines measurements[END_REF]. Successful demonstration of the combination of high-speed PIV and high-speed OH-PLIF was also performed to study flame-vortex interaction inside a novel Trapped Vortex Combustor (TVC) [START_REF] Xavier | Analysis of the Flame Structure in a Trapped Vortex Combustor[END_REF]. Reasons explaining the rare existence of detailed studies on real combustors running in high-pressure conditions are primarily related to the insufficient knowledge of pressure, temperature and species effects (i.e. energy transfer between the energy states of molecules) on the measured signals. Understanding the photophysics of molecules of main interest (OH, NO, CO…) is consequently a necessary condition to enable robust, reliable, and quantitative optical diagnostics. A second limitation lies in the complex, transient and highly three-dimensional properties of the turbulent flame. Finally, another challenge is associated to the reduced optical access available in conventional high-pressure test-rigs. The laser beam and the measured signals can then propagate through optical windows which can be degraded in time from the heat load imposed on the glass surface and from pressure effects which contribute to significant decreases of the intensity of the measured signals.

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Objectives & Organization

The present work is focused on the full characterization of three lean staged injection systems in elevated pressure reactive conditions. The Lemcotec injector is one of the numerous variants developed within the LEMCOTEC framework and constitute the reference injection system. The two MSFI variants were built with precise technological modifications from the Lemcotec design as potential ameliorations, with the two main focus points being the operability and the pollutant emissions. Some results such as the exact lean blow off limit or flame visualizations could not be presented within this manuscript for confidential reasons. The reference operating point has been chosen as a downscaled version of the cruise operating point so it can be operated on the HERON test rig. The manuscript is organized as follow:

 Chapter 2 will regroup and present the experimental configurations used  Chapter 3 is dedicated to the Planar Laser Induced Fluorescence (PLIF) diagnosis, with a particular emphasis on the NO-PLIF with was carried out quantitatively  Chapter 4 will detail the Phase Doppler Anemometry, used to characterize the spray in terms of droplet size and velocity. The first part will present the theoretical principles, and the second, an extensive bibliography. The third part will present preliminary results on a pressure swirl atomizer at atmospheric pressure, and measurements on the LP injector from Safran Helicopter Engines.  Chapter 5 will show the results of the full campaign on the three injection systems.

Results from high speed PIV, high speed tomography, PDA, simultaneous NO-Kerosene-OH PLIF and exhaust gas analyzer will be presented for an in-depth study on the differences between the three injection systems.  Chapter 6 will present some results on different operating points to extend the conclusions drawn from the main campaign, and propose different links between the studied technological modifications and their impact on operability and pollutant emissions.

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-Experimental Setup

Heron test rig

Experiments were conducted on the HERON (High prEssuRe facility for aerO-eNgines combustion) facility that was designed to investigate novel low-NOx full-scale injection systems for helicopter combustors up to their full operating ranges and airplanes combustors up to 2/3 of their nominal ranges. The test rig is mounted on rail and divided in modules to ease the maintenance and potential modification. A schematic of the test facility can be found in Fig. 16.

The first module is a stainless-steel pipe with a coarse honeycomb grid to breaks the largest structure, and homogenize the flow entering the combustion chamber. An insulation system ensures the minimization of heat loss. The second module hold the injector and position it with respect of the combustion chamber. The fuel supply line is shaped to reduce its impact on the airflow, and is water cooled to keep the fuel temperature constant during the experiments. Thermocouples (K-type) measure air and kerosene temperature. Static pressure is monitored with a KELLER PR-33X sensor, and dynamic pressure fluctuation is measured with a KISTLER 7001 sensor.

The combustion chamber is of square section, of dimensions 100 x 100 x 230mm 3 , and is fitted with four fused silica windows for optical access. Both lateral optical windows are 80 mm long and 100 mm high so that the entire region of the flame can be observed in the test chamber. Two upper and lower optical windows of 80 mm long and 30 mm wide complete the optical access. The inner surface of the windows is protected by pre-heated air films injected by thin slots located around the burner faceplate in which the injection system is set. The whole combustion chamber is cooled down by a pressurized water flow. Further downstream the visualization segment, a sensor module equipped with several measurement transducers is placed. Static and dynamic pressure are measured with the same sensors as in the second module and several thermocouples of type K are used to control the temperature of the reactive flow and the walls of the combustion chamber.

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Then a dilution segment designed to convert the square section of the combustion chamber toward a circular section, takes place. Additional dilution air at room temperature is injected through eight radial pipes positioned on the dilution segment for cooling the hot gases by mixing. Finally, the outlet of the combustor is equipped with an adaptive sonic throttle which can be gradually obstructed by a water-cooled needle in order to regulate the pressure inside the combustor. Exhaust gases are then collected by a fan to be extracted outside the building. At the beginning of the exhaust pipe, a sampling of burned gases can be made to study their composition with a gas flow analyzer (HORIBA PG250 and HORIBA MEXA 1170HDFID). By this mean, concentration of O2, CO2, HC, CO and NOx can be measured. The combustion chamber is supplied with compressed air preheated electrically. The combustion facility is designed for operation of fuel injectors at up to 2.0 MPa and 900 K, with a maximal air flow rate of 300 g/s stream.

PIV

Particle Image Velocimetry (PIV) is a well-known optical tool to measure planar velocity fields. It requires the flow to be seeded with low Stokes number particles, so they perfectly follow the flow fluctuations. The particles are illuminated with a short pulse high frequency laser sheet. The delay between laser shoots is inversely proportional to the mean flow velocity and is chosen so that group of particles can be followed between the two shoots. High speed cameras are mounted with bandpass filters around the laser wavelength to collect the Mie scattering of particles. For each instantaneous velocity field, two snapshots are taken and the velocity field is reconstructed by comparing the position of particles and estimating their displacement. As particles cannot be tracked individually, images are separated in N interrogation areas (IA), so that group of particles are compared instead of separate particles. A cross-correlation algorithm is applied on each IA to estimate a mean vector displacement. This leads to an instantaneous displacement vector map of N vectors. A calibration image and the knowledge of the time interval between laser shots allow the conversion of pixel displacement to an actual velocity. Here, a dual cavity Nd:YLF laser (Quantronix Darwin Dual 80) has been used, with a wavelength of 527nm, operating at a repetition rate of 10kHz with a pulse energy of 2.4 mJ. The time between the independent pulses was fixed at 5µs. This value has been found to be an optimum to correctly capture both the slowest velocities near the external recirculation zone (~ 5 m.s -1 ) and highest ones in the jet zone (~ 100 m.s -1 ). The emitted laser beam passes through sheet forming optics constituted of a convergent spherical lens of focal length f = 1000mm to focus them inside de combustion chamber, a cylindrical length of focal length f = -20mm for sheet forming, and then a convergent cylindrical lens of focal length f = 500 mm for laser sheet parallelization.

Seeding was performed with Zirconium Oxide ZrO2 particles of 5µm diameter, mixed with Silicium Oxide particles of 0.2µm diameter to avoid clustering at a mass proportion of 1%. Zirconium oxide fusion temperature is near 3000K and is thus well above any temperature encountered in the combustion chamber. The seeder is a cyclonic fluidized bed provided in part by ONERA French lab (Office National d'Études et de Recherches Aérospatiale) as part of the PERCEVAL industrial chair. The seeder is constituted of two air circuit. The first one is connected to a crown of small injection pipe that create a strong swirling motion, mixing the particles with air and suspending them. The second airflow pass through the middle of the fluidizer and through a nozzle, creating a depression that suck particles toward the outlet. The total air mass flow is controlled by a mass flow meter and the air repartition between the two circuits is set by a balancing valve. The seeded flow is mixed after the air heater. The proportion of air passing through the seeder is around 7% of the total airflow. The Mie scattering signals generated by particles were collected with a Phantom V2512 camera, mounted with a Zeiss Milvus 2/100M camera lens of length focal f = 100mm and an opening of f/# = 2.8. A pass band filter centered at 527nm of 10nm bandwidth is mounted in front of it to remove any parasitic signal. The CMOS sensor has a 12bits depth, and is composed of 1280 x800 pixels, each pixel having a size of 28µm. For each measurement, 10 000 dual shots were recorded at 10kHz, leading to 5 000 instantaneous vector fields.

The cross correlation has been done with Dantec Dynamic Studio, with a variable IA size from 10x10 pixels up to 64x64 pixels. At the first step maximum IA size is used, then, if the particle C2 -Confidential concentration is high enough, the size is automatically decreased, until either the minimum IA size is reached or not enough particles are found. A fixed Grid step size of 10x10pixels is fixed and define the number of computed vectors. Thus, an IA overlap occurs if the IA is larger than the grid step size. The minimum number of particles per IA is user defined and set here at 8, with a target of 15 particles per IA. Furthermore, velocity gradient is also considered to determine the ideal IA size, with a limit on each of their magnitude limited to 0.15, and their maximum combined norm limited to 0.30.

The resulting vectors are subject to two validation methods. First, it has been chosen that the correlation needs to meet a peak height of 0.3 and a signal over noise ratio of 3.0. Secondly, the universal outlier detection [Westerweel & Scarano, experiment in fluids, 2005] is applied. First, each surrounding vector residual, 𝑟 𝑖 , is calculated with (1), comparing the magnitude of those vectors to the median's one. Next, the residual of the checked vector is compared to the median of the neighbor's residual, see [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF]. The result, R, is compared to a user defined threshold value, rejecting of validating the result of the cross-correlation.

𝑟 𝑖 = |𝑈 𝑖 -𝑈 𝑚 | [START_REF] Nicol | The importance of the nitrous oxide pathway to NOx in lean-premixed combustion[END_REF] 𝑅 = 𝑟 0 𝑟 𝑚 + 𝜀 [START_REF] Malte | Combustion Science and Technology The Role of Energy-Releasing Kinetics in NO x Formation : Fuel-Lean , Jet-Stirred CO-Air Combustion The Role of Energy-Releasing Kinetics in NO : c Formation[END_REF] Here, the universal outlier detection is done with a square of 9x9 vectors around the checked one, a minimum normalization of 0.20. If R is superior to the threshold limit, the vector is considered invalid and the three next highest peaks of the cross-correlation are tested. If one of them pass the check it is used as the final vector, if not, the median vector is used instead. This set up lead to a replacement of ~ 1% of the vectors, mostly near strong shear zones.

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OH-PLIF/kerosene-PLIF/NO-PLIF

The flame structure was studied by means of the OH-PLIF diagnostic operating at low frequency (3.33 Hz). The laser beam was generated with a Nd-YAG (Quanta-Ray Pro Series) laser operating at 1064nm and a frequency doubling crystal gave a laser beam at 532nm, with an energy of 500mJ per pulse. It was used to pump a tunable dye laser (SIRAH Precision Scan), supplied with Rhodamine 590 dye. An oscillator and an amplification unit change the wavelength to 565nm. Then a BBO crystal double the frequency and bring the wavelength around 283nm. Two Pellin-Broca prisms are used to suppress potential 565nm residuals. The resultant output pulse energy was about 20mJ per shot in the probe volume.

A dye laser has also been used to generate the laser beam at 225nm for the NO-PLIF measurement. It consist of a Nd:YAG laser (Quanta Ray Pro Series), generating a 1064nm laser beam, that is split so that a part can be double in frequency to obtain a 532nm laser. The two beams are then combined to form a 355nm beam which is used at the end of the dye laser. The latter is a Sirah Precision scan pumped with a 532nm laser beam, using a mix of Rhodamine 640, Rhodamine B and ethanol. An oscillator and an amplification module rise the 532nm laser beam to 614nm. This wavelength that can be finely adjusted through a grating. Finally, the 614 and the 355nm beams are mixed to create a 225nm beam, with an energy of around 20mJ. Both lasers wavelengths were adjusted before any experiment to ensure the maximum fluorescence signal, on the Q1(5) transition for the OH, and on the Q1(29.5) for the NO. To do so, a Bunsen burner have been put inside the combustion chamber by removing the lower optical window. A CH4/air premixed flame doped with NO is thus created, allowing a wavelength sweep of both lasers. The fluorescence is collected, and averaged temporally and spatially. The collected fluorescence signal is then compared to simulated one to identify the transition and perfectly wavelength tune both lasers.

A periscope designed with two prisms was used to transport the laser beams above the HERON test rig. A slight angle was set on the 225nm beam so that it passes right above the second prism of the periscope carrying the 283nm beam and superpose with the latter in middle of the combustion chamber. The angle is small enough (<0.2 mrad) so that in the combustion chamber their separation is inferior to 8µm.

Both laser sheets were formed with the same optics, consisting of a converging spherical lens (f = 1000mm), a cylindrical diverging lens (f = -15 mm), and a cylindrical converging lens (f = 500mm). To ensure the superposition and the verticality of the laser sheets, two plate with slits of 750nm width were mounted during the set up on both the upper and lower optical window.

The fluorescence is collected with three cameras PIMAX4 emICCD from Roper Scientific, at a frequency of 3.33Hz and a resolution of 1024x1024 pixels with 13x13µm pixels. The gate is set to 20nm to optimize the signal over noise ratio. Each camera is mounted with a Cerco objective (f=100mm, f/# = 2.0), and a Tokina square filter holder of 50x50mm. The OH-ICCD camera is equipped with a band-pass filter Semrock Brightline, of 20nm width and centered on 315nm. The camera dedicated to kerosene fluorescence is mounted with two filters, a Sagem and a Schott WG 305 to cut off both OH and NO emission. Finally, the NO camera is mounted with Semrock and Asahi filters, allowing signal collection between 230nm and 266nm, collecting fluorescence signal from A-X (0,1-5) bands. Filters were carefully chosen to have a null transmission in the 300 -400nm spectral band to ensure the removing of the kerosene fluorescence.

PDA

In the current work, measurements were carried out with a commercial system Dual PDA provided by Dantec, coupled with two continuous 1W diode laser (Coherent Genesis MX488-10 0 0 SLM and Genesis MX514-10 0 0 SLM) providing laser beams at 514 nm and 488 nm respectively. Each laser beam was divided in two parts with one part passing through a 40 MHz Bragg cell. Beams are then transported through fiber optics and transmitted towards the probe volume via a Dantec FiberFlow optical system. A beam expander of 1.98 ratio has been used to increase beam diameters and spacing in the emission probe. With an emitting focal length of 350 mm and a beam spacing of 40 mm, the probe volume dimensions in the flame are δx = 367 µm, and δy=δz= 44.5 µm. The receiving probe system was set to φ = 31° in the forward direction with a focal length of 350 mm and a slit aperture of 100 µm to cutoff the effective measurement volume and reduce the gaussian beam effect. Within these specifications, the optical setup was optimized to detect droplets within the range of diameter of 1-60 µm. The transmitting and receiving probes were mounted on a computer-controlled 3D mechanical system and the measurements were performed at various positions along the axial and radial directions downstream from the nozzle exit. PDA measurements were specifically carried out at two planes at an axial distance from the nozzle of Z/D = 0.04 and Z/D = 0.64. At Z/D = 0.04, a quarter of the flow exit cross-section was probed adopting a Cartesian grid with X = 0.25 mm and Y = 1 mm while at Z / D = 0.64, polar coordinates with θ = 10 °and Y = 1 mm were selected to mesh the grid. The PDA signals were processed with a P800 Dantec processor, with a signal-to-noise ratio of 3 dB and a first to second peak ratio of 4:1 as burst validation criterion. For each measuring location, it has been chosen to collect up to 10000 droplets, or during 30 s, whichever comes first, and to limit the area investigated to where a validated data rate of at least 100 Hz can be reached. This configuration leads to a validation rate of 90%, and a peak data rate of 55 kHz. In this section, the conditions for using laser-induced fluorescence to measure reaction intermediates in the flames generated in the HERON combustion bench are introduced. As a reminder, Laser-induced Fluorescence (LIF) is a well-established and versatile tool for detection of chemical species in a broad range of combustion applications. Typically, this optical diagnostic is based on the measurement of the light emitted spontaneously by a probed molecule after its excitation by the absorption of a laser radiation most often tuned in the UV-visible spectral range. One interest to use LIF in flames is its ability to detect minor and/or transient chemical species with concentration levels in the ppm range. This diagnostic has also attractive properties for recording fluorescence signals with a spatial resolution appropriate to the scales of the flow being analysed. For instance, spatially-resolved data can be obtained in regions with large concentration gradients to provide a detailed analysis of the flame structure, to measure the concentration of gaseous pollutants, and to acquire a description of the fuel/air mixing zones at the outlet of the injection system.

In addition, LIF measurement offers high selectivity, experimental signals with large signal-tonoise ratio as well as simplified experimental implementations compared to other spectroscopic diagnostics, such as nonlinear optical diagnostics like coherent anti-Stokes Raman scattering (CARS) and Degenerate Four Wave Mixing. This laser diagnostic is also non-intrusive and therefore does not disturb the flow at the time when measurements are performed. Typically, LIF provides point or one-dimensional (1D) measurements. In addition, Planar Laser-induced Fluorescence (PLIF), which is a derivative of LIF, enables two-dimensional (2D) measurements. In this particular case, PLIF involves the illumination of the flow with a thin laser sheet while an C2 -Confidential intensified charge-coupled device (ICCD) camera is used for collecting the fluorescence emitted in the area of the flow illuminated by the laser sheet. PLIF has been in the past applied to reactive flows, by the successful detection of various atoms as well as transient molecules such as OH, NO, CH, CO, C2, CN, O, H, CH2 and HCO, but also larger molecules such as aromatics and ketones.

Within the framework of this study, the detection of several molecules such as hydroxyl radical (OH), aromatics (mono-and di-aromatics) present in the formulation of the kerosene fuel (Jet A1) as well as nitrogen monoxide (NO) will be performed.

The processing of the fluorescence signals to get concentration and temperature remains however a complex task, requiring a detailed knowledge of the simulation of the fluorescence signals with the operating conditions in which measurements are performed. Quantification of the measurement signals also requires the use of appropriate calibration methods.

Before presenting the basic principles of fluorescence signal modelling, it may be of interest to provide the reader with some information on the key physical parameters required to understand the principle of laser-induced fluorescence diagnostic.

Introduction to spectroscopy

Energy levels

Internal energy of atoms can be changed through the modification of electron orbital and spinning motion. For diatomic molecules, electronic energy level modification may also apply, along with vibrational energy between both atoms, and the rotational motion around each other. Quantum mechanics states that energy transitions are quantified and that the possible changes are subject to selection rules. The total internal energy of a molecule is:

𝐸 𝑡𝑜𝑡 = 𝐸 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + 𝐸 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + 𝐸 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐 [START_REF] Rink | Influence of fuel composition and spray characteristics on nitric oxide formation[END_REF] With the following energy spacing:

𝛥𝐸 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 < 𝛥𝐸 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 < 𝛥𝐸 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐 [START_REF] Heberling | Prompt NO measurements in high pressure[END_REF] Thus, electronic energy transition are the most energetic ones. To attain the required energy, laser in the UV spectrum are needed.

Atom & Diatomic molecule structure

To understand the transitions used in a laser-induced fluorescence measurement, it is first necessary to understand the basics of the structure of an atom, and a diatomic molecule.

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Four quantum numbers are needed to summarize the state of an electron. The principal quantum number n, is a non-zero integer, and roughly represent the distance of the electron from the center of the atom. The orbital angular momentum quantum number, l, describes the general shape of the orbit and the shell the electron occupies, l = 0,1,2 …, n-1. Specific values of l can be represented with letters, such as s for l=0, p for l=1, d for l=2. The spin angular momentum, s, describes the direction of rotation of the electron. Only two values are allowed s = +/-½. The fourth quantum number, j, is the sum of the orbital and the spin angular momentum numbers. To describe an atom, the total angular momentum is defined.

𝐿 = ℎ 2𝜋 √𝑙(𝑙 + 1) (14)
𝑆 = ℎ 2𝜋 √𝑠(𝑠 + 1) (15)
𝐽 = 𝐿 + 𝑆, 𝐿 + 𝑆 -1, … , |𝐿 -𝑆| [START_REF] Lefebvre | Atomization and Sprays[END_REF] With those three numbers, a term symbol can be used to describe the state of an atom, such as below.

𝐿 𝐽 2𝑆+1

A similar notation can be used for diatomic molecules.

𝛸 𝛬 𝛺

2𝑆+1 [START_REF] Mcguirk | The aerodynamic challenges of aeroengine gas-turbine combustion systems[END_REF] Where, Λ is the projection of orbital angular momentum 𝐿 ⃗ on the internuclear axis. A symbol is associated to each value.

Λ 0 1 2 Symbol Σ π Δ
 S is the total electronic spin angular momentum, i.e. the sum of electron spin in unfilled shell. It is defined only if Λ≠0, and the possible values are S, S-1, …, -S.  Ω is the sum of projection of the electron spin and orbital angular momentum on the internuclear axis. Values possible follow the same logic as eq. ( 5).  𝛸 defines the electronic level of the molecule, and is represented with capital letters (A-Z), with X used for the ground state. As an example, both OH and NO ground state is 𝛸 𝜋 2 .

Ω=3/2, 1/2; which gives two spin sub-states 𝛸 𝛱 1/2 2 and 𝛸 𝛱 3/2 2 .

For diatomic molecules, two more quantum numbers have to be defined. N the quantum number associated to the nuclear angular momentum and the rotational quantum number J. Selection rules impose that ∆𝐽 = -1, 0, 1; with the respective associated symbols P, Q, R.

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Transition denotation can now be detailed:

𝑍 𝛼𝛽 (𝐽") 𝑌 [START_REF] Mongia | GE aviation low emissions combustion technology evolution[END_REF] Y represents the ∆𝑁, the variation of nuclear angular momentum. Z represents the ∆𝐽, the variation of rotational quantum number. α and β respectively give the spin multiplicity at the ground and excited state. If Y=Z then Y is suppressed, likewise when α=β, β is suppressed.

As an example, in the current study detailed within this manuscript, the OH-PLIF used the rotational transition 𝑄 1 (5) , and the NO-PLIF the 𝑄 1 (29.5) transition in the 𝐴 2 𝛴 + ← 𝑋 2 𝛱 band. 𝑋 2 𝛱 means that molecules in the ground electronic state will be the target of the excitation, and will undergo a transition towards the A electronic state. The second symbol of the ground state is 𝛱 which means that the projection of the orbital angular momentum on the internuclear axis is Λ = 1. In the excited state it becomes 𝛱 which implies that Λ = 0 and thus decreased.

The superscript of 𝑋 is equal to 2, then the total electronic spin S = 1/2. Λ and S are necessary to determine which model can be used to describe energy levels of the molecule. The description of those models and the expressions of those energies is beyond the scope of this manuscript, but can be found in [START_REF] Aquilanti | Hund's cases for rotating diatomic molecules and for atomic collisions: angular momentum coupling schemes and orbital alignment[END_REF][START_REF] Huber | Molecular Spectra and Molecular Structure I[END_REF]. Vibrational numbers can be added in parenthesis after the main symbols, with 𝜈 ′ for the excited state, and 𝜈 ′′ for the ground state.

The symbol 𝑄 implies that there is no modification of the rotational quantum number associated with those transitions. No superscript is given, then ∆𝑁 = ∆𝐽 = 0. The number in parenthesis is the rotational number of the ground state (and thus the excited state since ∆𝐽 = 0) and is equal to 5 in the case of the OH LIF and 29.5 in the case of NO-LIF. Only one subscript is given, then 𝛼 = 𝛽 = 1 and no change of the spin multiplicity occurs.

The energy of a photon allowing a certain transition between two states is thus the difference between electrical, vibrational and rotational energy. The Planck's law gives the following relation:

𝛥𝐸 = 𝐸 𝑢𝑝𝑝𝑒𝑟 𝑠𝑡𝑎𝑡𝑒 -𝐸 𝑙𝑜𝑤𝑒𝑟 𝑠𝑡𝑎𝑡𝑒 = ℎ𝜈 (20)
With h the Planck's constant, h=6.62607 x10 -34 J.Hz -1

Population of energy levels

By assuming a thermodynamic equilibrium in the probe volume, the population distribution on the different quantum states follow the Boltzmann distribution:

𝑓 𝑏 = 𝑁 𝑖 𝑁 = 𝑔 𝑖 𝑒 - 𝐸 𝑖 𝑘 𝑏 𝑇 𝑄 (21)
With 𝑁 𝑖 the number of molecules on the energy level 𝑖, N the total population, 𝐸 𝑖 the energy of the rovibrational level, 𝑘 𝑏 the Boltzmann constant, 𝑇 the temperature, 𝑔 𝑖 the degeneracy of the level 𝑖, i.e. the number of individual states with a common energy, and Q the partition function. For diatomic molecules, and T>300K, the rotational partition function can be simplified as:

𝑄 𝑟𝑜𝑡 = 1 𝜎 𝑘 𝑏 𝑇 ℎ𝑐𝐵 𝜈 (23)
With 𝜎 a symmetry number, 𝜎 = 1 for heterogeneous molecules, such as NO and OH, and 𝜎 = 2 for homogeneous molecules sur as 𝑂 2 .

The expression of the vibrational partition function depends on the vibration model used. The most common and simple model for diatomic molecules is the Simple Harmonic Oscillator (SHO) approximation, which consider the diatomic molecule as two mass oscillating around an equilibrium distance as if they were connected by a linear spring. This leads to the convenient result that a SHO has a constant energy spacing between adjacent quantum states, and that the vibrational quantum number can only change by 1, 𝛥𝜈 = 𝜈 ′ -𝜈 ′′ = 1.

The rotational partition function can then be expressed as :

𝑄 𝑣𝑖𝑏 = [1 -𝑒𝑥𝑝 ( -ℎ𝑐𝜔 𝑒 𝑘 𝑏 𝑇 )] -1 𝑒𝑥𝑝 ( -ℎ𝑐𝜔 𝑒 2𝑘 𝑏 𝑇 ) (24)
At last, by considering that electronics levels other than the ground configuration have sufficiently high energies to not contribute to the electronic partition function in cases studied here, the electronic partition function can be approximated as the degeneracy of the electronic configuration:

𝑄 𝑒𝑙𝑒𝑐 = (2𝑆 + 1)(2 -𝛿 0,𝛬 ) (25)
With 𝛿 0,𝛬 = 1 if 𝛬 = 0, and 𝛿 0,𝛬 = 0 otherwise, as for the OH and NO ground state.

Spectral Line shapes

The spectral line associated with a transition is, in reality, not a Dirac peak, but this one is broadened, due to the uncertainty of Heisenberg, that relates the uncertainty of energy levels to their lifetimes.

∆𝐸 𝑖 ≥ ℎ 2𝜋𝜏 𝑖 [START_REF] Nickolaus | Development of a Lean Direct Fuel Injector for low emission aero gas turbines[END_REF] with 𝜏 𝑖 the lifetime of level 𝑖. There exists an uncertainty for both the upper and lower state lifetimes, thus, by combining both, the range of energies over which the transition can occur is given by:

C2 -Confidential 𝛥𝜈 = 1 2𝜋 ( 1 𝜏 ′ + 1 𝜏 ′′ ) (27)
The main mechanism responsible for this broadening are described thereafter.

Natural Broadening

If no interaction with other atoms occurs, the lifetime of an atom in a particular state is determined by spontaneous emission. It is described by a Lorentzian lineshape function derived by modelling the atomic system as a damped oscillator.

𝜙 𝐿 (𝜈) = 1 2 𝛥𝜈 (𝜈 -𝜈 0 ) 2 + ( 𝛥𝜈 2 ) 2 (28)
With 𝛥𝜈 the Full Width at Half Maximum (FWHM) of the lineshape.

Doppler Broadening

The random motion of a molecule can induce a shift in the frequency at which it will absorb a photon. The velocity has a non-null component is the same direction as the laser beam propagation axis. The distribution of the random velocities of molecules in a gas are described by the Maxwellian velocity distribution, which means that molecules of same velocity can be grouped in velocity classes. This leads to a line-shape function with a gaussian form:

𝜙 𝐷 (𝜈) = 2 𝛥𝜈 𝐷 ( ln 2 𝜋 ) 1 2 exp (-4 ln 2 ( 𝜈 -𝜈 0 𝛥𝜈 𝐷 ) 2 ) (29)
with 𝛥𝜈 𝐷 the Full Width at Half Maximum (FWHM) of the line-shape, defined as:

𝛥𝜈 𝐷 = 𝜈 0 ( 8𝑘 𝑏 𝑇 ln 2 𝑚𝑐 2 ) 1/2 (30)

Collisional Broadening

Collisions between molecules is another way of deexcitation that leads to further decreasing the lifetime 𝜏 𝑖 of the upper state which, according to Eq. 26, this broaden the line-shape. The number of collisions of a molecule B with all molecules A is given by

𝑍 𝐴𝐵 = 𝑛 𝐴 𝜋𝜎 𝐴𝐵 2 𝑐̅ (31)
With 𝑛 𝐴 the number density of species A. 𝑐̅ has been defined previously with eq. 17, and so is 𝜎 𝐴𝐵 with eq. 16.5.

By summing this number of collisions over all the possible species (index 𝑎), the total collision frequency can be expressed such as:

𝑍 𝐵 = ∑ 𝑛 𝐴 𝜋𝜎 𝐴𝐵 2 ( 8𝑘 𝑏 𝑇 𝜋𝜇 𝐴𝐵 ) 1/2 𝐴 (32) C2 -Confidential
The FWHM due to collisional broadening is given by: ∆𝜈 𝑐 = 𝑍 𝐵 𝜋 [START_REF] Johnson | Combustor Mixer Having Plasma Generating Nozzle[END_REF] With the ideal gas law, the FWHM can be expressed as :

∆𝜈 𝑐 = 𝑃 ∑ 𝑋 𝐴 𝐴 𝜋𝜎 𝐴𝐵 2 ( 8 𝜋𝜇 𝐴𝐵 𝑘 𝑏 𝑇 ) 1/2 (34)
Thus, the FWHM is directly proportional to the pressure and species concentration. The collisional broadening coefficient 2𝛾 𝐵-𝐴 can be introduced such as

2𝛾 𝐵-𝐴 = 𝜋𝜎 𝐴𝐵 2 ( 8 𝜋𝜇 𝐴𝐵 𝑘 𝑏 𝑇 ) 1/2 (35)
This coefficient is determined experimentally for specific transitions and collisional partners.

Voigt Profiles

In most cases, the Doppler and the Collisional broadening are the two major components of the line-shape broadening, and the natural broadening can be neglected. By making the hypothesis that those two effects can be decoupled, it is possible to apply the collisional broadening to each velocity class described by the doppler broadening, leading to a convolution of the two broadening effects:

𝜙 𝑉 (𝜈) = ∫ 𝜙 𝐷 (𝑢)𝜙(𝜈 -𝜈 0 )𝑑𝑢 +∞ -∞ = 𝜙 𝐷 (𝜈 0 )𝑉(𝑎, 𝜔) (36) 𝑉(𝑎, 𝜔) = 𝑎 𝜋 ∫ exp (𝑦𝑟) 𝑎 2 + (𝜔 -𝑦) 2 𝑑𝑦 +∞ -∞ (37)
where 𝑉(𝑎, 𝜔) is the "Voigt function", and 𝜙 𝐷 (𝜈 0 ) the line-center magnitude, defined from eq. 28 at 𝜈 0 . 𝑎 is called the Voigt parameter and is a ratio of collisional and doppler broadening, b is the nondimensional line position, and y is an integral variable. They are defined as:

𝑎 = √ln 2 𝛥𝜈 𝐶 𝛥𝜈 𝐷 (38) 𝜔 = 2√ln 2 (𝜈 -𝜈 0 ) 𝛥𝜈 𝐷 (39) C2 -Confidential 𝑦 = 2𝑢√ln 2 ∆𝜈 𝐷 (40)
Quantification of LIF signal requires the resolution of the Voigt profiles. It is done with the algorithm proposed by [START_REF] Humlicek | Optimized computation of the voigt and complex probability functions[END_REF].

3.2 Laser Induced Fluorescence

LIF Modeling: Quasi-Two-level System

Fluorescence is the spontaneous emission of radiation by which an atom or molecule relaxes from a higher energy level to a ground state level. The higher energy level can be excited optically, thermally or chemically. With laser-induced fluorescence (LIF), a given rotational and vibrational level in a higher electronic state is populated with a pulsed narrow-band laser excitation, which makes feasible a selective and instantaneous excitation of a target molecular species. Such a state is unstable and the population is rapidly redistributed by rotational energy transfer over the neighboring rotational states, before it decays by emitted a spontaneous light to the rotational and vibrational levels in the ground electronic state. The fluorescence signal does not only occur at the excitation wavelength (resonance fluorescence), but also at other wavelengths essentially shifted to longer wavelengths. Conveniently, spectrally shifted fluorescence is usually recorded, in order, for instance, to suppress spurious signals from laser light scattered from the walls of the combustion chamber as well as Mie scattering from liquid fuel droplets or soot particles. Unfortunately, fluorescence radiation is not the only path of deexcitation, and other non-radiative mechanisms must be considered. Indeed, the molecule can also return to its initial energy level via collisional energy transfer processes with the molecules constituting the medium to study. This energy transfer process is called collisional de-excitation or "quenching". A second non-radiative process to be considered is the dissociation of the molecule. When this process occurs between a stable electronic state and a repulsive electronic state, the phenomenon is called predissociation.

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Based on these common considerations, Laser-induced Fluorescence can be viewed as a quasitwo-step process, an excitation of the probed molecules by absorption of resonant photons, followed by an emission of a spontaneous radiation. A schematic of the quasi-two-level model is displayed in Fig. 19 for a typical diatomic molecule with transitions occurring across two electronic states. This physical approach enables a link between the fluorescence signal and the physical quantities of the medium to be analysed by using key spectroscopic parameters such as absorption coefficients and collisional de-excitation rates. In the case of the absorption of photons from a monochromatic laser beam of low energy (i.e. avoiding the perturbation of the ground electronic state), the fluorescence emission can be viewed as the result of several independent physical processes for which several energy transfer pathways govern their efficiency. Thus, the fluorescence signal (Sf) can then expressed as the product of the four terms specified in the following equation:

S 𝐹 ∝ { 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 × 𝐴𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 × 𝐸𝑚𝑖𝑡𝑡𝑒𝑑 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 × 𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛
The second term includes the absorption process directly coupled with the laser/molecule excitation. It is a function of the absorption coefficient describing the pumping efficiency of a specific molecule. This coefficient is defined by the Beer-Lambert law [START_REF] Hanson | Spectroscopy and Optical Diagnostics for Gases[END_REF] which states that the intensity of a monochromatic radiation of frequency () and spectral intensity (𝐼 𝜈 ) after passing through a uniform gaseous medium of length 𝐿 (cm) and concentration of absorbing species Cabs (cm -3 ) can be expressed by

𝐼 𝜈 (𝐼 𝜈 ) 0 = 𝑇 𝜈 = 𝑒𝑥𝑝(-𝐶 𝑎𝑏𝑠 𝜎 𝜐 𝐿) (41)
Fig. 19 : Ground and excited energy states, vibrational energy levels and the OH Q1( 5) transition

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𝜎 𝜐 (cm 2 ) is the effective absorption cross-section of the absorbing species and 𝑇 𝜈 is the transmittivity of the medium.

The third term is relative to the radiative and non-radiative de-excitation mechanisms. It is defined as the fraction of the excited molecules returning to their ground electronic energy state by emitting fluorescence. The radiative de-excitation process is relative to the radiative decay which constitutes the main mechanism for LIF signal production. This one can be expressed via the Einstein 𝐴 21 coefficient (s -1 ) representing the fluorescence rate. Molecules in specific excited rovibrational energy levels can also relax down to the ground electronic state by collisions with all the molecules of the medium. The overall rate of de-excitation by collisions 𝑄 21 (s -1 ), also called quenching, can be expressed as

𝑄 21 = ∑ 𝜒 𝑖 𝜎 𝑖 𝑣 𝑖 𝑖 = 𝑛 𝑇 𝑣 𝑎 ∑ 𝜒 𝑖 𝜎 𝑖 √(1 + 𝑚 𝑎 𝑚 𝑖 ) 𝑖 (42)
The term 𝜎 𝑖 is the quenching cross-section upon collision with the species i. 𝑣 𝑖 is the relative average velocity of the collisional partners, 𝑣 𝑎 is the average molecular velocity of the absorbing species while 𝑚 𝑎 and 𝑚 𝑖 are the masses of the absorbing molecular species and collisional partners respectively. The density of the medium (𝑛 𝑇 ) is function of the ratio P/T (P is the pressure and T the temperature) and the molecular velocity has a dependence on (T) 1/2 . Assuming that the effective cross-section of collisions is independent of temperature, the collisional quenching rate is linearly dependant on pressure and its temperature dependence varies as 1/(T) 1/2 .

In addition to the energy transfer processes described above, other non-radiative de-excitation mechanisms act on the relaxation of the population of the excited states. Absorption of additional photons for molecules in the upper state can result in photoionization or photodissociation of the molecules (𝑄 𝑝ℎ𝑜𝑡𝑜 in s -1 ). Excitation of the molecule into a rovibrational level of a bonding state above the dissociation energy or onto a potential surface crossing to an anti-bonding state can also result in a dissociation of the molecule. The last effect is referred to the predissociation factor 𝑄 𝑃 (s -1 ). Furthermore, intersystem-crossing occurring between two overlapped potential surfaces can influence the dynamics of LIF. These mechanisms are refereed with the internal conversion coefficient 𝑄 𝑖𝑐 (s -1 ) and the intersystem crossing coefficient 𝑄 𝑖𝑠𝑐 (s -

). All these mechanisms result in a loss of fluorescence efficiency, which yields the following expression for the fluorescence efficiency F:

Φ 𝐹 = 𝐴 21 𝐴 21 + 𝑄 21 + 𝑄 𝑝ℎ𝑜𝑡𝑜 + 𝑄 𝑃 + 𝑄 𝑖𝑐 + 𝑄 𝑖𝑠𝑐 [START_REF] Eckbreth | Laser Diagnostics for combustion temperature and species[END_REF] With the knowledge of the different physical parameters detailed above, the quasi-two-step model can be used to predict the fluorescence signal. Although this modelling is relevant because it introduced the effects of various physical quantities on the behaviour of the fluorescence signal, the use of a monochromatic radiation with low energy for the excitation of molecules reduces its use. Moreover, this model cannot include others physical mechanisms C2 -Confidential such as, for instance, the stimulated emission. To improve the fluorescence modelling, it is therefore important to develop more realistic models such as the multilevel models. Among these models, the simplest model to use is the two-level model because effects of stimulated emission and saturation mechanisms can be included.

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LIF Equation: Two-level Derivation

A schematic of the two-level model in which the medium is described by two electronic energy levels is displayed on Fig. 20 The rate constants for the optical and collisional processes connecting the upper and lower energy levels E1 and E2, are displayed.

The essence of the model consists of the non-transient rate analysis of the energy balance occurring between both energy levels. The basic energy transfer processes considered in this model are absorption, stimulated emission, collisional quenching and spontaneous emission. If required, the non-radiative de-excitation processes detailed above can also be included. The rates of absorption and stimulated emission are respectively 𝑏 12 𝐼 𝜈 and 𝑏 21 𝐼 𝜈 , where 𝑏 12 and 𝑏 21 (cm 2 .cm -1 .J -1 ) are the Einstein coefficients and 𝐼 𝜈 is the spectral intensity of the laser beam (J -1 . cm -1 . s -1 ). 𝑄 21 (s -1 ) is the collisional quenching rate while 𝐴 21 (s -1 ) is the Einstein coefficient representative of the spontaneous emission rate.

Two following assumptions are considered in a non-transient approach.

-First, the excitation laser beam has a uniform temporal profile with a duration 𝜏 (s). Its integrated intensity is expressed from the density of energy (𝐸) of the laser beam (J.cm -2 ) by

𝐼 = 𝐸 𝜏 (44)
-Second, assuming a spectral width of the laser beam to be broader than the absorption width of a rotational transition, the spectral intensity 𝐼 𝜈 of the laser beam is expressed from the spectral width of the laser beam (Δ𝜈 𝐿 ) as However, these assumptions are not relevant in many cases. Indeed, the overlap between the absorption line transition and the laser shape must be computed explicitly by convolving the two profiles. 𝐼 𝜈 is then expressed as 𝐼 𝜈 = 𝐼 ∫ 𝜙(𝜈, 𝜈 12 , Δ𝜈 12 )𝑔(𝜈, 𝜈 𝐿 , Δ𝜈 𝐿 )𝑑 𝜈 = 𝐼𝐺(𝜈 𝐿 , 𝜈 12 , Δ𝜈 12 , Δ𝜈 𝐿 ) +∞ -∞ [START_REF] Meyer | Simultaneous PLIF/PIV investigation of vortex-induced annular extinction in H2-air counterflow diffusion flames[END_REF] 𝜙(𝜈, 𝜈 12 , Δ𝜈 12 ) (cm) is the absorption line profile centered at 𝜈 12 while 𝑔(𝜈, 𝜈 𝐿 , Δ𝜈 𝐿 ) is the laser line profile centered at 𝜈 𝐿 . 𝐺(𝜈 𝐿 , 𝜈 12 , Δ𝜈 12 , Δ𝜈 𝐿 ) (cm) is then the spectral profile resulting from the overlap of both profiles. The integrals of the absorption line and the laser beam profiles over the frequency range are normalized.

𝐼 𝜈 = 𝐼 Δ𝜈 𝐿 (45)
The rate equations for the population number densities of the two energy states, 𝑁 1 and 𝑁 2 can be expressed by

𝑑𝑁 1 𝑑𝑡 = -
𝑁 𝑡𝑜𝑡𝑎𝑙 = 𝑁 1 + 𝑁 2 = 𝑁 1 0 (49)
𝑁 1 0 is the population of the lower energy level before the excitation by the laser beam. 𝑁 2 is assumed to be negligible, meaning that level 2 is not populated by energy transfer through collisions between molecules and/or chemical reaction before the laser excitation. In the case where the laser pulse is temporally slow with respect to the non-radiative de-excitation mechanisms, the solving of the rates equations enables the determination of the rate of change of the population of molecules in level 2 (𝑁 2 ) as:

𝑁 2 = 𝑁 1 0 𝑏 12
𝑔 1 and 𝑔 2 are the degeneracies of the two energy levels. Two limits of operation can be envisioned using this type of steady-state analysis.

When the laser excitation energies are weak (i.e., representing a weak field regime, 𝐼 𝜈 << 𝐼 𝜈 𝑠𝑎𝑡 ), the expression for the fluorescence signal per unit time and per unit volume can be simplified in

𝑆 𝐹 = 𝑁 1 0 𝑏 12 𝐼 𝜈 ( 𝐴 21 𝐴 21 + 𝑄 21 ) (54)
We then find the result of the absorption/fluorescence model previously described with a signal evolving linearly with the laser energy. The term

𝑆 𝐹 = 𝑁 1 0 𝑏 12 𝐴 21 𝑏 12 + 𝑏 21 (55)
The fluorescence signal is then function of the population of the ground energy state. This particular case is of special interest because the fluorescence signal can be recorded without knowing the quenching rate (𝑄 21 ). However, its practice remains limited because the saturation of the excited energy level is never reached. Indeed, the quasi-Gaussian spatial distribution of the energy profile of the laser beams delivered by common laser sources leads to a reduction of energy at the periphery of the laser beam profile. This contributes to the attenuation of the saturation of the excited level in the whole measurement volume. For this reason, the weak field regime remains the most efficient operating regime for laser-induced fluorescence measurements. This regime has been selected for my experiments.

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To simulate the fluorescence signals, it is also necessary to integrate the fluorescence signal over the duration of the laser pulse and to use the optical characteristics of the detection system i.e. the solid collection angle () and the optical efficiency () of the detection system. The collected fluorescence signal representing the number of fluorescence photons emitted during the duration of the laser pulse t and in a probed volume 𝑉, then becomes: By introducing the line strength of the molecular absorption transition 𝑘 𝑉 " 𝐽 " 𝑉 ′ 𝐽 ′ (cm 2 .atm -1 ) with the Einstein coefficient 𝑏 12

𝑆 𝐹𝑐𝑜𝑙𝑙 =
𝑏 12 𝑁 1 0 = 1 ℎ𝑐 𝑘 𝑉 " 𝐽 " 𝑉 ′ 𝐽 ′ 𝑃 𝑎𝑏𝑠 (57)
The fluorescence signal collected on the detection system is:

𝑆 𝐹𝑐𝑜𝑙𝑙 = 1 ℎ𝑐 𝑘 𝑉 " 𝐽 " 𝑉 ′ 𝐽 ′ 𝑃 𝑎𝑏𝑠 𝐸𝐺(𝜈 𝐿 , 𝜈 12 , Δ𝜈 12 , Δ𝜈 𝐿 )𝑉 ( 𝐴 21 𝐴 21 + 𝑄 21 )   4𝜋 (58)
Pabs being the partial pressure of the absorbing species (atm -1 ).

