Isabelle Flour

Bruno Audebert

Olivier Pouliquen

Lounes

Vladimir Paul

Clément Colas

Riccardo Milani

Germain Davy

Sarah Chatenet

Li Ma

Keywords: scientifiques, professionnelles, aussi humaines, j

Remerciement

Ces trois années de recherche est une très

Résumé

Cette étude rapporte le développement d'un modèle numérique basé sur les simulations à grandes échelles en vue de simuler des panaches de feu bien-ventilés et sous-ventilés et entièrement contrôlés par les forces de flottabilité. Un nouvel algorithme de résolution pour les d'écoulements à densité variable à faible nombre de Mach, précis au second ordre en temps et espace, a été développé et vérifié afin de décrire avec précision les processus instationnaires forts liés à la nature purement contrôlée par les forces de flottabilité des écoulements. Des modèles de sous-maille à l'état de l'art pour la turbulence, la combustion et le rayonnement ont été développés, incluant les modèles non-adiabatiques SLF (steady laminar flamelet) et FPV (flamelet/progress variable). Le RCFSK a été considéré comme référence pour la modélisation des propriétés radiatives des gaz et la performance de modèles plus simples, tels que des versions non-grises et grises du modèle à somme pondérée des gaz gris, a été évaluée. Le modèle numérique a été intensivement validé en simulant plusieurs panaches de feu non-réactifs et réactifs. Le modèle s'est avéré capable de reproduire avec une grande fidélité les structures dynamiques et radiatives de ces panaches dans des configurations ventilées et sous-ventilées sans ajuster aucun des paramètres du modèle. En particulier, le modèle FPV permet de traiter des conditions d'oxydant dilué en oxygène proches de l'extinction. Plusieurs problèmes de modélisation spécifiques ont également été étudiés au cours de ce travail. L'importance de capturer directement les instabilités laminaires thermo-convectives, qui se forment périodiquement à la base des panaches de feux et croissent pour générer les larges tourbillons régissant le processus de mélange, a été démontrée. La formation et la croissance de ces instabilités sont significativement affectées par la géométrie du brûleur qui doit être scrupuleusement cohérente avec les expériences pour obtenir une description précise de la dynamique du panache de feu. Les versions grises des modèles à somme pondérée de gaz gris doivent être évitées tandis que les versions non-grises de ce modèle peuvent être une alternative raisonnable pour la simulation de feux de nappe sans suie. Finalement, les interactions rayonnement-turbulence de sous maille ont un effet [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF][START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22 Radial profiles of mean temperature at different heights. The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF], except the radial profile at z a H:T m that is taken from Ref. [START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF]. . . . . . . . . . . . . . . . . . .

Radial profiles of rms value of temperature at different heights.

The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24 Radial profiles of mean axial velocity at different heights. The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Radial profiles of rms value of axial velocity at different heights.

The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF] The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF] and total heat feedback measured at 13 mm elevation above the fuel surface. The experimental data are taken from Ref. [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF], except those for the radiative feedback that are taken from Ref. [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF]. . . . . . . . . . . . . . . 112 6.30 Snapshots of temperature at different times for the baseline case.

The time between two snapshots is 0.06 s. Refs. [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] and [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF], respectively. 119 7.3 Effects of TRI closures on R . The experimental value for F1D is 0.19 [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF] Electricité de France (EDF), as an operator, is responsible for the safety of the nuclear power plants and defines, in agreement with the regulation, the means and the organisation implemented to ensure that its facilities do not present risks for the public and the environment.

Unwanted fires in nuclear plants is the most important risk for internal damage.

Besides the safety of its staff and the operation of its facilities, EDF has an economical interest, related to the loss of material and immobilization, to reduce the fire risk by ensuring a high level of safety without using oversized firefighting means. The need for demonstration of the efficiency of fire safety with numerical models, thanks to the improvement of knowledge and computational resources, is increasing. Addressing these issues, with their complex economic and environmental implications, requires a better understanding of the fundamental physics of fire spreading and the development of reliable methods for modelling and analysing fire safety systems.

The modelling tools developed by EDF R8D have been used for several years to answer questions relative to fire safety. Today, the two-zone code MAGIC is used as reference. It is based on the assumption of a stratification of the hot gases above the fresh gases and can provide rapid answers due to its low computational cost. Although this approach is efficient, it is not able to capture all the processes of the fire phenomena. In addition, it becomes less relevant in absence of stratification. 2 1.2. PHYSICAL PROCESSES GOVERNING A POOL FIRE The development of a large eddy simulation (LES) based CFD model of fire spread in nuclear plants is a natural way to circumvent these limitations. The present work is related to these developments. This model has to describe the fire dynamics and the related physics with fidelity. Before discussing the objectives of the present work, the next section will use the example of a pool fire to illustrate the main physical processes occurring during a fire and that has to be accurately modelled.

Physical processes governing a pool fire

Pool fires are natural canonical scenarios in fire safety science since they contain most of the coupled physical processes involved in fire problems, namely, buoyancy-controlled flows, buoyancy-induced turbulence, turbulent combustion, thermal radiation, soot generation, and burning rate. trates the main features which control the combustion of a pool fire. The condensed phase (liquid or solid) is transformed into gaseous fuels either by pyrolysis (thermal degradation of solids or heavy liquids) or by vaporization (for light liquids). The gasification processes occur due to the influence of the convective and radiative fluxes transferred from the flame to the condensed fuel surface and, the heat transfer mode controlling this process depends on the fire size: at small-scale the heat transfer is dominated by convection whereas for fires with diameter greater than about 0.2-0.3 m thermal radiation prevails [START_REF] De Ris | Fire radiation -a review[END_REF]. This highlights the importance of radiation in real-sized fire problems. Gaseous combustion products (hydrocarbon fuels, CO P , H P O, CO) and soot are responsible of the radiative heat flux from the flame. Gaseous species emit radiation in some discrete bands, whereas the radiation from soot Li MA CHAPTER 1. INTRODUCTION 3 1.2. PHYSICAL PROCESSES GOVERNING A POOL FIRE is continuous, covering the entire thermal spectrum. The contribution of soot depends widely on the combustible.

The flame depicted in Fig. 1.1 is a diffusion flame characteristic of unwanted fires. The word diffusion means that fuel and oxidant are not initially premixed. The flow evolves within the buoyancy-driven regime with the fuel injection velocity being significantly lower than that induced by the gravitational acceleration, g. In fire problems the Froude number, F r a u P inj =gL, is typically in the range IH T IH P [START_REF] Li | Combustion fundamentals of fire[END_REF]. In the previous expression, L is a characteristic length of the problem, the flame length or the diameter of the pan in the present example. The purely-buoyant nature of the flow leads to the formation and the growth of non-dissipative non-linear laminar instabilities near the edge of the pan that develops to become energy containing turbulent structures [START_REF] Tieszen | Chapter 2. Transport phenomena that affect heat transfer in fully turbulent fires[END_REF]. These structures develop periodically to form energy containing large-scale toroidal vortices that govern the flow pattern, the air entrainment as well as the mixing and combustion processes. As discussed in Ref. [START_REF] Tieszen | Toward the development of a LES-SGS closure model for buoyant plumes[END_REF][START_REF] Tieszen | Chapter 2. Transport phenomena that affect heat transfer in fully turbulent fires[END_REF], these instabilities are initially non-dissipative and the associated mixing cannot be captured by conventional subgrid-scale turbulence models that are dissipative in nature. The near field flow is thus rather complex. It exhibits a rapid transition from laminar to fully turbulent regime in the few first inlet diameters and a puffing motion characterized by a repetitive shedding of these coherent vortices [START_REF] Cetegen | Experiments on the periodic instability of buoyant plumes and pool fires[END_REF].

Fires in nuclear plants are characterized by an additional difficulty related to the high level of enclosure, leading to conditions for combustion generally different from those encountered in well-ventilated fires. Air feeding the flame becomes rapidly vitiated, which leads to partial and/or total extinction and then to incomplete combustion. Gaseous combustibles, carbon monoxide, and soot escape then from the fire and are released into the surroundings. This can lead to extremely dangerous hazards, especially if the gaseous combustibles mix with air and encounter points with a sufficiently high temperature to ignite this mixture. Idealized experiments of under-ventilated fire plumes were performed at the University of Maryland (UMD) [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF] an oxygen anchor is considered to suppress the lift-off, the flammable domain is extended to 0.130 for both fuels. Such extinction mechanism, characterized by the detachment of the base of the flame from the burner rim, its oscillation and eventually its extinction, as the oxygen is gradually diluted into a co-flowing oxidizer stream was also observed in laminar co-flow diffusion flames [START_REF] Bibliography Takahashi | Extinguishment mechanisms of coflow diffusion flames in a cup-burner apparatus[END_REF][START_REF] Guo | A numerical study on the effects of CO P /N P /Ar addition to air on liftoff of a laminar CH R /air diffusion flame[END_REF].

Objectives of the thesis

This thesis is a part of a research effort devoted to the development of a LESbased fire simulator including state-of-the-art submodels for the different physical processes involved in fire scenarios in nuclear plants. It focuses on gas phase processes with the main objective to develop a CFD model simulating with high fidelity non (or weakly) sooting well-ventilated and under-oxygenated fire plumes. The strategy adopted in these developments can be summarized as follows:

• As discussed previously, the near field of fire plumes is characterized by strongly unsteady processes related to the purely buoyant nature of the flow. The development and the implementation in Code_Saturne, the in-house code of EDF, of a new second-order Navier-Stokes solver for low-Mach variable-density flows will be then a natural prerequisite. Methods of manufactured solutions will be considered for the numerical verification step.

• The second step will focus on the implementation of variable-density subgrid-scale (SGS) stress tensor and scalar flux models. The dynamic Smagorinsky model for SGS stress tensor and advanced SGS scalar flux models, including the dynamic eddy diffusivity model, dynamic tensorial diffusivity models and gradient models. These models will be applied and assessed in LES of the 1 m helium plume investigated in Ref. [START_REF] O'hern | Experimental study of a turbulent buoyant helium plume[END_REF].

• The third step will consider the SGS combustion and radiation models. This latter is important to account for SGS turbulence-radiation interaction (TRI). They will be based on non-adiabatic steady flamelet [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF] to select the most appropriate choice.

• Based on these conclusions, the capability of the model will be extensively assessed by simulating several medium-scale and large-scale fire plumes of the literature, including the 30 cm methanol pool fires investigated experimentally by [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF], the 50 cm ethanol pool fires investigated experimentally by [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF], the 1 m methanol pool fire investigated experimentally by [START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF], and the line fires investigated experimentally at UMD [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. Beyond the validation exercise, each experimental configuration will lead to the investigation of a given process. Burner boundary conditions, which were found to be of significant importance to capture the fire plume dynamics during the course of this work, will be investigated on the 30 cm methanol pool fire. The importance of SGS TRI will be discussed by considering the 50 cm ethanol pool fire. The UMD turbulent line fires, that consider methane as fuel burning either in air or nitrogen-diluted atmospheres up to flame extinction, will serve to evaluate the limitation of the SLF and the capability of the FPV to capture the weakening of the flame while reducing the oxygen concentration in the oxidizer stream and the subsequent flame extinction.

• Finally, several radiative property models, including the rank-correlated full-spectrum-k-distribution model (RCFSK) [START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF], and grey and non-grey versions of the weighted-sum-of-grey-gases (WSGG) [START_REF] Modest | Radiative Heat Transfer[END_REF], have been implemented. Their modelling capability will be investigated in the 1 m methanol pool fire. This fire plume will be also considered to investigate the effects of the angular discretization of the radiative transfer equation solver on radiative outputs.

6 CHAPTER 1. INTRODUCTION Li MA 1.4. LITERATURE SURVEYS

Literature surveys

This section will discuss the current developments of LES-based fire simulator as well as the modelling of SGS combustion and radiation in the framework of fire modelling that are the core of this work.

Background

LES has become the standard for numerical modelling of turbulent fire over the last twenty years [START_REF] Branley | Large Eddy Simulation of a turbulent non-premixed flame[END_REF][START_REF] Ma | Numerical simulation of axi-symmetric fire plumes: accuracy and limitations[END_REF][START_REF] Desjardin | Modeling of conditional dissipation rate for flamelet models with application to large eddy simulation of fire plumes[END_REF][START_REF] Xi | Fire dynamics simulation of a turbulent buoyant flame using a mixturefraction-based combustion model[END_REF][START_REF] Cheung | A fully-coupled simulation of vortical structures in a large-scale buoyant pool fire[END_REF][START_REF] Wang | Large eddy simulation of fire plumes[END_REF][START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF]Maragkos et al., 2017;[START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF][START_REF] Maragkos | On the use of dynamic turbulence modelling in fire applications[END_REF]. In LES, the governing equations are spatially filtered [START_REF] Pope | Turbulent flows[END_REF][START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]; the large-scale dynamics (large turbulent eddies whose transient behaviour can be resolved numerically) are captured explicitly, while the effects of unresolved (subgrid) scales are modelled. LES is expected to be more accurate and general compared to Reynolds averaged Navier-Stokes (RANS) approaches, since at least some of the fluctuations (the largest, most energetic ones) are captured explicitly and only the (presumably) more universal small-scale dynamics require modelling. LES is also expected to capture phenomena that are difficult to accommodate in RANS, such as large-scale unsteadiness as observed in fire plumes [START_REF] Wang | Large eddy simulation of fire plumes[END_REF].

The filtered Navier-Stokes equations contain the unclosed SGS tensor stress and scalar fluxes. In LES-based fire simulators, the standard Samgorinsky and linear eddy diffusivity [START_REF] Pope | Turbulent flows[END_REF] models are usually used to close these terms, although dynamic versions of these models [START_REF] Moin | A dynamic subgridscale model for compressible turbulence and scalar transport[END_REF] were used to simulate fire plumes [START_REF] Maragkos | On the use of dynamic turbulence modelling in fire applications[END_REF]. Nevertheless, the linear eddy diffusivity model assumes a linear alignment with the large-scale scalar gradient which is inconsistent with the physics of turbulent convection and cannot correctly reflect the local geometric property of the SGS scalar flux vector at each time step [START_REF] Peng | On a subgrid-scale heat flux model for large eddy simulation of turbulent thermal flow[END_REF][START_REF] Higgins | The heat flux and the temperature gradient in the lower atmosphere[END_REF][START_REF] Wang | A general dynamic linear tensor-diffusivity subgrid-scale heat flux model for large-eddy simulation of turbulent thermal flows[END_REF]Wang et al., 2007a). To correct this behaviour, other approaches were considered in the literature, such as those based on tensorial forms of the diffusivity for the SGS scalar flux modelling [START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF] 2011).

In reacting radiating flows such as fires, the resulting set of filtered equations contains additional unclosed terms which are manifestations of turbulencechemistry interactions (TCI) and turbulence-radiation interactions (TRI) [START_REF] Poinsot | Theoretical and numerical combustion[END_REF][START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]. In particular, the LES of combustion processes introduces an important modelling issue since the combustion process occurs essentially at the smallest scales of the subfilter level, and has to be modelled entirely [START_REF] Pitsch | Large-eddy simulation of turbulent combustion[END_REF]. In addition, the importance of SGS TRI is not fully elucidated and has to be addressed (Liu et al., 2020a).

Combustion models and fire extinction

Current approaches

A significant part of the LES of fire scenarios reported to date used the eddy dissipation concept (EDC) proposed by [START_REF] Ertesvåg | The Eddy Dissipation Turbulence Energy Cascade Model[END_REF], to model the turbulence-combustion interaction [START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF]Maragkos et al., 2017;[START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF][START_REF] Maragkos | On the use of dynamic turbulence modelling in fire applications[END_REF]. In EDC, the reactions are assumed to occur only in the smallest eddies along the turbulence energy cascade, which are called fine structures. To account for detailed chemical kinetics, these fine structures are typically treated as a perfectly stirred reactors (PSRs) in which chemical reactions depend on the molecular mixing between the reactants. In fire research, the detailed kinetics is generally not retained and an infinitely fast chemistry single-step irreversible reaction is assumed. Due to non-linearity of the combustion process, the filtered reaction rate is modelled by introducing model constants. However, no universal values of these constants exist [START_REF] Panjwani | Turbulence combustion closure model based on the Eddy dissipation concept for large eddy simulation[END_REF][START_REF] Bösenhofer | The eddy dissipation Concept-Analysis of different fine structure treatments for classical combustion[END_REF]. Fire extinction models were also developed in the framework of EDC approach with simplified chemical kinetics. In the available fire extinction studies, critical flame temperature based models [START_REF] Hu | Towards large eddy simulations of flame extinction and carbon monoxide emission in compartment fires[END_REF][START_REF] Vaari | CFD simulations on extinction of co-flow diffusion flames[END_REF][START_REF] White | Modeling flame extinction and reignition in large eddy simulations with fast chemistry[END_REF] or Damköhler number based models (Narayanan et al., 2011;[START_REF] Li | Local extinction of diffusion flames in fires[END_REF][START_REF] Snegirev | Treatment of local extinction in CFD fire modeling[END_REF][START_REF] Snegirev | Perfectly stirred reactor model to evaluate extinction of diffusion flame[END_REF][START_REF] Vilfayeau | Numerical simulation of under-ventilated liquid-fueled compartment fires with flame extinction and thermally-driven fuel evaporation[END_REF][START_REF] Vilfayeau | Large eddy simulation of flame extinction in a turbulent line fire exposed to air-nitrogen co-flow[END_REF] are coupled with the EDC approach to perform LES of fire extinction and reignition problems. the UMD line fires [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. These studies considered a classical presumed FDF, parametrized by the filtered mixture fraction and SGS mixture fraction variance, to characterize the mixture fraction statistics at the subgrid scale.