Multi-level Fluorescence

At this level of knowledge, the two-level model therefore is a useful tool for processing the fluorescence signals. However, more than two energy levels are involved in the LIF process. In particular, when the higher energy level corresponds to an excited vibrational state, vibrational energy transfer (VET) can also lead to population redistributions over the different rotational states of the excited electronic state, causing a fluorescence emission from and to a large number of sub-levels. The same reasoning can be extended to the case of rotational population redistributions arising from rotational energy transfer (RET). A multi-level model with a number of energy states larger than 2 is then required to simulate the fluorescence signals with accuracy.

As shown previously, the collisional molecules can induce energy transfer that broaden the fluorescence line-shape. This means that rotational or/and vibrational energy transfer occurs, that will be denominated RET and VET respectively. The rates at which those transitions occur are similar to the one defined previously in eq. 16:

𝑄 𝑅𝐸𝑇 = 𝑝 𝑘 𝑏 𝑇 ∑ 𝑥 𝑖 𝑐̅ 𝜎 𝑅𝐸𝑇,𝑖 𝑖 (59)
The VET rate is defined the same way. This leads to a more detailed rate equation for population 𝑛 2 .

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With these two mechanisms, molecules undergo transitions that put them in energy levels adjacent to the upper or lower one, which adds more deexcitation paths, but at the same time, those adjacent states undergo transition back to the upper and lower levels. The associated rate equation needs to consider both gains and losses of those new transitions. For the sake of clarity, fluorescence and the different quenching, electronic, vibrational, and rotational, will be denoted with the symbol 𝑍. The rate equation of the level k is the cumulation of all excitation and deexcitation processes that can occur on the 𝑁 energy levels considered, with 𝑖 the index associated to those levels, and 𝑙 the index associated to the lower states.

𝑑𝑛 𝑘 𝑑𝑡 = ∑ 𝑍 𝑖→𝑘 𝑛 𝑖 𝑖 + ∑ (𝑊 𝑖→𝑘 𝑛 𝑙 -𝑊 𝑘→𝑙 𝑛 𝑘 ) 𝑖𝑝𝑢𝑚𝑝 -∑(𝑍 𝑘→𝑖 𝑛 𝑘 ) 𝑖 (60)
For the two-level model, the hypothesis of stationarity has been made, yet, it is rarely satisfied in experimental conditions, since laser pulses are often shorter than the characteristic time of deexcitation processes. This means that populations of considered levels are time dependent, and thus, the previous equation needs to be time integrated over the pulse duration.

∫ 𝑑𝑛 𝑘 𝑑𝑡 𝑑𝑡 = ∑ 𝑍 𝑖→𝑘 ∫ 𝑛 𝑖 𝑑𝑡 𝑖 + ∑ (𝑊 𝑖→𝑘 ∫ 𝑛 𝑖 𝑑𝑡 -𝑊 𝑘→𝑖 ∫ 𝑛 𝑘 𝑑𝑡) 𝑖𝑝𝑢𝑚𝑝 -∑ (𝑍 𝑘→𝑖 ∫ 𝑛 𝑘 𝑑𝑡) 𝑖 (61)
This equation can be written with a matrix formalism, with 𝒁 the transfer mechanism matrix, 𝑵 the integrated populations, and 𝑾 the induced excitation and deexcitation due to laser interaction. Matrix are written as 𝒁(𝑖, 𝑗) = 𝑍 𝑖→𝑗 𝒁 × 𝑵 = 𝑾 [START_REF] Humlicek | Optimized computation of the voigt and complex probability functions[END_REF] The inversion of this system gives the integrated populations of all the considered states. This allow the computation of the fluorescence signal emitted by molecules undergoing a transition from an excited state 𝑖 𝐴 towards a ground state 𝑖 𝑋 . The total fluorescence signal is obtained by summing the fluorescence from all the considered states.

∫ 𝑆 𝑓𝑙𝑢𝑜 (𝜆)𝑑𝑡 = ∑ ∫ 𝑆 𝑓𝑙𝑢𝑜 (𝑖 𝐴 → 𝑖 𝑋 )𝑑𝑡 𝑖,𝑘 (63)
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OH laser-induced fluorescence

Under reactive conditions, the presence of radicals within the flame front allows the position of the reaction zone and its intensity to be identified using optical measurement techniques. The determination of the hydroxyl radical (OH) makes it easy to visualize the flame front and the distribution of the burnt gases within the reactive flow. Indeed, this chemical species is present in large quantities in the reaction zone but also in the burnt gases. As the life time of this radical is relatively long in high-temperature environments (of the order of several ms), it is possible to probe this molecule in the flowfield.

A simple diagnostic, easy and quick to implement, consists in recording the chemiluminescence (natural emission of the flame) of the OH radical and allows a qualitative analysis of the flame shape and the reaction zone. This diagnostic, even if it was used during this thesis, will not be developed in this section. Indeed, this measurement method has the disadvantage of integrating the OH radical emission signal along the line of sight. As a result, the exploitation of these images is very limited and does not allow to obtain a sufficient spatial resolution to detect and analyze detailed structures of the flame.

In contrast, the laser-induced fluorescence diagnostic in a measurement plane allows for a good spatial resolution at a given time. Indeed, PLIF can be considered as an instantaneous measurement technique because the decay time of the fluorescence is very short compared to the characteristic times of chemical reactions. Moreover, as the density of OH molecules in these regions is high, the fluorescence signal is generally very intense, which makes it possible to easily exploit and discretize the different areas of the images acquired by OH-PLIF. The visualization of the OH radical by the PLIF diagnostic can also be used to obtain quantitative information, in particular on the concentration of the species probed and the temperature of the medium, when the spectroscopic data of the latter are known, which is the case of the OH radical.

The objective of this section is not to re-demonstrate the usefulness of OH-PLIF but to directly use this technique, whose effectiveness has been widely used in combustion systems, and to focus on the interpretation of the images obtained. A reminder of the state of the art on the advantages and post-processing of PLIF is proposed in order to present the bases on which this work rests.

Our efforts will then focus on the development of effective tools to qualify the combustion resulting from the fuel/air mixture described above, in a qualitative and quantitative manner. Indeed, there are few studies allowing to analyze the combustion process in a precise way at the outlet of the LP injection system as well as the effect of the physical input parameters governing this process, in the aeronautical conditions of high pressure/high temperature two-phase injection. The methodologies developed will also be critically analyzed with respect to measurement accuracy.

The use of PLIF on the OH radical is very common in combustion studies. This laser measurement technique is relatively simple and inexpensive to implement in simple optical configurations, while providing images with large signal-to-noise ratio (SNR) and good spatial resolution. Indeed, the OH radical is mainly produced in the reaction zone but also persists in the burnt gases in premixed flames. The particularly long high-temperature lifetime (about 3 ms [START_REF] Sadanandan | Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor[END_REF]) of this radical makes it possible to detect it in burnt gases far from the reaction zone, even though its reactivity C2 -Confidential with oxygen is very efficient. The analysis of OH-PLIF images allows us to discriminate the fresh gas zone from the burnt gas and to trace the usual and important quantities of combustion models such as the advancement variable or the probability function of the flame surface density [START_REF] Deschamps | SURFACE DENSITY MEASUREMENTS OF TURBULENT PREMIXED FLAMES IN A SPARK-IGNITION ENGINE AND A BUNSEN-TYPE BURNER USING PLANAR[END_REF].

In this study, this laser diagnostic will be applied in a high-pressure, high-temperature reactive environment. As the fluorescence efficiency is reduced when increasing the pressure, it is therefore necessary to select the absorption line of the OH radical with the best fluorescence yield to optimize the SNR. Many studies using OH-PLIF to describe the flame structure in combustion chambers have been conducted. There are also several possible strategies to excite and collect the fluorescence signal of the OH radical. However, in this current study, the Q1(5) rotational line with excitation at 282.75 nm (vacuum wavelength) will be used. Indeed, this absorption line has several advantages. Firstly, the Q1 transitions (ΔJ=1) of the 𝐴 2 𝛴 + ← 𝑋 2 𝛱 electronic transition are the most intense in terms of absorption. Secondly, the absorption intensity of the rotational lines around J' = 5 are the least sensitive to temperature [START_REF] Bresson | Techniques d ' imagerie quantitatives : fluorescence induite par laser appliquée aux écoulements et aux combustions To cite this version[END_REF], which allows us to consider, as a first approximation, that the absorption and fluorescence yields are weakly function of temperature variations. Finally, this rotational line is spectrally isolated, which ensures that the fluorescence from the OH molecule arises from a single rovibronic transition. Numerous spectroscopic studies were performed on the OH radical because of its importance in many areas of physics. A detailed study has been carried out by Bresson et al [START_REF] Bresson | EXPERIMENTAL INVESTIGATION OF A SUPERSONIC ROCKET ENGINE PLUME USING OH * -EMISSION[END_REF] allowing the absorption, emission and fluorescence spectra to be calculated precisely, particularly for the Q1(5) transition. The knowledge of these quantities allows a quantitative analysis of the fluorescence signals and enables the molar fraction of the OH radical [START_REF] Bresson | EXPERIMENTAL INVESTIGATION OF A SUPERSONIC ROCKET ENGINE PLUME USING OH * -EMISSION[END_REF] or also the temperature of the burnt gases [START_REF] Heinze | PLIF Thermometry Based on Measurements of Absolute Concentrations of the OH[END_REF] to be determined, depending on the electronic transition studied.

In the reaction zone, the amount of OH radical produced increases significantly and makes this chemical species useful for flame front detection. The position of the flame front can therefore be extracted from the OH-PLIF images. Generally, this position is obtained by thresholding the images and then binarizing them to obtain the interface separating the fresh and burnt gases. However, this technique is not optimal and sometimes leads to erroneous flame front detections. On the other hand, the gradient obtained on the OH-PLIFPLIF-OH images is a more accurate criterion for improving this detection. A good example of this is the work of Hartung et al [START_REF] Hartung | Effect of heat release on turbulence and scalar turbulence interaction in premixed combustion[END_REF], who use this methodology to obtain accurate flame contour detection and analysis. In this study, flame front detection is achieved after applying a non-linear scattering filter and a binning operation to improve the SNR. The flame front is then defined by the maximum of the OH gradient, determined using the Canny boundary detection algorithm. These optimized optical configurations provide very good contour resolution and quantities such as flame curvature can also be calculated.

In order to evaluate the accuracy of the flame front position by analyzing the OH-PLIF images, many numerical and experimental studies were performed [START_REF] Paul | PLANAR LASER-INDUCED FLUORESCENCE IMAGING OF FLAME HEAT[END_REF][START_REF] Ayoola | Spatially resolved heat release rate measurements in turbulent premixed flames[END_REF][START_REF] Pfadler | Flame front detection and characterization using conditioned particle image velocimetry[END_REF][START_REF] Mulla | Heat release rate estimation in laminar premixed flames using laserinduced fluorescence of CH 2 O and[END_REF]. A methodology for postprocessing based on the calculation of the image gradient and obtaining the reaction zone is proposed by Sadanandan et al [START_REF] Sadanandan | Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor[END_REF]. It would therefore seem that the calculation of the OH gradient image is a promising method for discriminating reaction zones from zones of simple mixing between fresh and burnt gases. This process has already been used to analyze the interactions between turbulence and flame [START_REF] Boxx | Temporally resolved planar measurements of transient phenomena in a partially pre-mixed swirl flame in a gas turbine model combustor[END_REF][START_REF] Steinberg | Flow -flame interactions causing acoustically coupled heat release fluctuations in a thermo-acoustically unstable gas turbine model combustor[END_REF][START_REF] Stopper | Experimental study of industrial gas turbine flames including quantification of pressure influence on flow field , fuel / air premixing and flame shape[END_REF].

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Kerosene laser-induced fluorescence

As presented earlier, the equivalence ratio (or mixing fraction) is the scalar quantity that controls a large part of the combustion phenomena, including pollutant emissions. In the case of a twophase combustion, the local equivalence ratio can vary greatly from one place to another in the combustion chamber. Even if the injection is in a lean regime, droplets are still present at the outlet of the injector and in certain areas, there may be significant pockets of kerosene in vapor phase. The combustion regime in these positions will then be locally rich even if the target of the lean-burn injection systems is globally weak (less than 0.7). It is therefore interesting to know the distribution of kerosene vapor within a combustion chamber. This concentration distribution thus makes it possible to better characterize the operation of these new injection systems and to compare them. In addition, these measurements can be correlated with other observed physical parameters such as the position of the reaction zone (instantaneous or average), the intensity of the combustion reaction, or pollutant emissions… One way of obtaining this kerosene mass concentration field is to use the Planar Laser Induced Fluorescence lase diagnostic on fluorescent molecules naturally present in the fuel composition following an optical excitation in the mid-UV range. From the fluorescence signals, it is possible to visualize the areas where vapor and liquid phases fuel are present. Using a precise study of the evolution of the fluorescence signal as a function of the thermodynamic conditions performed by B. Rossow [START_REF] Rossow | Processus photophysiques de molécules organiques fluorescentes et du kérosène applications aux foyers de combustion : applications aux foyers de combustion To cite this version[END_REF], it is proposed in the framework of this study to measure the kerosene distributions at the injector outlet and to correlate this kind of information with similar distributions obtained on the OH radical and the NO pollutant under real operating conditions.

The application of PLIF with mid-UV fuel excitation has been extensively studied in the automotive and internal combustion engine fields [START_REF] Zhao | [END_REF]. These studies have been supported in the context of compliance with the pollutant emission standards that have long been present in this sector. For example, two-color PLIF allows the temperature and concentration field of a specific species in the fuel/air mixture to be obtained simultaneously using dual laser excitation at two different wavelengths [START_REF] Thurber | Simultaneous imaging of temperature and mole fraction using acetone planar laser-induced fluorescence[END_REF]. These studies are usually carried out with substitute fuels or fluorescent tracers and not with real fuels (such as gasoline or diesel).

In the context of the study of the operation of aeronautical combustors, many studies were performed on the characterization of the fuel injection. These studies generally focus on droplet size distribution and velocity, air temperature or the impact of fuel flow distribution [START_REF] Michou | EXPERIMENTAL STUDY OF LEAN PREMIXED AND PREVAPORISED TURBULENT SPRAY COMBUSTION[END_REF][START_REF] Becker | GT-2002-30102 LIQUID FUEL PLACEMENT AND MIXING OF A GENERIC AEROENGINE PREMIX MODULE[END_REF][START_REF] Renaud | Flame shape transition in a swirl stabilised liquid fueled burner To cite this version[END_REF][START_REF] Lantz | High-speed imaging of fuel/OH distributions in a gas turbine pilot burner at elevated pressure[END_REF], but they concentrate on the liquid phase of the fuel. Some work has focused on the spatial distribution of the vapor phase of the fuel [START_REF] Kramer | OPTIMIZATION OF THE MIXING QUALITY OF A REAL SIZE GAS TURBINE BURNER WITH INSTANTANEOUS PLANAR LASER-INDEC FLUORESCENCE IMAGING[END_REF][START_REF] Stufflebeam | Quantifying Fuel / Air Unmixedness in C2 -Confidential Premixing Nozzles Using an Acetone Fluorescence[END_REF][START_REF] Galley | Experimental Study of the Dynamics of a LPP injection System[END_REF] but most of this research uses alternative fuels. However, evaporation, diffusivity or combustion may be biased due to the different chemical composition of the fuel compared to paraffin. For example, Chong et al [START_REF] Chong | Spray flame structure of rapeseed biodiesel and Jet-A1 fuel[END_REF] compare the structure of a lean swirling two-phase flame fed with paraffin in one case and Rapeseed Methyl Esters (RME) in the other. The average gas flow structure in the combustion chamber measured is identical in both cases. However, a significant difference in the flame structure is observed and is directly related to the slower evaporation in the case of RME.

Few studies have been devoted to the study of the distribution and evaporation of kerosene in an aeronautical combustor under reactive conditions. The pioneering study by Löfström et al [START_REF] Löfström | VISUALIZATION OF FUEL DISTRIBUTIONS IN PREMIXED DUCTS IN A LOW-EMISSION GAS TURBINE COMBUSTOR USING[END_REF] shows that it is possible to visualize the vapor phase of kerosene following a laser excitation C2 -Confidential at 266 nm while a laser excitation at 532 nm allows a simultaneous visualization of the droplet position. Images of a Kerosene-PLIF campaign combined with OH-PLIF are also presented by Orain et al [START_REF] Orain | Simultaneous measurements of equivalence ratio and flame structure in multipoint injectors using PLIF[END_REF] and allow a more detailed analysis of the operation in terms of kerosene vapor distribution in a high-pressure combustion chamber (bench M1, ONERA) equipped with a SAFRAN AE (formerly SNECMA) multipoint staged injection system. In parallel, Meier et al [START_REF] Lefebvre | Atomization and Sprays[END_REF], applied Kerosene-PLIF to obtain images of the kerosene distribution in a high-pressure combustor equipped with a Rolls-Royce LDI injector.

The quantitative study of the spatial distribution of kerosene vapor is almost non-existent in the literature. Indeed, obtaining quantitative values from the fluorescence signals requires a precise knowledge of the fluorescence spectroscopy of paraffin in the vapor phase under the thermodynamic conditions of a combustion chamber. However, the studies carried out by Löfström et al [START_REF] Löfström | VISUALIZATION OF FUEL DISTRIBUTIONS IN PREMIXED DUCTS IN A LOW-EMISSION GAS TURBINE COMBUSTOR USING[END_REF] on the characterization of the mixing and evaporation of paraffin in a combustor equipped with an LPP injector using PLIF on kerosene (Jet-A1) are noteworthy. Indeed, an atmospheric pressure study of paraffin spectroscopy over the temperature range 300 -900 K was carried out in an attempt to quantify the fluorescence signal in the combustor. However, an obvious lack of information, particularly on the dependence of the fluorescence signal on oxygen quenching, did not allow quantitative measurements to be made.

The development of this measurement technique was reported by P. Baranger [START_REF] Baranger | Détection du kérosène par imagerie de fluorescence induite par laser , pour application sur foyer aéronautique To cite this version[END_REF] and B.

Rossow [START_REF] Rossow | Processus photophysiques de molécules organiques fluorescentes et du kérosène applications aux foyers de combustion : applications aux foyers de combustion To cite this version[END_REF]. The work of P. Baranger has made it possible to identify the chemical species responsible for the fluorescence of kerosene. He also proposed a measurement strategy for the excitation of kerosene that allows us to obtain the average and instantaneous concentration and temperature field of kerosene in a heated kerosene-air jet. B. Rossow completes this study by performing a detailed study of the fluorescence signals of Jet-A1 kerosene, as well as the fluorescent organic species that compose it. The study detailed the fluorescence behavior of Jet-A1 over a temperature range of 300-900 K and a pressure range of 0.1-3.0 MPa. A methodology for post-processing the fluorescence signals collected simultaneously over two different spectral ranges was proposed in order to trace the simultaneous value of the molar concentration of kerosene vapor and temperature. Recently, P. Malbois et al. [START_REF] Malbois | Quantitative measurements of fuel distribution and flame structure in a leanpremixed aero-engine injection system by kerosene/OH-PLIF measurements under high-pressure conditions[END_REF] applied this measurement strategy with success on high/pressure kerosene/air flames produced with a Lean-premixed injection system developed by Safran Helicopter Engines. In the current study, the aromatics present in the chemical composition of kerosene (mono-and di-aromatics) are excited by a laser beam at 266 nm and the resulted fluorescence located in the 300 -450 nm spectral range was detected by an ICCD camera. Suitable optical filters were used to isolate the kerosene fluorescence from fluorescence signals or spurious signals arising from the laser reflections on the walls of the combustion chamber and the light diffusion produced by the interaction of the laser beam with the optical windows.

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NO laser-induced fluorescence

This section aims to resume previous studies that successfully applied the NO-LIF diagnostic in flames.

NO has a relatively high absorption cross-section and good fluorescence efficiency, making LIF signals easy to detect. The fluorescence wavelength is spectrally shifted from the laser wavelength, which allows the use of efficient optical filters to completely eliminate Rayleigh scattering and self-absorption. In addition, the NO molecule does not exhibit radiative or collisional de-excitation parameters that are dependent on the rotation number J, which could lead to systematic measurement error. A high-fidelity numerical simulation can therefore be performed, in order to tabulate the fluorescence spectra as a function of local thermodynamic parameters such as temperature, pressure, and species concentration. This simulation, coupled with an experimental calibration, can lead to a quantitative evaluation of the NO concentration.

Pioneering work

NO-LIF was studied for a long time and numerous studies were published over the years, both on academical and semi-industrial applications. In 1980, Grieser and Barnes [START_REF] Grieser | Nitric oxide measurements in a flame by laser fluorescence[END_REF] were among the first to study the NO fluorescence in flame using LIF. The CH4/O2 flame was doped with a high NO concentration to compensate the low laser energy (~ 3µJ). Excitation spectra were measured though the fluorescence from the (𝐴 2 𝛴 + ← 𝑋 2 𝛱) transition in the 𝛾 band. Rovibronic transitions have been identified in both reactive and non-reactive medium, and showed that the presence of the flame decreased significantly the SNR. A year later, Mc Kenzie and Gross [START_REF] Mckenzie | Two-photon excitation of nitric oxide fluorescence as a temperature indicator in unsteady gasdynamic processes[END_REF] used a two-photon excitation of the (𝐴 2 𝛴 + ← 𝑋 2 𝛱) transition in the 𝛾 band to make single shot temperature measurement in cold, low density, turbulent flow seeded with low NO concentration. One of the advantages of this novel technique was the sub millimeter probe volume, while other techniques in those conditions required spatial averaging. Results were compared to thermocouple measurements, and agreed to within ±2% while averaged, while single shot measurements varied between 2.5 -4.0% from the reference value.

In 1982, Chou et al. [START_REF] Chou | Laser induced fluorescence and absorption measurements of NO in NH 3/O2 and CH4/air flames[END_REF] demonstrated the capability of the NO-LIF to measure NO concentration in reactive conditions, exploiting the same transition as the two previous works. In a NH3/O2/N2 flame of equivalence ratio 𝜙 = 0.59, absolute concentration was obtained via integrated absorption measurements. The lean condition was chosen to maximize NO and reduce NH3 concentration. The results served as calibration for quantification of NO concentration through LIF in richer CH4/air/O2 flames. After showing the practicability of the OH PLIF, Kychakoff [START_REF] Kychakoff | Flow visualization in combustion gases using nitric oxide fluorescence[END_REF] successfully set up the NO PLIF in 1984. Measurements were carried out on two atmospheric CH4/air flames doped with NO, a laminar flat flame, and a turbulent premixed flame of Reynolds number ~ 5000. The high energy laser required was produced by a Nd:YAG laser coupled with a dye laser, with a mixing of a 570nm and a 1064 beams in a KDP crystal to obtain wavelength near 225nm at 1mJ. The 𝑅 1 [START_REF] Lefebvre | Atomization and Sprays[END_REF] transition was specifically targeted to reduce the sensibility of temperature. The collected signal was filtered with a Schott glass UG-5 and an interference filter centered at 𝜆 = 252𝑛𝑚 of 24nm bandwidth. The results showed the capacity of NO PLIF to bring to light large vortex structures and variation of NO concentration due to mixing with the surrounding air.

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In 1985, Alden et al. [START_REF] Aldén | Simultaneous spatially resolved NO and NO_2 measurements using one-and two-photon laser-induced fluorescence[END_REF] a two-photon excitation at 452𝑛𝑚 to simultaneously detect both NO and NO2 in a cell. The two-photon excitation was chosen to significantly reduce the LIF lifetime of NO2 and thus, the quenching, that can be severe in the case of a one photon excitation. The fluorescence was collected in a 1D line by a diode-array detector, and results suggested the possibility of an application in flame. The same year, Seitzman et al [START_REF] Seitzman | Instantaneous temperature field measurements using planar laser-induced fluorescence[END_REF] published one of the first 2D temperature measurements by the mean of NO PLIF with a single laser frequency. Only the 𝑄 1 [START_REF] Arttic | NEWAC Publishable Final Activity Report[END_REF] transition was exploited to get a temperature dependent LIF signal, but the authors suggested that 𝑄 1 [START_REF] Arttic | NEWAC Publishable Final Activity Report[END_REF] could lead to better accuracy. Measurements were carried out on a rod stabilized CH4/air flame doped with NO and the 2D temperature map showed a maximum temperature in flame of 1830K, which is in reasonable range of the adiabatic flame temperature of 1740K. Finally, Gross and McKenzie [START_REF] Gross | Measurements of fluctuating temperatures in a supersonic turbulent flow using laser-induced fluorescence[END_REF] pursued their previous works of LIF NO thermometry [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF] by measuring temperature in a supersonic cold turbulent flow seeded with NO, with both one and two-photon 𝛾(0,0) rotational line excitation. The one photon excitation in particular showed great agreement with pitot-probe measurements with less than 2% difference. In 1988, Wodtke et al. [START_REF] Wodtke | High-C2 -Confidential sensitivity detection of NO in a flame using a tunable ArF laser[END_REF] worked on improving the sensibility of NO-LIF by using a tunable ArF excimer laser providing 100mJ at 193𝑛𝑚 to excite the rotational branches of the 𝐷 2 𝛴 ← 𝛸 2 𝛱(0,1) NO branch. A detection limit of 1ppm has been reached in a non-NO-seeded flame, and the authors suggested that this limit could be reduced down to the 1-ppb range by improving the optical filtering. Cattolica et al. [START_REF] Cattolica | Laser-fluorescence measurements of nitric oxide in low-pressure H2/O2/No flames[END_REF] proposed a method to measure both quenching rates and NO concentration in a low pressure no doped H2/O2 flame. By using laser pulses shorter than the characteristic lifetime of the excited state, the fluorescence signal can be temporally resolved. This has been done with a 225.6nm laser beam to excite de 𝐴 -𝑋(0,0) band. By knowing the 𝐴 21 constant or supposing that the quenching rate is much larger than the fluorescence rate, the quenching rate can be extracted from the photon flux decay after the laser pulse. A second measurement was made by probing the de 𝐴 -𝑋(0,1) band with a 213.3nm laser beam to quantify NO concentration.

The first 2D velocity measurement by NO PLIF were reported in 1989 by Paul et al. [START_REF] Paul | Molecular velocity imaging of supersonic flows using pulsed planar laser-induced fluorescence of NO[END_REF], in a supersonic flow at Ma ~7 seeded with NO. The velocity field was deduced from the fluorescence detection of Doppler-shifted absorption. The use of a broadband excitation instead of a narrow band allowed the use of laser with short timed UV pulses for flow stopping measurement, and efficient fluorescing species such as NO.

Many other works have been carried out and those cited here are a mere fraction of the application of NO-LIF. The next section is dedicated more specifically on the NO-LIF in reactive conditions in internal combustion engines and high-pressure flow.

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NO-LIF in high pressure / reactive conditions

In 1990, Andressen et al.. [START_REF] Andresen | Fluorescence imaging inside an internal combustion engine using tunable excimer lasers[END_REF] used different fluorescence-based techniques to measure 2D distribution of molecular species density in a linear internal combustion engine. Laser Induced Predissociative Fluorescence (LIPF) has been carried out to visualize OH and O2.

The main advantage is that the predissociation rate does not depends on the surrounding gas properties as the quenching does, so the specie concentration is directly proportional to the LIPF intensity. This is possible by working with high energy UV lasers and short-lived states so quenching does not have the time to happen or be impactful.

It is important to note that LIPF signal is inferior by a factor 10 in comparison of LIF. LIF was used for NO and iso-C3H8 visualization. While NO could not be observed in the combustion process and shortly after due to a strong absorption of the laser, 2D imaging were performed and showed inhomogeneities in the NO repartition, suggesting that the spark tends to increase the NO production, and that from cycle to cycle the spatial repartition is not constant. The authors concluded by showcasing the capability of quantitative measurement with the LIPF technique.

Brugman et al. published the first study of NO-LIF in an internal combustion engine fuel with actual diesel in 1993 [START_REF] Brugman | Laserinduced-fluorescence imaging of NO in an n-heptane-and diesel-fuel-driven diesel engine[END_REF]. Commercial grade diesel is composed of numerous species, favorizing soot formation, a lot of them showing strong absorption and fluorescence properties. This leads to a significant absorption of the laser beam and a broadband of parasite fluorescence signal. To compensate for this absorption, a high energy pulse is required, and has been obtained by the mean of an ArF laser. The energy at the engine entrance is ~20mJ at 193nm, but the real energy inside the combustion chamber is lower due to a soot formation on the optical access. Furthermore, a significant absorption of the laser beam by the O2 occur and further decrease the measurements effective energy. The LIF signal is collected from the fluorescence at 208nm of the 𝐷 2 𝛴 + (𝜈 ′′ = 0) → 𝑋 2 𝛱(𝜈 ′ = 3) transition. The results showed a notable difference of the NO production through the combustion process between n-heptane and diesel combustion, underlying the importance of carrying such measurement with realistic fuel for the study of pollutant formation in semi-industrial configuration.

A solution to the strong absorption of the UV laser beam by the O2, has been proposed by Schulz et al. in 1995 [101]. The goal was to exploit the 𝐴 -𝑋(0,2) band with a laser around 247.8nm. Measures were carried out on a Bunsen flame fueled with a C2H4/Air mixture with a KrF laser providing a laser beam of 120mJ of which 10mJ are used. With spectral filters, NO fluorescence from 𝐴 2 𝛴 + (𝜈 ′ = 0) → 𝑋 2 𝛱(𝜈 ′′ = 3) transition is collected at a wavelength shorter than the emission's one. This choice has been made to avoid potential interferences with OH and O2 fluorescence at higher wavelength. Another significant advantage of this transition is the low dependence of the LIF signal on the temperature at 2000K, with a change of 5% of the ground population per 100K. This is particularly convenient for the NO is mostly created at high temperature in C2 -Confidential combustion engines. Finally, for the vibrational excited state (𝜈 ′ = 0), the quenching rate is independent of the burned gas composition. For all those reasons, the authors highlighted the interest of using this particular transition in realistic internal to study NO formation mechanism. And so, one year later in 1996, the same authors published a practical application of the use of this transition in a semi-transparent internal combustion engine working with propane and iso-octane [START_REF] Schulz | Quantitative 2D singleshot imaging of no concentrations and temperatures in a transparent SI engine[END_REF]. The same KrF laser has been used to produce a 247.9nm laser beam to excite the 𝐴 -𝑋(0,2) NO band. The LIF measurements were combined with Rayleigh Scattering from the same laser pulse in order to get temperature distribution. The use of this wavelength instead of the most common 225nm and 193nm paths allowed an absorption of ~20% of the laser beam, instead of ~90% for the two lower wavelengths. A known amount of NO was introduced into the intake to rend possible a quantification of the NO produced. The uncertainty of this method has been estimated of ~30% by the authors, mainly due to the calculation of the quenching cross section and the overlap integral of the laser and absorption line.

The results showed that even small differences in temperature could lead to significant changes in the rate of production of NO via the Zeldovich mechanism.

Also in 1996, Knapp et al. [START_REF] Knapp | Crankangle-resolved laser-induced fluorescence imaging of NO in a spark-ignition engine at 248 nm and correlations to flame front propagation and pressure release[END_REF] applied the NO-LIF in a semi-transparent spark-ignition engine combustion chamber. As in [START_REF] Maughan | A dry low NOx combustor for the MS3002 regenerative gas turbine[END_REF], they showed the advantage of using a 247.9nm laser beam from a KrF laser instead of a 193 or 226nm one, by measuring the transmission of a broadband light emitted by a deuterium lamp. During the combustion phase, the transmission of the 226nm light is ~40%, while at 248nm it is ~80%. The results showed that NO is mainly formed in the post-flame gases, where conditions of high temperature and high residence time are reached. Measurements were made for different air fuel ratio (AFR), which pointed out that the maximum NO production is attained with an AFR~1.1 and not at stoichiometry. While the slight dilution with air can reduce combustion temperature and soot formation, the decrease of temperature is overcompensated by the increase of O presence that encourage NO production. In 1997, Brugman et al. [START_REF] Brugman | Imaging and post-processing of laser-induced fluorescence from NO in a diesel engine[END_REF] published their work on image processing of NO PLIF and potential photo-chemical effect from the use of high energy excimer lasers. They worked on a semitransparent diesel engine running on standard diesel fuel. An ArF excimer laser providing 60mJ at 193nm to probe the (𝐷 2 𝛴 + (𝜈 ′ = 0) ← 𝑋 2 𝛱(𝜈 ′ = 1)) band, a Nd:YAG laser is used to produce a 226nm beam to probe the (𝐴 2 𝛴 + ← 𝑋 2 𝛱). The two beams are temporally separated by a 40ns delay, the two LIF signals are then compared to check is differences coming from photo-dissociation are present. Previous works have a shown that soot deposit on optical access perturb the transmission of the laser sheet which can lead to underestimation of NO. To correct this bias, measurement of laser intensity distribution was made before and after the experiment to estimate the loss of transmission, and an absorption computation have been carried out. Both corrections were combined to create a correction profile. The results showed that the application of correction significantly impacted the NO repartition, and the authors underlined the importance of such correction for future studies.

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In 1998, Dec and Canaan [START_REF] Dec | PLIF imaging of no formation in a DI diesel engine[END_REF] published NO-LIF measurements in a direct injection diesel engine. The goal was to accurately study the NO formation through the whole process of combustion and exhaust. They came back to an excitation at 226.035nm to excite the 𝐴 -𝑋(0,0) band, with a collection of the fluorescence from the (0,1) to (0,4) vibrational bands between 237nm and 276nm. This fluorescence scheme has been selected to, in conjunction with optical filters, effectively collect NO fluorescence without interferences from O2 fluorescence, laser induced incandescence (LII), elastically scattered light and polyaromatics hydrocarbons (PAH). Optical filtering consisted of one or two UG5-Schott filters, which possess a sharp transition between 226 and 237nm, and a bandpass filter centered at 246nm with a bandwidth of 45nm and a 40% transmission rate, to block wavelength above 280nm. The high absorption of the 226nm laser beam was compensated with a high energy laser system, which consisted of a tripled Nd:YAG laser pumping a narrow line Optical Parametric Oscillator (OPO) whose output was doubled. The output energy was ~ 12-15mJ per pulse. Because the measurements were made at both high and low temperature and pressure, a correction of the NO signal was needed to make any comparison. To do so, a spectroscopic simulation has been carried out to estimate collisional-quenching rate and line broadening. The pressure was directly measured, but hypothesis on both temperature and composition were needed during the different phases. The temperature drop due to expansion was calculated with polytropic relations, and different scenarios of mixing were used to define limit cases of full or non-mixing. The results showed that the NO form mostly in the lean part of the diffusion flame and in the hot gases after the combustion, in which the spectroscopic correction was the most impactful.

In 1998, Schulz et al. [START_REF] Schulz | Comparative Study of Experimental and Numerical NO profiles in SI combustion[END_REF] made a numerical and experimental comparison of the 225nm and 148nm excitation scheme for NO PLIF, based on their previous works [START_REF] Schulz | Quantitative 2D singleshot imaging of no concentrations and temperatures in a transparent SI engine[END_REF] [107]. The main focus was on the temperature, quenching and O2 interference sensibility. Experiments were carried out in a semi-transparent internal combustion engine. The simulated flame was considered laminar and one dimensional, and was computed with MixFla which considered fifty-six reactive species. The computed laminar flame speed was estimated ten times slower than the estimated turbulent flame speed from a previous work, and thus, calculated NO profiles after the flame front are stretched to fit the experimental flame development. The experimental and numerical comparison showed a better agreement between the 248nm excitation LIF signal and the numerical results than with the 225nm. The later overestimate by ~5% the NO production near the flame front due to the interference of O2 fluorescence at 232nm, 241nm and 248nm. The authors concluded that the 248nm excitation scheme was more suited for NO investigation in reactive conditions, and that the 1D simulation showed promising results, but underestimated the prompt and thermal NO production. The same year, Hildenbrand and Schulz [START_REF] Hildenbrand | Laser spectroscopic investigation of flow fields and NO-formation in a realistic SI C2 -Confidential engine[END_REF] applied the 248nm excitation NO-LIF on an industrial spark-ignited engine in order to obtain quantitative information for numerical code validation. A KrF excimer laser produced a narrow band 248nm laser beam of C2 -Confidential 45mJ/pulse, and was tuned to consider the pressure broadening and the shifting of rotational lines at 7bars. The fluorescence signal was filtered through a combination of four dielectric mirrors with a narrow spectral reflection centered around 230nm, and a 0° mirror centered at 248nm with a FWHM of 10nm, notably to remove the O2 fluorescence and Rayleigh scattering. The quantification of NO concentration required several corrections of the raw signal. First the LIF was corrected with the laser intensity of each pulses, then, temperature effect was considered, authors estimated that a change of 5% of the probed rotational level occurred with a 100K difference. Quenching rate has been considered constant within 5% in the burned gases. Finally, the detection efficiency and calibration were made by introducing a precisely known NO quantity inside the chamber. The results compared the NO production through different ignition scenario, and showed that in case of an early ignition, 30° Before the Top Dead Center (TDC), the NO production was maximum, while the minimum has been found with an ignition 10° before the TDC. The authors associated this behavior with the Zeldovich mechanism.

Still in 1998, Akihama et al. [START_REF] Akihama | Laser-induced fluorescence imaging of NO in a port-fuel-injected stratified-charge SI engine-correlations between NO formation region and stratified fuel distribution[END_REF] studied the impact of the fuel stratification on NO production inside a four valve single-cylinder spark ignited engine, running with isooctane/air mixture. The engine consisted of two horizontal injectors, and a flow with a tumble ratio of 0.45 with almost no swirling motion. This allowed a homogeneous fuel distribution with the use of both injectors, or a horizontally stratified combustion when only one injector is used. The excitation at 247.9nm with a KrF excimer laser was selected. The optical filtering of the LIF signal consisted of a bandpass filter centered at 223nm of 12nm bandwidth, and an acetone based liquid filter. The resulting transmittance was around 10%. The results showed that most of the NO is produced in the 0.8≤ϕ≤1.1 equivalence ratio range, and that the exhaust NO concentration is directly proportional to the amount of fuel burning at this equivalence ratio.

In 1999, Hildenbrand et al. [START_REF] Hildenbrand | In-cylinder NO-LIF imaging in a realistic GDI engine using KrF excimer laser excitation[END_REF] compared the detection scheme of NO fluorescence excited at 248.94nm in a transparent internal combustion engine. Stratified and homogenous conditions were considered, at equivalence ratio of 0.77 and 1.0 respectively, that lead to peak average pressure of ~19bars. As in previous works, a narrowband KrF excimer laser was used to excite the 𝐴 -𝑋(0,2) band. The blue shifted NO-fluorescence was filtered through a set of four dielectrics mirrors with narrow spectral reflection centered around 230nm, and two short pass filters at 248nm to remove any interferences from Rayleigh scattering and O2 fluorescence. Furthermore, the absorption of the fluorescence signal was considered during the correction process.

For the red shifted fluorescence detection, a quartz cuvette filled with butyl-acetate was used to suppress elastically scattered light, and a set of two dielectric mirrors centered at 272nm with a bandwidth of 8nm removed other interferences. The two fluorescence signals were recorded simultaneously and separated by a dichroic filter of high reflectance above 272nm, ±8nm, and high transmission below 245nm. The results showed strong interferences of the O2-LIF with the red shifted fluorescence, making this C2 -Confidential detection scheme ineffective to study the NO production. The authors concluded that the blue shifted detection scheme, coupled with this filtering process seemed, was a far more effective technique.

In 2001, Jamette et al. [START_REF] Jamette | Laser induced fluorescence detection of NO in the combustion chamber of an optical GDI engine with A-X(0,1) excitation[END_REF], published their work on a new excitation scheme for NO-LIF in internal combustion engine. To avoid the high absorption problems of the 193nm and 226nm excitation schemes, and relatively low sensibility of the 248nm excitation from excimer lasers, and excitation scheme at 236nm was tested. The use of an OPO based laser pumped by a Nd:YAG laser allowed good energy levels, ~15-20mJ, with a narrow linewidth and a tunability from 220nm to 250nm. While any rovibronic transition in the 𝐴 2 𝛴 + ← 𝑋 2 𝛱 NO band could in theory be tested with this laser system, the authors focused their work on the yet unexplored 𝐴 -𝑋(0,1), with a particular attention on the avoidance of O2 parasite fluorescence. To do so, absorption, excitation and fluorescence spectra of NO and O2 have been computed thanks to a simulation code based on the work of Hanson's team [START_REF] Battles | Laser-induced fluorescence measurements of NO and OH mole fraction in fuel-lean, high-pressure (1-10 atm) methane flames: Fluorescence modeling and experimental validation[END_REF]. Pressure broadening and shifting are considered, and collisional processes are estimated by estimating the composition of the burned gases. The resulting simulation showed that a group of lines around 236.221nm could suit the high-pressure NO-LIF needs quite well. The involved transitions are 𝑄 1 (8), 𝑄 2 [START_REF] Mcguirk | The aerodynamic challenges of aeroengine gas-turbine combustion systems[END_REF], 𝑅 1 (2), 𝑄 12 [START_REF] Mcguirk | The aerodynamic challenges of aeroengine gas-turbine combustion systems[END_REF], 𝑅 21 [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF] and 𝑄 21 [START_REF] Zel | The Oxidation of Nitrogen in Combustion Explosions[END_REF] at 225.221nm. The 𝑃 2 (29.5) line at 225.909nm was also considered for its good isolation from any O2 lines, but its intensity decreases too rapidly with pressure. NO PLIF was applied in a direct injection internal combustion engine, and results showed the capacity of this excitation scheme to measure NO during the whole combustion process without any O2 interaction, including the high-pressure period near the TDC.

Bessler et al. [START_REF] Bessler | Quantitative in-cylinder NO-LIF imaging in a direct-injected gasoline engine with exhaust gas recirculation[END_REF] studied the impact of Exhaust Gas Recirculation (EGR) on NO production with NO PLIF based on the 𝐴 -𝑋(0,2) scheme. The novelty of their work was the simplification of the quantification of NO concentration by associating the ideal gas law and the four-level model LIF intensity equation. The vibrationally excited ground state from which NO molecules are excited presents a Boltzmann repartition monotonously dependent on temperature. By using 𝑁 𝑁𝑂 = 𝑝 𝑁𝑂 /𝑘𝑇, and thus calculating the NO mole fraction instead of the number density, the temperature dependency is nullified over a large range. Quenching and spectral overlap broadening are considered with various models. The authors estimated the systematic error due to this simplification being around -14%/+8% for T>1900K and by assuming a constant temperature Tmean = 2200K. The results showed that EGR with a high heat capacity can effectively reduce the NO production by lowering the temperature combustion if it is introduced in a significant amount.

Hildenbrand et al. [START_REF] Hildenbrand | Quantitative laser diagnostic studies of the NO distribution in a di diesel engine with PLN and CR injection systems[END_REF] applied this method in a direct injection diesel internal combustion engine. An excimer KrF laser provided a 247.94nm laser beam to excite the 𝐴 -𝑋(0,2) NO band. The blue shifted part of the fluorescence is collected to minimize C2 -Confidential the interference of PAH and O2 fluorescence. This excitation method allowed a measurement as close as 20° from the TDC. The accuracy of the quantification of NO concentration was estimated around ±50%. An external calibration was necessary to do so, with a NO-doped lean CH4/air McKenna flat burner. The atmospheric calibration was transferred to in cylinder pressure to consider pressure effect on line-broadening and shifting via a simulation code.