Flamelet models

Flamelet models coupled to presumed FDF represent a relevant alternative to the EDC models usually used in fire simulations and their development will be an important part of this work.

In the flamelet modelling approaches, the turbulent reaction zone is represented by a collection of canonical laminar flames corresponding usually to 1D counterflow diffusion flames [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF]. These simple 1D flames can be simulated with detailed chemical kinetics and molecular transport properties for a set of parameters that characterize the effects of the flow on the flamelet structure.

Both steady and unsteady flamelet approaches were developed [START_REF] Pitsch | A consistent flamelet formulation for nonpremixed combustion considering differential diffusion effects[END_REF][START_REF] Poinsot | Theoretical and numerical combustion[END_REF], the steady approaches assuming that the flame structure immediately adjusts to local flow field conditions. Under this assumption, the SLF model provides state relationships for temperature, species mass fractions and, more generally all thermochemical variables as a function of two parameters: the mixture fraction and scalar dissipation rate [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF][START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. This latter parameter measures the degree of departure from the equilibrium state and can be interpreted as the inverse of a characteristic diffusion time or as a diffusivity in the mixture fraction space [START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. The main limitation of the SLF model in view of under-ventilated fire applications is that, although it provides state relationship for the stable burning flamelet between equilibrium and ex- et al., 2005;Ihme and Pitsch, 2008).

The flamelet progress variable (FPV) offers an attractive alternative to the SLF for modelling the fire extinction and reignition phenomena. As the SLF, the FPV is based on the steady assumption and is capable of incorporating detailed chemistry kinetics. The main differences is that a flamelet parameter based on a reactive scalar is used rather than the scalar dissipation rate in order to get access to both stable and unstable burning flamelets. This enables, in principle, prediction of extinction and reignition effects [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Ihme | Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach[END_REF]Ihme and Pitsch, 2008;[START_REF] Shunn | Large-eddy simulation of combustion sysytems with convective heat-loss[END_REF].

As discussed previously, radiative loss plays an important role in fire problems and its effects on the flamelet structure have to be taken into account. This is generally achieved by introducing an additional parameter characterizing the enthalpy loss. This leads to the non-adiabatic SLF or FPV models (Ihme and Pitsch, 2008;[START_REF] Carbonell | Flamelet mathematical models for non-premixed laminar combustion[END_REF].

In LES of non-premixed combustion, many important mixing and reaction processes occur at scales that are unresolved on the computational grid. As a consequence, the SGS fluctuations of the flamelet parameters and their statistical distribution need to be considered. In practice, once state relationships have been determined in terms of the reduced set of parameters, filtered thermochemical scalars are obtained using a statistical averaging based on presumed FDF to account for subgrid fluctuations of these parameters [START_REF] Poinsot | Theoretical and numerical combustion[END_REF][START_REF] Echekki | Turbulent Combustion Modeling. Advances, New Trends and Perspectives[END_REF].

In particular, SGS fluctuations of mixture fraction are characterized by subgrid mixture fraction variance and its dissipation rate. The subgrid variance quantifies the level of unmixedness at the subgrid scales and the filtered scalar dissipation rate describes the rate at which scalars relax towards a fully mixed state. The modelling of both quantities has received significant attention over the last years and several models using different assumptions can be found in the literature. [START_REF] Cook | A subgrid model for equilibrium chemistry in turbulent flows[END_REF] proposed a model based on the scale similarity assumption for the subgrid variance. [START_REF] Bibliography Jiménez | A priori testing of subgrid models for chemically reacting non-premixed turbulent shear flows[END_REF] showed that the model constant depends on the exponent of the scalar spectrum function in the large Reynolds number limit. [START_REF] Cook | Determination of the constant coefficient in scale similarity models of turbulence[END_REF] [START_REF] Balarac | Development of a dynamic model for the subfilter scalar variance using the concept of optimal estimators[END_REF] modified the dynamic procedure of [START_REF] Pierce | Dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar[END_REF] to include certain leading-order terms in the Taylor series expansion of the left hand side of the dynamic closure.

Although being computationally efficient, it was reported that the dynamic local equilibrium-based models produce inaccurate estimation of scalar mixing and hence scalar dissipation rate [START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF][START_REF] Kaul | A posteriori analysis of numerical errors in subfilter scalar variance modeling for large eddy simulation[END_REF][START_REF] Knudsen | Modeling scalar dissipation and scalar variance in large eddy simulation: Algebraic and transport equation closures[END_REF][START_REF] Kaul | Large eddy simulation of a lifted ethylene flame using a dynamic nonequilibrium model for subfilter scalar variance and dissipation rate[END_REF].

An alternative to equilibrium models is to consider a transport equation for the subgrid scalar variance. This approach, initiated by [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF], will be referred to as the VTE model hereafter. [START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF], Kaul and[START_REF] Kaul | A posteriori analysis of numerical errors in subfilter scalar variance modeling for large eddy simulation[END_REF][START_REF] Kaul | Large eddy simulation of a lifted ethylene flame using a dynamic nonequilibrium model for subfilter scalar variance and dissipation rate[END_REF] proposed to solve the transport equation for the filtered second-moment of mixture fraction, referred to as the STE model hereafter, instead of directly solving the VTE. After the STE is solved, the subgrid mixture fraction variance is then computed by subtracting the square of filtered mixture fraction from the filtered second-moment of mixture fraction. The two models are identical at the level of their continuous equations, but give different results when discretized. [START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF] claimed that the calculation of production term of VTE introduces discrepancies due to the chain-rule of product operation at the discrete level, especially in high Reynolds number flows. The STE model does not consider this term and thus is not affected by this discrete chain-rule caused error. They also mentioned that another advantage of STE is to recover the maximum subgrid variance in the case where dissipation is negligible. Based on these reasons, they argued that the use of the STE model is more accurate than the VTE model. [START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF] and [START_REF] Jain | On the non-equilibrium models for subfilter scalar variance in large eddy simulation of turbulent mixing and combustion[END_REF] also addressed the differences between both models at the discrete level. Both two models require modelling for the scalar dissipation rate. Closure may be achieved through algebraic expressions using a turbulent time-scale [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF][START_REF] Knudsen | Modeling scalar dissipation and scalar variance in large eddy simulation: Algebraic and transport equation closures[END_REF]. Alternatively, closure can also be performed by solving a transport equation for the filtered square gradient of mixture fraction [START_REF] Knudsen | Modeling scalar dissipation and scalar variance in large eddy simulation: Algebraic and transport equation closures[END_REF].

Studies of these models for SGS scalar variance and dissipation rate were limited to momentum-driven turbulent jet flames and investigations of these models on purely buoyant fire plumes were not reported to date.

Radiation heat transfer

Experimental studies evidenced that radiation is the dominant mode of heat transfer to fuel surface for scales larger than about 0.2-0.3 m (De Ris, 1979).

More quantitatively, [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] reported that radiation contributes to the total heat feedback for 55, 80, and 967 in the case of 0.3 m diameter methanol, heptane and toluene pool fires, respectively. This contribution generally increases with the fire size and the fuel sooting propensity [START_REF] Ditch | Pool fires-An empirical correlation[END_REF].

Consequently, the development of LES-based fire simulators over the last 20 years raised naturally the issue of the radiation modelling for fire applications [START_REF] Xi | Fire dynamics simulation of a turbulent buoyant flame using a mixturefraction-based combustion model[END_REF][START_REF] Wang | Large eddy simulation of fire plumes[END_REF][START_REF] Chen | Large eddy simulation of a medium-scale methanol pool fire using the extended eddy dissipation concept[END_REF]Chen et al., 2014a;[START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF]Maragkos et al., 2017;Maragkos and Merci, 2017;[START_REF] Snegirev | On soot and radiation modeling in buoyant turbulent diffusion flames[END_REF][START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF][START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF][START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF]. At the same time, the radiative heat transfer community reported significant progresses in the modelling of radiation in turbulent flames [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]. This issue can be divided into three separate sub-problems, namely the RTE solver, the spectral dependence of radiating (RTE) solver mainly due to their compatibility with the discretization of the fluid dynamics equations, ease to code, fairly good accuracy, and often acceptable computational efficiency [START_REF] Coelho | Advances in the discrete ordinates and finite volume methods for the solution of radiative heat transfer problems in participating media[END_REF]. Moreover, Monte Carlo techniques were also used in some recent simulations of flames (Wang et al., 2008a;[START_REF] Gupta | Turbulence-radiation interactions in large-eddy simulations of luminous and nonluminous nonpremixed flames[END_REF][START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a Monte Carlo method coupled to large eddy simulation[END_REF][START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF]. Secondly, the absorption coefficient of the radiating combustion gases is strongly dependent on the wavenumber. The modelling of gas radiative properties has benefited from the development of accurate high-temperature high-resolution spectroscopic databases such as HITEMP 2010 [START_REF] Rothman | HITEMP: the high-temperature molecular spectroscopic database[END_REF]. Line-by-line (LBL) has then emerged as the reference for decoupled radiative heat transfer calculations and was combined to Monte Carlo techniques for coupled simulations (Wang et al., 2008a). However, LBL/Monte Carlo calculations are limited to academic configurations. Alternative global gas property models were developed based on the k-distribution concept, such as the full-spectrum correlatedk (FSCK) or the spectral line based weighted-sum-of-grey-gases (SLW). These models were found to provide a good compromise between accuracy and computational efficiency and to be adequate for engineering applications [START_REF] Denison | A spectral line based weighted-sumof-gray-gases model for arbitrary RTE solvers[END_REF]Denison and Webb, 1995c;Denison and Webb, 1995a;[START_REF] Denison | The spectral-line weighted-sum-ofgray-gases model for H2O/CO2 mixtures[END_REF][START_REF] Modest | The full spectrum correlated-k distribution for thermal radiation from molecular gas particulate mixtures[END_REF][START_REF] Cai | Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database[END_REF][START_REF] Solovjov | The rank correlated SLW model of gas radiation in non-uniform media[END_REF][START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF]. Although having similar features with the WSGG model, these methods do not suffer from its limitation, namely its restriction to spatially constant absorption coefficients [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]. FSCK/SLW models can be also applied to mixtures of gases and nongrey soot [START_REF] Modest | Assembly full spectrum k-distribution from a narrow band database: effects of mixing gases, gases and non-gray absorpting particles and non-gray scatters in non-gray enclosures[END_REF]. In addition, the development of look-up tables has significantly improved their computational efficiency [START_REF] Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2 and CO at variable temperature, mole fraction, and total pressure[END_REF]Wang et al., 2016c;Wang et al., 2016a;[START_REF] Wang | Full-spectrum kdistribution look-up table for nonhomogeneous gas-soot mixtures[END_REF][START_REF] Wang | Efficient full-spectrum correlated-k -distribution look-up table[END_REF][START_REF] Wang | Full-spectrum correlated-kdistribution look-up table for radiative transfer in nonhomogeneous participating media with gas-particle mixtures[END_REF]. Finally, TRI results from the highly non-linear coupling between turbulent fluctuations in temperature, composition, and radiative intensity [START_REF] Coelho | Numerical simulation of the interaction between turbulence and radiation in reactive flows[END_REF]. Accounting for TRI requires the modelling of two terms, namely absorption TRI and emission TRI. In LES, the contribution of turbulent fluctuations to TRI is decomposed into resolved-scale fluctuations and SGS fluctuations and only these latter require modelling. A consensus was reached that the SGS absorption TRI is negligible in both luminous and nonluminous flames [START_REF] Gupta | Turbulence-radiation interactions in large-eddy simulations of luminous and nonluminous nonpremixed flames[END_REF][START_REF] Coelho | Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows[END_REF][START_REF] Consalvi | On the modeling of the filtered radiative transfer equation in large eddy simulations of lab-scale sooting turbulent diffusion flames[END_REF]. On the other hand, the importance of SGS emission TRI was found to depend on [START_REF] Coelho | Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows[END_REF][START_REF] Consalvi | On the modeling of the filtered radiative transfer equation in large eddy simulations of lab-scale sooting turbulent diffusion flames[END_REF] and its modelling is strongly dependent on the turbulent combustion model. An appropriate way to model SGS emission TRI is to consider FDF approaches [START_REF] Gupta | Turbulence-radiation interactions in large-eddy simulations of luminous and nonluminous nonpremixed flames[END_REF][START_REF] Miranda | Study of turbulence-radiation interactions in large-eddy simulation of scaled Sandia flame D[END_REF][START_REF] Miranda | Large-eddy simulation of a bluff-body stabilised nonpremixed flame with radiation heat transfer[END_REF].

One of the objectives of the fire simulators is to forecast the growth of fires at the scale of industrial fires. This introduces considerable challenges for the modelling of the physical processes involved and compromises between accuracy and computational resources have to be found. Consequently, less sophisticated radiation models were generally considered in most of the LES of fire plumes reported to date. These studies have mainly considered DOM and FVM as RTE solver [START_REF] Xi | Fire dynamics simulation of a turbulent buoyant flame using a mixturefraction-based combustion model[END_REF][START_REF] Wang | Large eddy simulation of fire plumes[END_REF][START_REF] Chen | Large eddy simulation of a medium-scale methanol pool fire using the extended eddy dissipation concept[END_REF]Chen et al., 2014a;[START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF]Maragkos et al., 2017;Maragkos and Merci, 2017;[START_REF] Snegirev | On soot and radiation modeling in buoyant turbulent diffusion flames[END_REF][START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF][START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF]. Since the computational cost related to the solution of the RTE is relatively high, simplified radiative property models, assuming either an optically thin medium with the emission specified to reproduce the measured radiant fraction [START_REF] Xi | Fire dynamics simulation of a turbulent buoyant flame using a mixturefraction-based combustion model[END_REF][START_REF] Wang | Large eddy simulation of fire plumes[END_REF]Maragkos and Merci, 2017) or a grey medium [START_REF] Chen | Large eddy simulation of a medium-scale methanol pool fire using the extended eddy dissipation concept[END_REF]Chen et al., 2014a;[START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF][START_REF] Snegirev | On soot and radiation modeling in buoyant turbulent diffusion flames[END_REF][START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF], were generally retained.

Nevertheless, non-grey radiative property models, based on non-grey versions of the WSGG or box models [START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF], were also applied in LES of medium-scale methanol and ethanol pool fires. Concerning the modelling of SGS TRI, several works [START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF][START_REF] Snegirev | On soot and radiation modeling in buoyant turbulent diffusion flames[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF] have extended the moment-based closure, proposed by [START_REF] Snegirev | Statistical modeling of thermal radiation transfer in buoyant turbulent diffusion flames[END_REF] for RANS, to model SGS emission TRI. This approach consists in approximating the filtered emission term by a Taylor series expansion and retaining only the lower-order moments. Nevertheless, momentbased methods introduce empirical adjustable coefficients that are generally tuned to match the experimental data [START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF][START_REF] Snegirev | On soot and radiation modeling in buoyant turbulent diffusion flames[END_REF][START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF]. [START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF] presented DNS of a 7.1 cm heptane pool fire using a very fine description of radiative heat transfer. In their work,

14 CHAPTER 1. INTRODUCTION Li MA
1.5. ORGANISATION OF THE MANUSCRIPT radiation was modelled using the LBL/Photon Monte Carlo proposed by Wang et al. (2008a), and the contributions of CO P , H P O, CO, CH R , C P H R and soot were taken into account.

Organisation of the manuscript

The present manuscript is organised as follow.

• Chapter 2 will be devoted to the description of the mathematical model.

In particular, a special emphasis will be paid on the formulation of the SLF and the FPV models and on the description of the gas radiative property models used in the present study.

• Chapter 3 will present the LES filtering governing equations. The modelling procedures considered in the present study for the unclosed SGS stress and scalar flux will be described. The presumed FDF approach to close SGS combustion and TRI will be also discussed in details.

• Chapter 4 will describe in details the development and the verification of a new second-order Navier-Stokes solver for low-Mach variable-density flows.