In 2004, Suck et al. [START_REF] Suck | NO laserinduced fluorescence imaging in the combustion chamber of a spray-guided direct-injection gasoline engine[END_REF] used the same method to study NO formation in a direct injection engine, running with commercial gasoline. The use of a realistic fuel induced further difficulties in the quantification of NO concentration, as the aromatics present show both strong absorption and fluorescence. The temperature of the LIF signal was made via the assumption of adiabatic flame temperature during the reaction process up to 75% of the mass fraction burned point. After a crank angle of 100°, full gas mixing of burned gases is assumed. For intermediate values, temperatures were linearly interpolated. With pressure measurements, the broadening and shifting of line shapes was simulated with the code LIFSim. The attenuation of the laser light and the LIF signal was estimated with photodiodes with an engine filled with fuel or burned gases. Finally, a calibration has been made at similar conditions as real measurements, by putting a flat burner in place of the spark-plug. The calibration flame was seeded with NO, and the temperature was measured with multi-line NO-LIF thermometry. The LIF signal was filtered through eight dielectric mirrors with a high reflection at 220-240nm, and two short pass edge filters at 245nm to remove the elastically scattered light. This procedure allowed a semi-quantification of NO concentration, and the results showed that 30% of EGR could reduce up to ten times the NO production. An exhaust gas analyzer has been used and the same trend as the LIF was found.

Selection of the excitation scheme

The previous section showed that the choice of the excitation scheme is not obvious. Three main difficulties arise from NO-LIF in high pressure reactive conditions: absorption of the laser beam and potentially LIF signal from the medium at low wavelength (< 220nm), interference with fuel-LIF, PAH-LIF and LII of soot particles at wavelength above 280-300nm, and parasitic O2 LIF from the 𝐵 3 𝛴 -← 𝑋 3 𝛴 + Schuman-Runge bands, for wavelength between those two limits.

The excitation of the 𝐷 2 𝛴 ← 𝛸 2 𝛱(0,1) branch at 193nm has been ruled out for high pressure measurement since the laser absorption is too severe. The excitation of the 𝐴 -𝑋(0,0) band at 226nm present a high fluorescence yield and well isolated lines, but can interact with O2 and also suffers from significant laser absorption from fuel and combustion products, notably CO2 and H2O. Two higher excitation wavelengths have been successfully used, the 236nm and 248nm to exploit the 𝐴 -𝑋(0,1) and 𝐴 -𝑋(0,2) bands, with a blue shifted LIF collection to avoid fuel, PAH and soot interferences. Yet, their fluorescence yields are significantly inferior to the 𝐴 -𝑋(0,0) band, and their low sensibility thus requires high laser energies.

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In 1997, Thomsen et al. [START_REF] Thomsen | Background corrections for laserinduced-fluorescence measurements of nitric oxide in lean, high-pressure, premixed methane flames[END_REF] studied potential interferences in a lean premixed high pressure flat flame. Measurements were carried out on a McKenna flat burner at an equivalence ratio of 𝜙 = 0.6 and a dilution ratio of 2.2, fueled with a CH4/O2/N2 mixture. Pressure varied from 1 to 14.6 bars, which lead to flame temperature from 1745K to 1845K. The authors chose those conditions to be one of the worst-case scenarios possible in terms of O2 LIF interaction. The 𝑄 2 (26.5) transition of the 𝐴 -𝑋(0,0) band at 225.57nm has been selected for its isolation from other transitions, and for its relatively low dependence on temperature with respect to its Boltzmann fraction. The fluorescence is collected in the red shifted region between ~234.4nm and ~237.4nm to avoid the O2 fluorescence near 233.5nm and the Stokes branch of the spontaneous Raman signal of N2 at ~238nm. This method is quite effective to avoid interferences at atmospheric pressure, but due to pressure broadening, is not sufficient at high pressure.

Thus, an off-line excitation is suggested to evaluate the background contribution to the NO-LIF signal. The possible off-line detection wavelengths are selected from a spectral scan with a argon diluted flame, and two values arise: at the right of the 𝑃 2 (34.5) NO transition, at 255.51nm, and between the 𝑅 1 (16.5) and the 𝑆 21 (11.5) transitions at 225.61nm. While those wavelengths can be used at atmospheric pressure, the authors underly that at high pressure, wavelengths further away from NO transitions should be chosen due to pressure broadening. This can induce a significant uncertainty while making calibrations, as the reference background made at atmospheric pressure is very different from the one at high pressure. The authors concluded that in order to minimize O2/NO interferences at high pressure, the excitation transition must be carefully selected, but in addition, a calibration at the same pressure as the measurements is preferable, with a collection scheme that maximizes the SNR.

Between 2002 and 2003, Bessler et al. published three papers thoroughly comparing the three excitation schemes mentioned above [START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames. I. A-X_0,0_ excitation[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames III Comparison of A-X excitation schemes[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames II. A-X(0,1) excitation[END_REF]. Measurements were made on a laminar premixed CH4/Air flat burner at pressures from 1-60 bars, at equivalence ratios of 𝜙 = 0.83, 0.93, 1.03 and 1.14. The flame was seeded with 300ppm of NO. A Nd:YAG pumped frequency doubled dye laser was used to produce laser light at 224-227nm for 𝐴 -𝑋(0,0) band (Coumarin 120 dye), 233-238nm for 𝐴 -𝑋(0,1) band (Coumarin 102 dye), and 245-248nm for 𝐴 -𝑋(0,2) band (Coumarin 307 dye). While the authors tested an important number of excitation schemes, only three will be reported here, as they have been designated as the most efficient in the case of high pressure lean premixed combustion. In the 𝐴 -𝑋(0,0) band, the 𝑃 1 (23.5), 𝑄 1 + 𝑃 21 (14.5), 𝑄 2 + 𝑅 12 (20.5) transition at 226.03nm originally studied by [START_REF] Battles | Laser-induced fluorescence measurements of NO and OH mole fraction in fuel-lean, high-pressure (1-10 atm) methane flames: Fluorescence modeling and experimental validation[END_REF]120]. In the 𝐴 -𝑋(0,1) band, the 𝑃 1 (25.5), 𝑅 1 + 𝑄 21 (11.5), 𝑄 1 + 𝑃 21 (17.5) at 235.87nm. And finally, the 𝑂 12 (7.5 -10.5) bandhead at 247.94nm in the 𝐴 -𝑋(0,2) band. Similarly, only the results in lean combustion, 𝜙 = 0.83 will be reported here, as it is the closest equivalence ratio to experimental conditions encountered in the present work. The results are presented in Table 1. With the 235.87nm and 247.94nm excitation, both red shifted and blue shifted detection are possible and reported here.

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Excitation 𝐴 -𝑋(0,0) 𝐴 -𝑋(0,1) 𝐴 -𝑋(0,1) 𝐴 -𝑋(0,2) 𝐴 -𝑋(0,2) Detection (0,1) + (0,2) (0,0) (0,2) + (0,3) (0,0) + (0,1) (0,3) + (0,4) Temperature is of primal importance for LIF quantification as it influences the fluorescence quantum yield through collisional quenching, ground state population, and spectral overlap. In most of the quantification application, the local temperature distribution is unknow, a homogeneous temperature is supposed, and an arbitral value is chosen. Therefore, it is necessary to choose an excitation scheme that present the lowest dependence on temperature for it induce a direct inaccuracy in the NO concentration measured. The simulated temperature dependence at 10 bars is reported in Table X. Depending on the use of the ideal gas law in the LIF signal expression (as in [START_REF] Bessler | Quantitative in-cylinder NO-LIF imaging in a direct-injected gasoline engine with exhaust gas recirculation[END_REF]), the temperature dependence can be significantly modified. Both temperature for maximum LIF signal and signal variation between 1700 and 2500K, which correspond to relevant temperature variation in lean premixed combustion, are reported.

T
The signal strength is another characteristic to consider. To maximize it, transitions with a large ground state population at relevant temperature and high oscillator strength must be selected. For the three transitions considered here, the (0,0) possess the largest ground state population, while the (0,1) transition has a maximum population at ~2700K, and the (0,2) at 5400K. This mean that for typical lean premixed conditions, the (0,0) transition is to favor. The signal strength from narrow-bandwidth excitations at 10 bars and 1900K have been both simulated and measured experimentally. Both methods showed a good agreement, and demonstrated that the (0,0) transition possess the strongest signal strength. In comparison, the signal strength from the (0,1) transition is ~17%/35% of the (0,0) one with a blue/red shifted collection, and the signal strength of the (0,2) transition is ~3.5%/2% from the (0,0) one. While this show a clear advantage of the use of the (0,0) transition, the lower wavelength induces a stronger absorption of the medium, and doesn't include the differences in available laser energy. Thus, the authors simulated the transmissions properties of both the laser and the LIF signal at 40bars and 1900K, representing a high pressure lean premixed burner, and 50bars and 2400K to represent typical diesel engine conditions. In both cases, an equivalence ratio 𝜙 = 0.9 has been considered, with species concentration at equilibrium. This approach is well suited to evaluate the significant absorption of H2O and CO2 that is both temperature and pressure dependent, and responsible for a large part of the absorption C2 -Confidential [START_REF] Hildenbrand | Measurements and simulation of in-cylinder UVabsorption in spark ignition and diesel engines[END_REF][START_REF] Schulz | Ultraviolet absorption spectra of shock-heated carbon dioxide and water between 900 and 3050 K[END_REF]. For both laser and LIF transmission, a path length of 𝑙 = 4𝑚𝑚 is considered.

For the lean premixed high-pressure burner case, the results demonstrate a 75% total transmission for the (0,0) transition, 73%/85% transmission for the (0,1) transition with a blue/red shifted collection, and 82%/92% transmission for the (0,2) transition with a blue/shifted collection. The resulting NO-LIF signals, considering both signal strength and transmission, are reported in table X, and show that even with a lower transmission, the (0,0) excitation scheme is better suited for typical lean premixed combustor conditions.

Finally, the last comparison made was the interaction with others species that could present a fluorescence in the same spectral region. The two species considered were CO2 and O2, for they are the most difficult to suppress with optical filters, as it can be done with PAH or soot interaction. The results are based on measurements done on a seeded (300ppm NO) laminar high-pressure burner at pressure from 1 to 60 bars and equivalence ratio of 0.83 and 1.13. Signal purities, defined as the ratio of NO-LIF and the sum of total emission from the three species, are reported in Table X for lean conditions and a pressure of 10 bar. In the case of the (0,0) excitation, it is ~99%, for the (0,1) excitation, values of 94%/97% are reported with a blue/red shifted collection, and for the (0,2) excitation, the signal purity is ~94%/77%.

In 2007, Lee et al. [START_REF] Lee | Experimental evaluation of strategies for quantitative laser-induced-fluorescence imaging of nitric oxide in high-pressure flames (1-60 bar)[END_REF] pursued the work of Bessler et al. by further improving the separation of NO-LIF and interferences from O2 and CO2, with the goal of providing practical guidelines for PLIF at numerous experimental conditions. The experimental setup is the same as in [START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames. I. A-X_0,0_ excitation[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames III Comparison of A-X excitation schemes[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames II. A-X(0,1) excitation[END_REF], experiments are made on a laminar flat flame burner with pressures ranging from 1 to 60 bars, and equivalence ratio of 𝜙 = 0.9 and 𝜙 = 1.1.

Three detection windows are used to recover signals that could allow the differentiation of the NO, O2 and CO2 signals. The first one used a bandpass Asahi filter centered at 254nm and with a FWHM of 17.5nm, the second window used a Lambda Research Optic filter centered at 280nm with a FWHM of 12nm to collect the O2 fluorescence, and the third, an Asahi filter centered at 340nm with a FWHM of 10nm, was used to collect the CO2 fluorescence. The collection schemes were chosen to maximize the signal over noise ratio of the three species fluorescence with spectrally resolved 1D line-imaging measurements. Temperature measurements were also carried out to simulate temperature dependent properties of the different LIFs by the mean of 2-line thermometry of NO-LIF. The resulting procedure allowed a precise quantification of NO-LIF with a correction of temperature and pressure effect, and the suppression of interferences from O2 and CO2 LIF.

More recently, in 2019, Mulla et al. [START_REF] Mulla | Quantitative imaging of nitric oxide concentration in a turbulent n-heptane spray flame[END_REF] proposed a new PLIF approach for quantitative NO concentration measurements in a turbulent diphasic atmospheric flame. The excitation was based on the 𝑄 1 (29.5) transition in the 𝐴 -𝑋(0,0) band at 225.12nm, with a collection from 235nm to 297nm, corresponding to the 𝐴 -𝑋(0,1) to (0,6) bands. This specific transition was selected for its high fluorescence yield while being well isolated from parasitic signal from other species, namely PAH, O2 and soot. The laser C2 -Confidential beam was generated by a frequency-doubled Nd:YAG laser pumping a frequency tripled dye laser (LDS798 dye). A calibration was setup with a McKenna premixed flat burner, fueled with a CH4/O2/N2/NO mixture, and working at equivalence ratio 𝜙 = 0.9. NO doping was made by replacing some of the N2 by a known amount of NO. The potential NO re-burn has been checked with a Cantera 1D simulation, using the NOMecha 2.0 mechanism. Temperatures computed by Cantera were validated by Coherent Anti-Stokes Raman Scattering (CARS) measurements. Finally, quenching processes were simulated by a five-levels code developed by [START_REF] Gautier | Dosage des polluants NO et CO par imagerie de fluorescence induite par laser dans les écoulements réactifs To cite this version[END_REF] based on [START_REF] Bessler | A versatile modeling tool for nitric oxide LIF spectra Physikalisch-Chemisches Institut[END_REF], that considers five species, namely NO, CO, CO2, N2, and Ar. In the main flame, species concentration and temperature field were computed by the mean of a LES of the burner with the AVBP solver and the ARC kinetic mechanism. Therefore, quenching and temperature effect could also be considered in the main flame for NO-LIF simulation and NO concentration measurements. The results showed low NO concentrations (<100ppm, ±30%) with a high dependence on mixing and thus combustion mode.

In 2020, Salaün et al. [START_REF] Salaün | Experimental investigation on NO pollutant formation in high-pressure swirl-stabilized kerosene/air flames using NO-, OH and kerosene PLIF and PIV laser diagnostics[END_REF] published their work on NO-PLIF in high pressure reactive conditions on a lean premixed industrial injection system in the HERON test rig running with Jet-A1 commercial kerosene. The authors used the 𝑄 1 (29.5) transition to quantify NO concentration in realistic operating conditions of a helicopter injector. This transition has been carefully selected for its very low temperature dependence and high fluorescence yield. Potential interferences from O2 are dealt with a set of optical filters and as in [START_REF] Mulla | Quantitative imaging of nitric oxide concentration in a turbulent n-heptane spray flame[END_REF], by detuning the laser and recording background images. Authors found that the ratio of NO-LIF over background images is around 95, which demonstrate the efficacy of the optical filtering and the low impact of O2 in these operating conditions, as it has been suggested by [START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames. I. A-X_0,0_ excitation[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames III Comparison of A-X excitation schemes[END_REF][START_REF] Bessler | Strategies for laser-induced fluorescence detection of nitric oxide in high-pressure flames II. A-X(0,1) excitation[END_REF]. NO-LIF signal is collected between 235nm to 297nm, corresponding to the 𝐴 -𝑋(0,1 -6) bands. Kerosene is excited at the with the same laser pulse, and its fluorescence is collected between 300nm and 425nm. The presence of gaseous kerosene induces a strong absorption and overlap the fluorescence signal from the 𝐴 -𝑋(0,4 -6) bands, thus, the authors set up a two-color detection strategy to isolate the kerosene fluorescence from the NO images. The calibration has been made by the mean of a McKenna flat burner placed in the center of the HERON combustion chamber, and fueled with a CH4/air/NO mixture. NO concentration post flame was simulated with Chemkin and the NOmecha2.0 mechanism. With the hypothesis of equilibrium in the Inner Recirculation Zone, and a five-level fluorescence model to consider pressure and temperature effect, the normalized NO fluorescence signal from the main experiment could thus be converted to NO concentration.

In the light of those results, it has been chosen to use the 𝑄 1 (29.5) transition with an excitation at 225.12nm, for its low temperature dependence and high fluorescence yield. The O2 and CO2 interactions have been proven to be negligible in those elevated pressure (<20bars) lean premixed combustion conditions, and only the kerosene fluorescence have to be accounted for.

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Kerosene interaction and filtering

The normalized fluorescence of NO and kerosene for an excitation at 225.12nm is presented in Fig. 21. As explained before, the collection of the 𝐴 -𝑋(0,1 -6) bands induce a collection of some kerosene fluorescence. To reduce this interference, it has been chosen to collect only collect NO signal from the 𝐴 -𝑋(0,1 -4) by adding a short pass filter. The overall NO-LIF signal amplitude is not changed by a significant margin since the (0,5 -6) bands are not the strongest, yet the signal purity is enhanced.

NO filtering is constituted with a longpass filter at 225nm (Semrock, razor edge), and Asahi shortpass filters with cutting wavelength at 265, 300 and 350nm. The total transmission is showed in Fig. 22.

Quantification procedure by a calibration approach

In the present work, the calibration flame was generated with a McKenna premixed flat burner fueled with a H2/Air/N2/NO mixture. The body of the burner is made of stainless steel with an outer diameter of 120mm and a height of 60mm. A sectional view of this burner is shown in Fig. 23. The porous plate for flame stabilization is water cooled with a cooling coil sintered into the plate. The annular co-flow wasn't use here. The flows of H2, air and N2/NO (9000 ppm of NO) were controlled with mass flowmeters, and were combined and thoroughly mixed prior to being supplied to the burner. To keep the exact same optical arrangement that in the high-pressure experiments, the McKenna burner was mounted inside the combustion chamber at a height so that the flame and above could be seen on camera without moving them. Operating conditions are presented in Table 2. Conditions were selected so that the flame temperature is ~1600K at 𝜙 = 0.5 [START_REF] Prucker | A flat flame burner as calibration source for combustion research: Temperatures and species concentrations of premixed H2/air flames[END_REF], which is within a reasonable range of the one encountered at high pressure in the inner recirculation zone, and low enough to avoid any re-burn effect of the seeded NO.

The production of NO from the calibration flame is considered negligible, so that the NO concentration in the burned gases can be assimilated to the seeding concentration. ) the collection fraction, (

𝑉 𝑐𝑜𝑙 𝐴 𝑙𝑎𝑠
) the length of the measurement volume, defined as the ratio of the collection volume over beam cross section, 𝐺 is a camera parameter for photon to electron conversion. Those terms depend only on the optical and geometric arrangement of the experiments, and will be regrouped as ε. (∫ 𝑠 𝑓𝑙𝑢𝑜 𝑠𝑖𝑚𝑢 . 𝑇(𝜆)𝑑𝜆) is the theoretical total fluorescence signal, computed numerically, with 𝑇(𝜆) the filters transmissivity. It will be denoted as 𝑆 𝑓𝑙𝑢𝑜 + .

We can thus express Eq. ( 64) such as:

𝐼 𝑓𝑙𝑢𝑜 𝐸 𝑙𝑎𝑠𝑒𝑟 = ε 𝑆 𝑓𝑙𝑢𝑜 + 𝑛 𝑁𝑂 (65)

C2 -Confidential

During the calibration experiments, 𝜀 is the only unknown, and can be thus determined. This method avoids the difficult determination of parameters such as the length of the measurement volume for the quantification of NO in the actual measurements. With 𝜀 known, the NO molar fraction during experiments can be determined with:

𝑋 𝑁𝑂 = 1 ε 𝐼 𝑓𝑙𝑢𝑜 𝐸 𝑙𝑎𝑠𝑒𝑟 1 𝑆 𝑓𝑙𝑢𝑜 + 𝑘 𝐵 𝑇 𝑃 (66)
The temperature impacts directly the molar fraction and the theoretical fluorescence signal.

Though the latter vary only by 5% per 100K for a temperature of 𝑇 = 1800𝐾.

Quantitative determination of the mole fraction of the NO by an absorption approach

We will see in the presentation of the results obtained in the HERON combustion chamber that the flames studied show, in most of the high-pressure operating points, a symmetrical structure, characterized by a certain homogeneity of the NO concentration in the burnt gases. We also found that the absorption of the laser sheet by NO molecules is not really significant due to the low molecular density of NO. By using an average image of the NO fluorescence, this finding allows us to obtain quantitative information on the NO mole fraction. As a reminder, the fluorescence signal is expressed for NO as:

𝑆 𝐿𝐼𝐹 𝑁𝑂 =
Indeed, if we consider a confined swirled flame attached to the inside of the injection system, the temperature field and the chemical species composition can be assumed to be quasihomogeneous in the burnt gases. In particular, it was observed that the fluorescence along the line in the center of the injector is almost constant, which supports this assumption.

If we consider the temperature, pressure and composition of the burnt gases as constant and uniform, the fluorescence signal 𝑆 𝐿𝐼𝐹 𝑁𝑂 is then only a function of the incident laser intensity and the density of NO molecules. Eq.( 67) can be simplified to the following form:

𝑆 𝐿𝐼𝐹 𝑁𝑂 ∝ 𝐼 𝑙𝑎𝑠𝑒𝑟 . 𝑁 𝑁𝑂𝐻 𝑡𝑜𝑡𝑎𝑙 . 𝐹 1 (𝑇, 𝑃). 𝐺(𝑇, 𝑃, 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛) [START_REF] Hartung | Effect of heat release on turbulence and scalar turbulence interaction in premixed combustion[END_REF] where 𝑆 𝐿𝐼𝐹 𝑁𝑂 is the fluorescence intensity, 𝐼 𝑙𝑎𝑠𝑒𝑟 is the incident laser intensity, 𝑁 𝑁𝑂 𝑡𝑜𝑡𝑎𝑙 is the NO molecular density (number of molecules per unit volume), 𝐹 1 describes the settlement distribution of the energy levels via the Boltzmann function and G is a function describing the non-radiative de-excitation processes (quenching). As the incident laser energy passes through the burnt gases, it is absorbed by the NO molecules present, as the laser sheet propagates. The laser intensity received by the molecules decreases according to their position in the flame and also the fluorescence emitted by the flame (always in the case of linear fluorescence regime). This decrease in the laser energy available to the burnt gases is expressed by the following Beer-Lambert law:

𝑆 𝐿𝐼𝐹 𝑁𝑂 (𝑥) ∝ 𝐼 0 𝑒 𝑘 𝑎𝑏𝑠 .𝑥.𝑃 (69)
C2 -Confidential where x is the length crossed inside the flame (cm), 𝑆 𝐿𝐼𝐹 𝑁𝑂 is the fluorescence intensity (proportional to the incident laser intensity in the linear fluorescence regime), 𝐼 0 is the initial incident laser intensity and 𝑘 𝑎𝑏𝑠 is the absorption coefficient of the medium studied (cm -1 .atm -

1

).

The procedure for correcting the absorption phenomenon visible on the average fluorescence image consists of iteratively calculating the absorption of the incident laser beam pixel by pixel, considering that the fluorescence signal is proportional to the concentration and the incident laser energy in the pixel located at a distance x from the optical window

𝑋 𝑁𝑂 𝑥 = 𝑋 𝑁𝑂 𝑥-𝑑𝑥 . 𝑒 (-𝑘 𝑎𝑏𝑠 .𝑑𝑥.𝑃) (70)
The value of the absorption coefficient 𝑘 𝑎𝑏𝑠 (in cm -1 ) is expressed using the effective absorption cross-section σ 𝑎𝑏𝑠 (cm 2 ) and the molecular density of the absorbing species 𝑁 𝑁𝑂 (number of NO molecules.cm -3 ):

𝑘 𝑎𝑏𝑠 = σ abs . 𝑁 𝑁𝑂 𝑥-𝑑𝑥 (71)
or as a function of the mole fraction of the NO molecule X 𝑁𝑂 𝑥-𝑑𝑥 :

𝑘 𝑎𝑏𝑠 = σ abs . 𝑋 𝑁𝑂 𝑥-𝑑𝑥 . 𝑁 0 (72)
with 𝑁 0 , the total number of molecules per unit volume (cm -3 ). The latter is estimated using the values at thermodynamic equilibrium. Indeed, in a perfect gas, the molecular density 𝑁 0 in the medium is expressed as:

𝑁 0 (𝑇, 𝑃) = 273 𝑇 . 𝑃 1 . 𝐶 𝐿𝑜 (73)
where P is in atm and T is in K, 𝐶 𝐿0 corresponds to the Loschmidt constant (= 2.687.1019 molecules/cm -3 ). The absorption coefficient is calculated using a home-made fluorescence simulation code. This quantity also depends on the pressure and temperature of the NO molecules and therefore on the temperature of the burnt gases. Nevertheless, the temperature dependence of the absorption is relatively small for the excited transition 𝑄 1 (29.5) which was chosen for this reason.

As the pressure is measured and assumed to be constant and homogeneous throughout the combustion chamber, this parameter is considered to be known. It remains to determine the temperature in order to estimate the molecular density in the burnt gases and thus obtain the absorption coefficient.

The temperature is obtained by calculating the thermochemical equilibrium assuming a complete and premixed combustion inside the injection system. The CANTERA simulation code

The quantitative study of the spray was performed by means of the Phase Doppler Anemometry (PDA) optical diagnostic. This technique is a pointwise optical measurement of droplet velocity and diameter. The basis of the measurement is the crossing of two laser beams to form a measurement volume with a size of approximately 100µm. PDA has been extensively used in the past to validate atomization models, or study injectors performances, especially in internal combustion engines. Commercial systems were developed to facilitate and extend the use of PDA, known as Phase Doppler Particles Analyzer (PDPA). This chapter is dedicated to the description of the PDA principles, its development and applications, and finally preliminary experiments on a Lean premixed Injector fueled with liquid kerosene will be presented.

C2 -Confidential

The Doppler Effect

As its name suggests, PDA is based on the Doppler effect: the frequency of a signal emitted by a transmitter and received by an observer is shifted if a relative motion between the two exists. If the motion has a non-zero velocity component towards each other, then the observed frequency is positively shifted, and reversely. In the case of PDA, the doppler effect is invoked twice. First a laser beam acts as a stationary transmitter, and a droplet acts as a moving receiver.

For instance, if the droplet moves away from the laser emission, then the frequency seen by the particle will be lower than that of the laser. The particle then scatters the light and acts as a mobile transmitter. The frequency of the scattered light received by a detector will be shifted according to the same logic. A diagram representing these two cases is shown in Fig. 26.

In the first case (left Fig. 26), the perceived frequency 𝑓 𝑝 of the particle moving with a relative velocity 𝑉 𝑝 with respect to the laser emitting a beam at a frequency 𝜆 𝑙 is given by:

𝑓 𝑝 = 𝑐 -𝑉 𝑝 . 𝑒 𝑙𝑝 𝜆 𝑙 = 𝑓 𝑙 - 𝑉 𝑝 . 𝑒 𝑙𝑝 𝜆 𝑙 (74)
Where 𝑒 𝑙𝑝 is the unit vector describing the laser -particle direction.

In the second case (right figure), the signal observed by the stationary receiver from the moving particle will see a frequency 𝑓 𝑟 such as:

𝑓 𝑟 = 𝑐 𝜆 𝑟 = 𝑓 𝑝 1 - 𝑉 𝑝 . 𝑒 𝑝𝑟 𝑐 (75)
𝑒 𝑝𝑟 is the unit vector defining the particle -receiver direction, and 𝑐 is the speed of light.

In the case of fixed laser beam and detector, and a moving particle, both Doppler effects can be combined to give: Assuming that |𝑉 𝑝 | ≪ 𝑐, the previous expression can be approximated as:

𝑓 𝑟 ≈ 𝑓 𝑏 + 𝑉 𝑝 . (𝑒 𝑝𝑟 -𝑒 𝑏 ) 𝜆 𝑏 [START_REF] Thurber | Simultaneous imaging of temperature and mole fraction using acetone planar laser-induced fluorescence[END_REF] The second term of the right-hand side is called the Doppler shift. For typical conditions, this one ranges between 1MHz and 100MHz, which is tremendously lower than that of the laser beam, 𝑓 𝑏 ~ 10 14 𝐻𝑧. Some techniques have been developed in the past to measure such a small frequency shift [START_REF] Paul | Rapid velocity sensor using a static confocal Fabry-Perot and a single frequency argon laser[END_REF][START_REF] Jackson | Measurement of supersonic velocity and turbulence by laser anemometry[END_REF][START_REF] Smeets | Michelson spectrometer for instantaneous Doppler velocity measurements[END_REF][START_REF] Komine | System for measuring velocity field of fluid flow utilizing a laser-Doppler spectral image converter[END_REF][START_REF] Komine | Real-time, Doppler global velocimetry[END_REF], but the most common method is to mix two laser beams to form a measurement volume in which it is possible to get rid of 𝑓 𝑏 in the expression for the observed frequency. Various configurations have been used over the years, such as the so-called dualscattered waves or reference beam configurations, which use only one laser beam. However, the most widely used strategy in most PDA applications today is the dual beam configuration, see Fig. 27.

In this configuration, the doppler frequency shift can be measured in the measurement volume, and the frequency observed is given by:

𝑓 𝐷 = 𝑓 2 -𝑓 1 = 𝑉 𝑝 . (𝑒 1 -𝑒 2 ) 𝜆 𝑏 (78)
𝜃 is the incident angle and (𝑒 1 -𝑒 2 ) = 2 sin

Homogeneous planar wave and Fringe model

To illustrate the interaction of the two laser beams and the resulting scattered light when a particle passes through the measurement volume, it is common to use the fringe model. This model is only valid for very small particles following the condition 𝐷 𝑝 ≪ 𝜆 𝑏 .

In this model, laser beams are considered as a homogeneous plane wave. The solution to Maxwell's equation, describing electric and magnetic fields strengths, are:

𝐸 = 𝐸 0 𝑒𝑥𝑝 (𝑗𝜔 𝑏 [𝑡 - 𝑒 𝑘 .𝑟 𝑐 ]) (80)
𝐻 = 𝐻 0 𝑒𝑥𝑝 (𝑗𝜔 𝑏 [𝑡 - 𝑒 𝑘 .𝑟 𝑐 ]) (81)
With 𝑒 𝑘 a unit vector defining the direction of propagation and 𝑟 a vector to an arbitrary point in the field.

The introduction of the two previous equations into Maxwell's equation leads to the conclusion that the electric field and the magnetic field are perpendicular to each other.

Let's introduce the wave number

𝒌 ⃗ ⃗ = 2𝜋 𝜆 𝑒 𝑘 ⃗⃗⃗⃗ (82)
Then the following relation can be shown:

𝑒 𝑘 × 𝐸 = √ 𝜇 𝜀 𝐻 𝑒 𝑘 × 𝐻 = -√ 𝜇 𝜀 𝐸 (83)
Which can simplify the calculation of the energy density, by considering only the electric field:

𝑤 = 1 2 (𝜀𝐸 2 + 𝜇𝐻 2 ) = 𝜀𝐸 2 (84)
with 𝐸 = |𝐸| and 𝐻 = |𝐻| Since energy is transported at the speed of light, energy fluctuations occur at a frequency too high to be perceived by optoelectronic detectors. This means that, for a PDA receiver, the electromagnetic field strength will be proportional to the time average of the energy density and consequently

𝐼 = 𝑐𝑤 ̅ = 𝑐 𝜀𝐸 2 ̅̅̅̅̅ (85)
From Eq. ( 83), the expression for the electric field of the waves considered in the dual beam system detailed above is, for a propagation in the z direction and a polarization perpendicular to the (x-z) plane:

𝐸 1 = 𝐸 0 𝑒𝑥𝑝 (𝑗 [𝜔 𝑏 𝑡 -𝑘 𝑏 (𝑥 sin 𝜃 2 + 𝑧 cos 𝜃 2 ) + 𝜑 1 ]) (86) C2 -Confidential 𝐸 2 = 𝐸 0 𝑒𝑥𝑝 (𝑗 [𝜔 𝑏 𝑡 -𝑘 𝑏 (-𝑥 sin 𝜃 2 + 𝑧 cos 𝜃 2 ) + 𝜑 2 ]) (87)
𝐸 0 is the amplitude of the electric field, 𝜔 𝑏 is the wave angular frequency, 𝑘 𝑏 is the wavenumber

𝑘 𝑏 = 2𝜋 𝜆 𝑏
, and 𝜑 is the initial phase. 𝜃 has been defined previously as the incident beam angle.

In the measurement volume, i.e., corresponding to the crossing of the two waves, the electric field is given by :

𝐸 = 𝐸 1 + 𝐸 2 (88)
The time-averaged calculation from Eq. ( 88) to determine the electromagnetic field strength in the measurement volume is, with identical phasing 𝜑 1 = 𝜑 2 ,

𝐼 = 𝑐 𝜀𝐸 2 ̅̅̅̅̅ = 𝑐𝜀𝐸 𝐸 * = 2𝑐𝜀𝐸 0 2 cos 2 (𝑘 𝑏 𝑥 sin 𝜃 2 ) (89)
This indicates that the electromagnetic field intensity in the measurement volume is directed in one direction only.

These results are shown in Fig. 28 On the top left and right corners are presented two incident planar homogeneous electromagnetic waves, representing two laser beams described by Eq.( 86) and Eq.( 87). The interaction of these waves, C2 -Confidential corner, in which horizontal interference fringes are observed. The intensity of these fringes follows Eq.( 89).

The distance between fringes, ∆𝑥, depends only on optical properties and is given by:

∆𝑥 = 𝜆 𝑏 sin 𝜃 2 (90)
From Eq.( 79), the velocity component perpendicular to the beam bisector, in the present case oriented in the x direction 𝑉 𝑝𝑥 , can be expressed as :

𝑉 𝑝𝑥 = ∆𝑥 𝑓 𝐷 = ∆𝑥 𝑇 𝐷 (91)
with 𝑇 𝐷 being the period of fringes crossing.

By introducing Eq. ( 82), ( 90) and ( 91) into Eq. ( 89), the temporal intensity from the scattered light from a small particle passing through the measurement volume and measured by a photodetector is given by:

𝑖(𝑡) ≈ 𝜀𝑐𝐸 0 2 [1 + 𝑐𝑜𝑠(2𝜋𝑓 𝐷 𝑡)] (92)
For a very small droplet, 𝐷 𝑝 ≪ 𝜆 𝑏 , passing through the interference fringes, the time evolution of the intensity of the scattered light detected by a photodetector will follow the intensity distribution described by Eq.( 93). The time-averaging process comes from the detector response time which will therefore measure a sinusoidal signal which will provide 𝑓 𝐷 . The knowledge of ∆𝑥 will then give the velocity component perpendicular to the fringes, 𝑉 𝑝𝑥 .

With this configuration, however, it is impossible to determine the sign of 𝑉 𝑝𝑥 , (i.e. particles crossing the fringes perpendicularly in opposite directions would produce the same signal and therefore identical velocity values). To overcome this limitation, a temporal change in the phase of one of the two laser beams is applied via an acousto-optic modulator (usually by positioning a Bragg cell on the propagation axis of a laser beam). This leads to a shift in the beam frequency of ±𝑓 𝑠ℎ𝑖𝑓𝑡 . Since the frequency is the time derivative of the phase, this leads to a time evolution of the phase as follows:

𝜙 = 2𝜋𝑓 𝑠ℎ𝑖𝑓𝑡 𝑡 (93)
The modification of the phase of one laser beam induces a displacement of the fringes in the perpendicular direction, in the presented example in the +x or -x direction. Following Eq.( 78)

and Eq.( 79), the measured frequency after the interaction of the two incident waves, 𝑓 𝑚 , is given by:

𝑓 𝑚 = 𝑓 𝑠ℎ𝑖𝑓𝑡 + 𝑉 𝑝 .(𝑒 1 -𝑒 2 ) 𝜆 𝑏 = 𝑓 𝑠ℎ𝑖𝑓𝑡 ± 2 sin 𝜃 2 𝜆 𝑏 |𝑉 𝑝𝑝 | = 𝑓 𝑠ℎ𝑖𝑓𝑡 ± 𝑓 𝐷 (94)
Thus, if the measured frequency is lower than the Bragg cell frequency, 𝑓 𝑚 < 𝑓 𝑠ℎ𝑖𝑓𝑡 , the droplet will move in the same direction as the fringes, and inversely.

C2 -Confidential

Phase Doppler Anemometry

The model described until now is valid only for very small droplets respecting the condition 𝑑 𝑝 < 𝜆 𝑏 . For larger particles, both the amplitude and phase of the incident wave vary across the diameter of the particle. This induces an additional phase shift in the expression of the electric field, Eqs. ( 86) and [START_REF] Baranger | Détection du kérosène par imagerie de fluorescence induite par laser , pour application sur foyer aéronautique To cite this version[END_REF], that is added to the initial phase 𝜑 1 and 𝜑 2 .

Fig. 29 shows a simplified but effective view of this effect with a particle of much larger diameter than the laser beams. The incident waves are considered to be an accumulation of rays of negligible diameter compared to that of the particle. Furthermore, only certain parts of the incident waves are collected on the detector (i.e. only particular paths allow the rays to reach the detector). These paths depend on the angles of the incident rays, the refractive index of the particle and the solid angle of the detector. The entering points of the rays inside the particle are called the incident points, and the outgoing points will be referred to as glare points. As the detector is not a point sensor, these glare points correspond to small areas. The differences in trajectory then induce a phase shift between the electric fields of the rays, as shown in Fig. 29.

With this configuration, the interaction of the rays does not occur directly in the measurement volume, but on the detector. The expression for the electric field is similar to the previous definition, and the intensity is also similar, see Eq. ( 92) and Eq. ( 92) respectively.

𝐸 𝑖 = 𝐸 0 𝑒𝑥𝑝 (𝑗 [𝜔 𝑏 𝑡 -𝑘 𝑏 (𝑥 sin 𝜃 2 + 𝑧 cos 𝜃 2 ) + 𝜑 𝑖 + 𝛹 𝑖 ]) (95)
𝑖(𝑡) ≈ 𝜀𝑐𝐸 0 2 [1 + 𝑐𝑜𝑠(2𝜋𝑓 𝐷 𝑡 + (𝜑 2 -𝜑 1 ) + (𝛹 2 -𝛹 1 ))] (96)
Compared to the expression of Eq. ( 92), an additional phase term is added, corresponding to a phase shift of the interference fringes. The first phase term (𝜑 2 -𝜑 1 ) depends only on the particle shape, while (𝛹 2 -𝛹 1 ), that appears only in the case of a refraction, depends on both particle refractive index and shape. By adding another detector used to measure the same area of interest, a phase difference can be obtained: The next section is devoted to determining the relationship between this phase difference and the droplet diameter, as is done by commercial PDA systems such as the one used in my thesis.

𝛥𝜙 12 =
The approach remains similar, as the expression of the incident electric field will be detailed first, followed by the calculation of the scattered electric field used to express the signal intensity collected by the detector. Then, the phase-diameter relation will be expressed in the case of a refraction for a droplet of average size.

Description of a Gaussian Laser beam

In the previous section, the laser beams were approximated to a homogeneous plane wave to obtain a simplified solution to Maxwell's equations. In reality, these laser beams have an inhomogeneous field distribution perpendicular to the direction of propagation, which leads to an intensity distribution close to a Gaussian distribution. Using the index b to denote the reference frame of the beam, a laser beam modelled as an electromagnetic wave propagating in the z-b direction and linearly polarized in the y-b direction will have an electric field given by: 𝐸 𝑏 = 𝑒 𝑦𝑏 𝐸 𝑦𝑏 + 𝑒 𝑧𝑏 𝐸 𝑧𝑏 [START_REF] Paul | Molecular velocity imaging of supersonic flows using pulsed planar laser-induced fluorescence of NO[END_REF] As the component in the z-direction makes no contribution to the scattering of light, it can be neglected. The field strength becomes:

𝐸 𝑏 =
with the Rayleigh length on the 𝑧 𝑏 axis defined as:

𝑙 𝑅𝑏 = 𝜋 𝑟 𝑤𝑏 2 𝜆 𝑏 (100)
The amplitude of the electric field in the center of the beam at the beam waist, 𝐸 0𝑏 , is given by the following expression, with 𝑃 𝑏 being the input laser power:

𝐸 0𝑏 = √ 2𝑃 𝑏 𝜀𝑐𝜋 𝑟 𝑤𝑏 2 (101)
Finally, the beam radius at a distance 𝑧 𝑏 from the beam waist:

𝑟 𝑚𝑏 (𝑧 𝑏 ) = 𝑟 𝑤𝑏 √1 + ( 𝑧 𝑏 𝑙 𝑅𝑏 ) 2 (102)
C2 -Confidential

Light scattering theories

The most common approach to describe light scattering by particles is the geometrical optics theory, further called GO. The incident light is considered as a superposition of parallel rays impacting a spherical particle. One part of this light is reflected, while the rest is transmitted through the particle, following the basics laws of geometrical optics. The intensity seen by the receiver is thus the sum of all scattering fields emitted by the particle. The ratio between the transmitted and the incident fields strengths is given by the Fresnel equations. In the case of a perpendicular polarization it is given by:

𝑡 ⟂ = ( 𝐸 0 (𝑡) 𝐸 0 (𝑖) ) ⟂ = 2 𝑛 1 𝜇 1 cos 𝜃 𝑖 𝑛 1 𝜇 1 cos 𝜃 𝑖 + 𝑛 2 𝜇 2 cos 𝜃 𝑡 (103)
𝜃 𝑖 is the incident angle, 𝜃 𝑡 is the transmitted angle, 𝑛 1 and 𝑛 2 are the refractive indexes of the medium and the particle respectively and 𝜇 1 and 𝜇 2 are the relative magnetic permeability. With the assumption of a dielectric media, 𝜇 1 = 𝜇 2 = 𝑐𝑡𝑒 and those terms disappear in Eq. ( 92).

In a same manner, the reflected ratio is given by:

𝑟 ⟂ = ( 𝐸 0 (𝑟) 𝐸 0 (𝑖) ) ⟂ = 𝑛 1 𝜇 1 cos 𝜃 𝑖 - 𝑛 2 𝜇 2 cos 𝜃 𝑡 𝑛 1 𝜇 1 cos 𝜃 𝑖 + 𝑛 2 𝜇 2 cos 𝜃 𝑡 (104)
This leads to the following expressions of the intensity of reflected and refracted waves:

𝑅 ⟂ = |𝑟 ⟂ | 2 𝑎𝑛𝑑 𝑇 ⟂ = 𝑚 cos 𝜃 𝑖 cos 𝜃 𝑡 |𝑡 ⟂ | 2 (105)
𝑚 = 𝑛 𝑝 𝑛 𝑚 ⁄ in the case of a transmittance through external surfaces, and 𝑚 = 𝑛 𝑚 𝑛 𝑝 ⁄ in the case of internal surfaces. Note that 𝑅 ⟂ + 𝑇 ⟂ = 1. We can then define the transmission factor for perpendicular polarization 𝛼 ⟂ (𝑁) :

𝛼 ⟂ (𝑁) = 𝑅 ⟂ 𝑓𝑜𝑟 𝑁 = 1 𝛼 ⟂ (𝑁) = 𝑅 ⟂ 𝑁-2 (1 -𝑅 ⟂ ) 2 𝑓𝑜𝑟 𝑁 > 1 (106)
As depicted in Fig. 30, depending on the incident point and the scattering order, a convergence or divergence of the rays can occur, leading to an amplification or an attenuation of the scattered field. This is represented by the gain function, defined as:

C2 -Confidential 𝐺 (𝑁) = | | sin 𝜃 𝑖 cos 𝜃 𝑖 sin 𝐷 (𝑁) √ 1 - sin² 𝜃 𝑖 𝑚 2 2(𝑁 -1) cos 𝜃 𝑖 𝑚 -2 √ 1 - sin² 𝜃 𝑖 𝑚 2 | | (107)
N is the scattering order and 𝐷 (𝑁) the associated deviation angle.

The scattering function can now be expressed, depending on the polarization:

𝑆 1 ⟂ (𝑁) = √ 𝑥 𝑚 2 𝛼 ⟂ (𝑁) 𝐺 (𝑁) 𝑒𝑥𝑝(𝑗𝜙 ⟂ (𝑁) ) 𝑆 2 ∥ (𝑁) = √𝑥 𝑚 2 𝛼 ∥ (𝑁) 𝐺 (𝑁) 𝑒𝑥𝑝(𝑗𝜙 ∥ (𝑁) ) (108)
The sum over all scattering orders gives the total scattering functions.

In a realistic case of Phase Doppler measurement, the droplet diameter is such that 𝜆 𝑏 ≪ 𝑑 𝑝 < 𝑅 𝑀𝑉 , 𝑅 𝑀𝑉 being the waist radius of the measurement volume, which coincides with the beam waist 𝑟 𝑤𝑏 (with the hypothesis of a well aligned and adapted setup). For small droplets of around 𝑑 𝑝 < 0.3 𝑟 𝑤𝑏 , the geometrical optics fails to accurately predict the scattered field. For all particle sizes, Lorenz [START_REF] Lorenz | Auffassung von Tondistanzen[END_REF] and Mie [START_REF] Mie | Sättigungsslrom und Stromkurve einer schlecht leitenden Flüssigkeit[END_REF] gave a solution for the scattering of a plane wave from a spherical, homogeneous and isotropic particle. The so-called Lorentz Mie Theory (LMT) is used here to compute the scattering function that transforms the incident field into the scattered field, coupled with a decomposition with Debye series [START_REF] Hovenac | Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series[END_REF]. The LMT is based on the decomposition of any incident plane wave into a sum of partial spherical partial waves, described in a spherical coordinate system with the particle center located at its origin. For every interaction of an electromagnetic wave with the particle, a part of the light is reflected, while the other is refracted. This means that after the first transmission of light inside the particle, there are an infinite number of refractions and reflections inside the particle. For instance, Fig. 30 shows the first four scattering orders.

The LMT is based on the resolution of the Maxwell's equations for each of the spherical waves that describe the incident wave. The incident field on the receiver is thus the sum of all the scattered waves coming from the glare points. The receiver is considered far enough from the measurement volume to be in the far field, which induces that the difference of length between the glare points and a receiver is negligible. Droplets are supposed spherical, homogeneous, and isotropic. The magnetic and electric fields of the incoming wave are considered independent and thus two solutions arise.

To express these solutions, two parameters, called Mie parameters, must be defined:

𝑥 𝑀 = 𝜋𝑑 𝑝 𝜆 𝑤 (109)
𝑦 𝑀 = 𝑛 𝑝 𝑛 𝑚 𝜋𝑑 𝑝 𝜆 𝑤 = 𝑚𝑥 𝑀 (110) 𝑚 = 𝑛 𝑝 𝑛 𝑚
is the relative refractive index.

The scattering functions [START_REF] Mie | Sättigungsslrom und Stromkurve einer schlecht leitenden Flüssigkeit[END_REF] for the electric and magnetic field are respectively:

𝑆 1 (𝜗 𝑠 ) = ∑ 𝑎 𝑛 𝜋 𝑛 (𝜗 𝑠 ) + 𝑏 𝑛 𝜏 𝑛 (𝜗 𝑠 ) ∞ 𝑛=1 (111)
𝑆 2 (𝜗 𝑠 ) = ∑ 𝑎 𝑛 𝜋 𝑛 (𝜗 𝑠 ) + 𝑏 𝑛 𝜏 𝑛 (𝜗 𝑠 ) ∞ 𝑛=1 [START_REF] Battles | Laser-induced fluorescence measurements of NO and OH mole fraction in fuel-lean, high-pressure (1-10 atm) methane flames: Fluorescence modeling and experimental validation[END_REF] with 𝑎 𝑛 and 𝑏 𝑛 the partial waves amplitudes, 𝜋 𝑛 and 𝜏 𝑛 the parameters to the associated Legendre function [START_REF] Born | Principles of Optics[END_REF], giving the angular dependence of the scattered light in the direction of the scattering angle 𝜗 𝑠 .

Based on the work of Debye, Hovenac and Lock [START_REF] Hovenac | Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series[END_REF] expanded 𝑎 𝑛 and 𝑏 𝑛 into series, whose terms could be interpreted as scattering orders. The partial waves amplitude are expressed as:

𝑎 𝑛 𝑏 𝑛 } = 2𝑛 + 1 2𝑛(𝑛 + 1) (1 -𝑅 𝑎 𝑛 𝑏 𝑛 (𝑀𝑀) -∑ 𝑇 𝑎 𝑛 𝑏 𝑛 (𝑀𝑃) (𝑅 𝑎 𝑛 𝑏 𝑛 (𝑀𝑀) ) 𝑝-2 𝑇 𝑎 𝑛 𝑏 𝑛 (𝑃𝑀) ∞ 𝑝=2 ) (113)
with 𝑅 and 𝑇 respectively the reflection and transmission coefficients, the superscript M for the medium, and P for the particle, so that 𝑅 (𝑀𝑀) represents the reflection from the outer particle surface, and 𝑅 (𝑃𝑃) the inner reflection. 𝑇 (𝑀𝑃) is associated to the transmission from the medium to the particle, and 𝑇 (𝑃𝑀) from the particle to the medium.

These coefficients can be expressed with the Riccati-Bessel function of the third kind, 𝜉 𝑛 and 𝜁 𝑛 , and their derivatives, denoted with the superscript ′.

The coefficients 𝛼 and 𝛽 allow the separation of electric and magnetic waves. For 𝑎 𝑛 : (𝛼, 𝛽) = (𝑚, 1), and for 𝑏 𝑛 : (𝛼, 𝛽) = (1, 𝑚).

Scattering function for a scattering of order 𝑝 can be expressed by summing all the partial waves 𝑛 :

The sum of these scattering functions over all the scattering orders gives back to Eq.( 111) and Eq.( 112):

𝑆 1 (𝜗 𝑠 ) = ∑ 𝑆 1 (𝑝) (𝜗 𝑠 ) ∞ 𝑛=1 , 𝑎𝑛𝑑, 𝑆 2 (𝜗 𝑠 ) = ∑ 𝑆 2 (𝑝) (𝜗 𝑠 ) ∞ 𝑛=1 (120)
Various scattered intensities as a function of scattering mode and angle are presented in Fig. 31, from [START_REF] Albrecht | Laser Doppler and Phase Doppler Measurement Techniques[END_REF] ,computed by LMT and Debye series decomposition. The physical parameters were chosen to represent a water droplet in air at room temperature, with a laser beam at 𝜆 𝑏 = 488𝑛𝑚 typically used in PDA application. For perpendicular polarization, the 25-45° range is dominated by the first order refraction, the 95-125° range by the reflection, and the 135°-180° domain is dominated by the second order refraction on small ranges. One of the main differences between LMT theory and GO theory based on ray tracing is the dependence of these intensities on particle diameter, which is why GO is not recommended for small particles.

These two theories have been modified and enhanced throughout the years. In particular, the Generalized Lorentz Mie Theory (GLMT) gave a solution in the case of a gaussian incident beam C2 -Confidential and have been largely used, as shown later in the state-of-the-art section. However, the computational cost of GLMT has limited its use to theoretical simulations, and commercial processors have used relations derived from GO although some extensions have also been proposed in the past to correct the scattered field of small droplets.

Intensity of the scattered field on the receiver

We consider now two laser beams mirrored about the (x-z) and (y-z) plane and crossing with an incident angle 𝜃, see Fig. 27. The center of the illuminated volume formed by the beam crossing defines the center of the coordinate system. The detectors are placed at an off-axis angle 𝜙 𝑟 (xrotation), and an elevation angle 𝜓 𝑟 . The notation for vectors and distances are depicted in Fig. 32 :

The index 𝑏 stands for beam/incident wave, 𝑔 for glare points and 𝑖 for incident points. As an example, 𝑟 0𝑟 denotes the vector origin-receiver, and 𝑟 𝑝,𝑏𝑟 (𝑔) designates the vector between the particle glare point 𝑔 for the incident beam 𝑏 and the receiver 𝑟.

Firstly, the incident field strength and phase must be expressed. We consider a gaussian beam polarized in the 𝑦 direction and propagating in the 𝑧 direction. The strength of the electric field given by Eq.( 99), can be transformed from the beam coordinate system to the main coordinate system with:

𝑥 𝑏 = 𝑥 cos 𝜃 2 ∓ 𝑧 sin 𝜃 2 (121)
𝑦 𝑏 = 𝑦 (122)

𝑧 𝑏 = 𝑥 cos 𝜃 2 ∓ 𝑧 sin 𝜃 2 + 𝑧 𝑡𝑏 ( 123
)
𝑧 𝑡𝑏 is the distance on the 𝑧 axis between the beam waist and the measurement volume, that can be non-zero in the case of some misalignment.

The magnitude and phase of the incident field from the beam 𝑏 on the incident point 𝑖 associated with the glare point 𝑔 visualized by the receptor 𝑟, are respectively the amplitude and phase of the field expressed in Eq.( 99), written in the coordinate system centered on the measurement volume :

𝐸 𝑏𝑟 (𝑖) =
The beam radius defined from Eq.( 102), is now given by:

𝑟 𝑚𝑏 (𝑧 𝑏 ) = 𝑟 𝑤𝑏 √ 1 + ( ±𝑥 0,𝑏𝑟 (𝑖) sin 𝜃 2 + 𝑧 0,𝑏𝑟 (𝑖) cos 𝜃 2 + 𝑧 𝑡𝑏 𝑙 𝑅𝑏 ) 2 (126)
The Rayleigh length 𝑙 𝑅𝑏 is expressed in Eq.( 100). The radius of the wavefront 𝑅 𝑏𝑟 (𝑖) is given by:

𝑅 𝑏𝑟 (𝑖) = (±𝑥 0,𝑏𝑟 (
The phase expressed in Eq.( 125) can be rearranged with the notation defined in Fig. 32 to highlight the physical meaning of each term, the notation 𝐫 𝑎𝑏 will be used for vectors, and 𝑟 𝑎𝑏 will denote their magnitudes :

𝜓 𝑏𝑟 (𝑖) = 𝜔 𝑏 𝑡 -𝐤 𝑏 . 𝐫 0𝑝 -𝑘 𝑏 𝑧 𝑡𝑏 + 𝜓 0𝑏 + 𝜓 𝐺,𝑏𝑟 (𝑖) -𝐤 𝑏 . 𝐫 𝑏𝑟 (𝑖) [START_REF] Prucker | A flat flame burner as calibration source for combustion research: Temperatures and species concentrations of premixed H2/air flames[END_REF] The first four terms, 𝜔 𝑏 𝑡 -𝐤 𝑏 . 𝐫 0𝑝 -𝑘 𝑏 𝑧 𝑡𝑏 + 𝜓 0𝑏 , describe the phase of a plane wave propagating in the beam direction with respect to its phase at the origin 𝜓 0𝑏 . More precisely, the first term is linked to the spatial displacement of the interferences fringes over the particle and is used for determining the velocity direction of the particle. 𝐤 𝑏 . 𝐫 0𝑝 corresponds to the change of the phase between the origin of the coordinate system and the center of the particle, C2 -Confidential and 𝑘 𝑏 𝑧 𝑡𝑏 describes the phase change between the beam waist and the origin. Note that for a perfect alignment, the crossing of the laser beams at their beam waists leads to 𝑧 𝑡𝑏 = 0. The fifth term, 𝜓 𝐺,𝑏𝑟 (𝑖) comes from the wavefront curvature beyond the beam waist, thus, if 𝑧 𝑡𝑏 = 0, this term disappears. The last term 𝐤 𝑏 . 𝐫 𝑏𝑟 (𝑖) corresponds to the phase change between the incident point and the center of particle.

The scattered fields observed by the receivers come from the application of the scattering function described in Eq.(120) on the incident field. In most of Phase Doppler applications, the laser beams are linearly polarized in the direction perpendicular to the scattering plane, and a polarization filter is placed in front of the detectors. Thus, only one scattering function needs to be used for each beam-receiver couple, usually 𝑆 1 (𝜗 𝑠 ), with an amplitude 𝑆 𝑏𝑟 , and a phase 𝜓 𝑆,𝑏𝑟 .

The received scattering strength from the beam (𝑏) on receiver (𝑟) is then:

𝐸 𝑏𝑟 (𝑟) =
As explained in previous section, the detector records the time average of the signal. With two laser beams, 𝑏 = 1,2, this leads to an expression of intensity on the receiver similar to Eq.( 89) :

𝐼 𝑟 = 𝑐𝜀 2 (𝐸 1𝑟 (𝑟) + 𝐸 2𝑟 (𝑟) ) (𝐸 1𝑟 (𝑟) * + 𝐸 2𝑟 (𝑟) * ) = 𝑐𝜀 2 [(𝐸 1𝑟 (𝑟) ) 2 + (𝐸 2𝑟 (𝑟) ) 2 ] +𝑐𝜀𝐸 1𝑟 (𝑟) 𝐸 2𝑟 (𝑟) 𝑐𝑜𝑠 [(𝜔 1 -𝜔 2 )𝑡 + (𝐤 1 -𝐤 2
This can be rearranged as:

𝐼 𝑟 = 𝐼 𝑟,𝐷𝐶 + 𝐼 𝑟,𝐴𝐶 cos (𝜔 𝑟 𝑡 + 𝜑 𝑟 )

The DC part corresponds to the envelope of the burst, i.e. the low frequency part, and the AC matches the high frequency part of the signal. For a correctly adjusted Phase Doppler system, some assumptions can be formulated. First, if the optical setup is symmetrical, the laser beams have the same beam waists.

𝑟 𝑤1 = 𝑟 𝑤2 = 𝑟 𝑤 (132)
Secondly, using a power meter, the power of each laser beam can be adjusted so that they are similar, which leads to the amplitudes of the electric fields in the beams being the same.

𝐸 01 = 𝐸 02 = 𝐸 0 [START_REF] Komine | System for measuring velocity field of fluid flow utilizing a laser-Doppler spectral image converter[END_REF] In time-independent terms of Eq.( 131), i.e., 𝐼 𝑟,𝐷𝐶 and 𝐼 𝑟,𝐴𝐶 , the previously introduced frequency shift 𝑓 𝑠ℎ𝑖𝑓𝑡 is irrelevant as it is negligible compared to the wavelength of light. Thus, the wavenumbers can be considered equal:

𝑘 1 = 𝑘 2 = 𝑘 𝑏 (134)
C2 -Confidential

Establishment of the Phase difference -Diameter relation

One of the main hypotheses of the Phase Doppler technique is to consider the particle diameter very smaller than the size of the measurement volume. The Gaussian intensity profile of the laser beams is then of negligible importance, as the distance between the glare points on the particle is very small. Thus, the intensity at the glare points of each beam can be considered identical, and only the phase of the modulated signal contains information on the particle size, i.e., the cosinus term of Eq. ( 131). This assumption is true for very small particles, but becomes less acceptable as the ratio of particle diameter to measurement volume increases. A common technique for maintaining this assumption is to use beam expanders to increase the measurement volume. From Eq. ( 130) and Eq. ( 131), the phase measured by a receiver, 𝜑 𝑟 , is given by:

𝜑 𝑟 = (
The physical meaning of most of the terms has been explicated previously with the definition of Eq. ( 128). One term is particularly interesting: 𝐤 1 . 𝐫 1𝑟 (𝑖) -𝐤 2 . 𝐫 2𝑟 (𝑖) , because it includes a phase difference resulting from the curvature of the particle. It can be approximated as: The fourth term, (𝜓 02 -𝜓 01 ) describes the initial phase difference, mostly due to the beam splitting and the difference in optical paths between the laser beams.

𝐤 1 . 𝐫 1𝑟 (
The term (𝑘 1 𝑟 𝑝,1𝑟 (𝑔) -𝑘 2 𝑟 𝑝,2𝑟 (𝑔) ) denotes the path-length difference between the glare points and the receivers, this one being also function of the particle diameter. Finally, the term (𝐤 1 -𝐤 2 ). 𝐫 0𝑝 comes from the movement of the interference fringes on the receiver, and is thus used to determine the direction of the particle velocity. It can be approximated by:

(𝐤 1 -𝐤 2 ). 𝐫 0𝑝 = 4𝜋 sin 𝜃 2 𝜆 𝑏 𝑥 0𝑝 (138)
The first four terms of Eq. ( 135) are function of the position of the particle and thus on its velocity. By using two receivers and calculating the phase difference, these terms can be suppressed. The part of the phase that is independent of time and position will then be denoted Φ 𝑟 . This leads to the following expression: The explicit description of Eq. ( 139) requires the definition of the incident angle, denoted 𝜏 𝑏𝑟 , and the transmitted angle 𝜏 𝑡,𝑏𝑟 ′ , as shown in Fig. 33.

𝐞 𝑏𝑟 (𝑖) is the unit vector associated to 𝑟 𝑏𝑟 (𝑖) .By definition, we have : Also, the law of refraction gives the following expressions:

cos 𝜏 𝑏𝑟 cos 𝜏 𝑡,𝑏𝑟 ′ = 𝑛 𝑝 𝑛 𝑚 = 𝑚 (141)
𝐞 𝑝,𝑏𝑟 (𝑔) . 𝐞 𝑏 = 𝑐𝑜𝑠 (2(𝜏 𝑏𝑟 -𝜏 𝑡,𝑏𝑟 ′ ))

For 𝑟 0𝑟 ≫ 𝑟 𝑝 , i.e. a receiver located in the far field, a simplification can be made 𝐞 𝑝,𝑏𝑟 (𝑔) ≈ 𝐞 𝑝𝑟 ≈ 𝐞 0𝑟 , that is, the unit vectors describing the glare points -receiver are almost the identical.

Which leads to :

sin 𝜏 𝑏𝑟 = √ 1 - 𝑚 2 (1 -𝐞 0𝑟 . 𝐞 𝑏 ) 2(1 + 𝑚 2 -𝑚√2(1 + 𝐞 0𝑟 . 𝐞 𝑏 )) (143)
The incident and glare points positions are then given by :

𝐞 𝑏𝑟 (𝑖) = 𝑚𝐞 𝑝,𝑏𝑟 (
From the far field assumption, we can also express the last two terms of Eq. ( 139) with glare points positions and unit vectors:

(𝑘 1 𝑟 𝑝,11

Finally, the definition of these two angles gives also the expression of the phasing due to scattering:

𝜓 𝑆,𝑏𝑟 = -𝑘 𝑏 𝑚𝑟 𝑡,𝑏𝑟 = -4𝜋𝑚 𝜆 𝑏 𝑟 𝑝 sin 𝜏 𝑏𝑟 ′ [START_REF] Smeets | Flow Diagnostics by laser Interferometry[END_REF] With the two previous equations, Eq. ( 139) can be rearranged as:

∆Φ 12 = -∆Φ 21 = Φ 1 -Φ 2 = (
Finally, , assuming the receivers are symmetrically positioned with respect to the y-z plane, i.e., 𝜓 1 = -𝜓 2 and 𝜙 1 = 𝜙 2 , Eq. ( 149) together with Eq.( 143)-Eq.( 146), gives the explicit Phase -Diameter expression for the first order refraction [ which we will call for convenience:

∆Φ 12 = 𝛽 Φ,12 𝑑 𝑝 = [Φ 1 (𝜓 1 , 𝜙 1 , 𝜃/2) -Φ 2 (𝜓 2 , 𝜙 2 , 𝜃/2)] (151)
In this expression, 𝛽 Φ is the diameter conversion factor, also known as geometrical factor.

Arrangement system with three receivers

From Eq. ( 150), Ambiguity may arise when the phase difference exceeds 2π, which limits the maximum measurable diameter. For common Phase Doppler arrangement with small incident and elevation angles, the maximum diameter can be calculated with:

𝑑 𝑝,𝑚𝑎𝑥 = 2𝜋 𝛽 Φ,12 (152)
To overcome this limitation, the most commonly used strategy is to add a third receiver in the same vertical plane as the two others, i.e., at the same off-axis angle (see Fig. 34). A third phase is measured, which makes it possible to determine two phase differences related to the same droplet diameter, but with different diameter conversion factors. The third detector is usually located close to one of the others, so as to obtain a small 𝛽 Φ,13 and thus, a large maximum diameter. The low inclination of the ∆Φ 13 = f(𝑑 𝑝 ) curve, i.e., the low sensitivity of particle diameter to phase difference produces small errors in phase measurements that can lead to large errors in particle diameter. For this reason, the closest receivers are used to approximate the diameter and remove the ambiguity of 2π, while the farthest detectors are used to define its accuracy. The phase difference -diameter conversion is shown in Fig. 34, on a so-called Phase Plot. The blue dotted line represents the ∆Φ 13 = f(𝑑 𝑝 ) curve, associated with the two closest detectors, while the red lines correspond to the ∆Φ 12 = f(𝑑 𝑝 ) curves. In this example, if taken individually, the phase difference ∆Φ 12 would lead to five possible diameters, represented by red dots, but the only solution corresponds to ∆Φ 13 , marked by a blue dot.

The addition of this receiver is not only vital for the range of measurable diameters, but also for the accuracy of the measurement, as it serves to create a validation criterion based on the cancellation of the sum of all phase differences, which is useful for estimating the validity of the droplet sphericity assumption.

C2 -Confidential

∆Φ 12 + ∆Φ 23 + ∆Φ 31 = 0 [START_REF] Naqwi | Extended Phase-Doppler System for Characterization of Multiphase Flows[END_REF] This expression is not always respected, as some of the assumptions that were used in the previous section may be partially wrong, notably that of the sphericity of the droplet. For this reason, a validation tolerance has to be defined around the ∆Φ 12 = f(𝑑 𝑝 ) curve, represented by grey doted lines (Fig. 34).

In the commercial system such as the FiberFlow from Dantec Dynamic, the elevation angle can be changed by installing masks inside the receiving probe. The topside of Fig. 35 shows a scheme of such a probe while the bottom side displays the three aperture plates. Distances shown for the different masks define the separation between the center line of the plate and the center line of the aperture. The ones connected to the receivers one and three being symmetrical with respect to the y-z plane, 𝐷 1 = 𝐷 2 . The maximum measurable diameter is linearly proportional to focal lengths and inversely proportional to the receivers separation, as given by the expression for the refraction of first order :

𝑑 𝑝,𝑚𝑎𝑥 = 2 𝜆 𝑏 𝑚 |cos 𝜙 𝑟 2 | √ 1 + 𝑚 2 -2𝑚 |cos 𝜙 𝑟 2 |
𝑓 𝑟 𝑓 𝑡 (𝐷 1 -𝐷 3 )∆𝑏 [START_REF] Bachalo | Method for measuring the size and velocity of spherical particles using the phase and intensity of scattered light[END_REF] In this expression, 𝑓 𝑡 and 𝑓 𝑟 are the transmitter's and receiver's focal lengths, and ∆𝑏 the beam spacing.

Gaussian beam and Slit effect

Two optical biases can lead to an erroneous phase measurement, and to a particle being rejected outside the tolerance zone plotted on the phase diagram. The first is called the Gaussian beam effect, or trajectory effect. As mentioned earlier, the phase Doppler technique uses the assumption of a dominant scattering mode. The dominant mode can vary with scattering angle, droplet size and polarization, as highlighted in Fig. 31. Another factor described above is the Gaussian profile of the laser beam intensity, which is more important the larger the droplet size, as the intensity at the incident points can vary. The Gaussian beam effect is a measurement bias that arises from these two factors. For some trajectories, the incident points may sample different incident intensities so that one detector receives scattered light from one scattering mode and another detector receives scattered light from another mode. This is more likely to occur when the diameter of the droplet is larger than the dimensions of the probed volume, as the intensity differences across the droplet increase. An example is shown in Fig. 36, where the C2 -Confidential intensity of the reflected beam may be greater than that of the refracted beam, due to a higher intensity of the incident beam, while the receiver is located at a scattering angle where refraction should be dominant according to scattering theories This is more likely to happen when the droplet diameter is large in comparison of the measurement volume, as the differences of intensities across the droplet increase. An example is shown Fig. 36, where the intensity of the reflected beam can be higher than the refracted one, due to the higher intensity of the incident beam, while the receptor is placed at a scattering angle where refraction should be dominant according to scattering theories.

The slit effect comes from the use of a slit in the receiving probe to cut the length of the effective measurement volume. The length of the measurement volume is reduced to limit the probability of several particles passing through it, and to more precisely estimated the cross section of the measurement volume. However, the presence of two droplets at the edges of the cut measurement volume, one scattering by refraction, the other by reflection, is still possible. The resulting error is similar to the Gaussian beam effect.

Velocity bias

Phase Doppler Anemometry has the particularity of being able to measure each droplet passing through the measurement volume, with a data rate imposed by the processor. In particular, it imposes that the velocity sampling rate increases with the velocity, i.e. faster droplets are more likely to be detected than slower ones. This causes a bias on the average velocities, but also on the average diameters by assuming a velocity-size correlation. To compensate this effect, the mean values should be weighted with a factor, that will be denoted 𝑔 𝑖 , this one being inversely proportional to the droplet velocity, such as :

𝑈 ̅ = ∑ 𝑔 𝑖 𝑢 𝑖 𝑁 𝑖=1 ∑ 𝑔 𝑖 𝑁 𝑖=1 (155)

Fig. 36: Example of beam effect, incident beam on a large droplet with parallel reflection and refraction

C2 -Confidential

One of the most obvious weighting factors would be the inverse of the velocity magnitude, defined by the following formula:

𝑔 𝑖 = 1 √𝑢 𝑖 2 + 𝑣 𝑖 2 + 𝑤 𝑖 2 (156)
This approach has been adapted for ellipsoidal measurement volume by McDougall in 1980 [START_REF] Mcdougall | Bias correction for individual realisation LDA measurements[END_REF].

𝑔 𝑖 = 1 √𝑢 𝑖 2 + 𝑣 𝑖 2 + 𝑎 0 2 𝑐 0 2 𝑤 𝑖 2 (157)
𝑎 0 is is the radius of the measurement volume, and 𝑐 0 its half-length. This weighting factor requires three velocity components and is thus rarely applied. McLaughlin and Tiederman [START_REF] Mclaughlin | Biasing correction for individual realization of laser anemometer measurements in turbulent flows[END_REF] have proposed in 1972 a weighting factor based only on the inverse of one velocity component. While for many applications this weighting may be wrong because a single velocity component cannot properly represent a turbulent flow, the authors showed that if the fluctuations of the unmeasured velocity components are at least twice as small in magnitude as the measured fluctuations, then 𝑔 𝑖 = 1 |𝑢 𝑖 | ⁄ effectively reduces the measurement bias on velocity. A weighting factor based on arrival time has also been widely used over the years, but it has been found to be valid with only when the data rate is high enough.

𝑔 𝑖 = 𝑡 𝑖 -𝑡 𝑖-1 [START_REF] Gréhan | Particle Trajectory Effects in Phase Doppler Systems: Computations and experiments[END_REF] 𝑡 𝑖 is the arrival time of the particle 𝑖. The condition of validity of this weighting factor is: 𝑁 ̇𝑇𝑢 > 5 [START_REF] Aizu | Particle sizing by planar phase doppler system[END_REF] 𝑁 ̇ is the mean data rate, and 𝑇 𝑢 the turbulence level, defined as:

𝑇 𝑢 = 1 3 √ 𝑢 ′ 2 + 𝑣 ′ 2 + 𝑤 ′ 2 𝑈 ̅ (160)
It should be noted that with an assumption of isotropic or partially isotropic turbulence, this equation can be reduced to two or one velocity component. Furthermore, in spray applications, it is very likely that the average data rates are always high enough to fulfil the conditions given by Eq. ( 159)

One last weighting factor commonly used is the transit time of the droplet through the measurement volume, 𝑔 𝑖 = 𝜏 𝑖 . Its estimation comes from the analysis of the burst length which will be detailed in the state-of-the-art section.

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State of the art 4.4.1 Phase Doppler Technique development

The first application of the laser doppler technique for particle velocity measurements was reported in 1964 by Yeh and Cummins [START_REF] Yeh | LOCALIZED FLUID FLOW MEASUREMENTS WITH AN He -Ne LASER MEASUREMENT OF LOCALIZED FLOW VELOCITIES IN[END_REF]. The objective was to measure slow velocities, i.e. <0.05 m.s -1 , of monodispersed polystyrene spheres of 557 𝑛𝑚 in diameter circulating in water.

The authors used a laser spectrometer to analyze the doppler shifts in the scattered light from a 632.8 𝑛𝑚 laser beam generated by a He-Ne laser. A laser beam is generated and then split by a beam splitter, so that one part passes through the measuring cell, while the other part passes through a single sideband modulator. The scattered light is then recombined in front of a photomultiplier via a semi-reflective mirror (see Figure 12), so that the measurement volume, or fringe pattern, is detected on the detector [START_REF] Sanjay | A Review of NO x Formation Under Gas-Turbine Combustion Conditions[END_REF]. With this device, they succeeded in measuring velocities of the order of 0.007 m.s -1 with a special resolution of 1.6 mm, corresponding to the diameter of the laser beam.

Their work was further developed in 1967 by Goldstein and Kreid [START_REF] Goldstein | Measurement of laminar flow development in a square duct using a laser-doppler flowmeter[END_REF]. They detailed the expression of Eq. ( 76) that gives the signal frequency observed by a receiver from a moving emitter, and subsequently, the doppler shift described by Eq. ( 78). Then, the authors modified the optical system proposed by [START_REF] Yeh | LOCALIZED FLUID FLOW MEASUREMENTS WITH AN He -Ne LASER MEASUREMENT OF LOCALIZED FLOW VELOCITIES IN[END_REF]. They proposed the so-called reference beam configuration, which is easier to set up. In this case, one beam is split in two, and the two resulting beams are transported with the same incidence angle in the measurement cell through mirrors and focal lenses. The receiver is placed in the path of the so-called reference beam(see Fig. 37.b) and collects the mixing of refracted and reference lights, process known as optical heterodyning. The adjustment of the incident angles is ensured by using mechanical apertures in front of the photomultiplier and checking that both signals passes through well. Measurements are performed on polystyrene particles of 557 𝑛𝑚 in diameter, inserted in water. Velocity is measured in transparent test section of known diameter, with a He-Ne laser at 632 𝑛𝑚 with a power output of 0.3 𝑚𝑊. The results are compared with the average velocity estimated using tanks and a balance, the measured filling time and theoretical simulations. They show a good agreement with the weighting technique and theoretical predictions in the center C2 -Confidential of the duct, with a difference of less than 0.1%. The velocity measured by laser Doppler differs slightly from the prediction at the duct entrance although the radial profile remains plausible.

In 1969, Vom Steim and Pfeifer [START_REF] Stein | A Doppler difference method for velocity measurements[END_REF] studied the limits of the velocity measurements with the laser doppler technique. With a 632.8 𝑛𝑚 wavelength, as used in previous works, they estimated that the highest frequency measurable was around 10 m.s -1 . This limitation is due to the technical specifications of the receivers, as the Doppler shift would be about 30 MHz with such a wavelength and particle velocity, and higher shifts would be impossible to measure with good accuracy. For higher velocities, they proposed the use of the crossing of two coherent beams, leading to the so-called dual beam configuration, depicted in Fig. 27. By doing so, the shift of frequency can be decreased by several orders of magnitude. The authors demonstrated the potential of the technique by measuring the velocity of particles on a rotating plate at 16.8m.s -1 , and showed good agreement with the theoretical value.

In the same year, Rudd [START_REF] Rudd | A new theoretical model for the laser Dopplermeter[END_REF] proposed a simplification of the laser Doppler-meter based on the observations of Vom Steim and Pfeifer, called the dual beam configuration. In this way, a real measurement volume is formed, as the light collected comes only from the scattering of the interferences formed there. As with Goldstein and Kreid, masks are used to control the position of the beams, which leads to an overall symmetrical configuration with fewer optical adjustments. The author also formulated conditions to achieve acceptable measurement accuracy, such as the number of fringes formed in the measurement volume as a function of wavelength and angle of incidence, without excluding the rules formulated in [START_REF] Maughan | A dry low NOx combustor for the MS3002 regenerative gas turbine[END_REF][START_REF] Lefebvre | Atomization and Sprays[END_REF].

In 1970, Jackson and Paul [START_REF] Dunn-Rankin | Lean Combustion: Technology and Control: Second Edition[END_REF] overcame the maximum velocity limitation by using a Fabry-Perot interferometer. The method uses a reference beam and a direct measurement of the frequency with the Fabry-Perot interferometer is made possible. Incoming light from one of the laser beams that have been shifted by the Doppler effect passes through the Fabry-Perot, creating concentric fringes that are used to evaluate the frequency of the incoming light. The authors studied the characteristics of such experimental setup for supersonic velocity measurements that lead to a measured frequency shift around 1Ghz. They imposed key parameters such as the target frequency range, the finesse and the laser wavelength, and proved analytically the advantages of a confocal instead of a plane Fabry-Perot. They applied this method on a supercritical flow, with ice particles of 500 𝑛𝑚 in diameter and a 300mW laser at 514.5 𝑛𝑚.

Results showed an accuracy of around 3% of the theoretical flow velocity at Mach 1.6. They later [START_REF] Paul | Rapid velocity sensor using a static confocal Fabry-Perot and a single frequency argon laser[END_REF] proposed a revised system of a Fabry-Perot coupled with apertures that does not require moving mirrors, and a theoretical procedure to visualize instantaneous velocity measurements.

In 1972, Farmer is one of the first researchers who reported droplet diameter measurements with laser interferometry [START_REF] Farmer | Measurement of Particle Size, Number Density, and Velocity Using a Laser Interferometer[END_REF]. Farmer expressed analytically the scattered intensity from a droplet passing through the crossing of two laser beams. He showed its dependence on scattering function, themselves related to the diameter of the particle. The development requires several assumptions, and notably the sphericity of the droplet, a diameter smaller than the width of the fringes, and an optical setup such as the measurement volume is capable to detect the bright fringe in the center of the illuminated volume and one of the adjacent dark fringes. Then, by comparing the intensity scattered by the particle passing through the dark and bright fringes, the diameter can be measured with the knowledge of the ratio of the AC and DC parts of the intensity signal on the receiver. The construction of the intensity-diameter relation C2 -Confidential led to some limitation: if the sampled particles had diameters proportional to the fringe's width, they would not lead to measurable doppler signals. If the velocity is known, the author also suggested the possibility of evaluating the number density.

After the review of Smeets [START_REF] Smeets | Flow Diagnostics by laser Interferometry[END_REF] in 1977 on various laser doppler velocity systems and their applications, Smeets and George [START_REF] Smeets | Michelson spectrometer for instantaneous Doppler velocity measurements[END_REF] proposed another optical system for instantaneous velocity measurements based on a phase-stabilized Michelson interferometer. The shifted light is collected by a combination of a converging focal lens and an optical fiber, and then transmitted through a polariser and a Pockels cell before entering the interferometer. The measured signal is proportional to the doppler shift and can follow unsteady features with a temporal resolution down to 1µs. The authors successfully measured velocities on numerous experimental cases, such as bullets or gas flows, but also carried out density measurements on a shockwave interacting with a steady flow using the variation of refractive index of the medium.

The same year, Yule et al. [START_REF] Yule | Particle size and velocity measurement in flames by laser anemometer[END_REF] pursued the work of Farmer on droplet size measurements by laser anemometry. They used a dual beam setup with an on-axis forward scattering to measure the diameter of droplets in the range 20 -250 µ𝑚 (see Fig. 38.a), the main objective being the future application on spray in combustion. The methodology was based on the amplitudediameter relationship, except that the average signal maximum was used instead of the envelope maximum (see Fig. 38.b). The main advantage of this method is that it is more reliable over a larger diameter range than the Farmer method. The relationship between the measured intensity and drop diameter has been used with the geometric optical approach, which is becoming ineffective for small drops and has imposed a lower limit on the measurable diameter. The optics were installed such as only the center of the illuminated volume was observed. Prior to the measurements, a calibration is performed with a rotating disc and glass spheres of different diameters. The calibration set-up also showed that the position-scattering intensity profile in the transverse direction, i.e. the energy profile of the laser beam, followed a Gaussian distribution. An argon-ion laser operating at 200mW was used for experiments, with an incident angle of 𝜃 = 6.81° and a collection angle of 𝜙 = 0.43°. The measured size distributions were compared to measurements made by the impaction technique coupled with the use of a microscope and computer assisted image analysis. The results showed reasonable agreement, although the Laser Doppler diagnostic showed significant errors for non-spherical particles or dense sprays, where several particles could penetrate in the measurement volume simultaneously. Under reactive conditions and depending on the brightness of the flame, the signal-to-noise ratio could in some cases not be exploited. For example, yellow soot flames produce too much spurious noise for the measurements to be processed. The authors then suggested the use of optical filters to reduce the light emission of the natural flame on the PDA signals.

In 1980, Bachalo [START_REF] Bachalo | Method for measuring the size and velocity of spheres by dual-beam lightscatter interferometry[END_REF] pursued the development of the dual beam light scatter interferometry by proposing a diameter measurement based on the ratio of the pedestral (see Fig. 38.b) and doppler component of the intensity. The advantages of the proposed methodology are that the measured signal is not dependent on incident intensity, droplet absorption or absolute scattering intensity. He also increased the measurable range by placing the collection probe off the transmission axis, thus eliminating the need for a beam blocker and its limitations. Bachalo started by defining the scattered field strength, with the same form as the first term in Eq. ( 130), C2 -Confidential and then decomposed it as the sum of an AC and DC parts, with the AC containing the information about the Doppler signal. Then the ratio of these two terms was defined as the visibility term, which was then interpreted as the average of the scattering function given by Van de Hulst's theory of geometrical optics [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF]. The resulting diameter-visibility relationship is therefore related to the phase shift of the light scattered by the incoming beams and not to the total amplitude. The author then compared this theoretical method with experiments performed on droplets of 5µm-2500µm diameter. Having found satisfactory results, Bachalo concluded this study by indicating that an uncertainty in the refractive index leads to a proportional uncertainty on the measured diameter, while also reminding that non-spherical droplets and dense sprays can increase the error on the measured distributions.

In 1984, Bachalo [START_REF] Bachalo | Phase/Doppler Spray Analyzer For Simultaneous Measurements Of Drop Size And Velocity Distributions[END_REF] enhanced the previous method by deriving a linear expression between droplet diameter and phase difference measured with three detectors. Due to the prohibitive cost of the Mie scattering calculation, the scattering functions were determined with geometric theory although the authors point out that the GO and LMT approaches give significantly different results for small diameter droplets, with a validity limit around 5µm. With this optical arrangement, two measurement of the phase difference could be carried out on the same droplet, resulting on the phase plot shown in Fig. 34.b, with a removal of the 2π ambiguity and a droplet rejection method based on Eq. ( 153). A specific computer processor was also developed for the determination of the period and phase difference between two bursts. It was used to count the number of periods of each signal and to check its usability by comparing it to a user-defined minimum number of periods. The elevation angles were made adjustable with masks having different spacings and apertures. This method was verified by measurements on reference sprays as the phase Doppler technique was able to effectively reproduce the droplet distribution of the spray with good resolution. The authors concluded that a significant error may arise from an uncertainty in the refractive index of the scattering medium. Thus, even with a weaker measurement signal, backscatter measurements can be an interesting choice, since the reflection does not depend on refractive index.

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In 1990, Naqwi et al. [START_REF] Naqwi | Extended Phase-Doppler System for Characterization of Multiphase Flows[END_REF] proposed an extended Phase Doppler Anemometry system to minimize the error due to the uncertainty on the refractive index of the measured particle. By using two receiving probes aligned on the same measurement volume, two phase differences can be measured on the same particle at different scattering angles. With the doubling of the phase difference-diameter relationship, given by Eq. ( 150), the diameter can be removed and the refractive index can be expressed simply as a function of the phase difference. The authors further showed that the accuracy of this method could be improved by using a significantly different off-axis angle, while maintaining the same dominant scattering mode. For combustion applications, the method of refraction is shown to be the most favorable, and applicable for offaxis angles varying from 25 to 70°, which represent a convenient range for the extended PDA.

The authors tested this technique first on a monodisperse spray of glycerin-water mixture, of refractive index between 1.33 and 1.41. For scattering angles of 30° and 60°, the linearity of the phase-diameter relationship was verified, as were the phase-refractive index relationships giving refractive index values close to those measured with a commercial refractometer. Then, the extended PDA was tested on a polydisperse spray generated by a Danfoss oil burner nozzle operating at an injector pressure of 4 bar and producing particles of diameter between 2µm and 100µm. For each particle, two diameter measurements are made, and the arithmetic mean is kept. The results showed a droplet distribution centered around 13µm, with minimum and maximum droplet sizes of 2-4µm and 86µm respectively. In particular, the scatter plot diameter -phase ratio is shown, and smaller particles showed a significantly larger variability, especially below 20µm. To test the method on larger droplets, the triggering level of the receivers have been then fixed to 200mV instead of 10mV to remove droplets below 30µm from measurements. This second set of data, with droplets in the 40-70µm range, showed a much more linear phase ratio -diameter relationship which is supported by LMT theoretical calculations. The authors suggested that this method was best suited to medium to large diameter particles, while recommending further development of the technique to correct uncertainties and extend its range of application to smaller droplets.

In 1991, Bachalo [START_REF] Bachalo | Method for measuring the size and velocity of spherical particles using the phase and intensity of scattered light[END_REF] proposed an improvement of the phase doppler technique by adding an amplitude validation criterion to reduce the gaussian beam effect, see Fig. 36, as well as the probability of validated erroneous measurements. The amplitude of the received signal being proportional to the square of the particle diameter, the author proposed first an empirical approach, consisting of measuring the amplitude range in function of the diameter of particle, and a theoretical approach, based on the Lorenz-Mie theory. For each size class, the maximum and minimum intensities can be fixed to define an acceptable diameter range. Next, Bachalo proposed a method to compensate for the measurement bias of the sample volume, which favours the measurement of large particles over small ones. To ensure that the particles actually pass through the center of the measurement volume, a minimum number of crossed fringes is defined to guarantee signal processing. Next, the cross-sectional area of the measurement volume is calculated for each size class, assuming that the mean trajectory remains orthogonal to the fringes. The classes of measured distributions are then multiplied by a correction factor to recover a distribution corresponding to an identical effective cross-section over the entire size range. Finally, Bachalo proposed to use of the signal amplitude as an estimate of the diameter to deal with the 2π ambiguity.

In 1992, Edwards and Marx [START_REF] Edwards | Analysis of the Ideal Phase-Doppler System: Limitations Imposed By the Single-Particle Constraint[END_REF] studied the impact of the single particle in the measurement volume hypothesis on the measured particle flux by PDA. They conducted a theoretical study on C2 -Confidential the effect of the rejection of signals produced by multiple particles in the measurement volume on the particle flux. For this purpose, they separated the rejection rate into a temporal and a spatial part, corresponding respectively to a too long residence time and a too high spatial concentration of droplets. From this separation, they constructed two Poisson-based filters, that characterized the rejection rate of particles and thus the decrease of the measured flux. To develop this approach, the authors considered an "ideal" PDA system, with no gaussian or slit effect able to perfectly measure droplet diameters and velocities. Then they expressed the interparticle time and spatial distribution in the case of a continuous, spatially and temporally homogeneous particle flux, and considered that those distributions were appropriate to describe inter-droplet statistics. Since the residence time is inversely proportional to the droplet velocity and proportional to the size of the measurement volume, the temporal filter was expected to preferentially suppress droplet population with low velocities or large effective measurement volume, thus deforming the values of the mean diameters. After developing relationships between the measured and "true" size distributions and data rate in a general case, the authors considered an ellipsoidal volume, and defined its efficient cross section depending on the particle trajectory. The, they expressed the ratio between the measured and "true" quantities, applied on the data measured on a kerosene spray flame. The spray had a size distribution span from 2 to 50µm with a mean size of 26µm. Droplet mean velocity was around 10m.s -1 for smallest droplet, up to 20 m.s -1 for largest droplet. For a droplet concentration of 730 cm -3 , the rejection rate by the Poisson filters are almost nonexistent and the flux is accurately determined within an accuracy of 1%. With a concentration ten times higher, the fraction of the measured flux dropped to 87%. With a concentration of 73000 cm -3 , the measured flux dropped down to 24% of the "true" flux, showing a significant increase in measurement inaccuracy. Even with such a high rejection rate, it is interesting to note that the average diameters measured remained within 5% of the true value, which shows that the flux measured by PDA is very dependent on the particle concentration, but not the average diameter.