• Chapter 5 will present LES of a non-reactive 1 m helium plume and predictions will be compared with measurements. The objective of this

Chapter will be twofold: first to provide a further validation of the numerical algorithm developed in Chapter 4 and, second, to evaluate the capability of SGS stress tensor and scalar flux models to describe the buoyant mixing characteristic of fire plume. The main interest of considering a non-reactive helium plume for this latter task is that the mixing process is similar to that observed in fire plumes while avoiding difficulties related to the combustion and radiation processes.

• LES of well-documented well-ventilated medium-scale methane and methanol fire plume will be reported in Chapter 6. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. This latter case will be also considered to investigate the effects of the burner boundary conditions on plume dynamics.

• Chapter 7 will focus on radiative heat transfer. The important of SGS TRI will be discussed by considering LES of well-ventilated synthetic ethanol pool fires of different diameters generated by scaling up the 50 cm pool fire investigated experimentally by [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF]. On the other hand, the gas radiative property models will be assessed from LES of the well-ventilated 1 m methanol pool fire investigated experimentally by [START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF].

• Chapter 8 will investigate the modelling of under-ventilated fire plumes by considering the UMD line flames under oxidizer-dilution quenching conditions [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. This configuration will serve to assess the capability of the FPV model to describe the weakening of the flame and its subsequent extinction as the oxygen concentration in the oxidizer stream is progressively reduced.

• Finally, conclusions drawn from the present study are presented and perspectives for future works and developments will be outlined in Chapter 9.

1.6 Publications in the context of this thesis Mass: @ @t C @u i @x i a H

(2.1) Momentum:

@u j @t C @u i u j @x i a @p @x j C @ ij @x i C @ I Ag j

(2.2) Species:

@Y @t C @u i Y @x i a @ @x i D @Y @x i ! C
! ; a I; :::; N s

(2.3) Energy: @h @t C @u i h @x i a @q i @x i r • q HH R (2.4)
Equation of state:

p t a Ns X aI Y M RT (2.5) 2.2. COMBUSTION MODELLING
where is the local density and u i a @u; v; wA is the flow velocity, p, p t are the hydrodynamic and thermodynamic pressure, respectively. @ ij A i;jaI;Q is the viscous stress defined as:

ij a @u i @x j C @u j @x i ! C P Q ij @u k @x k (2.6)
with the dynamic viscosity. @D A aI;Ns are the species diffusion coefficients, N s is the number of species and ! are the species reaction rates, which verify P Ns aI ! a H. h is the total enthalpy defined as:

h a Ns X aI Y h @T A a Ns X aI Y h H f; C Z T T0 Cp @T 0 AdT 0 ! (2.7)
where h H f; is the formation enthalpy at the reference temperature, T I , and Cp the heat capacity at constant pressure of species , @q i A iaI;Q the heat flux modelled by the Fourier law:

q i a @T @x i (2.8)
where is the heat conductivity, and r • q HH R is the radiative source term. T , R and M are the temperature, the universal gas constant and the molecular weight of the th species, respectively. Note that the system (2.1)-(2.5) can be transformed into a reduced dimension system in which scalar transport equations are solved for three quantities: the mixture fraction, Z, scalar dissipation, , or reaction progress variable, c, and total enthalpy h. All other thermochemical quantities are obtained from a property database as a function of these scalars.

Combustion modelling

The present study will consider flamelet approaches to model combustion of non-premixed flames. First, the mixture fraction is introduced. Second, the flamelet equations are described. Third, the SLF and FPV are presented. Finally, the incorporation of radiative loss into the flamelet models are discussed.

Li MA CHAPTER 2. MATHEMATICAL FORMULATION

Mixture fraction

Non-premixed flames are a specific class of combustion problems where fuel and oxidizer are initially separated and have to mix prior to combustion [START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. A proper description of the mixing process is then a natural prerequisite to the modelling of these flames.

In general, the structure of non-premixed flames can be described by means of a conserved scalar. This variable is the mixture fraction, Z, [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF], defined between 0 and 1. Z is a conserved scalar that accounts for the level of mixing between the oxidizer and the fuel and changes because of diffusion and convection, but not due to the chemical process. In the present study, following the formalism introduced by [START_REF] Pitsch | A consistent flamelet formulation for nonpremixed combustion considering differential diffusion effects[END_REF], the mixture fraction is defined from its transport equation:

@Z @t C @u i Z @x i a @ @x i D @Z @x i ! (2.9)
where D is the diffusion coefficient for mixture fraction. [START_REF] Pitsch | A consistent flamelet formulation for nonpremixed combustion considering differential diffusion effects[END_REF] recommend to use the thermal diffusion for D.

Flamelet equations

A flame can be viewed as an ensemble of thin locally one-dimensional structures embedded within the flow field. Each element of the flame front can then be viewed as a small laminar flame also called flamelet [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF][START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. Physically, the flame structure is considered locally onedimensional and only depends on time and the coordinate normal to the flame front. A coordinate system attached to the surface of stoichiometric mixture fraction can be introduced and the flamelet equations can be derived [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF]. A transformation from the physical spatial-temporal space @x; tA to the mixture fraction-temporal space @Z; A can be applied to the governing equations of species and enthalpy. It is assumed that through the flame front, the combustion behaviour is one-dimensional. Terms corresponding to the gradients along the tangent directions to the flame front are neglected in comparison to those normal to the flame front. For simplicity, unity Lewis numbers are adopted in the following description, implying that all chemical species and enthalpy have the same diffusivity, D. [START_REF] Pitsch | A consistent flamelet formulation for nonpremixed combustion considering differential diffusion effects[END_REF]. This coordinate transformation results in the flamelet equations: @Y @ a I P @ P Y @Z P C !

(2.10) @h @ a I P @ P h

@Z P r • q HH R (2.11)
, is the scalar dissipation rate and is defined as:

a PD @Z @x i @Z @x i ! (2.
12)

The scalar dissipation rate has the units of the inverse of a time and characterizes typical diffusion times in the flamelet [START_REF] Poinsot | Theoretical and numerical combustion[END_REF].

Steady laminar flamelet model

The SLF model views a turbulent diffusion flame as an ensemble of thin, steady laminar reaction zones embedded in a turbulent flow field. At sufficiently high Damköhler number or sufficiently high activation energy, the reaction zone is thinner than the surrounding turbulence eddies. In this regime, turbulent structures can deform and stretch the flame sheet, but are unable to penetrate the reaction zone and modify its structure. The combustion chemistry reaches a quasi-steady state and immediately adjusts to local flow-field conditions.

Neglecting the temporal derivative in Eqs (2.10) and (2.11) leads to the SLF model:

I P d P Y dZ P a !
(2.13)

I P d P h dZ P a r • q HH R (2.14)
In order to solve this set of equations, a distribution of @ZA should be determined. The 1D flame structure is generally assumed to be represented by a counterflow diffusion flame. [START_REF] Peters | Laminar diffusion flamelet models in non-premixed turbulent combustion[END_REF] and (2.9)) in the physical space for a laminar counterflow configuration. The thermochemical variables are subsequently mapped into the mixture fraction space and the profile of scalar dissipation rate, @ZA, is directly obtained from its definition (Eq. 2.12).

Numerical solutions of the SLF equations can be analysed by plotting the temperature at the stoichiometric mixture fraction, T st , as a function of st .

This plot is referred to as the S-shaped curve in the literature and an illustration is provided in Fig. 2

Steady flamelet/progress variable model

The FPV approach has been developed to incorporate both stable and unstable states into the steady flamelet model [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Ihme | Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach[END_REF]Ihme and Pitsch, 2008). In order to achieve this objective, a reaction progress variable, c, is used instead of the scalar dissipation rate. A reaction progress parameter, £, defined by evaluating c at stoichiometric conditions, £ a c st , is introduced to parametrize the chemistry tabulation. This definition ensures that the reaction progress parameter is independent of the mixture fraction [START_REF] Ihme | Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach[END_REF]. £ has to uniquely identify both stable and unstable branches in the library. Many definitions of the reaction progress variable are possible. It can be based either on the temperature or on linear combination of species mass fractions [START_REF] Ihme | Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach[END_REF]Ihme and Pitsch, 2008). In the present study, two definitions of progress variable, commonly used in the literature [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Shunn | Verification of variable-density flow solvers using manufactured solutions[END_REF][START_REF] Ihme | Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach[END_REF][START_REF] Ventosa | Numerical simulation of turbulent diffusion flames using flamelet models on unstructured meshes[END_REF], are considered:

c I a Y CO2 C Y H2O C Y CO C Y H2 c P a Y CO2 C Y H2O
(2.17)

The relationship between T st and the progress parameters c I;st and c P;st is plotted in Fig. 2.3. This figure illustrates the bijective mapping of the entire S-shaped curve in the FPV approach.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

c 1, st [ -] 1000
a fpv @Z; cA (2.19)
where c is readily obtained from its transport equation.

@c @t C @u i c @x i a @ @x i D @c @x i ! C ! c (2.20)
where ! c is the progress-variable reaction rate defined similarly to the progressvariable as a linear summation of species reaction rates.

The FPV tabulation provides a description of stable branch of the S-shaped curve that is identical to the SLF model, but also includes the unstable branch that characterizes intermediate or partially-burning states. As such, contrary to the SLF model, it is expected to capture local extinction and re-ignition processes (Ihme and Pitsch, 2008).

Figure 2.4 illustrates the temperature profiles as a function of the mixture fraction for the two branches. For the stable branch, the maximum temperature decreases when st increases, while an opposite trend is observed for the unstable branch. Temperature profiles for the two branches meet at extinction. (Ihme and Pitsch, 2008;[START_REF] Xu | Simulations of the coupling between combustion and radiation 178 BIBLIOGRAPHY Li MA BIBLIOGRAPHY in a turbulent line fire using an unsteady flamelet model[END_REF]. Based on this observation, several extensions of the SLF have been proposed (Ihme and Pitsch, 2008;[START_REF] Carbonell | Flamelet mathematical models for non-premixed laminar combustion[END_REF] to incorporate the radiative loss. On the other hand, [START_REF] Xu | Simulations of the coupling between combustion and radiation 178 BIBLIOGRAPHY Li MA BIBLIOGRAPHY in a turbulent line fire using an unsteady flamelet model[END_REF] developed an unsteady flamelet model to couple combustion and radiation in a turbulent line fire. In the present study, the methodology described by [START_REF] Carbonell | Flamelet mathematical models for non-premixed laminar combustion[END_REF] was used to incorporate radiative loss in the flamelet library. The idea consists in generating flamelet profiles for each strain rate with different degrees of heat losses. A volumetric radiative heat sink term based on the optically-thin approximation was introduced in the flamelet energy equation (see Eq. 2.14) and was multiplied by a constant to allow variation in the degrees of radiative loss. For each strain rate, the flamelet temperature and species equations were solved for a set of prescribed factors ranging from 0 (adiabatic) to the maximum value which was adjusted to be near quenching conditions. For illustration, Fig. 2.6 presents temperature profiles with increasing from 0 to its maximum for a stable burning flamelet at a strain rate of a a V s I . As increases, more enthalpy loses are introduced leading to a decrease in temperature. Figure 2.7 shows the same illustration in the case of the FPV. As expected, the same behaviour is observed as for the SLF model for the stable branch. An opposite trend is observed on the unstable branch with the temperature profile increasing with the degree of heat loss until it encounters the last non-adiabatic profile in the stable branch, which corresponds to the local quenching due to enthalpy loses (see Fig. 2.7).

Figure 2.8 shows the profiles of c I and c P as a function of the mixture fraction for different values of on both the stable and unstable branches. The strain rate was also a a V s I . The progress variable c I follows the same trend as the temperature with increasing in both stable and unstable branches. On the opposite way, the behaviour of c P in the stable branch as increases is much intricate. This confirms that this latter definition of the progress variable may be not appropriate for stable burning flames.

As the total enthalpy is no longer conserved in a non-adiabatic flame, a new parameter characterizing the degree of heat loss has to be introduced to gen- for SLF or FPV models are given by the following state relationships: 8 < : a slf @Z; ; X R A for SLF a fpv @Z; c; X R A for FPV

(2.23)

Radiation modelling

The present study focuses on non-sooting flames. As a consequence, water vapour and carbon dioxide are considered as the only radiating species since our simulations showed that the contribution of carbon monoxide is negligible.

For all the gas radiative property models, the line parameters of CO P and H P O are taken from HITEMP 2010 [START_REF] Rothman | HITEMP: the high-temperature molecular spectroscopic database[END_REF].

Four radiative property models, namely the RCFSK, the non-grey WSGG, and two versions of the grey WSGG based on different evaluations of the mean path length are considered in the present work. The RTE can be written in a generic form as follows [START_REF] Modest | Radiative Heat Transfer[END_REF]:

dI i ds C i I i a i a i I b Y i a I; N g (2.24)
where I i , i , and I b are the radiative intensity, non-grey absorption coefficient, and blackbody intensity, respectively. N g and a i are the number of quadrature points and the stretching function for the RCFSK or the number of grey gases and the weight factor for the non-grey WSGG, respectively. For the grey Li MA CHAPTER 2. MATHEMATICAL FORMULATION 29 2.3. RADIATION MODELLING models, only one RTE @N g a IA has to be solved and the weight factor is set equal to 1.

RCFSK model

The RCFSK model belongs to the family of the global k-distribution-based gas radiation models as the SLW or the classical FSCK. The RCFSK preserves the total emission and does not require any specification of a reference state [START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF]. In configurations relevant for combustion applications, the RCFSK was found to provide an overall very good agreement with reference LBL or narrow band calculations in both emission and absorption dominated problems [START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF][START_REF] Liu | Effects of the K-value solution schemes on radiation heat transfer modelling in oxy-fuel flames using the full-Spectrum Correlated K-distribution Method[END_REF]Consalvi et al., 2019). Consequently, it will be used as reference in the present study. Full-spectrum (FS) cumulative k-g distribution function is defined as [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF][START_REF] Modest | The full spectrum correlated-k distribution for thermal radiation from molecular gas particulate mixtures[END_REF]:

g@k; ; T p A a R I H Hk @AI b @T p Ad I b @T p A (2.25)
where a fx CO2 ; x H2O ; T g is an array of thermodynamic variables affecting the spectral absorption coefficient, , at the wavenumber . H is the Heaviside function, k a given value of the absorption coefficient, and I b @T p A the spectral blackbody intensity at the Planck temperature T p . In the present study, mixed FS k-g distributions (for mixtures of H P O and CO P ) are constructed from HITEMP 2010 [START_REF] Rothman | HITEMP: the high-temperature molecular spectroscopic database[END_REF] by using the procedure proposed by [START_REF] Modest | Assembly full spectrum k-distribution from a narrow band database: effects of mixing gases, gases and non-gray absorpting particles and non-gray scatters in non-gray enclosures[END_REF], and a 10-point Gauss-Legendre quadrature scheme @N g a IHA is used (see Eq. (2.24)). For each quadrature point, g i , the RCFSK scheme determines the absorption coefficient, i , and the stretching function, a i @T A, by solving [START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF]:

g@ i ; ; T p A a g i (2.26)
a i @T A a g i ; ; T g i I ; ; T ¡g Y ¡g a g i g i I

(2.27)

The divergence of the radiative flux is then computed from the following equation:

r • q HH R a Ng X iaI R i a i I b ¡g | {z } Q 000 emi Ng X iaI i G i ¡g | {z } Q 000 abs (2.28) 30 CHAPTER 2. MATHEMATICAL FORMULATION Li MA 2.3. RADIATION MODELLING
where G i a R R I i d with being the solid angle. As expected, predictions were found insensitive to the choice of T p . In the present simulations, T p was set equal to 1500 K.

WSGG model

The WSGG assumes that the total emissivity, , can be computed as follows [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]:

a Ng X iaI a i @T AI exp@ i LA (2.29) with i a @x CO2 C x H2O Ap H p;i and a i a K X kaH b i;k T k (2.30)
L is the mean path length. In the present study, p H = 1 atm, N g = 4 and K = 4. The constant WSGG parameters, p;i and b i;k , derived in Ref. [START_REF] Coelho | WSGG correlations based on HITEMP2010 for gas mixtures of H2O and CO2 in high total pressure conditions[END_REF] for x H2O =x CO2 a P, are used. These parameters were optimized to fit total emissivities determined from HITEMP 2010 [START_REF] Rothman | HITEMP: the high-temperature molecular spectroscopic database[END_REF] for a set of specified temperatures and optical path-lengths.

The non-grey WSGG solves 4 RTEs @N g a RA. For each RTE, i and a i are computed from Eq. (2.30). This approach will be referred to as WSGG-NG, hereafter. The divergence of the radiative flux is then determined as:

r • q HH R a Ng X iaI R i a i I b | {z } Q 000 emi Ng X iaI i G i | {z } Q 000 abs Y with G i a Z R I i d (2.31)
The grey WSGG consists in determining a grey absorption coefficient, , from the WSGG emissivity (Eq. 2.29):

a ln@I A L (2.32)
There is no theoretical procedure to determine L in Eqs. (2.29) and (2.32).