Following the description of the scattering functions of a spherical particle by a gaussian laser beam with the Generalized Lorenz-Mie Theory (GLMT) [START_REF] Gouesbet | Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation[END_REF][START_REF] Gouesbet | The localized interpretation to compute all the coefficients in the generalized Lorenz-Mie theory[END_REF], Gréhan, Gouesbet et al. [START_REF] Gréhan | Particle Trajectory Effects in Phase Doppler Systems: Computations and experiments[END_REF] studied the effect of particles trajectories passing through a gaussian laser beam on phase doppler measurements and published their work in 1993. The GLMT add two coefficients to the formulation of the scattering function from the LMT: 𝑔 𝑛,𝑇𝑀 𝑚 and 𝑔 𝑛,𝑇𝐸 𝑚 , with the subscript TE denoting the transverse electric wave, and TM the transverse magnetic wave. Those coefficients consider the wavelength and shape of the incident beam, and the position of the particle inside the illuminated volume. One of the main advantages of this formulation is the possibility to simulate the evolution of the signal in function of the incident points on the particle, theses ones depending on the trajectory of the rays arriving on the particle. In [START_REF] Gréhan | Particle Trajectory Effects in Phase Doppler Systems: Computations and experiments[END_REF], the authors first simulated the diameter -phase / visibility / scattered power relations in function of the particle position with respect to the beam waist, for both forward and backward refraction. Eight particle positions were tested, and a similar trend was denoted for both scattering modes. The trajectories leading to the particle being positioned such as the laser beam waist is located in the lower part of the particle in the opposite direction of the receiver (see point a on Fig. 39) lead to erroneous measurements, which is highlighted by the non-linearity of the three verified relationships. The basics of the optical setup together with the particle and beam waist positions are shown in Fig. 39.a). The phase-diameter relation computed from GLMT and for the detector C2 -Confidential D2 is shown in Fig. 39.b). The points in the bottom/bottom-left of the particle, b and h, displays a relatively non-linear phase-diameter relationship, while the point at the lowest part of the particle, a, shows a very different shape from that expected from 𝐷 𝑝 ≈ 12µ𝑚. For the other beam waist points, the linearity is quite well respected, and thus, the geometrical optics stays relevant, and the incident wave can be considered plane. Those results were first carried out for one detector, then the authors focused their work on the forward scattering. They calculated the diameter from a PDA measurement with three detectors in function of the actual particle diameter for the position a. As expected, for diameters below 12µm, the results are still valid, but a significant shift appeared for the largest droplets. A common validation criterion was to compare the ratio of both phase differences,

𝜙 12 𝜙 13
, to the theoretical range (2.765 -2.965), and to reject measurements outside this range. It was found that particles smaller than 32µm were all validated, so there was a range of diameters, 12-32µm, where very erroneous measurements could be validated.

To reject these erroneous measurements, the authors proposed another version of the extended phase Doppler anemometry, by adding a second receiver probe positioned symmetrically with respect to the emission plane, on an off-axis angle 𝜙 𝑟2 = -𝜙 𝑟1 . Thus, for each possible trajectory, at least one receiver can measure the phase difference on the "right side" of the particle. As shown in Fig. 39.b), the phase modulus of erroneous measurements is always smaller than those of correct measurements, so a simple comparison of the phase modulus is sufficient to select the correct receiver. From this optical arrangement, a comparison of SNR could also be done, and measurements from droplets passing at the edges of the illuminated volume could be rejected since one of the receivers would detect a poor signal. The authors then tested this experimental setup with a forward scattering geometry on a water spray producing droplets in the 2-80 µm diameter range. The resulting size distribution showed weak differences from measurements from the separated receivers. The extended setup showed weak deviations for droplets larger than 60µm. The authors concluded that the impact of the trajectory effect was moderate under these conditions, but that it could lead to significant errors when the droplets were larger than the measurement volume.

In 1994, Aizu et al. [START_REF] Aizu | Particle sizing by planar phase doppler system[END_REF] proposed another approach to deal with the gaussian beam effect on large particles based on the generalized Lorenz-Mie theory. The authors proposed a planar optical arrangement with the polarization and collection in the same plane, i.e. using the offaxis angle as the elevation angle. The simulated phase-diameter relationships showed a perfect linearity for all particle positions with respect to the laser beam waist, with the exception of the most backward positions as reported by Gouesbet et al. Compared to Fig. 39.b), all trajectories are perfectly linear except for the positions a, b and h, which underlines the ability of the planar arrangement to eliminate a high number of uncertainties. The simulated relationship between phase difference and diameter showed good linearity for medium and large diameter droplets when the beam waist is placed on the receiver side or in the center of the droplet. Some fluctuations similar to those previously found by Gouesbet et al. when the beam waist was on the other side of the illuminated volume were also observed. However, fluctuations also appeared in the small droplet size region (<12µm). To minimize this effect, the authors computed the standard deviation of the GLMT from the linear relationship given by Eq. ( 150) in function of the incident and off-axis angles. The results gave an optimal system with an incident angle 𝜃 = 5° and off-axis angles 𝜓 1 = 50° and 𝜓 2 = 60°. The authors concluded that this arrangement was well suited for larger droplet measurements to effectively reduce the gaussian beam effect. During the same year, Xu and Tropea [160] also proposed an improvement of a two component PDA system to correct errors due to the trajectory effect. By using GLMT, they calculated the theoretical signal from a droplet passing in the Y direction, i.e. from the backside of the measurement volume towards the frontside with respect to the collecting probe. For a droplet of 38µm and a measurement volume of 50µm in diameter, they showed that the scattered light reached a maximum amplitude at 18µm from the measurement volume center in the positive Y direction (front side). This region is considered by the authors being the most reliable for diameter measurements as the phase -doppler ratio should be constant. Inversely, at -10µm from the center, the signal deviated strongly from the expected value, and its amplitude was equivalent to the one of a 6µm droplet passing through the center. Preliminary measurements were made on a glass sphere of 125µm in diameter with a PDA system (Dantec) using a 58N10 processor and a measurement volume of 130µm. The signal data processing with this processor was of particular interest, as it didn't simply evaluate the phase from the maximum of the burst mean envelope, but instead from an average value estimated from the whole burst and centered on the highest covariance point. This method allowed to obtain a phase differencediameter relation without oscillations for small droplets, and thus comparable results between geometrical optics and GLMT calculation over the whole size range analyzed. The evolution of the measured diameter in function of the droplet position is given by the authors. The results showed that all backside positions led to a large underestimation of the diameter, up to ~32% at y=-65µm, a correct sizing around 10µm from the center of the measurement volume, and an underestimation for frontside positions up to 12% with a mean value of 5%. To reduce the contribution of parasite signals from the trajectory effect, slits apertures are often used to cutoff the effective measurement volume in the z direction. However, a new measurement bias appears. This one, called the slit effect, corresponds to the case where two particles located close to the edges of the slit, but partially outside the measurement volume, are measured as C2 -Confidential one. Some examples given by the authors are reported in Fig. 40. The slit effect occurs when two droplets are assimilated into one droplet, which is a consequence of the Gaussian profile of the laser beam, as one drop emits a scattering signal by reflection while the other scatters by refraction. The authors found in experiments with glass spheres, that for all possible trajectories, a large variation of the signal amplitude was visible with the slit effect, which could be used to get rid of it. They also tested their experimental setup on a polydisperse water spray oriented along the usual x direction and along the y direction. Results showed a large gaussian beam effect for the globally downward spray (x positive) as reported previously, but without any trajectory effect when the spray is oriented along the y direction. In particular, measurements of droplets larger than the measurement volume showed a larger standard deviation on both velocity components. The authors concluded that the use of the two-components PDA was of particular interest, in particular with the droplet moving in the (y-z) plane, towards the receiver. The gaussian beam effect has been reduced with a combination of a slit aperture, a planar arrangement for one of the components, and the signal processing by the 58N10 processor. They also suggested the coupled use of standard PDA (SPDA) and Planar PDA (PDPA) to further reduce errors from the trajectory and slit effects.

In 1996, Tropea et al. [START_REF] Tropea | Dual-Mode Phase-Doppler[END_REF] presented this setup as the Dual-Mode Phase Doppler Anemometer, a.k.a. Dual PDA, suited for turbulent flows and measurements of mass flux and droplet concentration. As previously suggested, the Dual PDA is a combination of a PPDA and a SPDA. Naturally, the PPDA proposes a much broader measurable diameter range (phase conversion factors usually 2 to 4.5 times lower) than SPDA, and is also used with this configuration to remove the 2𝜋 ambiguity. The SPDA has a much higher sensitivity and provides a precise estimation of the droplet diameter. The authors presented the diameter -phase difference relationship for the SPDA and PPDA with the geometrical optic approach and the GLMT, see Fig. 41.a). Both systems showed some divergences from the linearity with the GLMT calculation for droplet smaller than 5µm, which corresponds to the limit of the GO theory. To extend the validity of the geometric approach, the authors proposed to integrate the signal received by the photomultipliers over the entire aperture area with the intensity of the scattered light used as a weighting factor. This integration leads to the relation denoted as "GO int." on Fig. 41.a), which showed a better agreement with GLMT calculation in the case of PPDA. The oscillation of the phase difference -diameter curve below 10µm is much stronger in the case of PPDA and can lead to negative values that could be interpreted as very large particles. For this reason, the authors defined a maximum measurable phase difference of 320° for PPDA, corresponding to a threshold value determined with GLMT computation. The validation routine also included a minimum number of crossed fringes to limit the number of measurements from droplets passing at the edges of the measurement volume. They tested the Dual PDA on a water spray from a 15° As expected, the removal of the trajectory and slit effects by the coupling of SPDA and PPDA changes the size distribution significantly, since the largest droplets are the most impactful. The evaluation of the mass flux is also subject to a validation procedure. First the authors suggested the use of a modified version of the measurement length evaluation based on the burst length. By using both velocity components, a better accurate evaluation of the effective cross section of the effective measurement volume could be obtained. However, the removal of the gaussian and slit effects would also impact the effective measurement volume.

To take into account this effect, computation with GLMT were performed depending on the particle refractive index and the diameter to adjust the cross-sectional area normal to the x and y directions. This resulted in a trajectory dependent weighting of each droplet in the mass flux computation. Finally, the authors compared a SPDA at an off-axis angle of 70° and the Dual PDA at 30°. They showed that the latter had a better capacity to remove erroneous measurements from gaussian and/or slit effects, which changed significantly the shape of the volume distribution. They finally suggested the use of dual phase measurements to also evaluate the refractive index of the particles if their motion are oriented along the Y direction, as Naqwi reported [START_REF] Naqwi | Extended Phase-Doppler System for Characterization of Multiphase Flows[END_REF].

In 1998, Damaschke et al. [START_REF] Damaschke | Response of Phase Doppler Anemometer systems to nonspherical droplets[END_REF], studied the response of PDA to non-spherical droplets, because sphericity is an assumption on which all commercial systems are designed.. They examined the response of the planar PDA (PPDA), standard PDA (SPDA) and Dual PDA to droplets in the 70 -200µm range in volumetric diameter 𝑑 𝐺 = (6𝑉 𝐺 𝜋 ⁄ ) 1 3 ⁄ , oriented along the axis perpendicular to the fringes (prolate), or parallel to the fringes (oblate). A CCD camera was used to record the shape of the droplet passing through the measurement volume and its aspect ratio. With this, scattering computation were performed for the first refracted ray for elliptical particle with the same aspect ratio. The volumetric diameter was known through the mass flow rate passing through the droplet generator and its excitation frequency. Coupled with the images recorded by the camera, and assuming an ellipsoidal shape, the length of the minor and major axis could be determined. Precise control of the droplet trajectory through the probed volume eliminated much of the trajectory and/or slit effects by ensuring minimal spatial variation in droplet illumination. Thus, two effects were studied, the capacity of the systems to reject or not non- The period of droplet oscillations was estimated to be of the same order as the transit time, i.e. approximately 120µs for a 100µm droplet. The processor used to process the signals was a Dantec 58N10, and the phases were computed with:

ϕ ̅ = ϕ 0 + ∫ 𝜕ϕ 𝜕𝑡 𝑑𝑡 𝑡 0 +∆𝑡 2 ⁄ 𝑡 0 -∆𝑡 2 ⁄ (161)
(𝑥 0 , 𝑡 0 ) are the position and instant where the droplet is the closest to the center of the measurement volume and ϕ 0 the associated phase. Droplet visualization and their position on the phase plot are shown in Fig. 42 for SPDA and Dual PDA. Results showed that even for a large droplet distortion, SPDA couldn't effectively reject the measurements, as they are all close to the theoretical linear curve. With a criterion of sphericity deviation fixed at 3%, all measurements were validated, and it would have required to low it down to 1% to yield to acceptable rejection rate. On the other hand, the Dual PDA was able to assess sphericity and reject low aspect ratio particles. Indeed, the SPDA analyses a single curvature whereas the Dual PDA calculates the phase differences of two orthogonal curvatures. Fig. 42.a) shows the relative error of phase difference measurements with respect to the droplet diameter, for the SPDA configuration and PPDA component of a Dual PDA system. The authors denoted that SPDA yielded to irregular results for prolate droplets (aspect ratio <1), and systematic error leading to diameter almost always inferior to the expected value, even for spherical droplets. The authors pointed out that this systematic error was due to alignment errors and not to sphericity bias. PPDA (Fig. 42.b), showed a linear evolution of the error of phase measurement with respect to the aspect ratio in the range of +/-20%, while SPDA showed errors equal to +/-50%. Particles with distortion in the x-direction have always been measured as being smaller than they are, while oblate particles are estimated to be too large. The dual PDA arrangement kept the relative error in the ±5% range.

To visualize the evolution of 𝜕ϕ 𝜕𝑡 , the authors used a custom-made processor for various cases of oscillating droplets. The results showed that the temporal evolution of the measured phase was significantly different depending on the oscillation, and thus on the deformation of the droplet during its transit through the measurement volume, which could explain the scatter shown in Fig. 42.

The authors concluded that even with a specific configuration optimized to suppress the trajectory and slit effects, the Dual PDA arrangement was superior to the SPDA and PPDA as it effectively rejects non-spherical droplets. Depending on the aspect ratio, the two more classical approaches showed significant error on the measured diameter, up to 50% of the real value for the most extreme cases with SPDA.

In 1999, Wigley et al. [START_REF] Wigley | High power, high resolution LDA/PDA system applied to gasoline direct injection sprays[END_REF] presented an improved transmitter system to allow easier modification of the measurement volume dimensions while keeping an effective laser energy transmission. The proposed new Bragg cell provided a 50/50 beam splitting and the 40MHz frequency shift usually used in most PDA applications. More importantly, the cross-section of the output beams is circular, which was not necessarily the case before, leading to a better definition of the measurement volume. Also, a set of divergent and convergent focal lenses were placed right after the Bragg's cell, and can be translated to adjust the beam expansion and separation. An interchangeable lens was located prior to the Bragg's cell to yield larger beam expansion with larger focal length. With 300mm and 450mm transmitting focal length, the different setups led to measurement volume diameters ranging from 47µm to 142µm. The latter can be of particular interest for spray applications since the trajectory and slit effects are dependent on the droplet diameter / beam waist ratio, and can be negligible when this ratio is quite low. However, an increase of the measurement volume would also lead to a higher probability of multiple particles passing through the beam crossing at the same time.

In 1998, Qiu and Hsu [START_REF] Qiu | Minimum deviation of spatial frequency in large-particle sizing[END_REF] proposed a new approach for particle size measurements with PDA without the trajectory and slit effects. As these biases arise from reflection measurements interacting with refraction measurements, they suggested the use of a particular off-axis angle, such that the conversion factors for reflection and refraction are identical. Therefore, the scattered fringe pattern, i.e. the spatial frequency measured, would only depends on the diameter. To take advantage of this, a new measurement system has been designed and combined with a linear CCD detector located along the collection axis between the two classical photomultipliers, see Fig. 43.a). The signal recorded by the CCD detector was digitalized and processed with an FFT algorithm to determine its frequency, from which the diameter is deduced. However, it requires that at least half a period of the fringe patterns be measured, which sets a minimum size limit on the measurable diameter, estimated by the authors to be around 30µm. The standard PDA detectors were thus used to measure droplets smaller than this limit. To demonstrate the advantages of such a system, the authors simulated with the GLMT the signal obtained for a two-detectors PDA, and for a linear CCD detector. The optical arrangement led to a beam waist diameter of 100µm. In particular, they studied the phaseelevation angle correlation in function of the particle size and its position in the measurement volume. Results showed that in the first case, i.e. with two detectors, phases were consistent over all elevation angles for a droplet of 30µm in diameter located at the center of the measurement volume and at 𝑦 = -60µ𝑚, i.e. away from the receiving probe. For larger sizes of particles, the measured phases changed for the case of particles moving away from the center of the measurement volumeimplying a trajectory effect. With the CCD detector, for particles up to 45µm in diameter, the fringe pattern was independent from its position. With an off-axis angle of 30°, the fringe pattern becomes increasingly position-dependent. However, at an offaxis angle of 45.75°, corresponding to an angle where the phase conversion factors are equal for both refraction and reflection, the fringe pattern is independent of the particle position. The authors also simulated a moving particle passing in the x-direction and showed that the measured spatial frequency depended only on the size, see Fig. 43.b). They concluded that the proposed system could tremendously increase the range of measurable diameter of droplets, and significantly reduce the errors relative to the gaussian beam effect.

In 1999, the same authors proposed another version of the PDA system using both refraction and reflection for suppressing gaussian beam effects [START_REF] Qiu | Method of phase-Doppler anemometry free from the measurementvolume effect[END_REF]. They expressed the phase -diameter relations for both refraction and reflection, as given by Eq. ( 150) in the case of refraction. The phase difference was then computed as the difference between both detectors of the sum of the contribution of both scattering modes. As the detectors could not separate the contributions from reflection and refraction, this setup was not sufficient to measure a diameter. Another pair of detectors have been added with smaller elevation angles for another phase difference to be calculated. The receiver was located at the off-axis angle giving the same phase factor ratio for both reflection and refraction, as explained in [START_REF] Qiu | Minimum deviation of spatial frequency in large-particle sizing[END_REF]. Furthermore, with the two pairs of detectors located with elevation angles symmetric to the (XY) plane, see Fig. 43.a), and assuming the scattering amplitude similar for all four detectors by considering small elevation angles and with an adjustment of the photodiode sensitivities, the authors managed to express the diameter in function of the phase differences ϕ 14 and ϕ 23 . Then, they simulated the resulting measured signal with the GLMT from a droplet of 50µm moving in various direction in function of its position in the measurement volume. Results showed that this new approach gave almost constant phase during the crossing of particle through the measurement volume, while the classical system would yield to signals highly dependent on the particle trajectory. As denoted previously, those effects increase with the size of the droplet. Thus, this new PDA system could be of particular interest for applications requiring measurements of large droplets. The authors concluded that this setup could also be of great interest for mass flow estimation, since no signal validation is necessary except for the SNR, and thus much less droplet would be rejected. In 2000, Qiu and Sommerfeld [START_REF] Qiu | High accuracy optical particle sizing in phase-Doppler anemometry[END_REF] tested this new configuration with experiments on water droplets of diameter ranging from 60 to 90µm. The experimental setup and theory are the same as previously explained, and the experiments were carried out with a 125µm beam waist. Results for a droplet of 60µm in diameter are reported in Fig. 44. The left plot shows the case of a particle moving towards the receiver in the Y direction. It is observed that the measured diameter with this new approach is independent of the particle position in the measurement volume, while with the classical setup, a particle in the back side of the illuminated volume would give an incorrect diameter. The right plot shows the results obtained for a droplet moving downward in the back side of the illuminated volume. While the measured burst showed a weak SNR, the dual scatter method gave a good estimation of the real diameter, while the classic one underestimated it by about 25%. Thus, the authors concluded by indicating the ability of their system to provide Gaussian-free measurements.

Phase Doppler anemometry applications

This section is dedicated to PDA applications in reactive or/and high-pressure conditions. While the previous section showed the development of the technique and its possible optical biases, a particular interest will be expressed here in the evaluation of the refractive index, averaging methods, and uncertainties calculations.

In 1996, Bossard and Peck [START_REF] Bossard | Droplet sie distribution effects in spray combustion[END_REF] used the PDA in reactive conditions to study the impact of the droplet distribution on a two-phase flame. To keep identical fuel -air ratios (FAR), mean drop size and velocities, the authors used the combination of two plain-jet air blast atomizers, mixing in a spray tube, which was equipped with a fuel drain to evacuate excess fuel impacted on the walls, see Fig. 45. The quantity of liquid removed from the main flow by this procedure was measured to correctly estimate the FAR of the flame. A propane pilot flame at the exit stabilized the flame at a fixed position. The Sauter Mean Diameter (SMD) of each atomizer was estimated by the correlation from Rizk and Lefebvre [START_REF] Lefebvre | Atomization and Sprays[END_REF], and the droplet distribution was varied by changing the air -liquid ratio of each atomizers. The resulting distributions were compared with the span, defined as (𝐷 09 -𝐷 01 )/𝐷 05 . Measurements were carried out with a SPDA three detectors arrangement with a 300mm collimating lens, a 500mm transmitting lens and a 100µm slit aperture. The maximum measurable diameter was estimated to be around 100µm. Measurements were carried out for various locations on the spray centerline, and at different axial positions away from the injector. Three distributions were tested, with a measured span at the spray tube exist of around 1.1, 1.6 and 2.0. For this location, the measured SMD showed a relatively constant profile, with the smallest SMD located towards the center due to the presence of the flame, and near the edges, which the authors assumed to be due to shear layers developing in the spray tube. The velocity profiles were equivalent for the three distributions. In the flame, the span and SMD radial profiles were reported for four distances downstream from the injector, from 25mm to 100mm. The authors reported an increase of the SMD with an increase of the axial distance, due to the consumption of small droplets by combustion. However, these variations were of small magnitude, such as one could not conclude firmly on this effect. The temperature profiles were also measured with thermocouples, and a gas analyzer was used to evaluate the concentration of carbon monoxide (CO) and unburned hydrocarbons (HC) at the exhaust, at distances from 125mm to 514mm. The smallest span distribution showed a higher temperature profile with a maximum temperature of around C2 -Confidential density and the 𝜎 0.5 𝑢 𝑎𝑖𝑟 ⁄ ratio were kept identical to the ones of the gas turbine test. With such conditions, the authors conducted PDA measurements, and used the results as inlet conditions for the numerical spray simulation. The comparison between the simulation and experimental results of the spray evaporation rate and mean diameters were in good agreement, with a difference within 10%. The authors continued with a study on the change of spray characteristics made by the modification of parameters, such as air temperature, pressure, velocity and fuel loading. Interestingly, PDA measurements showed unphysical droplet measurements ranging from 55µm to 100µm, while the distribution closed off at much lower diameters. These biases were associated to gaussian beam effect, and suppressed from the distribution by a low-pass filter coupled with a curve smoothing.

In 2000, Araneo and Tropea [START_REF] Araneo | Improving phase doppler measurements in a diesel spray[END_REF] studied the impact of the laser power, photomultiplier highvoltage and burst SNR validation on the data rate, velocities and diameters measured on a diesel single hole nozzle injector. They first reminded general guidelines regarding the limitation of the trajectory and slit effects. For dense sprays such as generated by a diesel injector, reducing the measurement volume is of interest, as it decreases the probability of multiple particles inside the measurement volume at the same time, though the authors reminded that it would also increase the trajectory effect. They gave as a guideline a minimum measurement volume two times larger than the largest particles. For the slit effect, removing the slit is a trivial solution, but it dramatically decreases the accuracy of the flux measurements as the dimensions of the illuminated volume are tricky to estimate. The authors recommended a slit aperture at least several times the size of the diameter of the measurement volume. In terms of scattering angle, a recommendation of 30° was given for PPDA, Dual PDA, or conditions requiring high scattering intensity. Otherwise, especially if the refractive index is not well-known, a scattering angle of 70° seemed better for the authors, as it reduces the system sensitivity to refractive index. A Dual PDA system from Dantec was used coupled with an Argon-Ion laser. The laser beam had an initial diameter of 1.35mm, then expanded through a beam expander of 1.98 ratio. With a front lens focal length of 310mm, the probed volume dimensions were 77x77x1990µm. Its length was cut off by a slit aperture and the projected length was thus reduced to 330µm. The receiver was mounted with a 160mm focal length lens, and the diameter range with this setup was 0-95µm. Signals were processed with a 58N80 processor from Dantec. Measurements were carried out at 5 -50 and 100mm from the injection point. The laser power varied from 2mW to 35mW on each beam. The photomultiplier voltage varied from 700V to 900V, and the SNR validation threshold, from -6dB to 0dB. For each measurement location, measurements on 20 000 droplets were registered, which represent a time record of 120s. Firstly, the impact of the laser power was studied. While it doesn't impact directly the beam diameter, the effective measurement volume is increased with laser power if the threshold detection level is kept constant. Thus, the authors reported an increase of the data rate with an increase of laser power, because more small droplets were detectable and the increase of the effective measurement volume. However, the average velocities did not vary much, suggesting a weak dependence of the velocity on the drop diameter. A slight decrease was noticeable though, attributed to the higher number of small droplets detected, that were generally slower due to the higher impact of the air drag. Inversely, at weak laser power, the mean diameter showed a significant increase below 5mW per beam, up to around 80% at 2mW per beam. The tests on the photomultiplier's voltages showed the same trend but with much lower impact than for the laser beam energy. Finally, variations of the SNR validation threshold had a strong impact on the data rate, with a sharp C2 -Confidential overlap, and the incident and off-axis angles couldn't be perfectly monitored. This leads to an error on the phase conversion factor and thus, on the measured diameters. The validation rates were typically in the 50-55% range, with data rates from 1kHz to 50kHz and a validation level of -2dB. The authors also noted the uncertainty due to the droplet heating and the subsequent decrease of refractive index. Using Eqs. ( 162) and [START_REF] Wigley | High power, high resolution LDA/PDA system applied to gasoline direct injection sprays[END_REF], they also evaluate the potential variation of refractive index. From room temperature to boiling point (553K for jet A1 at 6bar), an overestimation of the droplet diameter of 10% is reported. Results showed a decrease of the mean diameter with the increase of air temperature, that were associated with the decrease of the surface tension of kerosene. While the volumetric flux was reported, The authors emphasized that the uncertainties in droplet heating, beam direction and soot radiation were too large to use them quantitatively.

In 2001, Carl et al. [START_REF] Carl | Experimental and numerical investigation of a planar combustor sector at realistic operating conditions[END_REF] studied a planar combustor sector in realistic turbine conditions. Multiple optical techniques were used to characterize the burner, in particular PDA and Mie scattering to study the spray. Results of PDA measurements performed at 6 bar were reported, with an inlet air temperature of 700K and a AFR of 90, at 6mm from the burner exit. The optical arrangement is similar to the one described by [START_REF] Behrendt | Optical measurements of the reacting two phase flow in a realistic gas turbine combustor at elevated pressure[END_REF]. The injection systems were similar to the pilot part of the lean staged injection system studied within this work. A pressure swirled atomizer was combined with a first radial air swirl for partial premixing. Part of the liquid kerosene was impacted on a prefilming wall, and atomized by a second swirled air flow, see Fig. 47.a). Four burners were located in line in a pressure vessel, see Fig. 46.b) and Fig. 47.b). SMD between 19 and 25µm were reported, with the largest droplets located in the center of the spray. The authors attributed this behavior to a faster evaporation of the small droplets in comparison of the large ones, effect that tends to translate size distribution towards large diameters. Volume concentration were also reported, and showed a spray dissymmetry. While the uncertainties were not given but obviously quite significant, the same trend was visible on the mean velocity fields computed with a RANS approach. The authors attributed this particularity to the burner position that was not perfectly central, and to the subsequent larger wall interaction with one side of the flow. PDA measurements were not possible at a lower AFR due to the high density of the resulting sprays. In 2004, Becker and Hassa [START_REF] Becker | Experimental investigation of spatial and temporal aspects of the liquid fuel placement in a swirl cup at elevated pressure[END_REF] studied the injector presented in Fig. 47.a) in a single sector high-pressure combustor. The objective was to better understand the behavior of the spray within the injection system, and in particular the conditions for prefilming. As the pressure increases, the spray cone from a pressure swirl fuel nozzle reduces in length, while at the same time the transfer of momentum from the swirled air flow increases, potentially opening it closer to the inside cup. Here, the pressure swirl fuel nozzle produced an 80° hollow cone spray. Two air swirlers were used, one with a swirl number of 1.04 (denoted HI, swirl angle of 46°) and a second one with a swirl number of 0.55 (denoted LO, swirl angle of 29°). The authors specified that the injection system used was slightly smaller than typical aero-engine injectors, which increased the centrifugal force applied on droplets. They quantified this effect by computing the ratio of the square of the tangential velocity over the radius of the filmer 𝑢 𝑡𝑎𝑛 2 /𝑟 𝑓𝑖𝑙𝑚𝑒𝑟 . In the case of the swirler HI, this ratio had a value of 10.6x10 +5 , while the swirler LO, a value of 3. 0x10 +5 . As a reference, the authors gave some characteristics of two aero-engine nozzles (exact origin not given). Both conditions had a swirl number around 1.04, a swirl angle around 46°, and a 𝑢 𝑡𝑎𝑛 2 /𝑟 𝑓𝑖𝑙𝑚𝑒𝑟 of 7.1x10 +5 and 5.7x10 +5 respectively. Thus, the swirler HI was very similar to them, with a higher centrifugal effect on droplet due to its reduced size. PDA measurements were carried out at 10mm from the cup outlet, at an air temperature 𝑇 3 = 700𝐾, a relative pressure drop ∆𝑃 𝑃 ⁄ = 3% and a pressure of 6-12-18 bar, leading to Reynolds numbers of 60 000 -130 000 -190 000 respectively. A commercial PDA system from Dantec was used, with a receiver positioned at an off-axis angle of 50°, and an effective measurement volume of 70µm in diameter and 130µm in length. The burst validation criterion was set to +3dB, and each measurement recorded 20 000 validated droplets or lasted 15 seconds, whichever came first. The authors estimated the error on the SMD being around ±5%, accounting for optical misalignments, refractive index variations during droplet heat up, and PM settings. Radial SMD profiles and normalized volumetric flux were reported (see Fig. 48) though the fluxes were used only for a qualitative interpretation due to the well-known associated large uncertainty. The configuration with the swirler HI showed no dependence of the droplets location with pressure variation, and a very high propensity to filming with virtually no droplet measured at relative radius r/R<0.5, see Fig. 48.a). Inversely, the swirler NO showed only a partial filming at 6 bar, with a normalized flux at the center around 0.4, see Fig. 48.b). At higher pressure, the proportion of filming seemed to reduce drastically, with a variation of the normalized flux from the center to r/R=1.0 of 20% at 12bars, and 10% at 18bar. Also, an observed narrowing of the spray with this swirler was reported, with a maximum flux positioned at r/R=1.15 at 6 bar, but located at r/R=0.7 at 18 bar. The authors attributed this difference with the swirler HI to a very late interaction liquid/wall for the 6 bar case, that disappeared rapidly with a higher-pressure condition, less favorable to filming. In terms of SMD, the swirler HI configuration showed mean diameters between 10 and 15µm at 6 bar, with a mean value of 11µm at the highest data rate. At 12 and 18 bar, the radial profiles were almost identical, with values 8 and 11µm with a slight increase towards the exterior. The swirler LO configuration produced larger droplets, with SMD ranging from 22 to 17µm at 6 bar, 14 to 17µm at 12 bar, and 13 to 16µm at 18bar. Interestingly, the radial evolution was inverted by comparison with the swirler HI, with a decrease of the mean C2 -Confidential diameters towards the exterior of the cup. Those net differences in SMD profiles suggested that the filming condition is of prime importance for LPP injectors.

Finally, the authors studied the inter-particles times to verify the existence of an autocorrelation between the arrival of droplets and thus, if they are transported in groups. To do so, they computed the auto-correlation function 𝑅(𝜏) of the measured inter-particles times ∆𝑡.

𝑅(𝜏) = lim 𝑇→ ∞ 1 𝑇 ∫ ∆𝑡(𝑡). ∆𝑡(𝑡 -𝜏)𝑑𝑡 𝑇 0 (166)
Then, they defined a droplet arrival time scale 𝜏 𝑐ℎ𝑎𝑟 , that the authors associated to the period of time over which the inter-particle times remain closely related.

𝜏 𝑐ℎ𝑎𝑟 = ∫ 𝑅(𝜏) 𝑅(0) 𝑑𝜏 𝑇 0 (167)
The ratio of 𝜏 𝑐ℎ𝑎𝑟 with the inverse of mean data rate, i.e. the average of inter-particle time, were between 2.3 and 6.4, showing the auto-correlation of inter-particle times. Finally, to verify the existence of a particular frequency of arrival of droplets, the authors computed the power spectral density by FFT, defined such as:

𝑃𝑆𝐷(𝑓) = ∫ 𝑅(𝜏). exp(-𝑖. 2𝜋. 𝑓𝜏) 𝑑𝜏 ∞ 0 (168)
Results showed a very small peak in the case of the swirler HI around 2kHz, that the authors hypothetically attributed to a processing vortex core. The authors concluded that liquid fuel placement was primarily due to airflow swirl angle. The presence of an inner recirculation zone within the inner cup is also a key parameter for the formation of a filming, that has been shown to produce a finer atomization than the pressure swirl injection alone. The transfer of tangential momentum from the second swirled air flow led to a reopening of the spray. In the case of a fuel filming, air pressure and the fuel nozzle flow number are negligible for the determination of the fuel location. Finally, the authors noted that favorizing the IRZ inside the cup to ensure filming could lead to a flame stabilization within the swirl cup.

In 2004 also, Wigley et al. [START_REF] Wigley | Imaging and PDA analysis of a GDI spray in the near-nozzle region[END_REF] coupled LDA and PDA and a CCD imaging technique to measure liquid velocity and droplet mean diameter in the spray produced by a pressure swirl gasoline direct injection injector. To do so, the authors took advantage of the P80 processor from Dantec that allows a duplication of the input signal so they can be processed by another processor, in the present case a 58N50 covariance processor from Dantec. The newly developed P80 processor analyzes the signals bursts in terms of amplitude and SNR, with a centering on the burst's peak. The P80 was used for PDA measurements, while the 58N50 was used for LDA. Beams of 120 and 250mW at 488 and 51nm respectively were focused through a 300mm focusing lens, producing probed volume of 42 µm and 45µm in diameter. The effective probed volume was reduced with the use of a slit aperture in front of the receiver, leading to an effective length of 100µm. The collecting optics were placed an off-axis angle of 70° to collect refracted light. For each measurement point, 20000 validated events were recorded, or an elapsed time of 125s was reached. Experiments were carried out from 0.5mm to 5mm from the nozzle exit. At the highest data rate peak, the authors reported a validation rate of 29% only, though they attributed this low value to the very dense spray and the good rejection from the processor. LDA and PDA comparison showed a good agreement excepted close to the injector where high rejection rates were observed. Ligaments and non-spherical drop conserved longer their momentum and hence, higher velocities.

More recently, in 2010, Heath and Locke [START_REF] Heath | Optical characterization of a multipoint lean direct injector for gas turbine combustors: Velocity and fuel drop size measurements[END_REF] used PDA to study a lean direct injection system for aero-engines in high-pressure reactive conditions. Nine injectors were placed in a 3x3 square to be more representative of performances encountered in real engines where flames interact with each other. Six helical axial vanes were oriented at 60° and surround a simplex injector. The swirl number was estimated to 1.0. Measurements were carried out around the central injector at two operating conditions, at pressures of 10 and 13.8 bar and inlet temperatures of 728 and 828K respectively. Three component PDA was performed with a commercial system from Dantec, combined with an Argon-Ion laser providing laser beams at 514.5, 488 and 476.5nm. The transmitting probe was placed at a 15° angle, and the receiver at a 30° off-axis angle. The velocities components were put back in the main coordinate system by a manual transformation. A 310mm focal length led to a measurement volume diameter of 47µm and of 76µm in length. The authors reported only the axial velocity in function of diameter as scatter plot, which showed no direct velocity diameter correlation. A tremendous velocity span was shown, from 100m.s -1 to more than 250m.s -1 at positions between the middle injector and the middle low one, and it has been specified that some suspiciously high velocities droplets were manually removed. Diameters ranged from 1 to around 25µm were measured, with mean values in the 10-15µm range. The authors reported that smallest droplets had higher velocities than the largest ones, though it is important to denote the low drop count (8000 at the best) with C2 -Confidential respect to the velocity span. They concluded that PDA was suited to characterize injectors in such conditions, and could give reliable information on the spray angle and mean droplet velocities.

In 2011, Freitag et al. [START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF] used PDA to study the performances of an in-house designed swirling air-blast injector in high-pressure reactive conditions. A schematic of the injector is reported in Fig. 49.a). The fuel is supplied by two channels to an annular chamber, from there it passes through a 36 holes circular array towards a circumferential slot of 500µm width. The end of the slot is inclined at 31° inward, and its width is reduced to 200µm (width values published in [START_REF] Meier | Spray and flame structure of a generic injector at aeroengine conditions[END_REF], presented thereafter). Two co-rotating air-swirlers ensure the atomization of the film. The injection system was mounted in a single sector test rig operated at 4 and 10 bar, with inlet air temperature of 550 and 650K respectively. The pressure drop of the burner is estimated at 3%. Inlet fuel temperature was monitored with a thermocouple, though it was not positioned at the end of the supply line and henceforth, the authors specified that the fuel was likely to heat up more before being atomized. AFR was set to 20, note that the kerosene stoichiometry is equal to 14.7, with air mass flux of 60 g.s -1 and 140 g.s -1 and fuel mass flux of 3.0 g.s -1 and 6.8 g.s -1 respectively for pressure of 4 and 10 bar. The authors noted the large beam steering that happens is conditions of strong density fluctuations. To minimize this effect, a long emission focal length together with a wide beam diameter can be used, but the subsequent decrease of energy density lowers the SNR. Also, the high-density spray would require the inverse setup to reduce the probability of multiple particles passing through the probed volume at the same time. Finally, the authors noted that the uncertainty on the droplet temperature and thus, the refractive index, would lead to a large uncertainty on the measured flux. To reduce the dependence of the phase conversion factor on the refractive index, they selected a collection angle near the Brewster's angle at 68°. As shown in Fig. 49.b), a variation of the refractive index between 1.35 and 1.45, which corresponds to the variation from room temperature to boiling point, does not lead to a significant modification of 𝛽 12 . Although at this collection angle there is almost no reflection, the intensity of the scattered light is lower than at 30°, which further reduces the SNR. Considering all these effects, the authors selected a focal length of 310mm coupled with a beam expander of 1.98 ratio, leading to a probed volume diameter of 76µm. The laser power at the cross point was estimated at 70mW. Interestingly, as seen before the Dual PDA configuration should be favorized for flux measurement, although it couldn't be used at such large collection angles. The authors preferred to use a SPDA configuration with two velocity components measured. To enhance the flux measurement accuracy, they modified the method reported by Brandt in 1998 [START_REF] Brandt | An Experimental and Numerical Study of Kerosine Spray Evaporation in a Premix Duct for Gas Turbine Combustors at High Pressure[END_REF] to use both velocities to better estimate the effective probed volume with respect to droplet trajectories. In this approach, the third velocity component is supposed to be negligible. A minimum burst length has been defined as a minimum residence time inside the measurement volume for each size class. Then, considering the cross section of the measurement volume as a parallelogram, the mean burst length passing at a distance 𝑞 has been given. Assuming all trajectories within a size class uniformly distributed, an effective measurement diameter, and thus a reference area, per size class could be computed. The authors specified that this approach didn't account for the particle diameter, only trajectories and thus, velocities. The effective measurement volume cross section should increase with particle diameter, henceforth, this model overestimate fluxes. Measurements were carried out in the y-z and x-z planes to visualize both axial and radial development of the spray. Data rates were moderate, except in one small area where they could exceed 100 kHz. Validation rates C2 -Confidential were quite low, especially in reactive zones where high-density fluctuations produced strong beam steering effects, decreasing the validation to ~50%. Also, high droplet density decreased significantly the laser power and SNR, and measurement points requiring the beams to cross the dense spray showed validation rates as low as 30%, while they were around 50% at the injector exit where the highest data rate were recorded. Velocity maps showed that in the main flow both small and large droplets followed similar trajectories, but small droplets would be more deviated by the recirculation areas. SMD were reported at two axial locations, z= 10 and 12.5mm, and within the spray similar values were found, while in the recirculation zones higher SMD were measured at 12.5mm. The authors attributed this increase of SMD to a higher evaporation rate of small droplets. One size distribution is presented, with the modified Rosin-Rammler distribution from Lefebvre [START_REF] Lefebvre | Atomization and Sprays[END_REF]. A good fit is found, although the tail of the experimental distribution shows some oscillations, which could be attributed to large droplets erroneously measured by the Gaussian beam effect. Lastly, the authors used a laser sheet to visualize the Mie scattering, and compared the spray positioning from both techniques. It appeared that PDA positioned the highest spray density around 3mm further away from the center than the Mie scattering. This has been attributed by the poor estimation of the concentration in this specific region, that was subject to both high droplet density and beam stirring effect. The authors thus concluded that the combination of Mie scattering and PDA was of interest in such conditions.

In 2012, Meier and al. [START_REF] Meier | Spray and flame structure of a generic injector at aeroengine conditions[END_REF] presented the same experiments with further results, and in particular OH-PLIF to visualize flame structure and temperature distribution. The optical set up for PDA and LDA measurements is identical to the one detailed in [START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF]. The authors added a confidence interval estimate for the LDA and PDA measurements based on the required confidence level, the accuracy of the processor at 0dB SNR, the width of the distribution and the sample size, following the guidelines of Dantec, leading to a confidence interval on the D32 of +/-0.6µm. [START_REF] Janus | Experiments on swirl stabilized non-premixed natural gas flames in a model gasturbine combustor[END_REF]. This approach is considering a perfect measurement case and only the random statistical error, which is very questionable considering the operating conditions and all possible biases reported previously. Spray penetration and opening didn't evolve much between operating points at 4 and 10 bar, although the flame appearance changed significantly, from blueish at 4 bar to an intense yellow at 10 bar, indicating an increased production of soot). It seemed that the increase of fuel flowrate in the 10bar case compensated the finer atomization due to the increase of pressure, and led to a diffusion flame. The heat release computed from the OH-PLIF showed a much stronger intensity in the case of the 10 bar operating conditions, with a peak closer to the spray formation, suggesting a high reaction rate in a rich region. Radial velocity profiles at 2mm from the burner exit were presented. Both axial and tangential droplet velocity magnitude ranged from 0 to 40m.s -1 . Velocity peaks are visible near the edge of the inner air channel, and small dips due to the wake of the prefilmer lip were present. Volume fluxes agreed with the flow pattern, but couldn't be measured closer than 10mm from the burner exit due to the saturation of the PDA processor. Despite the high saturation, the authors presented some velocities results at 7mm from the burner exit in function of the size class.

Results showed that within the spray, smallest droplets were the fastest, with velocities around 45m.s -1 , while largest droplet (D=32µm) were the slowest with velocities around 35 m.s -1 . Finally, in terms of SMD, profiles at multiple axial distances are given. All SMD lied in the 14-20µm range for the 4 bar case, and between 10-15µm for the 10 bar case. Slightly smaller diameter (around -2µm) were measured at 7mm from the burner exit than further away, where the mean C2 -Confidential diameters were close. The authors concluded that PDA was successfully applied for diameter and velocity measurements, but that fluxes should be considered valid only at a certain distance from the burner exit.

In the light of those results, it appears that the application of the Phase Doppler Anemometry in high-pressure reactive conditions can be subject of numerous sources of uncertainties, and that the choice of an adapted optical setup is of primary importance. Slit and trajectory effects have been studied extensively over the years, as highly biased measurements can occur. Specific configurations have been developed to remove these biases, such as the Dual PDA [START_REF] Tropea | Dual-Mode Phase-Doppler[END_REF] or the systems proposed by Qiu and Tsu [START_REF] Qiu | Minimum deviation of spatial frequency in large-particle sizing[END_REF][START_REF] Qiu | Method of phase-Doppler anemometry free from the measurementvolume effect[END_REF], but it is important to remember that these errors can be greatly reduced by choosing a measurement volume diameter significantly larger than the measured droplet. In particular, with a water spray and droplet diameter ranging from 2 to 80µm, Gréhan et al. [START_REF] Gréhan | Particle Trajectory Effects in Phase Doppler Systems: Computations and experiments[END_REF] showed that the deviation of the measured mean diameters due to the trajectory and slit effects was moderate, for droplets were small enough with respect to the size of the measurement volume. With this conclusion and the beam stirring effects encountered in gas turbine conditions, such as reported by Freitag et al. [START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF] and Becker et al. [START_REF] Becker | Experimental investigation of spatial and temporal aspects of the liquid fuel placement in a swirl cup at elevated pressure[END_REF], it is clear that an increase of the measurement volume diameter has some advantages. However, this would increase the probability of several particles passing through the measurement volume at the same time, which could lead to another type of uncertainties. Furthermore, an increase of the illuminated volume diameter would reduce the energy density and fewer small droplets would be measured. Coupled with the absorption of the high-pressure two-phase flow medium, the high laser energy required could be greater than the one provided by a standard PDA system. In gas turbine conditions with lean premixed injector, mean diameters in the combustion chamber are in the 8-15µm range, with the largest droplet around 25-30µm. For the minimization of the slit and trajectory effects, a ratio between the measurement volume diameter and the maximum droplet diameter between two to five has been advised in the past, which in this case would correspond to around 50-150µm. The slit width has been suggested to be chosen at least twice as large as the measurement volume diameter, and should so be in the 100-300µm range [START_REF] Araneo | Improving phase doppler measurements in a diesel spray[END_REF].

Another important source of error that has been reported in gas turbine condition is the uncertainty on the refractive index of kerosene. Specific setups have been used to minimize or suppress the dependency of the phase difference -diameter relationship on the refractive index, such as Naqwi et al. [START_REF] Naqwi | Extended Phase-Doppler System for Characterization of Multiphase Flows[END_REF], Qiu et al. [START_REF] Qiu | Minimum deviation of spatial frequency in large-particle sizing[END_REF] and Freitag et al. [START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF]. From the work of Brandt et al. [START_REF] Brandt | An Experimental and Numerical Study of Kerosine Spray Evaporation in a Premix Duct for Gas Turbine Combustors at High Pressure[END_REF] and Eq.( 162), the heat up of the droplet from room temperature to the boiling point, and a reference value of 1.46 taken at room temperature, would lead to an error of around 3.5% with an off axis angle of 52°. To minimize the dependency of the measured diameter on the refractive index, the off-axis angle should be increased up to the Brewster angle, which is also beneficial to reduce any contribution from reflection.

Finally, no practical solution has been reported to minimize the beam stirring effect, that can drastically reduce the validation rates, and the capacity of measuring small droplets. Also, no guidelines have been expressed on the choice of the specific Signal-to-Noise Ratio, Gain and Photomultiplier voltage of the detection system. As they also affect the validation rates and the smallest measured diameters, they have to be chosen with special care. The following section is devoted to preliminary experiments carried out to optimize PDA measurements under highpressure aero-engines conditions.

Preliminary studies

Before using PDA for the characterization of multipoint lean premixed injectors studied in the current thesis, experiments were first performed on two configurations to determine best practices and optimize the optical setup.

Impact of the viscosity on spray distribution and spherical validation criteria

The first tests were carried out on the pilot part of an aeronautical injector at atmospheric pressure and ambient temperature. The injector was a simplex producing a hollow cone spray.

The main objective was to quantify the response of the PDA system to ligaments and nonspherical droplets, as they can have an impact on the accuracy of measurements as shown in the previous section. For this purpose, PDA measurements were performed at 30mm from the nozzle exit, with the injector working at different injection pressures. The quality of atomization was visualized by means of a shadowgraphy technique. Three operating points are reported here, denoted A, B and C, and corresponding to an injection pressure of 1.50, 3.35 and 6.50 bar of Jet A1 with 𝜌 𝑘𝑒𝑟𝑜 = 743 𝑘𝑔. 𝑚 -3 and 𝜐 𝑘𝑒𝑟𝑜 = 2.21. 10 -6 𝑚 2 . 𝑠 -1 . The shadowgraphy optical method has been carried out with a Phantom V2512 high-speed camera, using a 12 bits CMOS sensor with pixels of 28µm of diameter. The resolution has been reduced to 512x384 pixels to allow high-speed measurements up to 99 000 fps with a time exposure of 4µs. Backside illumination was achieved with a Xenon lamp (100W Newport Low Power Q series).

PDA measurements were performed with a commercial system from Dantec, combining a P800 processor and Dual PDA probes. Laser beams were generated by an argon Coherent Innova 300 system, producing laser beams with a power of 0.6W at wavelengths of 512 and 488nm. Beams are then split in two parts and one of each laser beam passed through a Bragg cell at a 40kHz working frequency. A standard PDA configuration with perpendicular polarization combined with three detectors was used, with an off-axis angle of 68.5°. Emitting and collecting probes C2 -Confidential were installed on a three-dimensional mechanical displacement equipped with linear translation stages to allow measurements at various positions into the spray.

Fig. 50 resumes the results of particular interest for PDA response to non-spherical droplets. In the top part, the visualization of the spray for the three operating points are shown. The yellow arrow shows the distance from the injector exit for PDA measurements. This position was chosen to be close to the nozzle outlet while keeping the atomization sufficiently advanced for the PDA measurements to be relevant. In particular, droplets for the operating point A are large enough so that a rough estimate of their diameter could be done with the shadowgraphy images. At position of PDA measurements, the maximum droplet sizes were estimated to be around 150-200µm. Below these spray visualizations, the phase diagram from PDA measurements in the center of the spray for the three operating conditions are shown in Fig. 50. In contrast to the diagram shown in Fig. 34, the horizontal and vertical axes represent the ∆𝜙 12 and ∆𝜙 13 measurements respectively. Each point corresponds to the detection of a single droplet, and is represented in blue if the PDA measurement has been accepted or red otherwise. The validation zone corresponds to the area between the two doted lines, and the theoretical relationship between the phase difference and diameter is shown by the continuous line. The verticals black lines indicate the 2π modulo from ∆𝜙 12 measurements. From left to right, the measured droplet diameter increases, so the phase diagram can provide insight into the measured distribution and the quality of the measurements. For instance, the droplet size distribution associated with a measurement at the center of the spray C is given at the bottom right. It has a classical shape, with a majority of droplets below 15µm, a 𝐷 𝑣01 = 14.6µ𝑚, a 𝐷 𝑣05 = 28.5µ𝑚, and a span equal to 116%. As pointed by [START_REF] Damaschke | Response of Phase Doppler Anemometer systems to nonspherical droplets[END_REF] and other researchers, SPDA is not efficient in rejecting an ellipsoidal droplet, due to its spherical validation criteria based only on the analysis of a single droplet curvature. To minimize the bias of measurement on the mean diameter, the optical setup is optimized so that the cloud of points on the phase plot is symmetrically spread around the sphericity line. The angle of incidence and the height of the receiving probe are of particular importance here, as detectors one and two must be symmetrical to the scattering plane. In the opposite case, one of the assumptions used to get from Eq.( 149) to Eq.( 150) is not valid, and thus the slope of the phase difference -diameter relationship becomes erroneous, then favoring the cloud of points on one side of the theoretical sphericity line. Such a dissymmetry is slightly visible on the phase plot B, and such misalignment may be one of the reasons. On the case B and even more on the case A, a significant number of events are validated and rejected over the whole domain of the phase plot. As the collection angle is close to the Brewster angle, the reflection shouldn't be a major contribution to erroneous measurements. Those events could come from a combination of slit and trajectory effects, though they should mostly arise from rejected measurements from ligaments and strong distortion of droplets. As reported by Araneo et al. [START_REF] Araneo | Improving phase doppler measurements in a diesel spray[END_REF], tests on the laser energy, SNR, gain and photomultiplier (PMT) sensibility showed a significant impact on the mean diameter. As expected, smaller droplets require minimal laser energy or gain/PM sensitivity to be detected, and SNR decay would reduce the probability of detecting them. To find the correct parameters to use, the measurements at the center of the hollow spray cone are used as the reference condition, as this corresponds to the position where the smallest droplet size should be found. The four parameters were set so that the minimum droplet size measured was around 1µm, while keeping the burst validation and spherical validation at high values. With a laser power of 50mW, PMT high voltage around 900V (slightly tuned so that the mean signals intensity of the PMTs were as close as possible), SNR of 4dB and Gain of 10dB, burst and spherical validations of around 85-90% were reached in case C, with detected droplets as small as 1µm. Before the characterization of the sprays produced by lean staged injection systems (Lemcotec and MFSI), PDA has been applied on the LP injector from Safran Helicopter engines, which has been the subject to an in-depth study, both numerically and experimentally in the CORIA lab. The objective of this study was to add PDA measurements into the existing database and verify the accordance between all the measured parameters describing the flame. Quantitative measurements of fuel distribution under elevated pressure reactive conditions were reported by Malbois et al. [START_REF] Malbois | Quantitative measurements of fuel distribution and flame structure in a leanpremixed aero-engine injection system by kerosene/OH-PLIF measurements under high-pressure conditions[END_REF], NO formation under the same operating conditions were published by Salaün et al. [START_REF] Salaün | Experimental investigation on NO pollutant formation in high-pressure swirl-stabilized kerosene/air flames using NO-, OH and kerosene PLIF and PIV laser diagnostics[END_REF], and high-fidelity LES simulations were performed by Domingo-Alvarez et al. [START_REF] Domingo-Alvarez | Impact of Spray Droplet Distribution on the Performances of a Kerosene Lean/Premixed Injector[END_REF]. The accumulation of this work has allowed the creation of a unique database with results from PIV measurements, coupled OH-PLIF, kerosene-PLIF and NO-PLIF measurements as well as LES simulations. A summary of the results, including some data from the PDA measurements detailed in this section, was already published in a peer-reviewed journal [START_REF] Legros | Combustion for aircraft propulsion: Progress in advanced laser-based diagnostics on high-pressure kerosene/air flames produced with low-NOx fuel injection systems[END_REF]. In particular, the radial kerosene-PLIF measurements coupled with the results of the parametric study of the effect of the injected droplet distributions in the LES simulations are of particular interest when comparing these data with the PDA measurements. In order to use PDA measurements as a tool to evaluate the droplet distribution to be injected as boundary conditions in the LES simulations, the measurements were performed close to the outlet of the injection system. This implies the use of two vertical laser beams, as the horizontal beams could not be used because their optical path was blocked by the edges of the optical window. The experiments were then performed with the SPDA configuration.

The lean premixed fuel injection system operating with kerosene is based on a multi-hole jet in crossflow atomizer with a partial filming on cup. These two atomization processes are expected to give beneficial effects to the fuel injector for improving the fuel/air mixture and fuel evaporation, directly increasing the flame stability and reducing its length. A schematic of the LP fuel injection is depicted in Fig. 51. The fuel injection consists of a central nozzle pierced with eight holes on the side wall, identical in diameter to the central orifice. Part of the fuel is injected through the central hole and atomized by the central swirling airflow via an airblast atomization concept. This atomization creates a rich zone of liquid droplets that helps to the stabilization of C2 -Confidential the flame. The second part of the fuel flow is injected through the side ports. The resultant droplets produced via a liquid jet in crossflow process can either impact the internal wall or be carried away by the internal swirling air flow. The lateral/central fuel injection ratio is not controlled but remains constant thanks to the geometry of the injector. In the event that part of the kerosene droplets impacts the wall facing the lateral holes, a liquid film could be created and carried away by the air stream. The liquid film is then torn off and finely atomized, creating an external conical zone, where air and partially evaporated fuel mix together. The air/fuel mixing is finally sheared between two swirled air flows to be homogenized prior to be ejected into the combustion chamber. With this geometrical architecture, the aerodynamic swirl number was measured to be 0.6.

For PDA measurements, a Dantec Dynamics dualPDA system was used. The emitting system includes two 1 W diode lasers, model Genesis MX 488-10 0 0 SLM and Genesis MX 514-10 0 0 SLM. After that, the diameters of the laser beams at 488 and 514 nm were set to 2.2 mm with an expander ratio of 1.98, a shift frequency of 40 MHz was generated by passing one of the laser beams through a Bragg cell. Both laser beams are then conducted through fiber optics and transmitted towards the probe volume via a Dantec FiberFlow optical system. Adopting an emitting focal length of 350 mm and a beam spacing of 40 mm, the probe volume with a length of 350 μm and a diameter of 50 μm is generated. The noise and reflected light contributions able to disturb the detection of the burst signal were minimized by fixing the receiving optical system at 30 ° to the forward scattered direction. The detection system consists of a 112 mm diameter Dual PDA receiving probe in which the focal length of the receiver is 310 mm. Within its specifications, this setup was optimized to measure droplets within the range of diameters of 1-75 μm. The measurement signal is transmitted by optical fibers to the detector unit. The transmitting and receiving optics were mounted on a computer-controlled 3D traversing system and the measurements were performed at various positions along the axial and radial directions downstream from the nozzle exit. The location of the measurement planes in the spray were defined by the axial distance from the injector (Z) and the radius (Y), with an origin defined as the nozzle exit. For this study, PDA measurements were performed on X -Y planes at Z / D = 0.04 and 0.64 from the injector. At Z / D = 0.04, a quarter of the flow exit cross-section was probed adopting a Cartesian grid with ΔX = 0.25 mm and ΔY = 1 mm while at Z / D = 0.64, polar coordinates with Δθ= 10° and ΔY = 1 mm were selected to mesh the grid, see Fig. 52.a). For each measurement location, the PDA signals were processed with a Burst Spectrum analyzer (BSA) P80 processor. The acquisition time for each measurement was set to 30 s or by fixing a maximal data sampling of 10 000 droplets. In the current study, the PDA validation rate was ~90%, with a SNR set to 3dB, a gain of 12dB, a first to second peak ratio of 4:1, and a PMT sensitivity ranging from 900V to 1000V to equilibrate the signal intensity seen by the PMTs.

C2 -Confidential Fig. 52.a) shows the PDA data rates. Black lines with point and dash symbols represent the edges of the air swirler. Values as high as 106kHz were observed at Z/D=0.04, which is below the limits of saturations of the PDA processors that have been reported previously. Validation rates were homogeneous and around 90%, despite the high-pressure reactive conditions. At Z/D=0.04, most of the droplets are located above the air swirler, forming two pockets of higher concentration. The positioning of the spray near the outer edge of the air swirler suggests a prefilming atomization at the edge of the cup, and a strong interaction with the external airflow which transports the droplet radially outwards. At Z/D=0.64, measured data rates were around ten times lower, and followed again a pocket pattern. This reassure on the confidence level of the data rates measured at Z/D=0.04, and indicates that the droplets from the spray do not mix perfectly in the airflow. The high validation rates found close to the injector outlet indicates that C2 -Confidential the use of a small measurement volume seemed to have strongly reduced the probability of having multiple droplets measured at the same time, and that the slit and trajectory effects were reduced thanks to the small droplet size. Indeed, as mentioned in the previous work, the LP injector burns a large amount of fuel inside the dome, so that the drops arriving in the combustion chamber are already subject to high evaporation. This is visible on the 2D maps of the mean diameter, weighted by the transit time, presented in Fig. 53.

Mean diameters D10 from 6µm to 12µm were measured, with the smallest droplets in the center of the injector, where the IRZ is present and hot gases maximizes the evaporation. The largest droplets were observed at the edges of the measurement domain, indicating that large droplets are being propelled radially outwards. However, it is important to note that the data rates were particularly low at the edges of the mesh, and that these large droplets represent only a small fraction of the fuel in the spray. By combining the data rates shown in Fig. 52.b) and the 2D maps of the mean diameter, it can be seen that most of the spray has a homogeneous size distribution, with mean values of D10~8µm at Z/D=0.04, and ~7µm at Z/D=0.64. Examples of results of the droplet size distribution for two locations in the highest density region of the spray are shown in Fig. 54.

Both size distributions are similar, and follow a Rosin-Rammler like shape. The average diameters are quite close, although the density of the measured droplets has decreased by a factor of twelve. This demonstrates that evaporation is quite homogeneous over this restricted range of droplet sizes, and the conclusion that small droplets evaporate faster than large ones, as reported in previous studies [START_REF] Carl | Experimental and numerical investigation of a planar combustor sector at realistic operating conditions[END_REF][START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF], is not observed in the present study. The cumulative volume distributions show similar patterns, but it appears that the end of the profile is subject to a discontinuous evolution towards larger diameters. This is due to the impact of some large droplets on this section of the cumulative function, which will significantly affect the Dv09 parameter. These validated events can be considered as rare productions of large droplets, or, more likely, as erroneous measurements related to a slit and/or trajectory effect, or to the presence of multiple particles crossing the measurement volume, as reported by [START_REF] Heath | Optical characterization of a multipoint lean direct injector for gas turbine combustors: Velocity and fuel drop size measurements[END_REF]. The associated phase plots are shown in Fig. 55. Only a few events corresponding to a diameter of 29 µm were recorded after ∆𝜙 13 = 100°. As the droplet size distribution of this injection system is generally continuous, it is highly likely that these events are erroneous, and that a low-pass filter must be applied to eliminate them to determine the volumetric functions. For the X/D=1.02 position, a unique population of droplet is measured, centered on an average velocity of 30 m.s -1 , indicating a deceleration of droplets as they are transported from the main air stream to the external recirculation zone. As the measurement volume is small and two distinct droplet distributions over a few millimeters of surface are observed, this behavior cannot be explained by possible measurement biases related to the positioning of the measurement volume in a transition zone between the IRZ and the main flow. In our opinion, this phenomenon represents the mixing of droplets from two atomization processes, namely the low velocity droplets from the air blast atomization passing from the ZRI to the main stream, and the higher velocity droplets already existing in the transition and/or from the pre-film atomization.

With regard to the average values characterizing a spray, many weighting methods have been used in the past, such as [START_REF] Brandt | An Experimental and Numerical Study of Kerosine Spray Evaporation in a Premix Duct for Gas Turbine Combustors at High Pressure[END_REF][START_REF] Bergenblock | Averaging of particle data from phase Doppler anemometry in unsteady two-phase flow: Validation by numerical simulation[END_REF][START_REF] Bergenblock | Experimental estimation of particle flow fluctuations in dense unsteady two-phase flow using phase Doppler anemometry[END_REF][START_REF] Rodriguez | An innovative LDV data processing method for statistical error corrections. Application to homogeneous and nonhomogeneous seeding[END_REF]. For the results presented in the section describing the state of the art, it is clear that the PDA data rate under high-pressure reactive conditions may be subject to a non-negligible bias. Although it would in theory be appropriate to construct a characteristic mean diameter weighted by the data rate, it was considered in the current study that it would induce too much uncertainty. The average values were instead weighted by the transit time, and by the radius from the center of the injector. The average droplet distribution at Z/D=0.04 and Z/D=0.64 is shown in Fig. 58. Both distributions can be represented in fairly good agreement with Rosin-Rammler distributions.

Comparison with PLIF measurements/ LES simulation data base

As described above, a large number of results have already been published on this LP injector for this operating condition. A summary of the experimental set-up used by E. Salaun and P.

Malbois [START_REF] Malbois | Quantitative measurements of fuel distribution and flame structure in a leanpremixed aero-engine injection system by kerosene/OH-PLIF measurements under high-pressure conditions[END_REF][START_REF] Salaün | Experimental investigation on NO pollutant formation in high-pressure swirl-stabilized kerosene/air flames using NO-, OH and kerosene PLIF and PIV laser diagnostics[END_REF][START_REF] Legros | Combustion for aircraft propulsion: Progress in advanced laser-based diagnostics on high-pressure kerosene/air flames produced with low-NOx fuel injection systems[END_REF], as well as the numerical study carried out by P. Domingo-Alvarez [START_REF] Domingo-Alvarez | Impact of Spray Droplet Distribution on the Performances of a Kerosene Lean/Premixed Injector[END_REF], are presented before making a comparison with the PDA measurements. In these measurements, the signal emitted by the kerosene droplets was suppressed and replaced by the fluorescence signal of the vapor that surrounds them.

The quantitative measurements of kerosene vapor were performed with a one-photon excitation at 266 nm, and a collection of two distinct spectrally broadband fluorescence signals recorded in the spectral domains 270-310 nm and 310-410 nm, associated with the fluorescence of mono-aromatic and di-aromatic compounds respectively. The laser sheet was produced from a quadrupled-frequency Nd:YAG laser, with an output energy pulse of 60mJ. Two emICCD cameras (PIMAX 4, Roper Scientific) equipped with a 100 mm, f/2.8 UV-CERCO lens and a combination of colored glass filters were used to respectively collect the whole fluorescence signal and the signal from di-aromatics only. PLIF images were corrected by noise, laser sheet energy inhomogeneity, and distortion using the tools included in the DaVis 8.1 commercial software.

Large-Eddy Simulation (LES) was used to simulate the LP injector operating in the HERON test rig. LES has been proved to be particularly well suited for turbulent combustion. Indeed, even if chemical reactions mostly occur at the sub-grid scale, LES can resolve the large-scale dynamics, which is a key condition for combustion performances such as pollutant emissions. In LES, flow variables are filtered in space. The filtered balance equations, coupled with the sub-grid models are numerically solved to determine the instantaneous flow fields. The sub-grid Reynolds stresses are closed using the localized dynamic Smagorinsky model proposed by Germano, which is based on the Boussinesq assumption. The turbulent viscosity is modeled as a function of the turbulence integral length scale, the resolved shear stress and a model constant (CS) which depends on the flow configuration. Here, CS is automatically computed in space and time, that is, the eddy viscosity coefficient is locally calculated. The test-filtered quantities need spatial averaging in neighboring regions of space. In this case, the constant is averaged over small volumes that have a size of 6Δx, being Δx the local mesh size around the local control volume. This model was chosen because it takes advantage of the simplicity of Smagorinsky's model, while having an extended validity domain, in the presence of mean shear for instance. Detailed chemistry effects are modeled by a tabulated chemistry approach. All thermodynamic data are tabulated as a function of a progress variable built as a linear combination of species mass fractions (YC = YCO + YCO2 + YH2O), and the mixture fraction (Z), which estimates the local equivalence ratio. Thus, only those two variables are transported to deal with chemistry with this approach. To take into account the sub-grid flame-turbulence interaction the Presumed-Conditional Moments for Flame Prolongation of ILDM (PCM-FPI) method is used. The look-up table is built from laminar premixed flame archetypes to map the flame evolution. These flamelets are computed with Cantera using the skeletal LUCHE mechanism. Reaction rates and species mass fractions are tabulated as functions of YC and Z. This laminar look-up table is then convoluted by two beta Probability Density Functions (β-PDFs) to consider their sub-grid fluctuations. While the gaseous is described by an Eulerian approach, the liquid phase is represented by a Lagrangian frame. Each particle is independent and has its own properties (Temperature, Diameter, Velocity) and mass, momentum, and energy transfer apply between the two phases.

Coupling the two phases thus requires interpolation of gas properties to the droplet position or projection at the Eulerian mesh nodes. Droplet are considered spherical, and their motion is described by the kinematic of the points and the fundamental principle of dynamic equations. In the latter equation, the gravitational force is considered negligible in comparison of the drag force and thus neglected. So is the collisional force, for the spray is supposed dilute and thus, interaction between droplet are not considered.

The evaporation is modeled with the Abramzon and Sirignano approach, which is an extension of the classical mass transfer model of Spalding. Thanks to the hypothesis of sphericity, droplet mass is related to its diameter and is given by: 𝑑𝑚 𝑝 𝑑𝑡 = -𝜋𝑑 𝑝 𝑆ℎ(𝜌 𝑝 𝐷) log(1 + 𝐵 𝑀 )

With Sh the Sherwood number, 𝜌 𝑝 the droplet density, D the thermal diffusivity, Bm the Spalding mass transfer number. The Sherwood number is calculated with the Froessling equation, with Rep the Reynolds number associated to the particle, defined as:

𝑅𝑒 𝑝 = 𝜌 𝑝 |𝑈 𝑝 -𝑈 𝑏𝑢𝑙𝑘 | 𝜐
The temperature and composition of the gas surrounding the droplet are estimated using the 1/3 rule, and computed as a mean value of 2/3 of the surface value, and 1/3 of the far field one. Finally, Droplet temperature is considered uniform, and its evolution is obtained by integrating the energy conservation from the droplet surface to the far field.

Simulations were performed with the YALES2 code developed at CORIA based on the Finite-Volume Method (FVM). For the present study, a projection method for low-Mach number flows with variable density is used. Density, momentum and scalar equations are solved using a 4thorder centered scheme in space and a TFV4A 4th-order scheme in time. The numerical domain covers the entire HERON test rig, from the plenum to the combustion chamber outlet. The mesh is composed of 237 million tetrahedral cells, with a refinement near the injection channels and reaction area in which the largest cells have a size of 250µm. The film cooling of the lateral windows also requires a mesh refinement and cells have a maximal size of 150µm. The different refinements lead to y + values near the injector walls of around 10, with maximum values of 50.

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The atomization process is considered an air-blast atomization due to a liquid jet momentum greatly inferior to the airflow momentum. The spray is modeled as a cloud of droplet every time step near the injection holes, with a prescribed diameter PDF and a velocity profile supposing that droplet trajectories are dictated by the airflow. The diameter PDF is computed from the Rosin-Rammler distribution:

𝑓(𝐷) = 𝑞𝐷 𝑞-1 𝑋 𝑞 𝑒𝑥𝑝 [-( 𝐷 𝑋 ) 𝑞 ] 𝐷 32 = 𝑋 3 𝛤 ( 3 𝑞 + 1)
𝑋 2 𝛤 ( 2 𝑞 + 1) Where Γ is the gamma function. The parameter X is representative of the mean diameter, and q defines the width of the distribution. The choice of those parameters is essential to represent closely the real atomization and thus, four methods have been tried out, leading to four different simulations. The first distribution is based on data provided by SHE and extrapolated from experimental tests carried on at the DLR at 10bars. The associated set is D32 = 32µm, and q=3.1 and will be denoted LES D32. A second set of parameters is given by the empirical Rizk and Lefebvre correlation for plain-jet airblast atomizers, and lead to D32 = 16µm, and q=3.1, and will be referenced as LES D16. Due to the uncertainty on the leading atomization process, a third set is chosen from the Jasuja's prefilming airblast atomizers correlation, and lead to D32 = 12µm, and q=3.0, and will be denoted as LES D12. Finally, an arbitrary fourth distribution is built with D32 = 9µm, and q=3.0, denoted LES D9. Lagrangian data have been accumulated and projected at the mesh nodes, and then interpolated on the same planes used for PDA measurements. Statistics have been cumulated during at least five characteristic flow time, which was estimated to be around 5.3ms.

A comparison of the maps of mean diameter at Z/D=0.04 is presented on the topside of Fig. 59. It appears clearly that the LES D12 case, shows a good agreement with PDA measurements. Both experimentally and numerically, small diameters around 5-6µm are reported right above the cup. Mean diameters radially increase, and the LES D12 case shows mean values around 8µm above the air swirler, where most of the spray is located. At the same radius as the edge of the PDA measurement plane, a similar circle of larger droplet is found numerically. In the case using the smallest droplet distribution, LES D9, shows droplet distributions that are significantly smaller than the ones measured by PDA with mean diameters below 6µm. Inversely, LES D32 and D16 cases show mean diameters larger than the ones measured. This is consistent with previous findings that LES cases D12 and D9 more closely fit the experimental data recorded by PIV and PLIF. The bottom side of Fig. 59 gives a better detailed comparison between PDA measurements and results simulated in the LES D12 case. Firstly, the mean diameters, Fig. 59.e), are again in a good agreement with comparable droplet mean size. The pocket pattern is less pronounced numerically, but somewhat present. These comparisons suggest that the spray is numerically well reproduced, both in the distribution of the drops in space and in their diameters. Pannels f) and g) of Fig. 59 show the comparison of the vertical velocity at the two axial distances of interest. Again, good agreement is found between the measurements and the simulation. Although the simulated mean droplet velocities are slightly lower, a similar spatial distribution is still observed in both cases. The axial velocities are presented in Fig. 59.d). Similarly, the spatial evolutions remain similar, although disagreements on the intensity of the velocities remain.

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A qualitative comparison of the radial profiles of droplet diameters measured and simulated is shown in Fig. 62. At Z/D=0.04 on the Y=0 line, and at Z/D=0.64 on the 3π/4 line. Fig. 62.a), top left corner, shows the average diameter D10 evolution, and a nearly perfect agreement is reached between the LES D12 case and PDA measurements at both Z/D=0.04 and Z/D=0.64. This result highlights the ability of the Lagrangian model to predict the evolution of the mean diameter along the reaction zone, which, as already mentioned, started well upstream of the injector outlet. LES cases D9 and D16 show similar trends with a flat profile, but they also highlight differences of a few micrometers. As expected, the LES D32 case overpredicts by far the LES D10 case and shows a strong decrease near the edges of the profile, indicating that much smaller droplets recirculate in the ERZ. The quantitative comparison of the vertical velocity is performed on the bottom side of Fig. 62. At Z/D=0.04, good agreement is achieved between x/R=-0.9 and x/R=-0.5, which corresponds to the band where most of the droplets are detected by PDA. Close to the inner and outer edges, large differences between PDA and LES are observed. A measured velocity twice than the one found numerically for X/R = 0.95-1.0 is then noted. These differences are located in the area where clouds of drops transported at high velocity have been detected, as shown in the scatter plots in Fig. 57. The simulation of the gaseous fluid velocities also shows differences with the PIV velocity measurements [185], which may then explain the peculiar droplet behavior in the simulations.

The average droplet distribution is compared at Z/D=0.04 in Fig. 61. There is good agreement between the LES 12 case and the PDA measurements. The LES D9 and D16 cases show a similar distribution shape, shifted towards smaller and larger droplets respectively, while the LES D32 case shows large differences. This result was expected since the flame topology of the LES D2 case was very different from that observed experimentally, representing rather a diffusion flame than a premixed flame.

Finally, a qualitative comparison of the radial mean concentration profile of the kerosene vapor measured by PLIF and the PDA data rate is presented in Fig. 60 The same behavior is found, with a spray positioned around x/R~0.8, and areas structure of visible droplets that recall the jet in crossflow atomization. This is especially interesting since, as previously observed, the data rate may be subject to a strong bias under these operating conditions. Finding such good agreement with a proven measurement diagnostic such as kerosene-PLIF provides a high degree of confidence in the PDA application methodology under such conditions.

Meshing and velocity prediction

The mesh sizes used to measure the droplet diameter distribution by PDA were arbitrarily defined after exploring the spray. The velocity range to be used as input for PDA measurements was defined and adjusted at each measurement point after checking the velocity distribution manually. This method, which is time consuming, was modified for the measurement campaign focused on the MSFI injectors. The improved method, tested and used thereafter, consisted of an initial velocity measurement with a large velocity span intended to cover the velocity range ~-50 to 200 m.s-1 with a coarse mesh. The data were then interpolated to a finer mesh, and several high, medium, and low velocity meshes were defined with matched velocity means and spans. The interpolation was linear, weighted by the inverse of the distance of the measurement points. The PDA probes were then automatically moved to follow each mesh based on their average velocity, rather than following a Cartesian or polar displacement, which required frequent adjustments to the velocity setting. An example of the coarse measurements and subsequent interpolation is shown in Fig. 63.

Choice of the PDA configuration and optical optimization

Although the choice to perform the droplet diameter measurements was carried out with a SPDA configuration to have spray information close to the injector outlet, the dual PDA configuration was also briefly tested at Z/D=0.64 to obtain information on the possible impact of a better rejection of non-spherical droplets when determining the mean diameters. To also differentiate the impact of beam stirring and the droplet rejection method, both coincident and non-coincident SPDA configurations were tested. Like any SPDA configuration, the coincident approach is based on a sphericity criterion oriented only on the signal of the two beams, but droplets whose axial and vertical velocity measurements do not coincide in time are rejected.

The method consists of optimizing the signals from each pair of beams in the LDA configuration, until the data rates are maximized and equivalent for each pair of laser beams. The system is then adjusted in a non-coincident SPDA configuration. The data rate of the diameter channel would decrease due to the application of the more restrictive burst quality requirements and sphericity validation. A second optimization is then performed to maximize the burst and sphericity validation rates on the PDA channel, with the goal of achieving the LDA data rate. The system was then configured in SPDA coincidence mode. A decrease in data rate was observed, due to the non-overlapping of the two laser beam pairs. The LDA beams were slightly shifted to optimize the data rate to be close to the reference value, which varies between 40-50% and ~70%. This results in the Fiber Coincident configuration. A return to non-coincident mode would result in the reference configuration, noted as Fiber NC. Note that in order to optimize the superposition of the laser beams, both probes were installed on custom linear and rotational displacement systems, as the original ones were not accurate enough. Then, a third measurement was performed with the Dual PDA configuration. The optical filters as well as the polarization of the horizontal laser beams were modified accordingly, and their energy maximized and balanced back to the values used in the Fiber configuration.

The NC Fiber and Coincident configurations show the effect of the beam steering, while the comparison of Coincident and Dual Fiber configurations provides insight into the impact of an enhanced spherical validation criterion. These tests were performed on a reactive flow at atmospheric pressure at low power. As a result, a direct comparison of the results presented below with the measurements performed at high pressure is unfortunately not available.

The optical configuration used is similar to that described above, with one small difference: an increase in the off-axis angle from 30° to 34°. The reference data rate for LDA was 55 kHz. After performing the optimization steps, the average burst validation rates were 95.3% for the Fiber NC mode and 92.5% for the coincident modes. However, the data rates dropped from 55kHz to 15kHz, due to the low coincidence between the measured bursts from the two laser beam pairs. Note that the coincidence could be increased by setting a larger time gate for the coincidence validation, although this increases the probability of having erroneous measurements in the case of multiple particles inside the probed volume. The spherical validation for the Fiber modes was about 95%, while a value of about 68% was recorded in the Dual mode, which reduced the actual data rate to about 8kHz

The results are shown in Fig. 64. Following the results reported by [START_REF] Behrendt | Optical measurements of the reacting two phase flow in a realistic gas turbine combustor at elevated pressure[END_REF][START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF], the application of the coincidence criterion really increases the complexity of the measurement of small droplets, as highlighted in Fig. 64.a). The fiber coincidence configuration shows an increase in the average diameter of about 40%, due to the effect of beam steering that reduces the probability of having an ideal four-beam crossing combined with a maximum intensity located at the probed volume.

The Dual mode measured droplet distributions slightly shifted to the left compared to the coincident mode, because a better rejection procedure is used and reduces the probability of a poorly validated events, as for the results reported in Fig. 50.b)c). The axial velocity is modified by the decrease in small droplets measured, showing a clear correlation between diameter and C2 -Confidential velocity at these particular positions. The vertical velocity was also changed at some radial positions.

In light of these results, it appears that the effect of the beam steering significantly decreases the ability of PDA to measure small droplets. At Z/D=0.04, the optical paths allow only the use od the Fiber NC configuration with two vertical beams, and it was therefore chosen to perform measurements at Z/D=0.64 with the same mode to maintain an accurate comparison with measurements performed at Z/D=0.04. This chapter will present the results of the extensive campaign conducted on the reference operating point. All the measurements were carried out in reactive conditions, at 𝑃 = 0.75 𝑀𝑃𝑎 and 𝑇 3 = 725𝐾 with 𝑚 ̇= 252.0 𝑔. 𝑠 -1 and 𝐹𝐴𝑅 = 24‰. The fuel distribution injected through the pilot stage was set to the lowest possible value while ensuring a stable combustion. While testing the extinction limits of the three injection systems, the MSFI-V2 showed the least resistance to extinction when decreasing the pilot/main fuel ratio, while the Lemcotec showed the lowest extinction limit. However, the NOx emissions were the lowest with the MSFI-V2 (1.69g/kg EINO) and the highest with the Lemcotec (2.59 g/kg EINO). The objective of this chapter is thus to propose explanations for these differences, and try to link them to the technological modifications from the Lemcotec design.

Aerodynamic

Velocity fields were measured by the mean of PIV on the reference operating point, i.e., in reactive conditions at 𝑃 = 0.75 𝑀𝑃𝑎, 𝑇 3 = 725𝐾, 𝑚 ̇= 252.0 𝑔. 𝑠 -1 and 𝐹𝐴𝑅 = 24‰. Mean planar velocities are presented in Fig. 65. Here and in all subsequent figures, the Lemcotec injector will be presented on the left panel, the MSFI-V0 on the middle one, and the MSFI-V2 on the right one.

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Planar velocity magnitude up to 85m.s -1 were measured, with the highest velocities originating from the main air swirler near the burner outlet. With both the Lemcotec and MSFI-V0, the pilot and main flows follow a V shape, with the two flows being parallel and interacting right at the burner's outlet. In terms of symmetry, the MSFI-V0 and Lemcotec are quite equilibrated, with only small differences of velocity magnitude between the bottom and the top flow. A different behavior is visible on the MSFI-V2 velocity field. The pilot flow is much more separated from the main flow at the burner exit, changing the overall topology. Here, the main flow follows more of a cup shape, and the pilot flow comes out of the burner almost parallel to the burner's axis, before expanding radially outward and interacting with the main flow. More precisely, the interaction between the pilot and main jets happens around z/R~1.0 with the MSFI-V2, while the jets from the Lemcotec and MSFI-V0 seem to interact almost right at the burner exit. With the elevated swirl number of the pilot flow of the MSFI-V2, one could have expected instead a rapid radial expansion of the pilot flow and a deeper penetration of the IRZ within the injection system. This delayed interaction of the pilot and main jets, and possibly the subsequent limitation of the inner recirculation zone, might be due to the increased separator. Indeed, the high flow velocity of the main is likely to create a depression above the separator and create an aspiration effect of the pilot flow, but with a larger separator, this effect might be suppressed. With the higher swirl number of the MSFI-V2 pilot, the axial direction of the flow at the burner outlet is peculiar. One possible explanation is that the dilution holes are pushing the pilot flow inward, reducing the swirling motion and radial expansion of the flow. While it is not visible on the MSFI-V0 and Lemcotec because of the close interaction with the main flow, it seems that the increased separator width allows the dilution holes to have a greater impact on the pilot flow of the MSFI-V2, creating this particular flow field. On the V2 injector, a small dissymmetry is noticeable, with higher velocities on the bottom part of the main flow, and higher ones on the upper part of the pilot flow. As a consequence, a recirculation zone between the pilot/main jets on the topside of the pilot flow is visible, while on the bottom side the faster interaction between the two jets seem to have reduce the chance of a recirculation zone to develop. This tends to demonstrate the effect of the main flow velocity and the induced aspiration of the pilot flow: as Axial velocity measurements are presented on Fig. 66. The top panel shows the axial mean velocity and the associated standard deviation. As expected, the axial velocity of the pilot jet from the V2 injector is higher than the ones from Lemcotec and V0, as it exits the burner almost horizontally. It is interesting to note that the external recirculation zones (ERZ) seem to be almost nonexistent here. Though the standard deviation in the ERZ shows values of the same order of magnitude as the mean values, indicating strong fluctuations. Lemcotec and V0 injectors show similar trend, though a higher axial velocity is registered with the V0. The Inner Recirculation Zone (IRZ), is delimited by the zero axial velocity surface, and represented by a white line. The late pilot/main jets interaction makes the V2 IRZ much narrower than the one from Lemcotec and V0, which will impact the combustion process inside the burner. The standard deviation for both injectors is lower than the one seen on the V2, indicating less fluctuations, or a steadier velocity field. The radial profiles are presented on the bottom panel of Fig. 66, and follow the same conclusions. The radial standard deviation of the pilot jet of the V2 injector is much higher than the one of the two other injectors, further showing an unsteady flow.

A quantitative comparison of the velocity magnitude is presented in Fig. 67. The bottom panel is the closest to the injector while the top one is the furthest as Z/D=1.5. Velocity magnitude profiles are almost identical between the three injectors at Z/D=2.0 and are thus not represented here. The velocity profile from the V2 injector, in red, show a clear separation of the pilot and main jets at Z/D=0.2 and Z/D=0.5, with a maximum velocity separation of around 0.55D and 0.75D respectively. The V0 and Lemcotec injectors follow a very similar pattern, with velocity peaks located at the same radiuses, though a slightly larger jet separation is noticeable on the Lemcotec pilot/main jets at Z/D=0.5. The wider opening of the main jets of the V2 is quantifiable here, and around 25% larger than the opening of the two other injectors. Velocity magnitudes are comparable across the three injection systems, the largest differences being due to a small asymmetry of the flow field. The Fig. 66 present axial and radial velocity components, altogether with their standard deviations. The axial velocity plots provide a measure of the width and strength of the IRZ. While negative axial velocities are similar on the three systems, a clear difference in width is noticeable at Z/D=0.5, even between the Lemcotec and V0 injectors, indicating a difference in the pilot/main jets mixing. At Z/D=0.5, the IRZ on the V2 injection system is around 0.35D in width, 0.60D with the V0, and 0.85D with the Lemcotec, which will have a significant impact on the amount of recirculating gas reaching the burner where the combustion process starts.

Lemcotec MSFI-V0 MSFI-V2

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The standard deviations of velocities of Lemcotec and V0 follow a very similar trends, considering the small differences explained previously. The V2 injector shows much larger velocity fluctuations, especially close to the burner. At the center of the burner, y/R = 0.0, the standard deviation of the axial velocity is almost equivalent to the mean axial velocity, with values around 15m/s. Also, at the same location, the radial standard deviation is around 15m/s while the mean velocity is null. Those indicate that the flow field is highly turbulent, and that the recirculation zone is likely to strongly change in shape with time. The boundaries of the main jets, located around +/-0.7y/R and +/-1.2y/R, are also subject to strong radial oscillations, especially at an axial distance of z/R=0.5. The radial velocity standard deviation is equal or greater than the mean velocity, indicating the presence of radial oscillations of the main jets, which could be correlated to the fluctuations of the pilot jets and depict an alternance of strong and weak interactions between the two.

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Instantaneous velocity fields and streamlines give a better view of the flow field fluctuations, some are presented on Fig. 69. In the Lemcotec and MSFI-V0 fields, vortices appear between the main jets and the IRZ, following a helicoidal shape depicted with the grey dashed lines. This shape is characteristic of a precessing vortex core (PVC), which is likely to be responsible for the fluctuations reported above. The MSFI-V2 injector is also subject to such an aerodynamic structure, but the snapshot presented on Fig. 69 is not the clearest to drawn its structure. Instead, it has been chosen to show the vortices that appear between the pilot and main jets, around Z/R and X/R = 0.75. The development of those vortices is the reason behind the high velocity fluctuations shown on Fig. 68, but can also explains the particular shape of the mean flow field of the V2 in comparison with the Lemcotec and V0 injection systems. The presence of those vortices is not continuous, as it can be seen comparing the top and bottom sides of Fig. 69.c). An alternation between a fast and a late interaction between the pilot and main jets is then observed. The latter occurs when the vortices push the pilot jet to follow a horizontal trajectory, while the main jets are pushed radially away from the center. In contrast, on the bottom side of Fig. By putting vorticity probes at various locations, local temporal vorticity profiles can be plotted. A Fast Fourier Transform is used to extract the frequencies of the passage of vortices. Probes were placed with constant spacing at various locations, for a total of 88 probes. Results for the MSFI-V2 injector, from eleven probes are plotted in Fig. 70, at an axial location Z/D=0.38. Probes are numbered from one to eleven with decreasing y/R, i.e. from top to bottom on previous representations. Two clear frequencies appear, one at 1364Hz and a harmonic at 2728Hz on the probes between the pilot and main jets. The fundamental frequency is measured on every probe, except the first one which is placed deep inside the top ERZ. All results are not reported here, but the same frequency is found further away from the burner exit, in particular near the C2 -Confidential border of the IRZ, which would coincide with the passage of the PVC. As this fundamental frequency is measured on the whole height of the chamber at Z/R=0.38, it is clear that the PVC has a significant impact on the behavior of the pilot and main jets. The same study of the Lemcotec and V0 instantaneous vorticity fields shows that similar PVC frequencies can be observable, with a fundamental frequency of 1338Hz for the Lemcotec and 1374Hz for the V0, see Fig. 71. For the sake of brevity, only three probes near the lower main jet are reported for each injector. Interestingly those frequencies where detected on a much more restricted area than with the V2 measurements, indicating a release of vortexes in a much more limited area, which explains the steadier flow in comparison with the V2 system reported previously.