Two approaches were considered in the literature. The first approach consists in using a cell-based mean path length whereas the second assumes a domain or flame-based mean path length. The two options are considered in the present study. The cell-based approach is referred to as WSGG-G-CB hereafter and, following the recommendation of Refs. [START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF] 2014), L is determined as five times the characteristic local grid size. On the other hand, the flame-based determination of L is arbitrary since it depends on the way to define the flame. In the present study, L a RV =A, where V and A are the flame volume and surface, respectively. The flame shape is assumed to be a cone with a circular base of diameter equal to the pool diameter and a height defined by the iso-contour 600 K. This value was selected since it is expected to characterize the intermittent flame tip [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF]. Varying this temperature between 500 and 700 K was found to effect the radiant fraction by less than 1.57. This approach will be referred to as WSGG-G-FB, hereafter. @ " @t C @ " ũ i @x i a H

(3.1)

@ " ũ j @t C @ " ũ i ũj @x i a @ " p @x j C @ @x i P Sij C @ sgs u i u j @x i C @" H Ag i (3.2) @ " h @t C @ " ũ i h @x i a @ @x i " D @ h @x i ! C @ sgs u i h @x i " Q R (3.3) @ " Z @t C @ " ũ i Z @x i a @ @x i " D @ Z @x i ! C @ sgs u i Z @x i (3.4)
Note that Z can be represent the mixture fraction or passive scalar. The filtering process removes instantaneous small-scale fluctuations, but these subfilter quantities still exert a statistical effect on the resolved scales via subgrid-scale terms namely subgrid-scale stress, sgs u i u j , and subgrid-scale scalar fluxes, sgs u i h and sgs u i Z , which require modelling.

3.1. SUBGRID STRESS TENSOR

Subgrid stress tensor

The SGS momentum stress is modelled by extending the dynamic Smagorinsky model (DSM) of Code_Saturne [START_REF] Archambeau | Code_Saturne: a Finite Volume Code for the Computation of Turbulent Incompressible Flows[END_REF]Code_Saturne n.d.) to density-weighted test filtering as described by Moin et al. (1991a), [START_REF] Desjardin | Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume[END_REF], and [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF]. The subgrid term appearing in Eq. (3.2) can be expressed as:

sgs u i u j a " @ g u i u j ũi ũj A % PC s " ¡ P j Sj Sij (3.5) t a PC s " ¡ P j Sj (3.6)
where j Sj a q P Sij Sij is the norm of the resolved strain rate tensor and t is the subgrid-scale viscosity. The coefficient C s is dynamically adjusted to the local structure of the flow. The dynamic procedure consists in using a test filter with width ¡ t a P¡. The test-filtered stress is defined by analogy to the SGS stress, sgs u i u j :

T u i u j a " g u i u j b " ! ũi ! ũj % PC s " ¡ P t j ! Sj ! Sij (3.7)
where d @ • A represents the filtering test operation and ! a b " "

denotes a densityweighted test filtered quantity. Using Germano's identity, the Leonard stress, L u i u j , that represents the resolved SGS correlations at the test filter level can be expressed as:

L u i u j a T u i u j sgs u i u j a @ " ũ i ũj b " ! ũi ! ũj A (3.8)
Combining Eqs. (3.5), (3.7) and (3.8) yields:

C s M u i u j a L u i u j (3.9)
where M u i u j represents the difference between the SGS model expressed in terms of test filtered quantities and the SGS model test filtered:

M u i u j a P " ¡ P t j ! Sj ! Sij P" ¡ P j Sj Sij (3.10)
The least-squares approach [START_REF] Lilly | A proposed modification of the Germano subgrid-scale closure method[END_REF] is applied to Eq. (3.9) to calculate

the constant C s : C s a < L u i u j M u i u j > < M u i u j M u i u j > (3.11)
where < • > denotes a local spatial averaging operation. In addition to local smoothing, entropy limit condition @ C t ! HA are enforced to limit the amount of backscatter in the simulations.

34 CHAPTER 3. LARGE-EDDY SIMULATION Li MA 3.2. SUBGRID SCALAR FLUX

Subgrid scalar flux

In this section, closures for the subgrid-scale scalar flux terms are introduced by modelling the subgrid-scale diffusivity, D t . Depending on the subgrid scalar flux model, D t can be either scalar or tensor, thus will be expressed specifically according to each model. Furthermore, the subgrid-scale diffusivity is only computed through one scalar field, and then applied to all other scalars fields.

The Dynamic linear eddy diffusivity model (DEDM) model will be first presented, followed by two models based on tensorial forms of the diffusivity for the SGS scalar flux, namely the dynamic generalized gradient model (DGGM)

and dynamic full linear generalized gradient model (DLGGM) [START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF]. In addition, two mixed models, adding a gradient model (based on

Taylor expansion of the filtering operation) to either the DEDM or the DGGM, will be introduced.

Dynamic linear eddy diffusivity model

The DEDM proposed by Moin et al. (1991a) expresses the SGS scalar flux, sgs u i Z , appearing in Eq. (3.4) as:

sgs u i Z a " @ g u i Z ũi ZA % " D E @ Z @x i (3.12) " D E a C I " ¡ P j Sj (3.13)
The same dynamic procedure, as described above, gives:

C I a < L u i Z M u i Z > < M u i Z M u i Z > (3.14) with L u i Z a @ " ũ i Z b " ! ũi ! ZA (3.15) M u i Z a " ¡ P t j ! Sj @ ! Z @x i " ¡ P j Sj @ Z @x i (3.16)

Dynamic generalized gradient diffusion model

This model is inspired from the generalized gradient diffusion hypothesis (GGDH) proposed by [START_REF] Daly | Transport equations in turbulence[END_REF] in the RANS approach. In Li MA CHAPTER 3. LARGE-EDDY SIMULATION 35

3.2. SUBGRID SCALAR FLUX the LES context, Wang and co-workers [START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF] expressed the SGS scalar flux as follows:

sgs u i Z % " D ik;G @ Z @x k (3.17) " D ik;G a C II T sgs u i u k (3.18)
where D ik;G is the tensor diffusivity which is a homogeneous linear tensor function of sgs u i u k . T is a characteristic subgrid time scale, which can be evaluated by I=j Sj. The coefficient C II is calculated dynamically from:

C II a < L u i Z G u i Z > < G u i Z G u i Z > (3.19) with G u i Z a T u i u k j ! Sj @ ! Z @x k sgs u i u k j Sj @ Z @x k (3.20) L u i Y , T u i u k ,

Dynamic full linear generalized gradient diffusion model

The tensor diffusivity, D ik;E , for the DEDM is a function of S ij of the zeroth order while the tensor diffusivity, D ik;G , for the DGGM is a function of sgs u i u k of the first order. According to the theory of the polynomial functions, a DLGGM can be constructed by both zeroth and first parts of sgs u i u k . Therefore, the DLGGM can be regarded as a combining form of the DEDM and DGGM, expressed as [START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF]:

sgs u i Z % " D ik;L @ Z @x k (3.21) " D ik;L a C III T j sgs j ik C C II T sgs u i u k (3.22)
where j sgs j a q P sgs u i u j sgs u i u j .

By using the least squares method, the computation of the two model constants,

C III and C II , can be recast to solving a local linear system as [START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF]:

36 CHAPTER 3. LARGE-EDDY SIMULATION Li MA 3.2. SUBGRID SCALAR FLUX 2 4 < P u i Z P u i Z > < P u i Z G u i Z > < G u i Z P u i Z > < G u i Z G u i Z > 3 5 • 2 4 C III C II 3 5 a 2 4 < L u i Z P u i Z > < L u i Z G u i Z > 3 5 (3.23) with P u i Z a jTj j ! Sj @ ! Z @x i j sgs j j Sj @ Z @x i (3.24)
where jTj a q

PT u i u k T u i u k . L u i Y , T u i u k , sgs u i u k ,
and G u i Y are calculated by Eqs.

(3.15), (3.7), (3.5), and (3.20), respectively.

Dynamic Clark model

The gradient model is a classical structural-based model coming from the Taylor series expansion of the filtering operation [START_REF] Bedford | Conjective filtering procedures in surface water flow and transport[END_REF]:

sgs u i Z a " ¡ P k IP @ ũi @x k @ Z @x k C y@¡ R A (3.25)
where ¡ k is the filter width in the direction k. This model is known to have high correlation level with the SGS unknown term, but is numerically unstable due to an incorrect prediction of the dissipation and the amount of kinetic energy backscatter [START_REF] Vreman | On the formulation of the dynamic mixed subgrid scale model[END_REF][START_REF] Lu | A modulated gradient model for scalar transport in large-eddy simulation of the atmospheric boundary layer[END_REF]. As a consequence, the Clark model consists in combining a gradient model with an eddy diffusivity model to provide a relative accurate representation of the SGS term (gradient model) and the appropriate level of dissipation (eddy diffusivity model) [START_REF] Vreman | Large-eddy simulation of the turbulent mixing layer[END_REF]. The Eq. (3.25) becomes [START_REF] Fabre | Development of a new dynamic procedure for the Clark model of the subgrid-scale scalar flux using the concept of optimal estimator[END_REF]:

sgs u i Z a " ¡ P k IP @ ũi @x k @ Z @x k C " D ik @ Z @x k (3.26)
In the present study, DEDM and DGGM are considered to model the tensor diffusivity, D ik , leading to the M1 and M2, respectively. Recently, [START_REF] Fabre | Development of a new dynamic procedure for the Clark model of the subgrid-scale scalar flux using the concept of optimal estimator[END_REF] proposed a new dynamic procedure based on a Taylor series expansion of L u i Z to determine the model constants:

C a < @L u i Z H u i Z AN u i Z > < N u i Z N u i Z > (3.27) Li MA CHAPTER 3. LARGE-EDDY SIMULATION 37 3.3. SUBGRID CHEMISTRY CLOSURE
L u i Z is the Leonard term, Eq. (3.8) , N u i Z and H u i Z are written as

N u i Z a 8 > < > : " ¡ P t j ! Sj @ ! Z @x i for M1 Tu i u k j ! Sj @ ! Z @x k for M2 (3.28) H u i Z a " ¡ P t;k IP @ ! ũi @x k @ ! Z @x k
for both models (3.29)

Subgrid chemistry closure

In LES of non-premixed combustion, many important mixing and reaction processes occur at scales that are unresolved on the computational grid.

Combustion-related variables such as the mixture fraction fluctuating quantities and their statistical distribution need to be considered. Subgrid fluctuations of the combustion variables can have an important impact on the filtered properties due to the sensitivity and non-linearity of combustion processes.

To account for subgrid fluctuations in the combustion variables, filtered combustion variables are obtained by integrating the state equation over the joint FDF of either (Z, and X R ) for the SLF model or (Z, c and X R ) for the FPV model.

In the case of SLF, the filtered combustion variables are given by: a Z slf @Z; ; X R A P @Z; ; X R AdZddX R

(3.30)

Z; and X R are assumed to be statistically independent and the marginal FDFs are modelled by a beta distribution for Z and delta distributions for and X R : P @Z; ; X R A a @ZY Z; V Z A@ A@X R XR A

(3.31)

The beta distribution is given by: @ZY Z; V Z A a Z a I @I ZA b I @a C bA @aA C @bA

(3.32) with a a Z@ Z ZP V Z A V Z b a @I ZA@ Z ZP V Z A V Z (3.33)
where V Z is the subgrid mixture fraction variance.

In the case of FPV approach, the filtered combustion variables are given:

a Z fpv @Z; c; X R A P @Z; c; X R AdZdcdX R (3.34) 38 CHAPTER 3. LARGE-EDDY SIMULATION Li MA 3.3. SUBGRID CHEMISTRY CLOSURE
By assuming the statistically independent Z, c, and X R , the joint FDF is expressed as:

P @Z; c; X R A a @ZY Z; V Z A@c cA@X R XR A (3.35)
Once the flamelet library is computed and assumed FDF integrals are evaluated, a look-up table can be generated to provide the filtered thermo-chemical variables as functions of the quantities readily available from LES [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF]. 8 < : a slf @ Z; V Z ; ; XR A for SLF a fpv @ Z; V Z ; c; XR A for FPV

(3.36)

The filtered reaction progress variable, c, will be obtained by solving the following transport equation:

@ " c @t C @ " ũ i c @x i a @ @x i " D @c @x i ! C @ sgs u i c @x i C ! c (3.37) and, ! c a " Z ! fpv c @Z; c; R A fpv @Z; c; R A @ZY Z; V Z AdZ (3.38)

Sperious heat release

In variable density low-Mach Navier-Stokes solver, inconsistencies between changes in density from the pre-tabulated non-linear equation of state and changes in density from the continuity equations can induce spurious density fluctuations, which may result in large spurious fluctuations of velocity and pressure that destabilize the simulation [START_REF] Shunn | Large-eddy simulation of combustion sysytems with convective heat-loss[END_REF]. The same procedure to control spurious density fluctuations as proposed by [START_REF] Vreman | Premixed and nonpremixed generated manifolds in large-eddy simulation of Sandia flame D and F[END_REF] is adopted in the present study. It consists in removing the smallest resolved scale by explicit filter operations with a filter width equal to P¡. The density is filtered by the explicit filter after the retrievals from the flamelet look-up

Subgrid scalar variance and dissipation

Non-equilibrium models

The subgrid scalar variance V Z , required to retrieve solutions from the filtered flamelet look-up table, is defined in terms of a FDF as [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF]) :

V Z @x; tA a f Z P @x; tA ZP @x; tA

(3.40)
The appearance of the second moment f Z P makes the variance unclosed. The subgrid scalar variance can be computed by using its transport equation (VTE) [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF]:

@ " V Z @t C @ " ũ i V Z @x i a @ @x i " D C D t @V Z @x i ! C " (3.41)
where the production term is defined as:

a P" D C D t @ Z @x i @ Z @x i (3.42)
Another way to compute the subgrid-scale variance consists in solving the transport equation for the second moment of the mixture fraction f Z P , (STE) [START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF][START_REF] Kaul | A posteriori analysis of numerical errors in subfilter scalar variance modeling for large eddy simulation[END_REF]. Once the second moment f Z P and Z are resolved, the subgrid scalar variance V Z thus can be computed from its definition, Eq. (3.40).

@ "

f Z P @t C @ " ũi f Z P @x i a @ @x i 0 @ " @ D C D t A @ f Z P @x i 1 A " (3.43)
In both VTE and STE models, the filtered scalar dissipation rate term, , is unclosed and requires modelling. [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF] proposed to model the filtered scalar dissipation rate by relating the subgrid-scale scalar mixing time scale to the subgrid-scale turbulent time scale. The characteristic mixing time is assumed proportional to the turbulent characteristic time. In LES, the subgrid-scale scalar mixing time can be defined as:

I Z a V Z (3.44)
An equivalent subgrid-scale turbulent characteristic time "

is introduced as the ratio between subgrid-scale kinetic energy, sgs a I=P@ g u i u i ũi ũi A, and the 40 CHAPTER 3. LARGE-EDDY SIMULATION Li MA 3.5. SUBGRID-SCALE TRI filtered kinetic energy dissipation rate, " a @u i @x j @u i @x j . Given a proportionality between both time scales, the model for, , is derived as:

V Z a I Z % C a C " sgs (3.45)
The filtered kinetic energy dissipation rate and the subgrid-scale kinetic energy are modelled as:

" a P @ C T A Sij Sij (3.46) sgs a PC I ¡ P Sij Sij (3.47)
As proposed by [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF], C a I=Sc, and C I can be computed by dynamic approach. Therefore, the filtered scalar dissipation rate can be written as:

a D C D t C I ¡ P V Z (3.48)

Algebraic equilibrium model

If production and destruction of the SGS mixture fraction variance is assumed to be in equilibrium [START_REF] Pierce | Dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar[END_REF], the scalar dissipation rate can be computed as:

a P D C D t @ Z @x i @ Z @x i (3.49)
With the local equilibrium assumption (LEA), the SGS mixture fraction variance is computed through a scale similarity model [START_REF] Pierce | Dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar[END_REF], denoted hereafter as the algebraic model:

V Z a C v ¡ P @ Z @x i @ Z @x i (3.50)
where C v is calculated dynamically.

Subgrid-scale TRI

The filtered RTE can be written in a generic form as follows [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]:

dI i ds C i I i a i a i I b Y i a I; N g (3.51) Li MA CHAPTER 3. LARGE-EDDY SIMULATION 3.5. SUBGRID-SCALE TRI
The filtered absorption term, i I i , is closed by neglecting the SGS absorption TRI, leading to i I i % i I i . This approximation was found to be valid in LES of non-luminous and luminous lab-and large-scale diffusion flames in which about VH7 of the turbulent fluctuations are resolved [START_REF] Gupta | Turbulence-radiation interactions in large-eddy simulations of luminous and nonluminous nonpremixed flames[END_REF][START_REF] Coelho | Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows[END_REF][START_REF] Consalvi | On the modeling of the filtered radiative transfer equation in large eddy simulations of lab-scale sooting turbulent diffusion flames[END_REF]. The absorption coefficient and the emission term depend only on temperature and mole fractions of CO P and H P O. Therefore, the filtering operation for these terms can be performed accurately by using the FDF approach [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF]: tan et al., 2014). These mathematical approaches represent the basis of the CFD code and physical sub-models for turbulence, combustion and radiation are then implemented to achieve numerical simulations of practical problems [START_REF] Yuen | LES and Multi-Step Chemical Reaction in Compartment Fires[END_REF][START_REF] Yuen | Study of three LES subgrid-scale turbulence models for predictions of heat and mass transfer in large-scale compartment fires[END_REF]. An important step in the development of a CFD code is the verification and validation processes which establish the degree of confidence in a computational model for specific applications [START_REF] Oberkampf | Verification and validation in computational fluid dynamics[END_REF][START_REF] Shunn | Verification of variable-density flow solvers using manufactured solutions[END_REF][START_REF] Mullyadzhanov | Verification of a low Mach variable-density Navier-Stokes solver for turbulent combustion[END_REF]. The verification step checks that the numerical models are well implemented and satisfy the desired accuracy through analytical solutions or method of manufactured solutions (MMS). In this chapter, a new low-Mach variable-density Navier-Stokes equations solver is developed. Its verification through method of manufactured solutions is realised in order to prove the second-order accuracy in both time and space. Then this solver is used to simulate the Rayleigh-Taylor instability case and results are compared with those reported in the literature [START_REF] Desjardins | High order conservative finite difference scheme for variable density low Mach number Li MA BIBLIOGRAPHY 167 BIBLIOGRAPHY turbulent flows[END_REF][START_REF] Mullyadzhanov | Verification of a low Mach variable-density Navier-Stokes solver for turbulent combustion[END_REF].

i a " Z @ i A slf @Z; ; R A slf @Z; ; R A @ZY Z; V Z AdZ (3.52) i a i I b a " Z @ i a i I b A slf @Z; ; R A slf @Z; ; R A @ZY Z; V Z AdZ (3.

NUMERICAL ALGORITHM

Numerical algorithm

The governing equations in the low Mach number form for mass, momentum, and scalar, as well as the equation of state are expressed in a compact form:

@ @t C div@uA a Q (4.1) @u @t C div@u uA a rp C div @PSA C Q u (4.2) @Y @t C div@uY A a div@DrY A C Q Y (4.3) a f@Y A (4.4)
where S is the strain rate tensor given by:

S a I P @ru C r T uA I Q div@uAI (4.5)
The variables , u, p, and Y denote the density, velocity vector, u a @u; v; wA, pressure, and additional scalar field, respectively. and D are the dynamic viscosity and kinematic diffusivity, respectively. The source terms @uA nC 1 2 ;H a P@uA nC 1 2 I @uA nC 1 2 P (4.6)

Q , Q u ,
p nC 1 2 ;H a p nC 1 2 I (4.7) @ Q u A nC 1 2 a Q P Q n u I P Q n I u (4.8) @ Q Y A nC 1 2 a Q P Q n Y I P Q n I Y (4.9)
The source terms Q u and Q Y are extrapolated by the Adams-Bashforth method which provides a second-order estimation. The density is initialized as nCI;H a Q n Q n I C n P to be compatible with @uA nC 1 2 ;H and ensure the conservation. div@ nCI;k u nCI;kCI A div@ nCI;k u nCI;kCI A a div@¡t∇p nCI;kCI A (4.11)

The first term of Eq. (4.11) can be extrapolated by the Adams-Bashforth scheme:

div@ nCI;k u nCI;kCI A a Q P div@uA nC 1 2 ;kCI I P div@uA n 1 2 (4.12)

In addition, mass conservation implies:

nCI;kCI n ¡t C div@uA nC 1 2 ;kCI a H div@¡t∇p nCI;kCI A a div@ nCI;k u nCI;kCI AC Q P Once the pressure increment is obtained, the variables are updated as:

p nC 1 2 ;kCI a p nC 1 2 ;k C p nCI;kCI (4.15) @ nCI;k u nCI;kCI A a ¡t∇p nCI;kCI C @ nCI;k u nCI;kCI A (4.16)

u nCI;kCI a @ nCI;k u nCI;kCI A nCI;k (4.17)

7. Reconstruct the mass flux:

@uA nC 1 2 ;kCI a P Q @ nCI;k u nCI;kCI A C I Q @uA nC 1 2 I (4.18)
8. The process is repeated from step 2 until convergence is achieved. The sub-iteration process is assumed to be converged when " iter < IH V with " iter defined as:

" iter a jjX nCI;kCI X nCI;k jj L2 jjX n jj L2

(4.19)
where X represents the numerical solution for both the scalar and velocity components and jj • jj L2 is the L P -norm.

Algorithm verification by MMS

This section presents the verification tests for the numerical algorithm proposed above. Three manufactured solutions proposed by [START_REF] Shunn | Verification of variable-density flow solvers using manufactured solutions[END_REF] are considered. In these three cases, the L P -error for the variable X is defined as:

E L2 @XA jjX e Xjj L2 jjX e jj L2 (4.20)
where X and X e are the numerical and exact solutions, respectively.

1D-D value 2D-AD value 2D-ODF value ( H , I ) (20,1) ( H , I ) (20,1) ( H , I ) (5,1) w H 50 (u f ,v f ) (1,0.5) u f a v f 0.5 D a 0.03 D a 0.001 D a 0.001 @k I ; k P A (4,2) (k,!) (R,1.5) k a ! 2 (a,b) (0.2,20)

One-dimensional Diffusion

We consider firstly a one-dimensional diffusion manufactured solution (1D-D):

Y @x; tA a exp@ k I tA cosh@w H xexp@ k P tAA exp@ k I tA@I H = I A cosh@w H xexp@ k P tAA @x; tA a Y @x; tA Q Y , appears in the scalar transport equation. This pseudo-diffusion problem is actually controlled by the relative values of the parameters k I and k P , and is independent of the physical scalar diffusivity, D. The diffusivity still appears in the MMS problem, but only affects the shape and magnitude of the source term, Q Y . The computational domain for this problem is x P H; P and t P H; I. The inlet boundary conditions are Dirichlet condition with a prescribed value of 0 for the velocity and Neumman condition for pressure and scalar. At the domain exit, an outflow boundary condition is used. and 6400 uniform control volumes and time steps ranging from 0.04 (number of time steps n t = 25) to 0.00125 (n t = 800) are considered. In Fig. 4.1(a), the time step is fixed at 0.00125 (n t = 800). At such a time step the error with respect to the spatial resolution exhibits a second-order convergence up to n x = 3200 where the temporal error starts to contaminate the solution. Similarly, for the highest grid resolution (n x = 6400) a second-order convergence is observed in Fig. 4.1(b) up to n t = 400 (¡t = 0.0025). For larger n t , the spatial error dominates and the temporal convergence starts to stall.

Two-dimensional advection and diffusion

The second MMS problem involves a two-dimensional corrugated front with advection and diffusion (2D-AD):

Y @x; y; tA a I C tanh@b xexp@ !tAA @I C H = I A @I H = I Atanh@b xexp@ !tAA @x; y; tA a Y @x; tA

I C I Y @x; tA H ! I u@x; y; tA a I H @x; tA ! x C ! x u f exp@Pb xexp@ !tAA C I (4.22) C !log@exp@Pb xexp@ !tAA C IA
Pbexp@ !tA ! v@x; y; tA a v f p@x; y; tA a H where x@x; y; tA a u f t xCacos@k@v f t yAA. a, b, k, !, u f , and v f are constant parameters (see Table 4.1). The computational domain for this problem is

I x Q, I=P

Two-dimensional oscillating density field

The final MMS problem describes an oscillating density field convecting with a constant velocity through the domain (2D-ODF):

Y @x; y; tA a I C sin@k xAsin@k yAcos@!tA

I C 0 1 C I 0 1
sin@k xAsin@k yAcos@!tA @x; y; tA a Y @x; y; tA I C I Y @x; y; tA H ! I u@x; y; tA a @ I H A! @x; y; tARk cos@k xAsin@k yAsin@!tA

(4.23)
v@x; y; tA a @ I H A! @x; y; tARk sin@k xAcos@k yAsin@!tA p@x; y; tA a I P @x; y; tAu@x; y; tAv@x; y; tA where x a x u f t and y a y v f t. k and ! are constant parameters given in Table 4.1. The computational domain for this problem is H x P, H y P and H t I. This manufactured solution satisfies the continuity equation with Q a H. Non-zero source terms appear in the x-, y-momentum, and scalar transport equations. Similar to the problem 2D-AD, the scalar field convects through the domain with horizontal and vertical velocities u f and v f , respectively. The spatial and temporal periods of the variable-density structures are controlled by the wavenumber, k, and the frequency, !, respectively. All of the boundary conditions are periodic.

The spatio-temporal convergence of the algorithm is evaluated by using meshes (n x ¢ n y ) of QP ¢ QP; TR ¢ TR; IPV ¢ IPV; PST ¢ PST; SIP ¢ SIP; IHPR ¢ IHPR, and PHRV¢PHRV cells. For each mesh, the time step is selected to keep the convective CFL based on u f constant and equal to 0.2. Results in Fig. 4.3(a) show the second-order accuracy. The corresponding observed order are reported in Table 4.3. Figures 4.3(b), (c), and(d) show that the L P -error remains on the whole smooth over the simulation. In addition, these figures confirm that the secondorder accuracy of the algorithm is maintained over the entire simulation.

Rayleigh-Taylor Instability

The two-dimensional Rayleigh-Taylor instability problem is considered to check These plots are taken from [START_REF] Mullyadzhanov | Verification of a low Mach variable-density Navier-Stokes solver for turbulent combustion[END_REF].

Conclusions

The proposed numerical solver for low-Mach variable-density NS equations is verified by three MMS cases and numerical results confirmed the second-order spatio-temporal accuracy. Through the Rayleigh-Taylor instability case, the solver is capable of accurately capturing the complex variable-density instability features and yield results in good agreement with those of the literature.

In the next chapter, this solver will be validated by simulating a 1 m diameter turbulent helium plume with LES.

Li MA CHAPTER 4. NUMERICAL METHODS

Chapter 5

Helium plume

Turbulent buoyant fire plumes are a canonical configuration in fire safety science. The far-field region of these flows is rather-well understood and selfsimilarity correlations were reported for temperature and velocity profiles (see for example Ref. [START_REF] Shabbir | Experiments on a round turbulent buoyant plume[END_REF]). The near field flow is more complex, exhibiting a rapid transition from laminar to fully turbulent regime in the few first inlet diameters and a puffing motion characterized by a repetitive shedding of coherent vortices [START_REF] Cetegan | Experiments on the periodic instability of buoyant plumes and pool fires[END_REF]. Rayleigh-Taylor (RT) and Kelvin-Helmotz (KH) instabilities have been proposed to explain the nature of puffing. Some authors suggested that the KH instability is the dominant mechanism [START_REF] Ghoniem | Numerical simulation of the dynamics of large fire plumes and the phenomenon of puffing[END_REF] whereas other works [START_REF] Cetegen | Experiments on the oscillatory behavior of buoyant plumes of helium and helium-air mixtures[END_REF][START_REF] Desjardin | Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume[END_REF][START_REF] Tieszen | Toward the development of a LES-SGS closure model for buoyant plumes[END_REF] suggested that the RT instability prevails. These vortical structures control the air entrainment and the resulting turbulent mixing and combustion processes which in turn affect the density and scalar fields.

Before applying the low-Mach variable density NS equations solver to turbulent buoyant fire plumes studies, this chapter presents a LES of turbulent helium plumes for the purposes of validating the numerical solver. Similarly to density ratio between hot gases and cold air in fire plumes, the density ratio of helium and air is a factor of seven. Medium-to large-scale helium plumes (H:I m < d < I m) are also characterized by a rapid transition from a laminar to a turbulent flow and puffing motions. The principal difference arises from the fact that heat is released locally from the combustion process in fire plumes whereas, in helium plumes, the buoyancy is produced only near the source 5.1. EXPERIMENTAL SETUP where large density gradients are present. Large helium plumes are thus good candidate to mimic buoyant fire plumes while avoiding the complexities related to combustion and heat transfer processes [START_REF] O'hern | Experimental study of a turbulent buoyant helium plume[END_REF]. O'Hern et al.

(2005) made significant experimental efforts to provide comprehensive turbulent velocity and helium concentration statistics in the 1 m diameter helium plume. These data were intensively used as a validation step for LES-based CFD models [START_REF] Desjardin | Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume[END_REF][START_REF] Blanquart | Large-eddy simulation of a turbulent buoyant helium plume[END_REF][START_REF] Brown | Proceedings of the first workshop organized by the IAFSS Work-ing Group on Measurement and Computation of Fire Phenomena (MaCFP)[END_REF][START_REF] Chung | Buoyancy corrected k models and large eddy simulation applied to a large axisymmetric helium plume[END_REF][START_REF] Maragkos | Large Eddy Simulations of the Flow in the Near-Field Region of a Turbulent Buoyant Helium Plume[END_REF].

Experimental setup

O'Hern et al. ( 2005) investigated experimentally the characteristics of near field turbulent buoyant helium plumes. The main objectives of these experiments were to provide data to validate CFD models suitable for fire modelling.

The dimensions of the main chamber are T:I ¢ T:I ¢ U:Q m Q . The chamber converges to a square chimney outlet at the top with dimensions of 2.4 m on each side. The plume source, with a diameter of 1 m, is located in the centre of the chamber 2.45 m above the floor. A gas mixture is composed of 96.47 helium, 1.77 acetone, and 1.97 oxygen by volume, with a molecular weight of 5.45 g • mol I . This mixture is released into ambient air at a Reynolds-averaged velocity of 0.325 m • s I corresponding to a Favre-averaged velocity of 0.339 m • s I , approximately. Measurements were obtained using particle image velocimetry (PIV) for velocity field measurements and planar laser induced fluorescence (PLIF) for helium mass fraction. The experimental data include the puffing frequency, radial profiles of mean and rms of axial and radial velocities and mean and rms of helium mass fraction. The experimental uncertainty on the measured velocities, concentrations, turbulent statistics and concentration fluctuations is estimated to be around ¦20, ¦18, ¦30 and ¦217, respectively.

Literature survey

The experiments of [START_REF] O'hern | Experimental study of a turbulent buoyant helium plume[END_REF] were used to validate LES and RANS models. the DGGM. These two models will be denoted as M1 and M2, hereafter. All these SGS models are described in details in the Chapter 3.

Q ¢ Q ¢ Q m 3 ¡ a I:S cm

Computational details

The Favre-filtered governing Eqs. (3.1) to (3.4) are solved by the proposed solver described in the Chapter 4. The second-order Crank-Nicolson scheme is used for time advancement. The Total Variation Diminuishing (TVD) scheme, Superbee [START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF], is applied for convection in the mixture fraction transport to limit numerical oscillation whereas a secondorder central difference scheme (CDS) is used for diffusion. For the momentum equation both convective and diffusive terms are discretized by using a secondorder CDS.

Simulations were performed on a simplified cubic domain with dimensions @x; y; zA of R ¢ R ¢ R m Q . Helium was released through a 1 m diameter orifice with an inlet Favre-averaged velocity fixed at H:QQW m • s I . The molecular weight of helium was set to S:RS g • mol I to be consistent with the experiments. A low coflow was injected with a velocity of H:HI m • s I . These conditions for the helium and coflow velocities correspond to the test 4 in Ref. [START_REF] O'hern | Experimental study of a turbulent buoyant helium plume[END_REF] that was dedicated to get the Favre-averaged statistics. In order to examine the effect of the grid resolution on the LES results, two grid systems with IQS¢IQS¢IIR and IWP¢IWP¢IQP control volumes were used. A uniform grid refinement was used in the region where turbulent mixing is expected resulting in grid-spacing of I:T cm and I cm, respectively. For convenience we will subsequently refer to them as coarse and fine grid, respectively. A constant time step of S¢IH R s was considered to provide an averaged CFL equal to 0.5. Simulations were run for QH s and the density-weighted time-averaged mean and rms values were collected from IS to QH s. The first IS s of simulation were used to establish a statistically stationary flow. for mesh refinement comparable to that considered in this study, the puffing frequency is not affected by the SGS model.

Mean and rms velocities

Model predictions for radial profiles of axial and radial mean velocities and rms values of radial velocities at different heights are displayed in The largest discrepancies are observed for the mean radial at x a H:P m (Fig.

5.3

). Nevertheless, it should be pointed out that the experimental profiles are not perfectly symmetric and a much better agreement is observed at x a H:P m. Radial velocity controls air entrainment in pool fires and the resulting combustion process and, as such, should be accurately predicted.