Lemcotec MSFI-V0

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Finally, the shear stress is presented on Fig. 72. As the previous results have shown, there are little differences between the shear stresses from the Lemcotec and MSFI-V0, although the area between the pilot and main flow is slightly more pronounced in the case of the Lemcotec injector, with the two streams being slightly more separated than in the case of the MSFI-V0 injector. With both injection systems, the shear stress within the main flow is around 3000 s -1 , with a short peak around 9000 s -1 right at the burner outlet. A major difference appears with the MSFI-V2 pilot stream, as the high shear region is much longer than with the two previous injectors, with shear values superior to 6000 s -1 measured up to z/R=0.6, whereas for the two other injectors those values are only found up to z/R=0.25. Those high levels of shear stress are due to both the higher pilot swirl of the MSFI-V2, and the intermittent passage of eddies in this region, as depicted on Fig. 69.

In order to locate and visualize the spray in relation to these results, a tomography was performed with the PIV setup to record the Mie scattering signals of the kerosene droplets under the same operating conditions.

Spray characterization

High speed tomography

The average Mie scattering signal of the droplets is plotted in Fig. 73, together with the average velocity streamlines from the PIV measurements. The Lemcotec injector shows a noticeable dissymetry between the top and bottom side, which is not the case for the V0 injector. Also, the spray penetration is slighly larger with the Lemcotec, with a mean spray penetrating up to around 0.7D, while the V0 spray penetrates up to 0.55D. The V2 injection system shows the smallest spray penetration, around 0.3D, which implies a much faster evaporation of droplets inside the burner. On all three injection systems, droplet tend to follow the main flow stream,

MSFI-V0 MSFI-V2 Lemcotec

C2 -Confidential implying that the main spray opening of MSFI-V2 is probably slightly wider than that of the other two, although its small length renders this difficult to discern in these images. Fig. 74 shows some snapshots of the Mie scattering signals of kerosene droplets. While the acquisition frequency was 20 kHz (Δt=50µs), images shown have a time interval of 200µs to provide an insight of the visible spray oscillations. The deeper penetration of the Lemcotec spray is clearly noticeable, and so is the dissymetry reported on the averaged velocity fields. This increased presence of droplets inside the combustion chamber compared to the case of the MSFI-V0 injector is probably the reason for the higher soot production by the Lemcotec injector that is perceptible with a simple surveillance camera. On all three injection systems, the spray from the pilot injection is barely visible, but spray oscillations are noticeable, see for example the first and last shot of the Lemcotec and MSFI-V0 spray. As expected, it is with the V2 that the oscillations are the strongest, with some shots where almost no droplet are detected. The full length of the tomography measurement reveal a clear pulsation of droplet release from the V2 injector, which is strong enough to apply a similar method to the one used previously and extract the oscillation frequency.

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Using square probes on tomography images right at the burner outlet, it is possible to extract a temporal profile of local spatially averaged Mie scattering signal. With the same procedure used on the temporal vorticity profiles, a pulse frequency of 1360Hz is found, with a harmonic at 2720Hz, see Fig. 75.

It is thus clear that there is a strong interaction between the PVC and the liquid phase with the V2 injector. The separation of the pilot and main jets, and the subsequent development of vortices between the two jets makes the pulsing impact not only the pilot spray but also the main spray. This will have consequences on the flame behavior, that will be reported later.

Phase Doppler Anemometry

Phase Doppler Anemometry has been used to provide quantitative information on the spray of the MSFI-V0 and V2 injection systems. In addition to provide a better comparison between the two injectors, diameter measurements have been used as a reference to determine boundary condition of Large Eddy Simulations (LES) for those two cases. As detailed in the experimental setup description chapter, measurements were carried out on two radial planes (X-Y planes), at axial distances from the burner outlet of Z/R=0.04 and Z/R=0.64. Since the data rates were significantly higher than the ones observed during the preliminary campaign with the LP injector, the maximum number of droplets measured was increased to 200 000, so at least one second of measurement was registered for each position probed. Results at Z/R=0.04 are presented in Fig. 76. On this figure are presented the data rate (upper panel), the D32 size (middle panel) and the vertical velocity Vy (bottom panel) for the two injection systems. Dash-dotted lines represent the inner boundaries of pilot and main channels. Red crosses represent measurement points from which scatter plots will be presented later, M stands for Main, P for Pilot and S denotes points of special interest. Measurements points with less than 2000 droplets were removed for the convergence of the mean diameter was not ensured. This explains the unregular shape of the borders of the V2 mesh, and in particular, the separation of the pilot and main sprays that can be seen on Fig. 76. Mean radial values are also shown in Fig. 78.

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The PDA data rate from measurements on the MSFI-V0 injector shows a main spray located at a radial distance of 0.75R, and a pilot spray around 0.42R. The latter is visible right above the edge of the pilot channel, showing a rapid outward displacement of the spray, which is probably caused by both the swirl of the pilot flow and the rapid interaction with the main flow as previously observed. This rapid interaction between the two counter-rotating flows explains why the data rates were sufficiently high that enough droplets could be measured at each location, even in the 0.55-0.65R region, which is above the pilot-hand separator. This was not the case with the MSFI-V2 injection system, where almost no droplets were found in the 0.45-0.60R area. The main spray of the MSFI-V2 is located at the same radial distance than the one from the V0, but the pilot spray is closer to the center, measured around 0.35R. More noteworthy, the pilot spray is contained within the inner border of the separator, and the lack of interaction with the main spray (and the induced radial expansion) made the dilution holes noticeable, see the regular drop of data rates near 0.4R. The presence of droplets between those dilution jets indicate that at least a part of the spray was near, or directly on, the inner wall of the pilot. The fact that dilution holes are not noticeable on the MSFI-V0 data rate could show that either the pilot-main interaction is strong enough to completely mask the impact of the dilution holes on the pilot spray with a strong mixing effect. Or that the lower swirl of the pilot flow projected less fuel on the wall, limiting the impact of the dilution holes on the droplet repartition. Even though PDA data rates are complex to interpret quantitatively, the difference between the two injectors is evident, with two or three times more droplet measured on the MSFI-V0, which is consistent with tomography results. The decrease in the data rate on the MSFI-V0 measurements on the top part of the mesh is most likely due to a much less effective optical path, as data rates, mean diameters and validation rates decreased suspiciously above Y/R=0.75 and X/R=0. Similarly, the strong peak on the main spray data rate of the MSFI-V2 might be a measurement artefact. Radial kerosene PLIF will serve later as another source of information on the radial spray homogeneity.

Mean diameters D32 are presented in the middle panel of Fig. 76. While areas of high spray density are already clearly outlined with the MSFI-V2 injector due to the removal of points where almost no droplet are measured, the D32 distribution of the MSFI-V0 must be considered in conjunction with the data rate distribution to keep an overview of where the mean values are representative of the spray. For the MSFI-V0 injection system, D32 between 9 and 13.5µm were measured, or D10 between 6.5 and 8.5µm. Where the highest data rates were measured, D32 around 10µm, and D10 around 6.5µm were found for the pilot spray, while the main spray shows values around D32 = 11µm and D10 = 8µm, see Fig. 78. The largest droplets measured around 0.62R are located in-between the pilot and main jets and thus represent a very small number of droplets. The MSFI-V2 injector shows slightly larger droplet, D32~11µm and D10~7.5µm for the pilot spray, and D32~12µm and D10 ~9µm for the main spray. This difference is likely due to the higher evaporation rate that has been noticed with the V2 injector, the combustion process inducing a fast of disappearance of small droplet that is not compensated by the decrease in size of larger ones.

MSFI-V0 MSFI-V2

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Mean vertical velocities are shown on the lower panel of Fig. 76. The fact that it is the vertical velocity component that is measured makes the interpretation less straightforward than the one carried out on mean diameters. Velocities on the Y/R=0 line are azimuthal velocities, and radial ones on the X/R=0 line. In the middle of the MSFI-V0 main spray, the azimuthal velocity is about 55m.s -1 and radial velocity about 60 m.s -1 . With the MSFI-V2 injector, the azimuthal velocity is around 50 m.s -1 while the radial component is around 85 m.s -1 . For the pilot sprays, the average velocities are very similar, with azimuthal velocities around 20 m.s -1 and vertical ones around 25m.s -1 measured for both injection systems. The principal difference between the two injectors is thus the radial component of the main spray, which is higher in the case of the MSFI-V2 injector due to the weaker interaction with the pilot. This is consistent with the previously reported PIV mean velocity flow fields, which showed a cup-shaped evolution of the mean flow field for the MSFI-V2 instead of the V-shape observed on the Lemcotec and MSFI-V0 injectors.

Results at Z/R=0.64 are presented in Fig. 77. Data rates were very limited with maximum values of 5kHz, and average values below 1kHz (upper panel of Fig. 78), and thus only few points of the initial mesh are reported. The mean diameters D32 are shown in the upper panel of Fig. 77, and the radial average values in the lower panel of Fig. 78. Slightly smaller droplets were measured with the V2 injector, with mean diameters around 10µm, while with the V0 system values around 11.5µm were found. No droplet from the pilot spray were detected, indicating a full evaporation at this axial distance from the burner. As expected from the radial velocity difference seen at Z/R=0.04, the MSFI-V2 main spray expanded more radially, and is located around 1.25R while the V0 spray is close to 1.1R. Azimuthal velocities are comparable, with mean values around 30m.s -1 for the MSFI-V0 and around 25m.s -1 with the MSFI-V2. In both cases, axial velocities are in the range of 55 to 75 m.s -1 , with the smallest values measured at the outer edge of the mesh, near 1.3R in the case of MSFI-V2 and 1.2 in the case of MSFI-V0.

Droplet distributions as PDF and cumulative functions are plotted n Fig. 79. At Z/R=0.04, the pilot and main sprays have been differentiated since they come from different atomization process and that their segregation is clear with the data rates 2D maps. All distributions follow a similar shape, which can be modeled by Rosin-Rammler distributions with the parameters specified on each plot. The width parameter q appears to be more dependent on the atomization process than on the injection system itself, i.e., similar values of q are found for both pilot sprays, ~2.6, and for the two main sprays, ~2.9.

C2 -Confidential the possibility of a velocity bias, which should be compensated by the transit time ponderation

MSFI-V0 MSFI-V2

C2 -Confidential anyway. However, as it can be seen at some specific locations, such as the ones displayed in the

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Finally, velocity -diameter scatter plots are shown on Fig. 80. The location of the points used has been shown in Fig. 76. The points on the top and middle panels have been selected where the highest data rates were registered, in the middle of the main and pilot sprays respectively. Some specific points with peculiar behavior were also found in the pilot spray, and two examples are shown on the bottom panel. These scatter plots are not intended for a direct comparison between the two injection systems as the points are not at the same locations. Rather, the idea is to provide an insight of some representative velocity -diameter correlation and velocity span that can be found in the high data rates regions. Velocity and diameter spans are generally larger with the V2, as would be expected from the higher fluctuations measured by PIV and the larger Dv09 from mean droplet size distributions. Velocity spans near the area of interaction of the main and pilot jets are particularly large due to the contra-rotating flow. Near the outer part of the pilot spray, the dilution holes are likely to play a role in the wide range of droplet velocity. On most measurement points, the correlation velocity-diameter is moderate, which greatly reduces bottom panel of Fig. 80, there are regions where two populations of droplet with very different velocities coexist. Those specific cases were measured only in the pilot spray, and may show droplet populations with negative or/and positive velocities, and different diameter spans. These points are located in the upper part of the pilot spray, but the interpretation of the vertical velocity can be misleading and it is likely than the same phenomenon would be observed in the lower part with a pure radial velocity measurement. At specific points in particular, see Fig. 80, bottom panel, a population of small droplet is clearly separated from a population of larger droplet with diameter ranging from 1 to 25µm. The passage of the PVC could explain those two different populations, though with the method described thereafter, the frequency of the PVC could not be measured on those points. It could also indicate the presence of droplet coming from different atomization processes, i.e., this could be an indicator of a fuel filming created on the separator of the pilot air swirlers. The earlier atomization from the pressure swirl in comparison of the one from the prefilming could indeed explain the smaller diameter range.

Note that on some points of measurement, such as the specific point of the V0, bottom left, the velocity span, i.e. the range of velocity magnitude, is superior to 80m.s -1 while the mean velocity is around 0 m.s -1 . As such, some droplets with velocities below -40m.s -1 cannot be measured, as it is the lower limit of the PDA system. To the author's knowledge, there exists no configuration that could avoid this issue, though it was limited to a handful of the measuring points (~2%) and a small proportion of the spray.

MSFI-V0 MSFI-V2

PVC detection with PDA

One of the most important features of the PDA is its capacity of measuring diameter and velocities in high density medium with a high acquisition rate. As such, this measurement technique should be suited for spray oscillation analysis. From the tomography results, it is clear that sprays studied, and in particular those from the MSFI-V2 injection system, are subject to periodic oscillations from the PVC. From the tomography results, it appears that pockets of droplet are periodically released from the injector. To capture this effect, a signal was reconstructed with the transit time, see Eq. ( 169), which corresponds to the time during which a droplet is in the measurement volume. The passage of a pocket of droplet thus creates a peak in the reconstructed signal. A fast Fourier transform is then applied on this reconstructed signal which was sampled at 20kHz, i.e., ∆𝑡 = 50µ𝑠. Results are shown on Fig. 82. The measurement point in the main spray for the MSFI-V2 injection system is not the same as the one used for the scatter plots example shown on Fig. 80, as the SNRs were much lower with the MSFI-V2 measurements than with the MSFI-V0 ones. Instead, another point with an acceptable SNR has been selected within the main spray. The three other points correspond to particular points used until here.

𝜏(𝑡

) = ∑ 𝜏 𝑡𝑖 𝑡+∆𝑡 𝑡𝑖=𝑡 (169)
The methodology worked really well with measurements on the MSFI-V0 injector, and peaks with high SNR have been found. In both measurements on the main and pilot sprays, periodic pulsations of the spray were found around 1370Hz, which corresponds to the PVC fundamental frequency measured with PIV (1374Hz). Some harmonics can also be found, around 2730Hz, which also coincides with PIV results. With the measurements on the V2 injectors, particular frequencies have also been found, but for an unknown reason, the SNR is much lower. Peaks near the frequency of the PVC (1364Hz, from PIV), and the associated droplet pulse (1360Hz, from tomography) have also been found at 1379Hz. The measurements points used to find this frequency do not lie in the X-Z plane, which is the plane used for the PIV and tomography, which confirms that the instability observed is a three-dimensional phenomenon, and not a 2D oscillation around the Y plane.

On the pilot spray of the V2, a peak can also be found with PDA measurements around 5100Hz. The PIV sampling rate was not high enough to search for instabilities at such high sampling rates, and the tomography didn't reveal a peak at this frequency.

MSFI-V0 MSFI-V2

Planar Laser Induced Fluorescence

Simultaneous OH-, Kerosene-, and NO-PLIF measurements were performed to provide combined information on the flame topology, the fuel distribution and the NO production on the three injection systems. To ensure a proper comparison between all the experiments, the following procedure have been applied.

PLIF image energy correction

Firstly, raw PLIF images have been corrected for camera gain and laser energy. The temporal integration of the images stayed constant throughout the whole campaign, but cameras gains were modified to ensure a good signal-to-noise ratio while avoiding any saturation on the intensified cameras. The intensities of the fluorescence signals encountered with the three different systems were quite different, and the cameras gains were adjusted accordingly. In particular, the liquid fuel can produce an intense fluorescence which led to the use of a smaller gain on the kerosene camera depending on the operating conditions and injectors.

The laser energy was maximized on both laser systems, but a variation of a few mJ was observed during the campaign, which was conducted over several weeks. This decrease of energy is due to the natural degradation of the dies, and the attrition of the optical components constituting the lasers. In particular, the UV lens are subject to a loss of transmittance due to the appearance of a white veil, but their replacement requires a realignment of a significant part of the laser system, which takes a long time. Salaün et al. [186] showed that with a configuration similar to the one used here, the PLIF signal depends linearly dependent of the laser energy. Thus, the raw images can be corrected by a simple ponderation to ensure a consistent comparison between images recorded at different times. Laser energy of 1500 shots were recorded during the take of background images, right after each of the PLIF measurements, giving the average laser energy. The weighting of the average PLIF images is then as follow.

𝐼 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 𝐼 𝑟𝑎𝑤 . 𝐺𝑎𝑖𝑛 𝑟𝑒𝑓 𝐺𝑎𝑖𝑛 𝐼 . 𝐸 𝑙𝑎𝑠,𝑟𝑒𝑓 𝐸 𝑙𝑎𝑠,𝐼 (170)
With 𝐼 𝑟𝑎𝑤 the uncorrected PLIF mean image, 𝐺𝑎𝑖𝑛 𝐼 and 𝐸 𝑙𝑎𝑠,𝐼 the gain and the mean laser energy used during the measurements on the injector I. The reference gain value, 𝐺𝑎𝑖𝑛 𝑟𝑒𝑓 , and the laser energy reference value, 𝐸 𝑙𝑎𝑠,𝑟𝑒𝑓 , were defined as :

𝐺𝑎𝑖𝑛 𝑟𝑒𝑓 = 𝑚𝑖𝑛(𝐺𝑎𝑖𝑛 𝐿𝑒𝑚𝑐𝑜𝑡𝑒𝑐,𝑉0,𝑉2 ) (171)
𝐸 𝑙𝑎𝑠,𝑟𝑒𝑓 = 〈𝐸 𝑙𝑎𝑠,𝐼 〉 𝐿𝑒𝑚𝑐𝑜𝑡𝑒𝑐,𝑉0,𝑉2

To measure the energy of each laser shoot for instantaneous images correction, a BoxCar integrator has been used during each experiment to integrate in time the energy profile of each shot. The instantaneous laser energy could then be known with :

𝐸 𝑖𝑛𝑠𝑡 = 𝐸 𝑙𝑎𝑠,𝐼 . 𝐸 𝑏𝑜𝑥 〈𝐸 𝑏𝑜𝑥 〉 𝐼 (173)

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With 𝐸 𝑏𝑜𝑥 the instantaneous BoxCar sample, and 〈𝐸 𝑏𝑜𝑥 〉 𝐼 the average of all the samples recorded simultaneously with the PLIF measurements.

Laser sheet correction

Due to the gaussian profile of laser energy, laser sheets are prone to show an energy inhomogeneity, with a high intensity in the center and a reduced energy on the borders. One way of reducing this bias is to diverge the laser sheet before entering the measurement volume, so that the top the gaussian profile is flattened. This is usually done by keeping only about 20-30% of the original laser sheet. With the laser energies at disposal, this would not have been sufficient to obtain PLIF images with a satisfactory SNR. Instead, around 50-60% of the laser sheet was used, and the energy profile was measured with reference PLIF images prior to any measurements. To do so, the combustion chamber was filed with either acetone or NO/Air mixture to record the energy profile of the OH or NO laser respectively. The filters used were the same as for the OH and NO-PLIF recordings to ensure that only the PLIF signal was recorded.

A reference laser sheet and the associated energy profile averaged along the x-axis are presented in Fig. 83. To compensate for any laser sheet inhomogeneity, each PLIF images were corrected by the inverse of the appropriate laser energy profile.

Distortion correction and images superposition

As detailed in chapter X, for the axial measurement campaign, the intensified ICCD camera detecting the kerosene-PLIF signal was placed beside the camera collecting the OH-PLIF signal, at a slight angle with respect to the laser sheet normal. A Scheimpflug optical mount was used to ensure the sharpness of kerosene fluorescence images, but a distortion correction was still required in order to superimpose the kerosene images on the NO and OH images. This was performed with the commercial software Davis 10.2. From two calibration images, a transformation matrix is calculated and applied to all the C2 -Confidential kerosene images so they end up in the same reference frame as the NO images. The OH images only required translations and resizing operations. Fig. 84 shows an example of such a correction. The kerosene Images were flipped from left to right beforehand, as the collecting camera was placed on the other side of the combustion chamber. The calibration images are shown on the top panel, before and after the distortion correction. In the middle, an example of a raw and a corrected kerosene image. At the bottom of Fig. 84, an example of a superimposition of NO and kerosene is shown. In the raw NO-PLIF image, Fig. 84.e, a very strong signal in V shape is visible at the injector outlet, which correspond to the kerosene fluorescence collected by the NO camera. The combination of NO and kerosene measurements allows the kerosene signal to be removed from the NO images through a masking operation.

For the radial PLIF, the images from the three ICCD cameras were corrected to put them on the XY plane, that is, with a normal coincident with the injector axis. The distortion in this case was extreme, in particular for the kerosene ICCD camera with a tilt angle of around 70°. Examples of distortion corrections are shown on Fig. 85. For the three species, calibration images are projected onto a reference frame, as it is shown on Fig. 85,a-b) for kerosene, and Fig. 85,c-d) for NO. Superimpositions of those images are presented on Fig. 85 e-f) for OH -kerosene, and OH-Kerosene-NO respectively. It can be seen that the superposition of OH and kerosene is great, but some distortion remains when superimposed with the NO reference image. To enhance the PLIF image superimposition, a resizing of a few percent of the NO PLIF images was necessary, coupled with some translations. To do so, the presence of kerosene fluorescence on both kerosene and NO PLIF was made an advantage of. As the NO signal was so weak in comparison of the kerosene one, some very discernable structure of kerosene fluorescence on the NO images could be used as reference for the superimposition. With this post processing, the superimposition of the signals showed a really good agreement as it is shown on Fig. 86,e), especially considering the extreme original distortions of the images. It is believed by the author, that a dedicated dewarping procedure could be written to avoid the second post processing, but the writing of such an algorithm is beyond the scope of this work. An example of kerosene-OH superimposition, with a flame contour extraction with the Otsu method, is shown on Fig. 86,f). The result is highly satisfactory, as the flame front follows quite precisely the edges of high concentration kerosene pocket

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Absorption correction

The correction of the absorption of the OH and kerosene signals is performed by assuming a symmetry of the flame around the axis of the injector, as far as possible in the combustion chamber, within the limit of the OH detectivity. This correction is not intended to obtain a quantification of the measurement signals as the thermodynamic conditions of the flow are not precisely known (in particular, the local values of temperature and kerosene concentration), but to improve the visualization of the distributions of both species. For each species, it was chosen to apply the same absorption parameter on the instantaneous and mean images in order not to alter the variability of the measurement signals. The absorption parameters are defined from the mean images and then applied to the instantaneous images. However, the high variability of the OH and kerosene signals limits the absorption correction as pockets of high kerosene and OH concentration in the instantaneous images can be found and may cause a bias on the absorption correction. To overcome this problem, the absorption coefficient is reduced so that divergence in the corrected instantaneous images does not occur. As a result, the upper part of the mean axial OH images tends to shows a higher OH concentration than the lower part because the absorption correction is then underestimated.

The radial images were also corrected for laser beam absorption and showed high intensity positions and disparities between the left and right sides of the images. The strong distortion correction and the positioning of the measurement plane so close to the injector outlet can lead to a high uncertainty on the signal intensity correction, especially on the left side of the images. Therefore, the right side of the images should be analyzed preferably for inhomogeneities in the instantaneous OH and kerosene distributions. With 𝐼 𝑖 the i th PLIF image, 〈𝐼〉 the mean PLIF image, and N the number of PLIF images acquired.

The standard deviation of the OH signal on the Lemcotec and MSFI-V0 injection systems are similar, though the slightly longer and narrower flame topology of the Lemcotec is noticeable.

On the MSFI-V2, the standard deviation shows a W shape due to the intermittent change of flame topology, and stronger fluctuations of the exterior main flame in comparison to the other two systems. The normalized standard deviation is an interesting way to combine both mean images and standard deviations, as it shows relatives variations. As such, the fluctuations of the pilot flame can be put in light more easily. The V-shaped flames observed on Fig. 87 are discernable in between high fluctuation zones as darker areas. For the Lemcotec and MSFI-V0 injectors, it can be seen that the pilot flame fluctuations draw a V shape that follows the borders of the IRZ. The MSFI-V2 shows pilot flame oscillations spread in a wider area, covering the whole IRZ. Those shapes remind clearly of the standard deviations of the velocity measured by PIV, and reported in Fig. 66.

A superimposition of radial and axial velocity standard deviations is reported in Fig. 91 and Fig. 92 respectively. Both formulations of OH PLIF standard deviations are used to demonstrate different behaviors, the classical formation, defined by Eq.( 174) is shown in the left columns, while the normalized standard deviation is presented in the right ones. The colored lines represent iso-contours of velocity fluctuations, while the white line denotes the null axial velocity, i.e., the interface between the IRZ and the main flow. The velocity fluctuations isocontours are drawn at 10-12-14-16-18-20m.s -1 .

First, the fluctuations of the main flame will be observed. A clear superimposition of both axial and radial velocity fluctuations with the OH Standard deviation is visible. This shows the predictable direct relationship between the aerodynamics and the flame fluctuations. When comparing with the normalized OH standard deviation, a correlation is found only with the radial velocity fluctuations. Those high intensity regions highlighted by the NStd formulation show areas where high intensity flames are measured while the OH mean signal is low, and thus, denotes the appearance of strong spatial flame displacements, i.e., flames measured far away from the mean flame position. The clear concordance between OH-NStd and radial velocity fluctuations, but not axial ones, then shows the occurrence of flame opening. It could be explained by fluctuations in the ERZ, or the development of eddies between the ERZ and the main flow. But since the main flame opening behavior has already been detailed with the MSFI-V2 conditioned OH statistic, Fig. 89, it is likely that those radial flame and velocity fluctuations are due to the same kind of pilot/main interaction. This means that the intermittence of pilot C2 -Confidential and main interactions could also occurs with the Lemcotec and MSFI-V0 injectors, but at a smaller scale than with the MSFI-V2, such as it is much more difficult to show with conditioned statistics on the OH instantaneous fields.

The IRZ, depicted with a white line, is exactly superimposed on the V shapes formed by the pilot flame fluctuations on the OH-NStd images. This line marks the interface between the recirculating gases and the main flow, and the path of PVC development. Such a close correlation between OH and velocity oscillations highlights the close interplay between the PVC and the pilot flame fluctuations. In particular, the higher intensity of OH-NStd on the MSFI-V2 injector in comparison with the two systems coincide well with the conclusions drown previously with the PIV results on the difference in PVC strength between the MSFI-V2 and the other two injectors.

Finally, the superimposition of pilot flame and velocity fluctuations is presented. With the Lemcotec and MSFI-V0 injectors, the radial fluctuations are small and no direct link between the flame structure and the velocity field can be identified. Some axial fluctuations can be observed over the pilot flame region, but no high OH variations are observed. This is not the case with the MSFI-V2 injector. Strong radial velocity fluctuations are located right in between the pilot and main flame fluctuation regions, which agrees with the pilot and main flame separations described earlier. More interestingly, the axial velocity standard deviation coincides very closely with those of the OH flame, indicating an axial pulsation of the pilot flame. This is consistent with the liquid pulsation shown in Fig. 74 and the partial extinction of the pilot flame shown in Fig. 89.

Kerosene Fluorescence

The results of the kerosene fluorescence measurements are shown in Fig. 93 for axial measurements, and in Fig. 94 for the radial ones. A superposition of the kerosene signals with OH-PLIF images is proposed to locate the kerosene with respect to the flame. The flame front extracted with the Otsu method is highlighted on the OH images. The average kerosene fluorescence images are closely linked to the average tomography images shown in Fig. 73. They exhibit V-shaped sprays, with the longest fuel penetration into the combustion chamber for the Lemcotec, and the shortest for the MSFI-V2. With all three injection systems, the flame is located within the fuel coming from the main injection, while the fuel coming from the pilot injection is located between the pilot and main flame. The radial kerosene fluorescence images show significant azimuthal inhomogeneities. The difference between the intensities observed on the left and right side of the images is the result of the very strong laser absorption by the fuel, and the inherent difficulty to correct it. On the axial images, the absorption correction was based on the axial symmetry of the flame in the far field, assuming a good mixing of the different species, but also an absence of kerosene. At Z/R=1.0, the laser beams are strongly attenuated by the kerosene, and the correction absorption is therefore more complex. Nevertheless, eight pockets seem to be distinguishable on the Lemcotec average kerosene images, suggesting the presence of an azimuthal inhomogeneity of the spray, which probably comes directly from the jet in crossflow injection. This is adequation with the tomography results showing a higher concentration of liquid fuel in the chamber compared to the two other injection systems, and the longer flame observed thereafter. Furthermore, the fuel observed on the mean radial images is located slightly further radially outward in the case of the MSFI-V0 (r/R~1.4) in comparison with the Lemcotec (r/R~1.3). This could be explained by a deeper penetration of the jet in crossflow with the MSFI-V0, which would explain both the better mixing and the fuel positioning of the MSFI-V0 compared to the Lemcotec injection system.

In a similar way to the OH images analysis, the fluctuations in the kerosene fluorescence signal are reported in Fig. 95. Here, only the normalized standard deviation is shown, as it is less sensitive to temporal signal variation and in particular, the passage of liquid pockets. All three injection systems showed the same type of main injection standard deviation, taking the shape of a "2V", located around the region of high radial velocity fluctuations. With Lemcotec and MSFI-V0 injectors, the large pilot fuel variations are located in the same region as the large axial flow field variations, at the outside of the IRZ. With the MSFI-V2 injection system, the superposition is not as obvious, indicating the pilot fuel undergoes both axial and radial fluctuations. This is consistent with the previous observations made with the OH images, suggesting an axial pulse of the pilot spray, coupled with a change in flame topology from 2v to W. In addition, the fluctuations of the pilot spray with respect to the averaged field are significantly larger than with the other two injection systems, further highlighting the instabilities of the MSFI-V2 system shown on Fig. 74.

NO Fluorescence

NO production was assessed by two methods, the analysis of the PLIF signals and the exhaust gases by a gas sampling. In the PLIF images, the kerosene signal was masked so that its contribution was hidden. However, the absorption of the laser sheet by kerosene renders the quantitative interpretation of the NO fluorescence in the reaction zone difficult, thus reducing the quantitative analysis to areas where kerosene is not detected. This absorption of the laser sheet is the reason why the NO signal drops in the inner reaction zone, which is particularly visible with the MSFI-V0 and MSFI-V2 injectors. One would expect the Lemcotec to show the lowest NO signal in this region, as it contains the most liquid kerosene in the combustion chamber, but a significant amount of NO is actually measured. Although the absolute concentration cannot be obtained because the absorption of the laser sheet with kerosene is too high, the fact that so much NO is measured in the z/R=0.5-1.0 region with the Lemcotec injector is an indicator of significant NO production in the reaction zone. As the liquid fuel is present in the combustion chamber, combustion occurs partially in a diffusion regime, and partially in premixed regime, and thus, in a wide range of equivalence ratios. Therefore, combustion occurs partially near the stoichiometric regime, which creates the highest temperature zone and thus the highest NO production. Following the same argument, the NO production measured with the MSFI-V0 and MSFI-V2 injectors agrees with the fuel measurements presented in Fig. 73 and Fig. 93. The Lemcotec system showed the highest fuel penetration into the combustion chamber, and the MSFI-V2 the lowest. The same applies to their NO production, as shown in Fig. 98. The average NO concentration values were evaluated by two methods, as explained in Chapter 3. The reference flame method uses a scaling of the NO-PLIF signal of a reference flame, while the absorption method is based on the correction of the laser sheet absorption by assuming a remote axisymmetric field in the combustion chamber.

As the reference flame method does not consider the laser absorption, only the upper part of the images can be used for the evaluation of the average NO concentration. As the window film cooling penetrates the combustion chamber to about z/R~2.0, this limits the region of interest to about z/R = 2.3 -3.4, represented by a black box in Fig. 96. With the absorption method, a similar width is used, but the full height of the PLIF images can be used to extract an average NO value. This is represented by a red box in Fig. Finally, radial measurements of NO-PLIF were performed in parallel with OH-PLIF and Kerosene-PLIF measurements, and the results are shown in Fig. 97. As these measurements were performed at z/R=1.0, the interaction of the Kerosene-PLIF signals with those of NO-PLIF, as well as the absorption of the laser sheet, are far from negligible. As a result, these images show a very low amount of NO produced by the Lemcotec injector, and a higher concentration by the MSFI-V2 system. This is simply the result of the difference in fuel distribution, and in particular the proportion of liquid fuel injected into the combustion chamber, and the resulting difference in laser absorption. At present, these images are more a demonstration of the ability to perform radial PLIF measurements in such a complex configuration, than a productive measurement from which conclusions can be drawn about the NO production from these injection systems. In light of these findings, the superposition of the kerosene and NO signals is remarkable, despite the extreme distortion of the images, which is very encouraging for future applications, as the interaction between the kerosene and NO signals remains one of the main challenges for quantitative NO-PLIF measurements in the near-field analysis of the output of an injection system operating with a fluorescent liquid fuel. In conclusion of the present study, a synthesis of the different observations is proposed. Two characteristics are of primary importance for the aeronautical application and can differentiate the three injection systems studied : the operability and NOx production levels. The Lemcotec injector showed the highest resistance to extinction while the MSFI-V2 showed the highest extinction limit, which has determined the operating point for the whole study. At the same time, the MSFI-V2 injector produced the lowest amount of NO, while the NO production of the Lemcotec was 53% higher. The MSFI-V0 was in between the two in terms of both extinction resistance and NO production, with a 24% increase in comparison with the V0. As the Lemcotec and MSFI-V0 share the same geometry, with the only differences being the additive manufacturing and the subsequent modifications of the inner canals, this study shows the high sensitivity of the NO production in lean combustion. These findings, and in general all the results presented previously, are directly related to the atomization and the aerodynamic of these injectors.

As the operating point and the volume of the combustion chamber are unchanged, the temperature and residence time of the exhaust gases can be assumed to be identical between the three injection systems. This implies that the difference in NO levels produced is the result of the combustion regime. As the Lemcotec injector injects the largest amount of liquid fuel into the combustion chamber, the proportion of fuel burnt at or near the stoichiometry is higher than in the other two systems. The main difference with the MSFI-V0 illustrated by the tomography is in the main injection, which suggests that a particular attention should be given to the design of jets in crossflow, and to the additive manufacturing processes. The difference in NO production levels between the MSFI-V2 and the MSFI-V0 can be explained by two phenomena. Firstly, the difference in pilot and main flame interaction. In the case of the MSFI-V0 and Lemcotec injectors, the pilot spray and flame rejoin quickly the main reaction zone, which could locally increase the equivalence ratio and temperature. The latter interaction of the pilot C2 -Confidential and main flame in the case of the MSFI-V2 could help keep a lower local temperature and thus reduce the NO production. In addition, with the MSFI-V2 system, the kerosene PLIF, PDA, and tomography diagnostics indicated fewer droplets in the combustion chamber, and a shorter spray penetration from the main injection in comparison with the MSFI-V0, though the difference is less marked than with the Lemcotec. The main injection systems are identical between the two injectors and so is their fabrication process except for the separator. In the case of the MSFI-MSFI-V2, the separator between the main and pilot streams is larger and has been widened within the main channel. This can have a slight change in the flow inside the injector near the outlet, and so, on the liquid fuel that could have been projected on the inner wall, though the high-speed tomography didn't indicate a strong presence of fuel filming.

In terms of operability, it is clear that the high velocity fluctuations and intense PVC of the MSFI-V2 injection system have a negative impact on its extinction limits. The intense axial fluctuations of the pilot spray are likely to be responsible for the partial extinction shown by the OH-PLIF measurements, and to participate to the change of flame topology from 2V-shape to W-shape.

The separator also plays a role in this duality of flame shape. By definition, a higher swirl number favors the radial expansion of the pilot flow, but this is not what is observed on the MSFI-V2 injector. Instead, the pilot jets take a more axial direction than the MSFI-V0, showing that the pilot jets of the MSFI-V2 don't follow the natural expansion of a swirled flow. The dilution holes near the pilot outlet are likely to reduce, at least partially, the swirl number, thus encouraging the delayed interaction of the pilot and main flames. As the air flow repartition is quite similar between the MSFI-V0 and MSFI-V2 systems, this effect would also have been observed with the V0 if it was the sole responsible for the W shape flame topology, especially since the swirl number of the MSFI-V0 pilot is lower. Thus, it is likely that there exists a suction effect from the main flow on the pilot, pulling the pilot jets radially outwards. The wider separator of the MSFI-V2 system limits this effect, to a point where this combination of pilot swirl number, air flow repartition and separator width put the flame stability regime right in between those two flame topologies. Without any doubt, the precessing vortex core of the MSFI-V2 plays a role in this particular regime of oscillating pilot/main flame interaction. Its frequency has been measured on both the main and pilot spray and velocity fields, and a harmonic have been found right on top of the separator where double eddies would appear, likely with a W flame shape. This last chapter will try to extend the present study on more operating point and verify or qualify the partial conclusions drawn. The overall conclusions of the present work will then be presented.

Idle point -OH and Kerosene PLIF

Combined PLIF measurements on OH, Kerosene and NO were also carried out at the idle regime, when fuel is only injected through the pilot atomizer. At this operating point, a large amount of kerosene is present in the combustion chamber, which limits the PLIF measurements in several ways. Firstly, intense soot formation is observed with all three injection systems. As a result, NO-PLIF cannot give exploitable results, because the optical filters used did not completely suppress the optical signals above 450nm and the visible soot signal was significantly more intense than that obtained with NO-PLIF. On the other hand, as the fluorescence of the liquid kerosene was considerably higher than that emitted by the gas phase, the gain of the ICCD camera had to be set to a minimum to avoid the saturation of the sensor. Because of this limitation, a significant part of the kerosene gas phase signal could not be detected. Finally, some large droplets are also visible in some of the PLIF-OH images, indicating that the OH filter scheme was not fully optimized for this operating regime. Nevertheless, the combined use of PLIF-OH and PLIF-Kerosene provided interesting behavior, and the results are presented in Fig. 100 and Fig. 101.

The OH signals obtained on the Lemcotec and MSFI-V0 injection systems reveal V-shaped flames. On the Lemcotec injector, the flame front develops on both the inner and the outer part of the spray, with the outer flame front being fixed on the separator. This is not the case with the MSFI-V0 system where the outer branch of the flame front is not always present, as can be seen in the instantaneous field shown in Fig. 100. Also, the imprint of the fuel spray is clearly visible on the OH fields, as black pockets between the injector and the flame front.

C2 -Confidential

Right away on the OH images, the Lemcotec injector showed a deeper penetration of the spray. The kerosene images also showed this difference, but as mentioned above, as the sensitivity of the cameras was reduced, caution must be exercised in interpreting the kerosene distributions, as some of the kerosene vapor phase could not be detected. Nevertheless, as the deeper penetration of kerosene on the Lemcotec was observed in both the OH and kerosene images, it is very probable that this represents the truth and not a measurement artefact. Again, this result shows an effect of the additive manufacturing on the development of the liquid phase of the fuel, perhaps a positive effect on the atomization of the fuel filming inside the cup. Finally, the instantaneous OH field shows wrinkled flames, typical of diffusion regimes, which explains the intense soot production.

The OH distributions on the MSFI-V2 injector showed a flame topology close to an M-shape. The outer branches of the flame front are anchored on the separator, and the inner part is stabilized at about z/R~0.9. This can be correlated with the aerodynamic field shown in Fig. 65. With the pilot flow exiting the burner with an axial direction, the droplets are ejected parallel to the burner axis, but at this operating point, with all the fuel injected into the pilot stage, the mixture is too rich, and the flame cannot enter the injection system as shown in the reference point of Fig. 89.c). As suggested earlier, such a flame topology could be an indicator that the larger separator limits the suction effect of the main stream on the pilot stream, allowing the dilution holes to play a more important role with the MSFI-V2 than with the other two injection systems, pushing the droplets toward the burner axis. Again, the kerosene PLIF did not detect the entire gas phase, but the large pocket observed in the OH images is an indicator of a large amount of fuel in the combustion chamber. Since the flame is not stabilized within the injection system as is the case with the Lemcotec and MSFI-V0 injection systems, fuel evaporation and combustion occurs further downstream of the injector, resulting in a large amount of liquid and vapor fuel in the combustion chamber.

MSFI-V2

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Overall conclusions

A large set of experimental data on various scalar parameters has been measured and presented in the current study. The objective is here to summarize the different conclusions drawn from the analysis of these experimental results and to relate them to the different architectures of the injection systems studied.

On each operating point, the Lemcotec injector produces more NOx than the MSFI-V0 one. On the tomography results as well as the Kerosene-PLIF images, more fuel was detected inside the combustion chamber, and in particular more liquid fuel. As explained in the introduction of this study, fuel inhomogeneities must be avoided because rich pockets could burn near stoichiometry and produce high levels of NOx. The higher NOx produced by the Lemcotec injector compared to the MFSI-V0 injector is therefore probably driven by a different behavior in atomization and premixing. One particular geometric parameter that could be impacted by the manufacturing process that could cause such a difference in atomization is the geometry of the injection holes. As an extension of this study, special attention should be paid to the exact geometries of the shapes of these holes and their impact on atomization.

The flame and spray topology and overall dynamics of the MSFI-V2 injection system are more complex to explain. Firstly, at the reference operating point, it produces the lowest amount of NOx. On the other hand, the flame shows some instabilities and the extinction limit is larger than for the other two injection systems. Furthermore, at idle regime, almost twice as much NOx is produced compared to the MSFI-V0 injector. Moreover, by changing the pilot/main fuel split at the reference operating point, the MFSI-V2 injector pilot showed poor performances in terms of emissions. Two main differences between the MFSI-V0 and MFSI-V2 injection systems can explain these modifications in combustion: the separator width and the pilot airflow swirl. Determining the accurate impact of each of these differences remains an open challenge, as all the physical processes governing this type of combustion are closely interconnected. However, several of the following points can nevertheless be mentioned.

As explained in Chapter 1, the pilot injector used in the current study is based on a pressurized swirl atomization, with a partial projection on a wall to create a liquid film which is then atomized by a secondary air flow. Increasing the number of swirl within the pilot injector will affect the proportion of liquid sprayed onto the wall, as shown in [START_REF] Becker | Experimental investigation of spatial and temporal aspects of the liquid fuel placement in a swirl cup at elevated pressure[END_REF]. This will reduce the radial droplet homogeneity, but the impact on the evaporation rate isn't quantified. Secondly, it is also susceptible to modify the strength of the PVC created within the injector which will also have an impact on both the mixture and the proportion of fuel sprayed on the walls. In terms of operability, the precessing vortex core is likely to have a negative effect by locally creating rich and lean pockets. With the objective of operating at already a very lean mixture, creating even leaner pockets lead to local extinction of the flame, and should thus be avoided.

To do so, the swirl number of the pilot air flow should be diminished. This can be done by either reducing the angle of the nozzle guide vanes of both stages, or by keeping one high swirl and diminish the other one, creating after the mixing an intermediate swirl number. With this idea, increasing the swirl number of the first stage would increase the proportion of fuel onto the wall, and favorize the creation of a strong PVC right at the atomizer exit. The reduction of the swirl number of the second stage might not be able to reduce the impact of the PVC on the spray repartition though. Then, it could be preferable to keep a high swirl number for the second C2 -Confidential stage, with the goal of increasing the shear stress close to the pilot bowl for the airblast atomization, and a lower swirl number for the first stage to avoid the creation of a strong PVC.

The increase in separator size has an effect on the main and pilot injectors. As shown by the tomography, PDA, and Kerosene-PLIF results, the premixing process of the main injection was better with the MFSI-V2 injector compared to the MFSI-V0 one. Since the manufacturing process was identical on both injection systems, this premixing improvement was either due to an uncontrolled difference in the jet in the cross-flow injection holes or to the effect of the separator shape on the atomization of the fuel sprayed on the wall. In any case, with the same reasoning used to explain the difference in NOx production between the MFSI-V0 and Lemcotec injectors, the reduced amount of liquid fuel into the combustion chamber with the MFSI-V2 injector is without any doubt the reason for the NOx reduction. The other impact of the increased separator width is the lesser interaction between the pilot and main streams, as detailed previously and highlighted in Fig. 103. The delayed interaction between the main and pilot sprays and flames might be favorable to avoid rich reaction zones, but at the same time it also means that the air in the main stream may not participate in reducing the pilot flame equivalence ratio or bring burned gases to the main flame to stabilize the combustion. In order to keep an interaction between the pilot and main flow while still avoiding the creation of a rich area where the two sprays rejoin, it would be interesting to increase the penetration of the jet in crossflow, so that the main spray find itself positioned more radially outward. Also, reducing the mass air flow injected through the dilution holes might help keeping a strong interaction between the pilot and main flow. The keep the same equivalence ratio of the pilot injector, the mass air flow passing through the second stage of the pilot could be increased.

Future studies

In order to improve our understanding of these injection systems, various studies could be conducted in the near future. The air-fuel mixture is a key physical phenomenon that controls both NOx production and stability performance, the latter depending on two physical processes: the atomization and the aerodynamics. With regard to the atomization performances under reactive conditions at high-pressure, the absence of optical accesses inside the injection system remains an obvious, but critical limitation. To our knowledge, the only method available to date to observe the atomization process inside the injector is the X-ray tomography diagnostic, as reported by Sforzo et al [188]. Such a diagnostic requires a synchrotron particle accelerator, and is thus is extremely complex and expensive to carry out. Another way to extend our knowledge on the atomization process is the use of high-resolution numerical simulations with solvers such as Yales2 and/or AVBP, as presented by [188]. With the extensive experimental database developed in the PERCEVAL Industrial Chair, and in particular with the PDA measurements performed in the framework of this study, an optimization of such numerical codes is feasible, and constitutes an essential step towards detailed numerical simulations of atomization. The proportion of liquid splashed onto the walls, and the resulting atomization behavior, should be one of the main results to be investigated, as it depends highly on the case studied and cannot be estimated by correlations as it is the case with standard atomizers. The impact of the precessing vortex core on the mixing process is another key aspect to be investigated, as it can produce rich and poor pockets into the flow, leading to local NOx production and extinction, but it can also assist to the improvement of the overall mixing. Experimentally, this could be studied with PIV and high-speed tomography diagnostics, with a radial laser sheet located nearby the burner outlet. The main difficulty would be to manage the reflection of the laser beam on the metal walls of the injector and the combustion chamber, but a possible solution would be to cover them with a dark material. Compared to the kerosene radial PLIF performed in this study, the use of a high-power laser source with a high repetition rate and emitting in the near visible range would mitigate the absorption of the laser sheet and lead to a better signal to noise ratio.

To observe the potential impact of the PVC on flame stability, and the potential local quenching induced, the OH-PLIF diagnostic operating at high repetition rates could be used and combined with tomography/PIV in order to study the correlation between the dynamics of the flame and the aerodynamics as well as the transport of fuel droplets.

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 1 Strategies for cleaner combustion ....................................................................

2 2 - 3 -

 223 Rich combustor: the RQL strategy ....................................................................... 1.3.3 Lean Injection systems ........................................................................................ 1.3.4 Lean Staged Injection Systems ............................................................................ 1.4 Experimental tools for aircraft combustion ................................................................ 1.5 Objectives & Organization........................................................................................... Experimental Setup .......................................................................................2.1 Heron test rig .............................................................................................................. 2.2 PIV ............................................................................................................................... 2.3 OH-PLIF/kerosen-PLIF/NO-PLIF ................................................................................... 2.4 PDA .............................................................................................................................. Planar Laser-induced Fluorescence ................................................................

3 . 1 1 C2 -Confidential 1 -Strategies for cleaner combustion 1 -

 31111 Introduction to spectroscopy ...................................................................................... 3.1.1 Energy levels ........................................................................................................ 3.1.2 Atom & Diatomic molecule structure ................................................................. 3.1.3 Population of energy levels ................................................................................. 3.1.4 Spectral Line shapes ............................................................................................ 3.2 Laser Induced Fluorescence ........................................................................................ 3.2.1 LIF Modeling: Quasi-Two-level System ............................................................... 3.2.2 LIF Equation: Two-level Derivation ..................................................................... 3.2.3 Multi-level Fluorescence ..................................................................................... 3.3 OH laser-induced fluorescence ................................................................................... 3.4 Kerosene laser-induced fluorescence ......................................................................... 3.5 NO laser-induced fluorescence ................................................................................... 3.6 Bibliography ................................................................................................................ Strategies for cleaner combustion .................................................................... 1

4 26 1. 1

 4261 Lean Staged Injection Systems ............................................................................ 15 1.4 Experimental tools for aircraft combustion ................................................................ 24 1.5 Objectives & Organization........................................................................................... Climate change and pollutants emission 1.1.1 IPCC report on climate change

Fig. 3 :Fig. 4 :

 34 Fig. 3 : Example of pressure swirls atomizers designs and sprays. a) Left, from Lefebvre, schematics of pressure swirl injectors head. b-d) Different atomization regimes depending on the pressure drop imposed on the injector. Those ombroscopy images were taken during the preliminary phase of the PDA study carried out within this work.Increase of injection pressure

Fig. 6 :

 6 Fig.6: a) Top, Rolls Royce phase 5 combustor, from[START_REF] Mcguirk | The aerodynamic challenges of aeroengine gas-turbine combustion systems[END_REF]. b) Bottom Left, General Electric "LEC" RQL combustor concept, from[START_REF] Mongia | GE aviation low emissions combustion technology evolution[END_REF]. c) Bottom Right, Pratt & Whitney "TALON" RQL injector, from[START_REF] Mckinney | The Pratt & Whitney TALON X low emissions combustor: Revolutionary results with evolutionary technology[END_REF]

Fig. 7 :

 7 Fig. 7 : The « Lean-dome Dual Anular Combustor » developped by NASA/General Electric under the Energy Efficient Engine Program. Injectors are based on the Lean Direct Injection approach, with a swirled cup atomization in between two contra rotating air flows. From Mongia et al. [24]

Fig. 9 :

 9 Fig. 9 : Lean staged Injection systems, from CFD Research. a) Left, scheme of the system from the patent US 6,272,840. b) Right, the corresponding flame arrangement, from[START_REF] Nickolaus | Development of a Lean Direct Fuel Injector for low emission aero gas turbines[END_REF]. The pilot function can be made with an airblast configuration (left), or with a pressure swirled injector (right). This injection system uses separated pilot/main flames. The blue dotted line shows the « bifurcated pilot flow » that had a significant impact on the LBO

Fig. 10 :

 10 Fig. 10 : TAPS (Twin Annular Premixing Swirler), from US Patent 6,389,815, presented by Mongia in [29].

Fig. 13 :

 13 Fig. 13 : General Electric fourth and fifth generation TAPS injection system. a) Left, from the patent US 8,171,735 B2, an evolution of the pilot injection with a swirling air flow around the centerline and a filming pilot fuel injection. B) Right, from the patent US 6,453,660, a plasma assisted pilot injector. This injector was designed for natural gas, with the goal of dissociating it into other species, notably H2, to increase flame stability.

Fig. 14 :

 14 Fig. 14 : A handfull of lean staged injection system studied under the EU 7th Framework program

Fig. 15 :

 15 Fig. 15 : The Multi Staged Fuel Injector (MSFI) concept.

Fig. 16 :

 16 Fig. 16 : Heron test rig, photo & scheme

Fig. 17 :

 17 Fig. 17 : Sirah precision scan

Fig. 18 :

 18 Fig. 18 : Simulated and measured NO-LIF signal on a NO doped McKenna burner flame

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 3 Planar Laser-induced Fluorescence 3 -Planar Laser-induced Fluorescence ................................................................ 37

Fig. 20 :

 20 Fig. 20 : Fluorescence two-levels model

  is called the Stern-Volmer factor or fluorescence efficiency; it is generally much smaller than 1 when 𝐴 21 << 𝑄 21The other mode of operation considers large irradiance for the laser beam (i.e. situation of a strong field or saturation regime, 𝐼 𝜈 >> 𝐼 𝜈 𝑠𝑎𝑡 ). Stimulated emission and absorption mechanisms dominate the other physical mechanisms, giving for the fluorescence signal per unit of time and per unit of volume the following expression:

Fig. 21 :Fig. 22 :

 2122 Fig. 21 : Normalized NO and Kerosene fluorescence for an excitation at 225.12nm

Fig. 23 :

 23 Fig. 23 : McKenna flatflame burner

Fig. 24 :Fig. 25 :

 2425 Fig. 24 : NO-LIF signal in H2/Air/N2/O2 calibration flame

Fig. 26 .

 26 Fig. 26. Doppler effect: Left case, stationary transmitter and moving receiver. Right, moving transmitter and stationary receiver

𝜃 2 𝑒Fig. 27 .

 227 Fig. 27. Optical configuration of a Dual incident beam systems

  Fig. 28. Interaction of planar homogeneous electromagnetic waves. Top left and right: electric field of incident waves. Bottom left waves mixing. Bottom right, temporal average of the mixing waves, generation of interference fringes.

Fig. 29 :

 29 Fig. 29 : Phase shift arising from ray particle interaction and glare points/areas

Fig. 30 :

 30 Fig. 30 : Light path for different scattering orders

𝑆 1 (

 1 𝑝) (𝜗 𝑠 ) = ∑ 𝑎 𝑛 (𝑝)𝜋 𝑛 (𝜗 𝑠 ) + 𝑏 𝑛 (𝑝)𝜏 𝑛 (𝜗 𝑠 ) (𝜗 𝑠 ) = ∑ 𝑎 𝑛 (𝑝)𝜋 𝑛 (𝜗 𝑠 ) + 𝑏 𝑛 (𝑝)𝜏 𝑛 (𝜗 𝑠 ) ∞ 𝑛=1

Fig. 31 :

 31 Fig. 31 : Intensity of the first 10 th scattering orders for water droplet in air (m=1.33, 𝜆 𝑏 = 488𝑛𝑚, perpendicular polarization). a: 𝑑 𝑝 = 10µ𝑚; b: 𝑑 𝑝 = 100µ𝑚 (from)

Fig. 32 :

 32 Fig. 32 : Vector notations for incident and glare points, receiver, and center of the measurement volume

  𝐞 𝑏 . (-𝐞 𝑏𝑟 (𝑖) ) = sin 𝜏 𝑏𝑟 , 𝑎𝑛𝑑, 𝐞 𝑡,𝑏𝑟 . (-𝐞 𝑏𝑟 (𝑖) ) = sin 𝜏 𝑡,𝑏𝑟 ′(140)

Fig. 33 : 1 +

 331 Fig. 33 : Angles and vectors for a refracted ray passing through by a particle

Fig. 34 :

 34 Fig. 34 : Left, receivers positioning for the three detectors arrangement. Right, Phase plot and validation area

Fig. 35 :

 35 Fig. 35 : Top, FiberFlow receiving probe for three detector arrangement. (From Dantec Dynamic user guide). Bottom, Aperture plates (masks).

Fig. 37 :

 37 Fig. 37 : Optical setup. a) Left, from Yeh and Cummins [143]. b) Right, from Goldstein and Kreid[START_REF] Goldstein | Measurement of laminar flow development in a square duct using a laser-doppler flowmeter[END_REF]

Fig. 38 :

 38 Fig. 38 : From Yule et al. a) Left, Optical setup and calibration rotating disk. b) Right, LDA burst, original and mean envelopes, and noise level.

Fig. 39 :

 39 Fig. 39 : From Gréhan et al. a) Left, optical setup and position of the beam waist with respect to the particle. b) Right, diameter-phase relation in function of the beam waist position, computed with the GLMT

Fig. 40 :

 40 Fig. 40 : From Xu and Tropea. a) Left, Slit effect from a refracted ray of an incoming droplet, and the reflection of outgoing droplet. b) Middle, Slit effect from very close outgoing droplets. c) Slit effect from very close incoming droplets.

Fig. 41 :

 41 Fig. 41 : From Tropea et al. a) Left, Phase difference -diameter relations for PPDA and SPDA in the dualPDA configuration. b) Right, Volume distribution with and without the removal of trajectory effect related particles

Fig. 42 :

 42 Fig. 42 : From Damaschke et al. a) Left, relative error of the SPDA in function of the aspect ratio of deformed droplet. b) Right, same plot in the case of the PPDA component of a Dual PDA system

Fig. 43 :

 43 Fig. 43 : From Qiu et al. a) Left, Optical arrangement with CCD and Avalanche Photodiodes (APD). b) Right, Temporal evolution of the combined signal from reflection and refraction on the Photodiode array, from GLMT computation

Fig. 44 :

 44 Fig. 44 : From Qiu and Sommerfeld, burst signal from a 60µm droplet with the standard PDA system and the newly developped one. a) Left, case of a particule moving in the Y direction. b) Right, case of a particle moving in the x direction

Fig. 47 :

 47 Fig. 47 : From Carl et al. a) Left, Injection system. b) Right, optical paths and burner positioning

Fig. 48 :

 48 Fig. 48 : From Becker and Hassa SMD and flux measurements. a) Left, HI swirler. b) Right, LO swirler

Fig. 49 :

 49 Fig. 49 : From Freitag et al. a) Left, schematics of the injection system. b) Right, Evolution of the phase conversion factor in function of the off-axis angle

Fig. 50 :

 50 Fig. 50 : Shadowgraphy and PDA measurements on a pilot injector at atmospheric pressure. Top : spray visualization and height of PDA measurements. Middle and bottom left : Phase plots of the corresponding measurements at the center of the spray. Bottom right : PDF associated with the operating point C at the center of the spray.

Fig. 51 :

 51 Fig. 51 : Schematic view of the Heron test rig and injector positioning (Left). Schematic cross section of the NEWAC injector (Right)

Fig. 52 :Fig. 53 :

 5253 Fig. 52 : a) Left, Mesh used for PDA measurements. Blue points represent the mesh used at Z/D=0.04, red dots represent the mesh at Z/D=0.64. b) Right, 2D maps of Data rates

Fig. 54 :

 54 Fig. 54 : PDF and cumulative distribution associated with two individual measurements points. a) Left, Z=1 mm. b) Right, Z= 15mm.

Fig. 55 :

 55 Fig. 55 : Phase plot of punctual measurements near the highest data rates. a) Left, at Z/D=0.04. b) Right, at Z/D=0.64

Fig. 57 :

 57 Fig. 57 : Scatter plots Diameter -Velocity on the Y/D=0 line at different radial positions around the high velocity standard deviation area

Fig. 58 :

 58 Fig. 58 : Mean droplet size distribution and associated Rosin Rammler distributions. a) Left, at Z/D=0.04. b) Right, at Z/D=0.64

  Fig. 59 : PDA and LES comparison. Top, PDA and LES mean diameter comparison at Z/D=0.04. a-d) Left to right D4 to D1.

Fig. 62 :

 62 Fig. 62 : Comparison of radial profiles of PDA and LES. Top, mean diameter. a) Left, Radial profile on a X/R=0 line. b) Right, Radial profile on a 3π/4 line. Bottom, vertical velocity profile. c) Left, Radial profile on a X/R=0 line. b) Right, Radial profile on a 3π/4 line

Fig. 63 :

 63 Fig. 63 : Coarse and interpolated vertical velocity on a MSFI injector. a) Left, measurement on a coarse mesh. b) Right, interpolation of the coarse results on a refined mesh

Fig. 64 :

 64 Fig. 64 : Comparison of PDA configurations. a) Top left, mean diameter D10. b) Top right, vertical velocity Vy. c) Bottom left, Axial velocity Vz. d) Bottom right, example of droplet size distribution.

Fig. 65 :

 65 Fig. 65 : PIV Mean planar velocity fields. a) Left, Lemcotec. b) Middle, MSFI-V0. c) Right, MSFI-V2

Fig. 66 :

 66 Fig. 66 : Mean axial and radial velocity components and associated standard deviation.

Fig. 67 :

 67 Fig. 67 : Comparison of velocity magnitudes at different axial locations.

  69.c) the two jets interact earlier, which is more similar to the behavior observed in the upper part of the Lemcotec flow field shown in Fig. 69.a) for instance. The more horizontal development of the pilot jets altogether with the presence of the PVC right at the burner exit are the reason the IRZ is much narrower with the MSFI-V2 injection system.

Fig. 69 :

 69 Fig. 69 : Instantaneous velocity field represented with streamlines. a) Left, Lemcotec. b) Middle, MSFI-V0. c) Right, MSFI-V2. Grey dashed lines represent a possible Precessing Vortex Core (PVC)

  Fig. 70 : Single-sided amplitude spectrum from FFT of the temporal vorticity signal at various

  Fig. 71 : Single-sided amplitude spectrum from FFT of the temporal vorticity signal near the main jet on the

Fig. 72 :

 72 Fig. 72 : Average shear stress. The streamlines show the mean flow field, and the white line denote the IRZ. a) Left, Lemcotec. b) Middle, MSFI-V0. c) Right, MSFI-V2.

Fig. 73 :

 73 Fig. 73 : Mean droplet scattering and streamlines of mean velocity from PIV. a) Left, Lemcotec. b) Middle, MSFI-V0. c) Right, MSFI-V2

Fig. 75 :

 75 Fig. 75 : Signal amplitude spectrum from Mie Diffusion.

Fig. 76 :

 76 Fig. 76 : PDA results at Z/R=0.04. Left, MSFI-V0. Right) MSFI-V2. Top, D32. Middle) Data rate. Bottom) vertical velocity Vy. Red crosses give the location of the scatter plots presented further.

Fig. 77 :

 77 Fig. 77 : PDA results at Z/R=0.64. Left : MSFI-V0, Right : MSFI-V2. Top) D32. Middle) Vertical velocity Vy. Bottom) Axial velocity

Fig. 78 :Fig. 79 :

 7879 Fig. 78 : From PDA measurements, mean radial plots of data rates (top side), and D32 (bottom side). Left) MSFI-V0, in blue. Right) MSFI-V2, in red

Fig. 80 :

 80 Fig. 80 : Scatter plots Diameter -Velocity Vy at Z/R=0.04. Each point represents a valid droplet measurement. The black line gives the mean velocity in function of droplet diameter. Top panel, points in the main spray. Middle panel, points in the pilot spray. Bottom panel, points of specific interest. Left, MSFI-V0. Right, MSFI-V2.

  comparison between the average Mie scattering signals from tomography and PDA mean data rates is proposed. At detailed in the chapter 4, PDA data rates can be subject to numerous biases, and Freitag et al.[START_REF] Freitag | Measurement of initial conditions of a C2 -Confidential kerosene spray from a generic aeroengine injector at elevated pressure[END_REF] reported a non-negligible difference in spray positioning measured by Mie scattering and PDA. The normalized PDA mean data rates and average Mie scattering signals on the x=0 line are presented in Fig.81, and a really good agreement is found between the two diagnosis, despite the drop of PDA data rates in the upper part of the spray reported earlier. This comparison gives further confidence in the methodology used to apply PDA in such operating conditions.

Fig. 82 :

 82 Fig. 82 : Single-sided amplitude spectrum from FFT of the temporal vorticity signal at various probes location on the MSFI-V2 injector, at z/R=0.38.

Fig. 81 :

 81 Fig. 81 : Normalized mean PDA data rates and average Mie scattering signal on the x=0 Line. a) Left, MSFI-V0. b) Right, MSFI-V2

Fig. 83 :

 83 Fig. 83 : a) Left, NO mean PLIF Signal of the combustion chamber filled with a NO/Air mixture. B) Right, normalized mean PLIF profile.

Fig. 84 :Fig. 85 :Fig. 86 :

 848586 Fig. 84 : Distortion correction and images superimposition of axial PLIF. a) Top Left, Raw calibration image. b) Top Right, distortion corrected calibration image. c) Middle Left, Raw kerosene PLIF image. d) Middle Right, Distortion corrected kerosene PLIF image. e) Bottom Left, Raw NO PLIF image. f) Bottom Right, NO PLIF image with a mask to remove the kerosene fluorescence.

Fig. 87 :Fig. 88 :Fig. 89 :

 878889 Fig. 87 : Axial OH-PLIF for the three injection systems. a) Top Left, Lemcotec instantaneous OH PLIF. b) Top Right, Lemcotec mean OH PLIF c) Top Left, MSHI-V0 instantaneous OH PLIF. d) Top Right, MSFI-V0 mean OH PLIF. e) Bottom Left, MSHI-V2 instantaneous OH PLIF. f) Bottom Right, Lemcotec mean OH PLIF.

Fig. 91 :Fig. 92 :

 9192 Fig. 91 : OH PLIF and radial velocity fluctuations. The colored lines represent the isocontours of radial velocity standard deviation. The white line is the isocontour of null axial velocity and represent the IRZ.

Fig. 93 :Fig. 94 :Fig. 95 :

 939495 Fig. 93 : Superposition of axial OH and Kerosene PLIF. Left) Instantaneous fields. Right) Averaged fields

96 .

 96 Both methods lead to similar results, as shown in Fig.98.a). NO production was also measured with a Gas Exhaust Analyzer, and the measurements are compared with the PLIF signals in Fig.98.b). All three methods show consistent results, with Lemcotec being the highest NO generator with 530 ppm, MSFI-V2 the lowest with 396 ppm, and MSFI-V0 in the middle with 441 ppm. The respective EINO values extracted from the exhaust gas analyzer measurements are 2.59 g NO / kg fuel for Lemcotec, 1.69 g NO / kg fuel for MSFI-V2 and 2.1 g NO / kg fuel for the MSFI-V0 injector.

Fig. 96 :

 96 Fig. 96: Axial NO-PLIF. . Left Instantaneous, Right, average images. The red rectangle shows the area used for averaging the NO concentration with the absorption method. The black rectangle shows the area of averaging for the flame reference method.

Fig. 97 : 4

 974 Fig. 97 : Radial NO-PLIF. Left Instantaneous, Right, average images.

Fig. 98 :

 98 Fig. 98 : NOx emissions with the three injection systems. Left : PLIF absorption and flame reference methods. Right, PLIF average measured concentration vs measurement from gas exhaust analyzer.

Fig. 99 : 6 -

 996 Fig. 99 : Top : Mean OH and Kerosene PLIF. Bottom NO PLIF. Streamlines from PIV are superimposed and colored by velocity magnitude.

Fig. 100 :

 100 Fig. 100 : OH PLIF on the Idle regime. Left, mean images, right, instantaneous fields.

Fig. 103 :

 103 Fig. 103 : Schemes of the injection system and velocity magnitude from PIV measurements. The white line denotes the IRZ. Lef) MSFI-V0. Right) MSFI-V2

  Climate change and pollutants emission ...................................................................... 1.1.1 IPCC report on climate change .............................................................................. 1.1.2 NOx regulations ..................................................................................................... 1.1.3 Clean combustion for aircrafts: the PERCEVAL Chair ............................................ 1.2 NOx production mechanisms ........................................................................................ 1.2.1 Thermal Mechanism ..............................................................................................

	1.1

1.2.2 Prompt Mechanism ............................................................................................... 1.2.3 Nitrous Oxide Pathway .......................................................................................... 1.2.4 Pressure effect on NOx formation ........................................................................ 1.2.5 Effect of atomization on NOx formation ............................................................... 1.3 Injection systems for a cleaner combustion ............................................................... 1.3.1 Atomizers for aircraft combustors ...................................................................... 1.3.

  Climate change and pollutants emission ...................................................................... 1 1.1.1 IPCC report on climate change .............................................................................. 1 1.1.2 NOx regulations ..................................................................................................... 4 1.1.3 Clean combustion for aircrafts: the PERCEVAL Chair ............................................ 4 1.2 NOx production mechanisms ........................................................................................ 5 1.2.1 Thermal Mechanism .............................................................................................. 6 1.2.2

	1.1

Prompt Mechanism ............................................................................................... 7 1.2.3 Nitrous Oxide Pathway .......................................................................................... 7 1.2.4 Pressure effect on NOx formation ........................................................................ 8 1.2.5 Effect of atomization on NOx formation ............................................................... 8 1.3 Injection systems for a cleaner combustion ............................................................... 10 1.3.1 Atomizers for aircraft combustors ...................................................................... 10 1.3.2 Rich combustor: the RQL strategy ....................................................................... 12 1.3.3 Lean Injection systems ........................................................................................ 14 1.3.

2 -Experimental Setup ....................................................................................... 30

  PIV ............................................................................................................................... 32 2.3 OH-PLIF/kerosen-PLIF/NO-PLIF ................................................................................... 34 2.4 PDA .............................................................................................................................. 36

	2.1	Heron test rig .............................................................................................................. 30
	2.2
	This section summarizes the different optical setups used during the current activity, including
	Particle Image Velocimetry (PIV), Planar Laser Induced Fluorescence (PLIF) and Phase Doppler
	Anemometry (PDA).

  Energy levels ........................................................................................................ 38 3.1.2 Atom & Diatomic molecule structure ................................................................. 38 3.1.3 Population of energy levels ................................................................................. 40 3.1.4 Spectral Line shapes ............................................................................................ 41 3.2 Laser Induced Fluorescence ........................................................................................ 44 3.2.1 LIF Modeling: Quasi-Two-level System ............................................................... 44 3.2.2 LIF Equation: Two-level Derivation ..................................................................... 48 3.2.3 Multi-level Fluorescence ..................................................................................... 51 3.3 OH laser-induced fluorescence ................................................................................... 53 3.4 Kerosene laser-induced fluorescence ......................................................................... 55 3.5 NO laser-induced fluorescence ................................................................................... 57 3.6 Bibliography ................................................................................................................ 76

	3.1	Introduction to spectroscopy ...................................................................................... 38
	3.1.1

  𝑏 12 𝐼 𝜈 𝑁 1 + (𝑏 21 𝐼 𝜈 + 𝐴 21 + 𝑄 21 )𝑁 2

		(47)
	𝑑𝑁 2 𝑑𝑡	= -𝑏 12 𝐼

𝜈 𝑁 1 -(𝑏 21 𝐼 𝜈 + 𝐴 21 + 𝑄 21 )𝑁 1

[START_REF] Schuermans | A Detailed Analysis of Thermoacoustic Interaction Mechanisms in a turbulent premixed flame[END_REF]

  𝐼 𝜈 𝑏 12 𝐼 𝜈 + 𝑏 21 𝐼 𝜈 + 𝐴 21 + 𝑄 21

	C2 -Confidential
	This equation can be reformulated as
						(52)
	𝑆 𝐹 = 𝑁 1 0 𝑏 12 𝐼 𝜈 ( 𝐴 21 + 𝑄 21 𝐴 21	) ( 1 + 1 𝐼 𝜈 𝑠𝑎𝑡 𝐼 𝜈	)
	In this expression, 𝐼 𝜈 𝑠𝑎𝑡 is the saturation intensity defined as
	𝐼 𝜈 𝑠𝑎𝑡 = ( 𝐴 21 + 𝑄 21 𝑏 12 + 𝑏 21	) = (	𝐴 21 + 𝑄 21 𝑏 12 (1 + 𝑔 2 𝑔 1	)	)
						(50)
	The fluorescence signal per unit time and per unit volume emitted in the 4 steradians is
	expressed as follows
	𝑆 𝐹 = 𝐴 21 𝑁 2 =	𝑏 12 𝐼 𝜈 + 𝑏 21 𝐼 𝜈 + 𝐴 21 + 𝑄 21 𝑁 1 0 𝑏 12 𝐼 𝜈 𝐴 21	(51)

  𝑁 1 0 𝑏 12 𝐸𝐺(𝜈 𝐿 , 𝜈 12 , Δ𝜈 12 , Δ𝜈 𝐿 )𝑉 (

	𝐴 21 𝐴 21 + 𝑄 21	) 	4𝜋 	(56)

Table 1 .

 1 Summary of NO-LIF key properties for most common excitation/collection schemes in lean premixed high pressure conditions

	for maximum LIF	1645 K	2815 K	2815 K	3770 K	3770 K
	Signal var. 1700-2500K	± 6.8%	± 18%	± 18%	± 36%	± 36%
	Relative signal strength	75	11	24	2.5	1.2
	NO/(NO + O2 + CO2)	99	94	97	94	77

Table 2 .

 2 Summary of the flowrates used for the NO seeded flame calibrationFrom the NO fluorescence section, the fluorescence intensity collected is expressed as:

	H2 [slpm]	Air [slpm]	N2/NO [splm]	XNO [ppm]
	9.85	46.9	0.0	0
	9.85	46.9	0.063	10
	9.85	46.9	0.126	20
	9.85	46.9	0.253	40
	9.85	46.9	0.509	80
	9.85	46.9	0.767	120
	9.85	46.9	1.030	160
	9.85	46.9	1.290	200
	𝐼 𝑓𝑙𝑢𝑜 = ( 4𝜋 𝛺	) . ( 𝐴 𝑙𝑎𝑠 𝑉 𝑐𝑜𝑙	) . 𝐺 . 𝜂 𝑜𝑝𝑡 . (∫ 𝑠 𝑓𝑙𝑢𝑜 𝑠𝑖𝑚𝑢 . 𝑇(𝜆)𝑑𝜆) . 𝑛 𝑁𝑂 . 𝐸 𝑙𝑎𝑠𝑒𝑟	(64)
	𝛺 4𝜋 With (

  𝜙 1 -𝜙 2 = (𝜑 12 -𝜑 11 + 𝜑 21 -𝜑 22 ) + (𝛹 12 -𝛹 11 + 𝛹 21 -𝛹 22 )

	(97)

  ). 𝐫 0𝑝 + (𝑘 1 𝑧 𝑡1 -𝑘 2 𝑧 𝑡2 )

	+ (𝜓 02 -𝜓 01 ) + (𝜓 𝐺,2𝑟 (𝑖) -𝜓 𝐺,1𝑟 (𝑖) ) + (𝜓 𝑆,2𝑟 -𝜓 𝑆,1𝑟 )
	+ (𝐤 1 . 𝐫 1𝑟 (𝑖) -𝐤 2 . 𝐫 2𝑟 (𝑖) ) + (𝑘 1 𝑟 𝑝,1𝑟 (𝑔) -𝑘 2 𝑟 𝑝,2𝑟 (𝑔) )]

  𝜔 1 -𝜔 2 )𝑡 + (𝐤 1 -𝐤 2 ). 𝐫 0𝑝 + (𝑘 1 𝑧 𝑡1 -𝑘 2 𝑧 𝑡2 ) + (𝜓 02 -𝜓 01 )

	+ (𝜓 𝐺,2𝑟 (𝑖) -𝜓 𝐺,1𝑟 (𝑖) ) + (𝜓 𝑆,2𝑟 -𝜓 𝑆,1𝑟 ) + (𝐤 1 . 𝐫 1𝑟 (𝑖)
	-𝐤 2 . 𝐫 2𝑟 (𝑖) ) + (𝑘 1 𝑟 𝑝,1𝑟 (𝑔) -𝑘 2 𝑟 𝑝,2𝑟 (𝑔) )

  𝜓 𝑆,21 -𝜓 𝑆,11 ) -(𝜓 𝑆,22 -𝜓 𝑆,12 )

	+𝑘 𝑏 [𝑒 1 . (𝐫 11 (𝑖) -𝐫 12 (𝑖) ) -𝑒 2 (𝐫 12 (𝑖) -𝐫 22 (𝑖) )]
	+ 𝑘 𝑏 [(𝐫 21 (𝑔) -𝐫 11 (𝑔) ) . 𝐞 01 -𝑘 𝑏 (𝐫 22 (𝑔) -𝐫 12 (𝑔) ) . 𝐞 02 ]

Remerciements

C2 -Confidential

C2 -Confidential

1080K, while the widest span led to a maximum temperature of 750K at 125mm. Directly related to these results, the concentration of CO was highest for the smallest span distribution, with a peak located in the inner recirculation zone, while at the same time HC was the lowest of the three distributions, with two peaks located near the highest droplet concentration zones. The authors reported a shorter flame for the case of the widest span distribution, similarly to what have been reported by Breña de la Rosa [START_REF] Breña De La Rosa | The influence of fuel properties on drop-size distribution and combustion in an oil spray[END_REF] in 1986. Breña de la Rosa et al. studied the change in flame topology in function of the viscosity of the fuel, which changed directly the droplet distribution. They showed that for large droplets, the decrease in the overall evaporation rate could lead to a much less homogeneous flame with small diffusion flames breaking away from the main combustion zone and located around the large droplets. This scattered reaction zone would lead to a shorter flame as observed by an average image from droplet photographs. In 1998, Brandt et al. [START_REF] Brandt | An Experimental and Numerical Study of Kerosine Spray Evaporation in a Premix Duct for Gas Turbine Combustors at High Pressure[END_REF] studied the spray of a Lean Partial Premixed (LPP) injector under realistic turbine conditions. The inlet air had a temperature T3 = 750K, a velocity 𝑢 𝑎𝑖𝑟 = 120𝑚. 𝑠 -1 and the pressure in the chamber was fixed to 9 bar. A linear duct of 25x40mm cross section with an observable pathlength of 180mm was used to conduct optical measurements. Flat prefilming airblast atomizers were mounted on various axial position in the duct, while keeping the residence time low enough to avoid auto-ignition. SPDA measurements were carried out at an off-axis angle of 52°, leading to a ratio of refracted over reflected light of 180. The authors estimated the uncertainty due to optical alignment to be below 1%. Gradient of refractive index inside droplets were neglected, even though the kerosene is a multi-component fuel, and the refractive index has been evaluated at ambient temperature, 𝑛 = 1.46. Droplet heating within the air flow or the flame up to the boiling point would decrease this value. The authors estimated the error induced on the measured mean diameter to be 3.5% due to the increase in temperature, based on the work of Fitzky and Bohn [START_REF] Fitzky | Messung der komplexen Brechungsindizes von Heizöl EL und Kerosin in Abhängigkeit von Druck und Temperatur[END_REF] who gave a correlation between refractive index and temperature for jet A1 kerosene, reported by Rachner in [START_REF] Rachner | Die Stoffeigenschaften von Kerosin Jet A-1[END_REF] They estimated the droplet deformation with the Bond number, defined as the ratio between acceleration force and surface tension, given by:

𝑎 𝑝 is the droplet acceleration, 𝐷 𝑝 the droplet diameter, 𝜌 𝑝 the liquid density and 𝜎 the surface tension. From data from Clift et al. [START_REF] Clift | Bubbles, Drops, and Particles[END_REF], a Bond number ("Eotvos number" in the book of Clift et al.) of 2.9 was estimated for a droplet of 26µm in diameter, which led to an ellipsoidal particle of estimated aspect ratio of 0.7. By assuming PDA measurements randomly distributed between the minor and major axis of ellipsoids, the authors estimated that the mean diameters would be 7% too large. With the objective to evaluate evaporation rates, the measured mass fluxes were corrected with an estimation of the effective cross section of the measurement volume for each size class. This has been done by using the transit time as the burst length, though the authors noted that it was under evaluated due to the gaussian beam profile. To compensate this effect, each burst length was corrected by a factor depending on the scattering angle, slit aperture, and gaussian beam diameter. The latter corresponding to the width of the laser beams where the energy is larger to 1/e 2 , where e 2 is the maximum energy. The data rates were corrected following the approach of Edward and Marx [START_REF] Edwards | Analysis of the Ideal Phase-Doppler System: Limitations Imposed By the Single-Particle Constraint[END_REF] detailed in the previous section. Furthermore, the evaporation estimation was limited to areas with moderate droplet concentration. The authors estimated that the volume flux errors were between +10% and -14%. Measurements were carried out in several axis locations, from 30 to 150mm from the injection point. A maximum of 20000 measurements was recorded, and the maximum measurement time was set to 10 sec. Numerical simulations were carried out with a Lagrangian approach from an in-house code. Droplets are considered spherical and of homogeneous temperature, and no interaction between droplets was considered. The properties of the boundary layer of both gas and particles were estimated with the 1/3 rules, which is a weighted average of 1/3 from the infinite field and 2/3 from the droplet. Also, the kerosene was considered as a single-component. The determination of the inlet conditions for the Lagrangian simulation was of critical importance for the numerical/experimental comparison. The authors observed that at 30mm in the highpressure high-temperature regime, 77% of the initial spray volume was already evaporated, so that PDA distributions were not suited to be used directly as initial conditions. At room temperature, this evaporation process is practically non-existent and such use of PDA measurements could be made. Based on a correlation obtained from a previous work reported by the same authors, the experimental conditions at room temperature were modified, so PDA measurements could be carried out on a similar atomization than in the high-pressure hightemperature conditions. The SMD measured by such an atomizer in those conditions was estimated with: 𝑆𝑀𝐷 = 𝑙 0.3 . 𝜎 0.5 𝜌 𝑎𝑖𝑟 0.3 . 𝑢 𝑎𝑖𝑟 [START_REF] Qiu | Method of phase-Doppler anemometry free from the measurementvolume effect[END_REF] To produce a similar atomization at ambient temperature than in gas turbine conditions, the authors increased the pressure up to 3.4bar, and speed up the air flow up to 170m.s -1 . The air C2 -Confidential decrease with the SNR increase. This only impacted slightly the mean velocities, but increased the mean diameters significantly, as fewer small drop were validated. For the two lowest SNR values tested, -6 and -4 dB, the measured D32 and D10 varied from 14.7µm to 15.29µm, and from 9.17µm to 9.54µm respectively, which correspond to an increase of 4%. Following those results, the authors tested the variation of the effective measurement volume length in function of these parameters, by estimating it with the droplet velocity and the transit time. Interestingly, the maximum measurement volume size was reported for droplets between 20µm and 55µm independently of the laser energy, SNR or PMT voltage. However, for the highest energies, the effective measurement volume decreased for larger droplets, which the authors impeded to a saturation of the detection system. Another interesting result reported is the decrease of the validation rate with the increase of velocity, which couldn't be explained thoroughly but only by assuming a lower SNR. To estimate the actual data rate, a pure LDA configuration was used for only measuring droplet velocities. The highest data rates reported were around 150kHz, with was near the limit of the measurement system. The authors concluded that in PDA, the data rates are much lower than the actual rates, and that particular attention should be paid to the choice of validation parameters and laser energy, as this can add significant error to the mean diameters.

In 2000, Behrendt et al. [START_REF] Behrendt | Optical measurements of the reacting two phase flow in a realistic gas turbine combustor at elevated pressure[END_REF] applied PDA in a realistic gas turbine combustor at elevated pressure and reactive conditions. Schemes of the test facility and optical accesses are reported in Fig. 46. Measurements were carried out in front of the center of the pilot burner. The incident beams were transmitted with a 15° angle against the main flow direction, through the optical windows the closest to the pilot burner. The receiver was placed at the other side, with a scattering angle of 35° against the main flow direction. The refractive index of kerosene fuel was chosen at ambient temperature to be equal to 1.45. The test facility was operated at 6bar, two operating points were tested, at air temperature 𝑇 3 = 666𝐾 and 𝑇 3 = 820𝐾. The associated Main/Pilot fuel mass flow rates were respectively 4.17/2.42 g.s -1 and 6.86/2.03 g.s -1 .

The authors emphasized the additional difficulties of applying PDA under such harsh conditions, such as soot radiation interfering with the scattered light, beam steering effects, spray density and limited optical access. To obtain a sufficiently large overlap, the gaussian beam diameter was increased to 300µm, yet, a stochastic movement of the beams reduced the effective On these two locations, the scatterplots of measurements are symmetrical around the mean value of the velocity, indicating a lack of correlation between velocity and drop diameter. Scatter plots have a similar shape over almost all the measurements area, with a cloud of points quite symmetric around the velocity mean value. One area though shows a particular behavior, few millimeters outwards the external edge of the air swirler, where the highest standard deviation is registered, a transition of two droplet population is visible. For instance, Fig. 57 displays the scatter plots from four locations along Y/D=0 and ranging from X/D=0.75 to X/D=1.02., which correspond to the area between the outer edge of the vortex and the upper limit attainable by PDA measurements. The scatter plots velocity/diameter are similar at X/D=0.75 and X/D=1.02, with mean azimuthal velocities in the 20-30 m.s -1 range, and diameters ranging between 2 and 20µm. For X/D=0.85, corresponding to the dark red area on the velocity standard deviation distributions shown in Fig. 56, two droplet size populations with significant differences of velocity are observed, one with a mean azimuthal velocity around 20 m.s -1 , comparable to the one observed at the center of the flow, and another around 55 m.s -1 . At X/D=0.95, which is located at the outer limit of the high velocity standard deviation zone, the Z/D=0.04 Z/D=0.64

Flame Topology

The results of the OH-PLIF measurements are presented on Fig. 87 and Fig. 88 for the axial and radial configuration respectively. On the three injection systems, V-shaped flames were found, as shown by the mean images on Fig. 87.b-d-f. Pilot flames start within the injection systems bowls, and quickly interact with the main flame. On the Lemcotec and MSFI-V0 systems, this interaction occurs at a distance ~0.4 Z/R from the injector exit, while for the MSFI-V2 injector it happens slightly further away, at ~ 0.6 Z/R. Fig. 87.f, shows that the pilot flame of the MSFI-V2 exits the injector with more of an axial propagation, and then open up to interact with the main flame. This is also shown by the radial OH mean fields, see Fig. 88.b-d-f, where the reaction zone of the MSFI-V2 is wider than with the other two injection systems. In particular, the MSFI-V2 pilot flame is distinguishible as a circle of high OH intensity at a radius r/R~0.6, see Fig. 88.f, while this is not the case with the two other injection systems. This is in adequation with the PIV and Tomography results, see Fig. 73, that showed a similar trend, that is, a more axial flow field exiting the MSHI-V2 pilot stage and a delayed interaction with the main flow in comparison with the two other injection systems.

In terms of flame length, the Lemcotec shows a longer reaction zone, with some OH starting to recirculate in the IRZ. The MSFI-V2 flame seems to be the shortest, with a reaction zone contained in the Z/R=2.0 length, while the MSFI-V0 shows the detection of OH up to Z/D=2.5. Axial instantaneous images, see Fig. 87.a-c-e, reveal strongly wrinkled flames, with some significant disparities of OH intensity around the injector's axis, while the averaged fields show a quite homogeneous field. These spatial disparities in heat release may be linked to the presence of a PVC, that could create pockets of high and low kerosene concentration. This could be verified with a high speed OH PLIF, a coupled PIV/OH PLIF or the coupling of a high frequency pressure sensor with the OH PLIF as it was made in [187].

In the case of the MSFI-V2 injection system, those oscillations reached a point where two flame structures could be distinguished, with a partial extinction of the pilot flame with one of them.

A conditional averaging based on OH intensity probes is proposed on Fig. 89. Out of 2000 OH images, 78.1% of them showed a V shape flame similar to the one observed with the Lemcotec and MSFI-V0 injectors, see Fig. 89.a-b), 21.5% showed a W shaped with a highly delayed pilot/main flame interaction, Fig. 89.c-d), and 0.4% of them showed a W shape with a partial extinction of the pilot flame, that is, no OH found right above the pilot injection exit, Fig. 89.e).

The transition between the V and W shape can be explained by the apparition of the double eddy structure shown on Fig. 69. As the pilot jets are separated from the main ones, the fuel follows an axial direction, and so does the flame. NO production was measured at several points other than the reference point: idle, and a series of points where the fuel split between the main engine and the pilot is changed so that more fuel is injected by the pilot. At the four Pil20/Pil25/Pil30/Pil35 operating points, 20/25/30/35% of the fuel is injected by the pilot atomizer respectively.

For an idle regime, the MSFI-V0 injector emits the least amount of NOx, with 26.1 ppm measured, while the MSFI-V2 emits the highest amount of NOx with 46.3 ppm. The Lemcotec injector is in between the two systems at 35.5 ppm. This is consistent with the OH-PLIF and Kerosene-PLIF results, and the NO production pathways explained in Chapter 1. The Lemcotec injector had a deeper spray penetration, indicating less premixing and thus more of the fuel burned in the diffusion regime. Over the four operating points that change the fuel distribution, the Lemcotec injector always showed the greatest NOx production. Both MSFI injectors had an equivalent level of NOx at 20% of fuel in the pilot, but as this proportion increased, this trend changed, and at 35% of fuel in the pilot, the MSFI-V2 injector produced about 30% more NOx than the MSFI-V0 system.

The increase in NOx produced with the increase of the proportion of fuel passing through the pilot is to be expected. With a constant air distribution, this results in a richer reaction zone, favorable for NOx production. The interesting point here is that this increase follows a different trend between the three injection systems, as shown in Fig. 102. While the MSFI-V0 and Lemcotec injectors show an increase in NOx production of 65% and 85% respectively, the MSFI-V2 injector was much more sensitive and showed an increase of 170%.