0.4 0.2 0.0 0.2 0.4

0 1 2 3 4 w [m •s -1 ] z =0.1 m Exp. Num. 0.4 0.2 0.0 0.2 0.4 0 1 2 3 4 z =0.2 m 0.4 0.2 0.0 0.2 0.4 0 1 2 3 4 w [m •s -1 ] z =0.3 m 0.4 0.2 0.0 0.2 0.4 0 1 2 3 4 z =0.4 m 0.4 0.2 0.0 0.2 0.4 x [m] 0 1 2 3 4 w [m •s -1 ] z =0.5 m 0.4 0.2 0.0 0.2 0.4 x [m] 0 1 2 3 4 z =0.6 m

Mean and rms helium mass fraction

5.7

). These discrepancies increase with the height (see Fig. 5.7). Figure 5.8 shows that the DEDM over-predicts significantly the helium mass fraction fluctuations whatever the height. The discrepancies reported in Figs. 5.7 and 5.8 for the DEDM simulations were also observed in previous studies [START_REF] Desjardin | Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume[END_REF][START_REF] Maragkos | Large Eddy Simulations of the Flow in the Near-Field Region of a Turbulent Buoyant Helium Plume[END_REF][START_REF] Chung | Buoyancy corrected k models and large eddy simulation applied to a large axisymmetric helium plume[END_REF]. As discussed previously, these discrepancies were attributed to deficiencies in the modelling of unresolved mixing due to buoyancy. [START_REF] Brown | Proceedings of the first workshop organized by the IAFSS Work-ing Group on Measurement and Computation of Fire Phenomena (MaCFP)[END_REF].

Conclusions

From these simulations the following conclusions can be drawn:

1. Predictions with DSM and DEDM capture the plume dynamics but underestimate the mixing rate at vicinity of the plume centreline.

Subgrid variance modelling

As discussed in the introduction, estimation of subgrid quantities is a crucial procedure for LES. Among other subgrid quantities, the subgrid variance of the mixture fraction is particularly important for LES of non-premixed combustion because of the role it plays in the prediction of mixing at the molecular level. This section focuses on investigating existing SGS mixing models for mix- Following [START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF] the comparison of LES solutions will be based on the convergence properties of their statistics with respect to the filter size, ¡. As discussed by Pope and co-workers [START_REF] Pope | Turbulent flows[END_REF][START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF], characterizing the dependence of predicted statistics on ¡ must be part of the overall LES methodology and gains importance in combustion problems since the rate-controlling processes such as reactant mixing and chemical reactions are modelled at the subgrid-scale level. Such convergence studies are rare in the LES modelling of fire plumes and were limited to mean temperature and/or axial velocity [START_REF] Sikic | Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires[END_REF][START_REF] Koo | Model sensitivities in LES predictions of buoyant methane fire plumes[END_REF]. The present study reports a more exhaustive convergence study on grids of different resolution involving mean and rms of mixture fraction, temperature and velocity.

Numerical resolution

Throughout this chapter, the Favre-filtered governing Eqs. 3.1 to 3.4 are solved by the proposed solver described in the Chapter 4. The second-order Crank-Nicolson scheme is used for time advancement. The Superbee-TVD scheme [START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF] is applied for convection in the mixture fraction transport to limit numerical oscillation whereas a second-order CDS is used for diffusion. For the momentum equation both convective and diffusive terms are discretized by using a second-order CDS. The subgrid contribution to the momentum stress and scalar flux is computed using DSM and DEDM model, respectively. The non-adiabatic SLF model is used to obtain the local thermochemical state relationship, slf , as a function of the mixture fraction, Z, the scalar dissipation rate, , and the enthalpy defect, X R a h h ad . The flamelet library was generated by solving the governing equations of counterflow diffusion flames in the physical space at a series of specified strain rates using the CHEMKIN code [START_REF] Lutz | OPPDIF: a fortran program por computing opposed-flow diffusion flames[END_REF]. Starting with a low strain rate, 72 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.1. SUBGRID VARIANCE MODELLING flamelet was calculated by gradually increasing the strain rate until the local quenching occurs. Radiative loss was incorporated in the flamelet library by using the methodology described in the Chapter 2. The Favre-filtered thermochemical quantities are then obtained from the state relationships by convolution with the presumed FDF. The RCFSK model is retained for radiative property modelling, its detailed description is found in the Chapter 2.

Computational domain and boundary conditions

Five methane fire plumes in open conditions, investigated experimentally by [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF], were simulated. These flames were generated from a burner of H:Q m in diameter with heat release rates (HRR, Q) of 14.4, 21.7, 33, 45 and 57.5 kW , respectively. Simulations were performed in the rectangular domain

of Q ¢ Q ¢ Q m Q in a Cartesian coordinate.
Consistently with the experiments, the burner is lifted by 0.3 m above the floor to prevent the influence of the ground. The flamelet library was generated using the full chemical kinetic scheme developed by [START_REF] Qin | Combustion chemistry of propane: A case study of detailed reaction mechanism optimization[END_REF]. The filtered RTE is solved by using the DOM with a S8 quadrature scheme [START_REF] Modest | Radiative Heat Transfer[END_REF]. This angular resolution was selected based on [START_REF] Jensen | On various modeling approaches to radiative heat transfer in pool fires[END_REF], which demonstrated that a S4 quadrature is sufficient within the flame, where radiation is isotropic, whereas a finer quadrature is required to predict accurately flux outside the fire.

In order to examine the effect of the grid resolution on the LES results, five grids G1, G2, G3, G4 and G5 have been used with progressively increasing the resolution in the three directions from 0.909 to 6.32 million cells. These grids are uniform in the region H:Q ¢ H:Q ¢ I:H m Q which results in a minimal grid spacing equal to I:S, I:PS, I:H, H:US and H:S cm, respectively (see Table 6.1).

Outside this region, all grids are progressively stretched in all the directions. Throughout all simulations, time step is set to S ¢ IH R s which corresponds to an averaged maximum CFL of 0.6. Simulations were run for 25 s and the time-averaged mean and root mean square values were collected over the last 19 s. The first 6 s of simulation were used to establish a statistically stationary flow. The convergence of statistics will be investigated for the 57.5 kW fire plume at heights above the burner of 0.1, 0.2, 0.4 and 0.6 m, respectively. The corresponding normalized heights are z £ a z= Q P=S a 0.019, 0.039, 0.079, and 0.119 m • kW P=S , showing that they are located close to the burner, in the continuous flame region (CF, z £ H:HV m , 1979). These locations cover all the reactive parts of the fire plume where SGS mixing and resulting combustion processes are expected to be the most important. In addition, it should be pointed out that the positions z a H:I and H:P m are located in the CF where the mixing is controlled by the formation and growth of laminar instabilities [START_REF] Tieszen | Chapter 2. Transport phenomena that affect heat transfer in fully turbulent fires[END_REF].

The Kolmogorov length scale, k , the Taylor length scale, , and the diffusive layer thickness, l d , at the CF tip can be estimated from the data of Mc-Caffrey (1979) and [START_REF] Cox | Some stochastic properties of fire plumes[END_REF] following the analysis proposed in

Refs. [START_REF] Williamson | Developing extinction criteria for fires[END_REF][START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF].The integral length scale, L t , is assumed to be one-half of the equivalent burner diameter [START_REF] Williamson | Developing extinction criteria for fires[END_REF][START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF], i.e. L t a H:IU m. Based on the turbulent intensity reported by [START_REF] Cox | Some stochastic properties of fire plumes[END_REF], the fluctuating velocity, w H , is assumed to be 30 7 of the mean axial velocity at the CF tip, w CF . This latter is estimated from the correlation of [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF], leading to w CF a I:WQ Q I=S a R:QQ m • s I and w H a I:Q m • s I , respectively. The kinematic viscosity is computed from a I @I C ¡T CF =T I A I:U a IVU ¢ IH T m P • s I [START_REF] Williamson | Developing extinction criteria for fires[END_REF] where I a IS ¢ IH T m P • s I is the kinematic viscosity at the ambient temperature, T I = 293 K, and ¡T CF = 1000 K is the temperature rise at the continuous flame tip [START_REF] Cox | Some stochastic properties of fire plumes[END_REF].

The turbulent Reynolds number is then computed as Re t a w H L t = a IIUS, leading to a Kolmogorov length scale, k a L t Re t Q=R , of about 0.82 mm.

The Taylor length scale can then be deduced from a p IHL t Re t I=P % IS mm [START_REF] Pope | Turbulent flows[END_REF]. In addition, the diffusive layer thickness is estimated as l d a q D st = st % IR mm, where D st and st are the molecular diffusivity and the scalar dissipation rate at stoichiometry, respectively. The scalar 74 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.1. SUBGRID VARIANCE MODELLING dissipation rate is determined from st a a t 'exp P@erfc I @PZ st AA P with ' a Q@T st =T I A I=P C I P =@RP@T st =T I A I=P C I P A. T st = 2000 K and Z st = 0.055 are the temperature and the mixture fraction at the stoichiometry, respectively.

The strain rate is calculated from a t a H:PV k [START_REF] Williamson | Developing extinction criteria for fires[END_REF] where k a k =w k and w k a w H Re I=R t are the Kolmogorov time scale and velocity scales, respectively. Table 6.1 shows that the present LES are resolved beyond the Taylor microscale and the diffusive layer thickness for all the grids. In addition, the finest filter size, G5, is about 6 times the Kolmogorov length scale.

Concerning the boundary conditions, an inlet velocity was imposed at the burner to ensure the specified HRR for each methane flame. Fixed values of fuel mass flow rate and enthalpy flow rate are maintained according to the specified HRR. Both convective and diffusive mass and enthalpy fluxes are accounted for at the inlet. Typical entrainment boundary conditions are used for lateral sides. At the domain exit, a convective condition was used:

@ @t C u c @ @n a H (6.1)
where a fu j ; Z; Z P ; V Z ; hg. Following Craske and Reeuwijk (2013), the convective velocity u c is given by a Gaussian profile, u c a u out H exp@ r P =b P A. The mean axial velocity on the plume centreline, u out H , and the plume radius, b, at the exit height were obtained from the Heskestad correlations [START_REF] Heskestad | Engineering relations for fire plumes[END_REF]. Homogeneous Neumann condition is used for the pressure. In the rest domain, the classical wall boundary condition is imposed.

Impact of grid spacing

The performance of algebraic equilibrium, VTE and STE models are examined based on the convergence of the LES statistics with respect to the filter width, ¡, in the 57.5 kW fire plume. For a consistent LES, time-averaged values must converge, root-mean-square (rms) resolved values must increase and the SGS variance diminishes when ¡ decreases [START_REF] Boudier | Effects of mesh resolution on large eddy simulation of reacting flows in complex geometry combustors[END_REF]. These three models differ by the calculation of the SGS mixture fraction variance. Consequently, the convergence properties of mean mixture fraction and mixture fraction variance are first analysed [START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF]:

hZi a h Zi C hZ HH i a h Zi (6.2)
Li MA CHAPTER 6. FIRE PLUME DYNAMICS 6.1. SUBGRID VARIANCE MODELLING hZ HP i a hZ P i hZi P a @h ZP i hZi

P A | {z } hZ V;ReS i C @h f Z P ZP iA | {z } V Z (6.3)
For sufficiently fine LES, it is reasonable to expect that the residual mixture fraction, hZ HH i, is equal to 0 [START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF]. Note that this assumption has been used to express Eq. ( 6.3). The SGS variance reaches on the whole maximum values along the plume axis at all the heights (see Fig. 6.2). As a consequence, the dependence of mean mixture fraction, SGS mixture fraction variance and total mixture fraction variance on ¡ along the plume centreline is further investigated in Figs. 6.15 in order to provide more quantitative information. Figure 6.1(g) and (h) and the diagram (b I ) in Fig. 6.15 show clearly that the STE model exhibits convincing convergence for mean mixture fraction on the different grids at z a H:R m and 0.6 m. In the CF region (z a H:I m and H:P m), the STE solutions are more grid-dependent, especially along the plume wings (see Fig. 6.1(e) and (f)).

Impact

Nevertheless, a reasonable convergence state is reached along the plume axis for these heights on the grids G3-G5 (see Fig. 6.15(b I )). On the other hand, the convergence of VTE is more complicated whatever the height at the vicinity of the fire plume axis (see Fig. 6.1 (a)-(d) and 6.15(a I )). In addition, similar grid effects as for STE are observed along the plume wings in the CF region (see Figs. 6.1(a) and (b)). Figure 6.1 shows also that the VTE model predicts higher mixture fraction close to the fire plume axis (see also Fig. 6.15) whereas the STE model predicts slightly wider radial distributions. The radial profiles of SGS mixture fraction variance are plotted in Fig. 6.2 for all the grids. In accordance with the criterion of consistent LES, hV Z i decreases continuously as ¡ is reduced for STE. This behaviour is further highlighted in the diagram (b P ) of Fig. 6.15. The rate of decrease for VTE is in overall slower than that of STE up to G4 and accelerates between G4 and G5 as illustrated in the diagram (a P ) of Fig. 6.15. Figure 6.2 shows also that, for a given grid up to G4, the z a H:I m and z a H:P m (see also the diagrams (a P ) and (b P ) of Fig. 6.15) and in the profile wings at all heights. This behaviour is consistent with the observations made in other flame configurations [START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF][START_REF] Kaul | A posteriori analysis of numerical errors in subfilter scalar variance modeling for large eddy simulation[END_REF][START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF][START_REF] Jain | On the non-equilibrium models for subfilter scalar variance in large eddy simulation of turbulent mixing and combustion[END_REF] that both models tend towards similar solutions. As expected, Fig. 6.3 shows that hZ V;ReS i increases as the grid becomes finer and finer up to G4 for both models. In addition, it can be observed that the corresponding rate of increase is higher close to the source (z a H:I m and H:P m). For G5, hZ V;ReS i keeps on increasing for both models at z a H:I m and H:P m whereas the shape of the radial profiles changes at z a H:R m and H:T m, to the centreline and wider radial extensions. STE reaches an approximate convergent state on G3-G5 grids for Z V;tot as shown in Fig. 6.4(e)-(h). This is confirmed by the diagram (b Q ) of Fig. 6.15. On the other hand, Fig. 6.4(a)- (d) and the diagram (a Q ) of Fig. 6.15 show that the convergence is less convincing for VTE whatever the height, especially close to the fire plume axis. A careful examination of the results shows that the differences between VTE and STE are substantially less important on G5 than on G4 and G3, which reinforces the feeling that both models converge towards the same solution when the grid becomes sufficiently fine. These differences between VTE and STE are the following. First at z a H:I m, the VTE model exhibits a more pronounced double-peak radial profile than the STE model. This behaviour is also observed for hV Z i and hZ V;ReS i (see Figs. 6.2 and 6.3). Second, close to the plume axis, the VTE model predicts slightly higher Z V;tot than the STE model (see the Li MA CHAPTER 6. FIRE PLUME DYNAMICS 6.1. SUBGRID VARIANCE MODELLING diagrams (a Q ) and (b Q ) of Fig. 6.15). Third, it appears also that the STE model predicts a slightly higher lateral spreading. In order to complete the resolution larger than 1 cm, predict a significantly narrower radial spreading of the fire plume than the grids G3-G5, especially in the CF region (z a H:I m and H:P m). These mesh effects are particularly remarkable for h¡Ti, T;tot and w whereas they are less pronounced for hwi. In addition, the radial extensions of hwi and w are insensitive to further grid refinement from G3 to G5 whereas those of h¡Ti and T;tot keep on being enhanced in the CF region. This latter behaviour is less pronounced for STE than for VTE. The mixing in the lower Li MA CHAPTER 6. FIRE PLUME DYNAMICS 6.1. SUBGRID VARIANCE MODELLING part of the CF region is driven by the formation and growth of initially laminar non-dissipative instabilities in the near source region of the flow [START_REF] Tieszen | Chapter 2. Transport phenomena that affect heat transfer in fully turbulent fires[END_REF]. These instabilities grow through the vortex dynamic and become nonlinear and energy bearing. This process is illustrated in Fig. 6.9 that shows instantaneous snapshots of the density field in the centreline x z plane over a sequence of 0.4 s. The formation and the growth of initially laminar bubble structures is clearly observed for the grids G3 to G5 whereas G1 and G2 are 84 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.1. SUBGRID VARIANCE MODELLING not sufficiently fine to capture these details. In addition, this figure shows that these instabilities become better resolved as the grid is refined from G3 to G5.

As discussed in Ref. [START_REF] Tieszen | Toward the development of a LES-SGS closure model for buoyant plumes[END_REF], the dynamic SGS turbulence model considered in the present study is dissipative in nature and is not capturing the mixing resulting from these structures. As a consequence, the present results evidence that grid resolution has a profound impact on the dynamic of the flow in the CF region and suggest that grid resolutions coarser than 1 cm are not able to describe adequately the formation of the initial base instability near the edge of the fire plume. These observations are in accordance with those of Tieszen et al. ( 2004) for 1 m diameter helium plume, who also showed that too coarse grids cannot capture the formation of the instabilities and their subsequent growth. In addition, Figs. 6.5 to 6.8 emphasize the main observations made previously. i) STE exhibits a better convergence behaviour than VTE on G3 to G5 for the temperature statistics (see Figs. 6.5 and 6.6). On the other hand, both models exhibits a reasonable convergence on G3-G5 for the axial velocity statistics (see Figs. 6.7 and 6.8). ii) A careful examination of Figs. 6.5 to 6.8 shows that the differences between VTE and STE are significantly more pronounced on G3 and G4 than on G5 where both STE and VTE predictions tend to become similar. This behaviour is further highlighted in Figs. 6.10 to 6.13 that compare VTE and STE solutions on G4 and G5. The observed differences are consistent with those described for mean mixture fraction and mixture fraction variances and are evidenced by the comparison of VTE and STE solutions on G4 reported in the Figs. 6.10 to 6.13. STE exhibits in overall a wider fire plume spreading at all heights. In addition, as for mixture fraction variances, the double peak radial profiles, observed for h¡Ti, T;tot and w at z a H:I m and resulting from the fact that close to the burner the hightemperature combusting sheets occur predominately close to the plume edge, are more pronounced for VTE than for STE. Finally, it can be also observed that T;tot is higher for STE than for VTE (see Fig. temperature gradient that drives the formation and the growth of the laminar instabilities in the near source region of the flow, are higher for STE than for VTE. This is evidenced in Fig. 6.10 that compares the temperature rise above the ambient for STE and VTE on G4. Therefore, the resolved-scale mixing is stronger for STE than for VTE. This is illustrated in the Fig. 6.14 that com-Li MA CHAPTER 6. FIRE PLUME DYNAMICS 87 This explains that STE predicts in overall wider radial profiles and lower centreline mixture fraction. This explains also why the double peak radial profiles observed for hV Z i, hZ V;ReS i, Z V;tot , h¡Ti, T;tot and w at z a H:I m are less pronounced for STE than for VTE. In addition, these interactions between SGS mixing and resolved-scale mixing in the CF region through the combustion process may explain that the differences between VTE and STE are higher in the present configuration than for the SANDIA flame D where, as stated by [START_REF] Kemenov | Modelling effects of subgridscale mixture fraction variance in LES of a piloted diffusion flame[END_REF], the flow is relatively insensitive to the incurred errors on the SGS variance. Table 6.2 shows the puffing frequency, f, predicted by both VTE and STE for the differ- ent grids. These predictions can be compared with the correlation of [START_REF] Cetegen | Experiments on the periodic instability of buoyant plumes and pool fires[END_REF], i.e. f a I:S q I=D eq a P:SV Hz where D eq a H:QR m is the equivalent burner diameter. It can be observed that the puffing frequencies computed with STE and VTE converge towards 2.5 Hz on G3-G5. The dependence of the radiant fraction and the optical thickness, defined as the part of the radiative emission reabsorbed within the flame (Nmira et al., 2020a)

Q abs =

Q emi , on ¡ are reported in Table 6.3. The radiant fraction computed with STE is nearly constant on G3-G5 with a value around 0.27 whereas that computed with VTE is more sensitive to the grid. It can be also observed that the differences between the two models are reduced as the grid is refined from G3. The flame optical thickness is less sensitive to the grid resolution than the radiant fraction and is around 0.47-0.48 on G3-G5 for the two models. These results show clearly that this fire plume cannot be assumed as optically-thin.

Impact of grid spacing on the algebraic model

The convergence behaviour of the algebraic model is investigated in Fig. 6.15(c). Figure 6.15c I shows that mean mixture fraction exhibits a good con- vergence for all the grids at z a H:R m and z a H:T m. On the other hand, the convergence is not established in the CF region (z a H:I m and z a H:P m) where the mean mixture fraction increases continuously as the turbulence resolution scale is reduced (see Fig. 6.15(c I )). The SGS mixture fraction vari- ance does not behave as expected for consistent LES along the plume axis at z a H:I m, z a H:P m and z a H:T m where it increases as the grid is refined up to G3 before reaching a nearly constant value as ¡ is further reduced (see Fig. 6.15(c P )). The mean total variance increases continuously as the grid size is refined and, therefore, does not convergence whatever the grid resolution more particularly in the CF region (see Fig. 6.15(c Q )). In addition, Table 6.2

shows that the convergence is not established for the puffing frequency whatever the grid. It can be also observed by comparing the diagram (c P ) and the diagrams (a P ) and (b P ) of Fig. 6.15 that, whatever the grid, the algebraic model z a H:P m than both STE and VTE models. This underprediction is in line with observations made in other simpler flames [START_REF] Jiménez | Subgrid scale variance and dissipation of a scalar field in large eddy simulations[END_REF][START_REF] Kaul | Numerical errors in the computation of subfilter scalar variance in large eddy simulations[END_REF][START_REF] Knudsen | Modeling scalar dissipation and scalar variance in large eddy simulation: Algebraic and transport equation closures[END_REF][START_REF] Jain | On the non-equilibrium models for subfilter scalar variance in large eddy simulation of turbulent mixing and combustion[END_REF] and, for the present fire plume configuration, can be related to the highly non-equilibrium nature of buoyancy-driven flows as described by [START_REF] Bakosi | Exploring the beta-pdf in variabledensity turbulent mixing[END_REF]. Figure

Comparisons with experimental data

Figures 6.17 and STE (Figs. 6.17(a I ), (a P ), (b I ) and (b P )). Both predictions are similar and follow the same scaling as the experiments. For velocity, the deviation of these models from the correlation near the burner surface shows a similar trend and magnitude as the deviation of the measurement points to the correlation. for thermocouple radiation [START_REF] Cox | Some source-dependent effects of unbounded fires[END_REF], and the data reported by [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF] for a 28 kW methane fire plume generated by circular burner with a diameter of 25 cm. The axial resolved and total temperature fluctuations are compared with the data of [START_REF] Cox | Some stochastic properties of fire plumes[END_REF] and [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF] (see the last two lines of Fig. 6.17). [START_REF] Cox | Some stochastic properties of fire plumes[END_REF] measured temperature fluctuations by using uncompensated 13 m thermocouples. An analysis performed by these authors showed that these probes can adequately follow signals with frequency up to 10 Hz. [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF] also measured temperature fluctuations by using 50 m thermocouple with a compensation technique based on a thermocouple time constant of 30 ms thorough the flame zone. They suggested that their data may significantly underestimate the temperature fluctuations. The data of [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF] follow quantitatively the same trends as those of [START_REF] Cox | Some stochastic properties of fire plumes[END_REF]. Both VTE and STE predictions reproduce correctly similar scaling and trends as the experiments (see Figs. 6.17(a Q ), (b Q ), (a R ) and (b R )). The total rms of temperature fluctuation overestimate the experi- mental data which can be attributed, as discussed above, to the measurement techniques (see Fig. 6.17(a R ) and (b R )). Nevertheless, it is interesting to note that the resolved temperature fluctuations predicted by both VTE and STE models are in better agreement with the experimental data (see Fig. 6.17(a Q ) and (b Q )). As for mean temperature and axial velocity, the algebraic model introduces more scatter for the different HRR than observed experimentally and predicted by both VTE and STE, especially in the CF region (see Figs.

6.17(c Q ) and (c R )). In addition, as expected from the discussion in section 6.1.3, the resolved and total fluctuating temperatures are significantly underestimated in the CF region, especially for HRRs higher than 33 kW (see Figs.

6.17(c Q ) and (c R )). Figure 6.18 compares VTE and STE radial profiles of mean temperature on G3 and G4 with data extracted from the experimental results

of McCaffrey for the 45 kW fire plume at different heights. The correlations of [START_REF] Cox | A study of the deterministic properties of unbounded fire plumes[END_REF] for a 47 kW fire plume are also reported. The two sets of experimental data are in good accordance at heights of 0.1, 0.5 and 0.6 m but present large discrepancies for heights of 0.2 to 0.4 m. On G3, the largest discrepancies between the model and the experimental data are observed at 0.2 and 0.3 m that correspond to regions where the intermittency is the largest.

At these heights, the model underestimates significantly the measurements for grid is refined to G4, a much better agreement is observed at these heights. As expected the STE predicts on the whole slightly wider radial profiles than the VTE.

Conclusions

The following conclusions can be drawn:

1. The simulations confirm the limitations of equilibrium algebraic model for purely buoyant flows.

2. The simulations with non-equilibrium transport models show that computational grids with resolutions coarser than about 1 cm cannot adequately resolve the initial base instability near the edge of the fire plume that plays an important role to describe accurately the mixing process,

Li MA CHAPTER 6. FIRE PLUME DYNAMICS 97 6.2. EFFECTS OF BURNER'S GEOMETRY especially in the continuous flame.

3. For resolutions finer than about 1 cm, the STE model is less sensitive to grid refinement than the VTE formulation and predicts a higher mixing that results in a slightly larger lateral expansion of the fire plume. The differences between the two models tend to be reduced with the turbulent resolution scale and comparable predictions are obtained on the finest grid.

4. For grids finer than 1 cm, both VTE and STE are in quantitative agreement with available experimental data in terms of axial velocity, temperature and rms of temperature fluctuations.

5. Following these conclusions, we adopt the STE model in the remaining of the simulations reported in this work.

Effects of burner's geometry

Experimental studies revealed the importance of burner boundary conditions on the fire plume dynamics. [START_REF] Orloff | Simplified radiation modeling of pool fires[END_REF] and [START_REF] Orloff | Froude modeling of pool fires[END_REF] observed that the fuel lip height, defined as the fuel level below the burner rim, has significant effects on the fire plume structure. A non-zero lip height was found to result in thicker and shorter fires [START_REF] Orloff | Simplified radiation modeling of pool fires[END_REF], which was attributed to effects on the transition to turbulence [START_REF] Orloff | Froude modeling of pool fires[END_REF][START_REF] Ditch | Pool fires-An empirical correlation[END_REF]. [START_REF] Weckman | The oscillatory behaviour of medium-scale pool fires[END_REF] considered different burner arrangements in medium-scale acetone pool fires in order to modify the air entrainment at the pool basis. They found that the puffing frequency as well as the behaviour of large-scale structures are extremely dependent on the burner boundary conditions and recommended to specify them in future experimental works. In most of the experimental studies of pool fires and fire plumes published to date, a great attention was generally paid to the design of the burner boundary conditions and a detailed description was provided. From the early works of [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF], [START_REF] Cox | A study of the deterministic properties of unbounded fire plumes[END_REF], [START_REF] Cox | Some stochastic properties of fire plumes[END_REF], and [START_REF] Cox | Some source-dependent effects of unbounded fires[END_REF] and [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF], the burner is generally set at least one-pool diameter above the floor to minimize the effects of surrounding surfaces on the fire behaviour. In the case of pool fires, the descriptions of the experimental set-up report also the fuel lip height. Nev-98 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.2. EFFECTS OF BURNER'S GEOMETRY ertheless, burner boundary conditions in most of the modelling studies of fire plumes are either not consistent with the experiments considered for comparison [START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF][START_REF] Wu | Detailed modeling of a small-scale turbulent pool fire[END_REF] or not specified. Exceptions are the works of [START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF] and [START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF]. [START_REF] Chatterjee | Application of a subgrid soot radiation model in the numerical simulation of a heptane pool fire[END_REF] simulated the 30 cm heptane pool fire investigated experimentally by [START_REF] Klassen | Structure and radiation properties of pool fires[END_REF]. The burner was 15 cm above the floor in line with the experimental setup but the authors did not mention if the experimental fuel lip height of 5 mm was reproduced. [START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF] simulated the 30 cm methanol pool fire investigated experimentally by [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. They reported that it was important to reproduce the experimental fuel lip height in order to capture the pool dynamics. However, they did not specify if they considered a burner located one-pool diameter above the floor as in the experiments.

As a consequence, the objective of this section is, on the one hand, to further assess the capability of the present LES-based numerical model to reproduce the dynamics of medium-scale methanol pool fires investigated experimentally by [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF] and, on the other to, investigate the effects of burner boundary conditions on the predictions. A special attention will be paid to the influence of the fuel lip height and on floor effects.

Experimental and computational details

The baseline configuration is the 30.5 cm diameter methanol pool fire investigated experimentally by [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. In these experiments, the methanol feed rate was maintained at 1.35 cm transfer. The uncertainties on mean temperature were estimated to be ¦ 57 whereas those on the rms values were found to be difficult to estimate mainly due to the uncertainty on the compensation method. However, uncertainties as high as 25-307 were reported using thermocouples to obtain measurements of fluctuating temperature [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF][START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF][START_REF] Weckman | Processing effects in simultaneous velocity and temperature measurements in an intermittent combusting flow[END_REF]. This set of experimental data is completed by those obtained by [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF], [START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF] and [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF] who also considered 30 cm diameter methanol pool fires. [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] measured the radiative feedback towards the fuel surface. [START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF] reported axial and radial distributions of mean temperature that were in good accordance with the measurements of [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. They also reported axial distribution of species volume fractions. [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF] measured radial (at the level of the burner rim) and axial (at a distance of 60 cm from the pool axis) distributions of radiative flux. In addition, they measured the local total heat flux in the downward direction at 13 mm above the fuel surface.

In the baseline case, consistently with the experimental configurations reported in Refs. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF][START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF][START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF], the burner stands at one-pool diameter above the floor and fuel lip height is set equal to 1 cm (see Fig. 6.19(a)). Two other configurations are considered by modifying the burner boundary conditions while keeping the pool diameter and HRR unchanged. In order to investigate the effects of the fuel lip height on the pool fire dynamics, the first configuration, referred to as NoLip hereafter, considers the same configuration as the baseline case, except that the fuel level coincides now with the pan rim (see Fig. 6.19(b)). The second configuration, referred to as NoLipNoEnt hereafter, modifies the NoLip configuration by considering that the burner rim is now floor flush, altering thus the air entrainment at the pool basis (see Fig. 6.19(c)). Simulations were performed in a computational domain @x; y; full chemical kinetic scheme developed by [START_REF] Li | A comprehensive kinetic mechanism for CO, CH2O, and CH3OH combustion[END_REF]. The Filtered RTE is solved by using FVM with an angular mesh of N ¢ N a IT ¢ PR control angles [START_REF] Modest | Radiative Heat Transfer[END_REF]. N and N represent the polar and azimuthal angles, respectively.

zA of Q ¢ Q ¢ Q m Q . The
The mean flame height was measured as 0.51 m [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF]. As discussed above, the grid resolution is lower than 5 mm in the flaming region and can be compared to estimations of the Kolmogorov length scale, k , and the Taylor length scale, . Applying the same analysis as used in the previous section, the Kolmogorov length scale and the Taylor length scale are found to be about 0.9 mm and 15.5 mm, respectively. This analysis shows that the present LES are resolved well beyond the Taylor scale and that the filter size is about 6 times the Kolmogorov length scale. Li MA CHAPTER 6. FIRE PLUME DYNAMICS 101 6.2. EFFECTS OF BURNER'S GEOMETRY Throughout all simulations, the time step is set to S¢IH R s which corresponds to an averaged maximum CFL of 0.6. Simulations were run for 25 s and the time-averaged mean and rms values were collected over the last 19 s. The first 6 s of simulation were used to establish a statistically stationary flow.

Quality of LES

The ratio of resolved temperature variance, h g T HP ReS i a h T P i h T i P , to the total temperature variance, h g T HP i a h f T P i h T i P , is considered to assess the quality of the present LES, where the term f T P is computed by integrating T P with FDF, then compiled into the look-up library. Figure 6.20 shows radial profiles of this ratio at different heights above the burner covering the flaming region. It can be observed that more than 807 of temperature variance is on the whole resolved by the present LES.

Comparison with available experimental data

Model predictions for the baseline case are compared to the available experimental data.

The predicted puffing frequency of 2.37 Hz was estimated from a Fast Fourier Transform of the time series of the axial velocity along the axis at z a H:P m. This location was selected because the corresponding experimental PSD was found to exhibit a distinct puffing frequency [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

This computed puffing frequency underestimates the experimental one of 2.80

Hz by about 187.

Figure 6.21 shows the mean temperature as a function of the normalized height, (z £ a z= Q P=S ), defined by [START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF] [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF][START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF]. ford et al., 1985;[START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF][START_REF] Weckman | Processing effects in simultaneous velocity and temperature measurements in an intermittent combusting flow[END_REF].

Consequently, the overprediction of the temperature fluctuations by the model observed in Fig. 6.23 is not surprising. However, Fig. 6.23 shows that the predicted temperature fluctuations follow the same trends as the experiments.

Figures 6.24 and 6.25 show that the radial profiles of mean axial velocity and rms values of axial velocity fluctuations exhibit an overall good agreement with the experimental data. However, close to the pool surface (z a P and z a R cm), the predicted axial velocity exhibits a M-shaped profile that is not observed experimentally and overestimates the measurements along the flame wings (see Fig. 6.24). On the other hand, axial velocity fluctuations are slightly overpredicted for z > IR cm (see Fig. 6.25).

Figures 6.26 and 6.27 show that the comparison is less satisfactory for radial velocity statistics than for axial velocity statistics. A careful examination of Fig. 6.26 shows that the experimental profiles of mean radial velocity are not symmetric with respect to the fire plume axis and that the values at the centreline are not equal to zero. This asymmetry, although less pronounced, is also observed for the radial velocity fluctuation at some heights (see Fig. 6.27)

and reflects the difficulty to make accurate measurements of radial velocity, especially at vicinity to the flame axis. Figure 6.26 shows a reasonable agreement with measurements in the wings of the profiles for z > V cm despite an overall 107 underprediction. For z V cm, larger discrepancies are observed.

At these heights, the experimental maximum value (in absolute), observed at about x a IH cm, is significantly underestimated and its predicted location is shifted away from the plume axis as compared to the measurements. Concerning the radial profiles of rms radial velocity, a good accordance between simulations and measurements is observed at the tails of the profiles (see Fig. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF], except the radial profile at z a H:T m that is taken from Ref. [START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF].

for CH Q OH, O P , CO P , CO and H P . The rapid decrease in the mole fraction of CH Q OH as well as the depletion of O P in the flaming zone are well predicted. The profiles of mole fractions of CO, CO P , and H P are also in good agreement with the experiments although the peak of CO mole fraction is overestimated.

Finally, the model underestimates significantly the H P O molar fraction over the entire fire plume axis. Surprisingly, such large discrepancies are only observed for H P O whereas, as discussed previously, a rather good agreement is observed The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

and Lock (2016) (see Fig. 6.28) shows that the experimental uncertainties associated to the measurements of H P O are significantly higher than for the other species. This can explain, at least partially, these large discrepancies.

The radiant fraction is computed by integrating the time-averaged divergence of the radiative flux over the computational domain, R a R r • q HH R dV= Q. The predicted radiant fraction is 0.261 and overestimates the measured one of 0.24

¦ 257 by about 8.757. Figure 6.29a shows the vertical distribution of radiative flux at a distance of 0.6 m from the fire plume axis. The distribution exhibits ray effects when computed with the 16 ¢ 24 angular mesh. A frozen field radiative calculation was performed with a finer angular mesh with 48 106 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.2. EFFECTS OF BURNER'S GEOMETRY [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

.0 z = 6cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ ⟨w⟩⟨[m. s -1 ] z = 8cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ z = 10cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ z = 12cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ 4 ⟨w⟩⟨[m. s -1 ] z = 14cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ 4 z = 16cm -0.2 -0.1 0.0 0.1 0.2 0 1 2 ⟩ 4 z = 18cm -0.2 -0.1 0.0 0.1 0.2 x⟨[m] 0 1 2 ⟩ 4 ⟨w⟩⟨[m. s -1 ] z = 20cm -0.2 -0.1 0.0 0.1 0.2 x⟨[m] 0 1 2 ⟩ 4 z = ⟩0cm
¢ 96 control angles in order to improve this behaviour and the corresponding prediction is plotted in Fig. 6.29a.

In this calculation, instantaneous fields of absorption coefficient and emission term, computed with the 16 ¢ 24 angular mesh, are saved during the statistically stationary state. These fields are used to solve the instantaneous RTE with the finer angular mesh and, subsequently compute the time-averaged radiative outputs. This approach is justified by the fact that the finer mesh was found to have no influence on the prediction of the radiative heat trans- The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

showed that radiative outputs within the flame, where radiation is isotropic, can be accurately computed with a coarse angular discretization whereas a finer one is required to avoid ray effects on radiative flux outside the flame. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF]. fluxes were also computed with the IT ¢ PR angular mesh. As discussed by [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF], the experimental radiative feedback decreases continuously from the centre towards the edge. This continuous decrease is also observed for the simulations although it is less pronounced. In addition, the model overestimates the experimental data. The model ignores the radiative contribution of the methanol whose the importance for estimating the radiative feedback is recognized [START_REF] Wakatsuki | Effects of fuel absorption on radiative heat transfer in methanol pool fires[END_REF]. This can explain at least The experimental data are taken from Ref. [START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

fuel lip height of 1 cm whereas that reported by [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] was 0.5 cm. Figure 6.29(c) compares the predicted total heat feedback at the fuel surface with that measured by [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF]. The experimental heat flux is approximately constant for x < T cm with a value around 26 kW • m P and, then, decreases with distance towards the pool edge to reach 16 kW • m P at x a IQ cm. As the distance towards the pool edge is further increased, a very sharp decrease of the total heat flux is observed to reach 3.7 kW • m P at x a IR:S cm. The model reproduces correctly this behaviour. Nevertheless, the computed total heat flux underestimates in overall the measurements with a maximum deviation of about 207 at x a IH cm.

Pool boundary condition effects

This section analyses the differences induced by the different burner boundary conditions on the fire plume structure. These differences can be attributed to effects on the formation and growth of the flame base non-dissipative laminar instability near the edge of the pool that develops periodically to form energy containing large-scale toroidal vortices [START_REF] Tieszen | Chapter 2. Transport phenomena that affect heat transfer in fully turbulent fires[END_REF]. These vortices govern the flow pattern, the air entrainment as well as the mixing and combustion processes [START_REF] Cetegen | Experiments on the periodic instability of buoyant plumes and pool fires[END_REF][START_REF] Cetegen | A phenomenological model of near-field fire entrainment[END_REF][START_REF] Hu | Flame necking-in and instability characterization in small and medium pool fires with different lip heights[END_REF]. Once generated near the edge of the source due to baroclinic and gravitational torques, these vortices convect upward and act like a pump, inducing the necking-in behaviour at the base of the flame beneath them where air is x a H:T m from the pool axis, (b) radial distribution of the radiative heat flux in the downward direction outside the burner, and (c) radiative heat feedback towards the fuel surface and total heat feedback measured at 13 mm elevation above the fuel surface. The experimental data are taken from Ref. [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF], except those for the radiative feedback that are taken from Ref. [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF].

engulfed. Figures 6.30, 6.31, and 6.32 show sequences of snapshots of temperature in the centreline (x-z) plane for the baseline case, the NoLip case and the NoLipNoEnt case, respectively. On each figure, the time step between two snapshots is 0.06 s. In these figures, the arrows indicate the directions of the density gradients, r, the pressure gradient, rp, and the gravity, g.

These snapshots are supplemented by, on the one hand, time-averaged axial and radial profiles at different heights of mean temperature and axial velocity (see Fig. 6.33) and, on the other hand, radial distributions of rms values of temperature and axial velocity (see Fig. 6.34).

Let us start to describe the sequence for the baseline case (see Fig. with the experimental observations made by [START_REF] Weckman | The oscillatory behaviour of medium-scale pool fires[END_REF].

The air entrained at the pool basis bypasses the rim and leads to a wrinkling of the flame sheet. This wrinkled flame sheet is subjected to a Rayleigh-Taylor instability as illustrated in Fig. 6.30(b). This results in the formation of a raising hot "bubble" structure whereas the air flows towards the fuel surface as evidenced in Fig. 6.30(c). The growth of the hot structure results in an-Li MA CHAPTER 6. FIRE PLUME DYNAMICS 113 The growth of this coherent structure and the characteristic necking behaviour beneath it are illustrated in Fig. 6.30(f) to (h). on the one-hand, the density gradient and, on the other hand, gravity and pressure gradient, although still presents, is lower than for the baseline case (see Figs. 6.31(a) and (b)). As a result, the formation of the hot "bubble" structure as described for the baseline case is not observed although Fig. 6.31(c) shows the appearance of smaller instabilities. As for the baseline case, the growth of this instability results in an unstable configuration (see Fig. 6.31(d)) and in the generation of a vortex that develops as it convects upward (see Figs.

6.31(e) to (h)).

This comparison illustrates the significant effects of the fuel lip height on the formation of instabilities. The inclusion of the fuel lip height modifies considerably the flow pattern due to the formation of hot "bubble" structures stemming from the interactions between the air entrainment at the pool basis, the burner rim and the flame sheet. These hot structures contribute to enhance significantly the mixing at vicinity of the burner as illustrated by the much wider

Li MA CHAPTER 6. FIRE PLUME DYNAMICS 115 6.2. EFFECTS OF BURNER'S GEOMETRY fire plume (see Figs. 6.33 and 6.34) and the higher temperature and velocity fluctuations (see Figs. 6.34) observed when the fuel lip height is considered. In addition, Figs. 6.33b-d and f-h show that the fire plume close to the burner is hotter and rises at a higher velocity in presence of the fuel lip height. As described by [START_REF] Orloff | Simplified radiation modeling of pool fires[END_REF] and [START_REF] Orloff | Froude modeling of pool fires[END_REF] from their experiments, this enhanced mixing results in a shorter flame, as illustrated in Fig. 6.33(a)

and (e) by the lower temperature and axial velocity observed along the axis in the intermittent flame region and in the inert plume (z £ > H:HS m • kW P=S ).

On the other hand, considering or not the fuel lip height has weak effect on the puffing frequency that is 2.27 Hz for the NoLip case instead of 2.37 Hz for the baseline case.

Comparing Figs. 6.31 and 6.32 shows that modifying the air entrainment at the base of the pool changes the flow pattern. In the case where the pool is located one-pool diameter above the floor (NoLip), the air entrained at the edge of the pan comes from both the sides and the bottom (see Fig. 6.31(a))

whereas, in the NoLipNoEnt case, the floor prevents the part coming from the bottom and air entrainment comes only from the sides (see Fig. 6.32(a)).

This stronger lateral air entrainment in the NoLipNoEnt case has two effects.

First, it pushes the flame sheet anchored to the pan edge towards the liquid surface and, as a result, reduces the misalignment between density gradient and both gravity and pressure gradient as compared to the NoLip case (see Figs. 6.31(a) and 6.32(a)). This reduces both gravitational and baroclinic torques and, in turn, the strength of the first instability. This is illustrated by comparing Figs. 6.31(c) and 6.32(b). Second, the coherent vortices, generated from the second instability, are significantly smaller than in the NoLip case as observed by comparing Fig. 6.31(e) to (h) with Figs. 6.32(e) and (f). As a consequence, the pumping capacity is lower in the NoLipNoEnt case which accelerates the formation of a new vortex [START_REF] Hu | Flame necking-in and instability characterization in small and medium pool fires with different lip heights[END_REF], leading to a flame puffing frequency of 2.97 Hz higher than that of 2.27 Hz observed for the NoLip case. Such increase in the puffing frequency as the air entrainment at the flame base is altered can be also observed in the PSD reported by [START_REF] Weckman | Processing effects in simultaneous velocity and temperature measurements in an intermittent combusting flow[END_REF] from their experiments in 30 cm acetone pool fires. The fact that air is entrained preferentially from the sides explains also why the fire plume is narrower in the NoLipNoEnt case (see Figs. 6.34). On an opposite way, the inclusion of the fuel Lip height in the simulations (baseline case) induces significant modifications in both radiative and total fluxes. As discussed by [START_REF] Babrauskas | Estimating large pool fire burning rates[END_REF], the presence of the fuel lip height produces a more emissive flame volume, resulting in a higher radiative flux (see Fig. 6.35(a)). This increase in flame emissivity can be explained by the enhanced mixing process produced by the fuel lip height that leads, as discussed previously, to a wider, hotter, and more turbulent fire at vicinity of the fuel surface (see Figs. 6.33 and 6.34). Figure 6.35(b) shows that, for the same reasons, the total heat flux for z < IH cm is higher for the baseline case. Close to the burner rim, the total heat fluxes predicted for the NoLip and NoLipNoEnt cases increases sharply owing to an increase in the convective flux. This increase in convection results from the higher proximity between the flame sheet and the fuel surface at the edge of the burner in these configurations. This is not observed for the baseline case where the flame sheet is anchored at the top of the burner rim, leading to lower temperature at vicinity 118 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.2. EFFECTS OF BURNER'S GEOMETRY 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 mm elevation above the fuel surface. The experimental data for radiative and total heat fluxes are taken from Refs. [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] and [START_REF] Kim | Energy balance in mediumscale methanol, ethanol, and acetone pool fires[END_REF], respectively.

of the fuel surface. This increasing importance of convection close to the edge of the pool in absence of fuel lip height is supported by the experimental results

of [START_REF] Hamins | Heat Feedback to the Fuel Surface in Pool Fires[END_REF] who observed a similar trend as the fuel lip height was reduced from 0.5 cm to 0.1 cm.

Conclusions

The following conclusions can be drawn:

1. An exhaustive comparison with experiments was carried out in terms of puffing frequency, mean and rms temperature and velocities, mean molar fractions of major species and radiative loss to the surrounding.

Numerical simulations were found to reproduce the experimental data with fidelity without introducing any adjustable constant.

2. Considering a non-zero lip height alters substantially the flow structure

Li MA CHAPTER 6. FIRE PLUME DYNAMICS 119 6.2. EFFECTS OF BURNER'S GEOMETRY and results in significantly wider fire plumes and shorter flames which affects both radiative and total heat feedback towards the fuel surface.

3. Considering that the burner rim is floor flush has a moderate influence on the fire pool structure as compared to the NoLip case but enhances significantly the puffing frequency owing to a reduction in the pumping capacity of the large-scale vortices.

4. These results show clearly that the experimental burner boundary conditions have to be reproduced scrupulously in the simulations for relevant validations of numerical models. 120 CHAPTER 6. FIRE PLUME DYNAMICS Li MA

Chapter 7

Radiation in fire plumes

The modelling of radiative heat transfer (RHT) in turbulent flows adds a difficulty since it requires a precise solution of the RTE, a proper modelling of the spectral dependence of radiating species and a proper evaluation of the TRI.

In the first section, the effects of SGS TRI in LES of ethanol pool fires ranging from medium to large sizes are studied.

In the second section, the modelling capability of several radiative property models, including the RCFSK model [START_REF] Solovjov | The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model[END_REF], and grey and nongrey versions of the WSGG [START_REF] Modest | Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications[END_REF] in 1 m methanol pool fire studied experimentally by [START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF] is investigated. This fire plume will be also considered to investigate the effects of the angular discretization of the radiative transfer equation solver on radiative outputs.

7.1 Assessment of subgrid-scale turbulenceradiation interactions

Numerical resolution

Throughout this chapter, the Favre-filtered governing Eqs. 3.1 to 3.4 are solved by the proposed solver described in the Chapter 4. The second-order Crank-Nicolson scheme is used for time advancement. The Superbee-TVD scheme [START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF] is applied for convection in the mixture fraction transport to limit numerical oscillation whereas a second-order CDS is INTERACTIONS used for diffusion. For the momentum equation both convective and diffusive terms are discretized by using a second-order CDS. The subgrid contribution to the momentum stress and scalar flux is computed using DSM and DEDM models, respectively. The non-adiabatic SLF model is used to obtain the local thermochemical state relationship, slf , as a function of the mixture fraction, Z, the scalar dissipation rate, , and the enthalpy defect, X R a h h ad . The flamelet library was generated by solving the governing equations of counterflow diffusion flames in the physical space at a series of specified strain rates using the CHEMKIN code [START_REF] Lutz | OPPDIF: a fortran program por computing opposed-flow diffusion flames[END_REF]. Radiative loss was incorporated in the flamelet library by using the methodology described in the Chapter 2.

The Favre-filtered thermochemical quantities are then obtained from the state relationships by convolution with the presumed FDF. The SGS mixture fraction variance required to retrieve solutions from the filtered flamelet look-up table is obtained by the STE model.

Experimental and computational details

The baseline configuration is the H:S m diameter ethanol pool fire investigated experimentally by [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF]. This pool fire is characterized by a heat release rate (HRR, Q) of UP:S kW . According to [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF], the measured fire plume is over-ventilated. Consequently, the configuration was simplified by considering a 3 ¢ 3 ¢ 3 m Q domain with free boundary condition on the domain sides. It was checked that the dimension of the domain does not affect the flow and heat transfer. Consistently with the experiments, the fuel pan is located on a solid support at I m above the floor and surrounded by a steel collar with an outside diameter of I:I m and a burner rim height of IH mm is considered. This latter point was found to play an important role on the pool dynamics. A structured mesh with Q:US ¢ IH T cells is considered.

The mesh is refined with cell size of 5 ¢ 5 ¢ 5 mm Q in a box of 0.6 ¢ 0.6 ¢ 0.6 m Q located above the fuel pan. The cell size is progressively stretched towards the boundaries. The simulations relative to this flame will be referred to as F1D hereafter. The flamelet library was generated using the mechanism tively, whereas g denotes the gravity. The PRI characterizes the quality of the resolution with higher values corresponding to a better resolution of the fire dynamics (Maragkos et al., 2017;[START_REF] Maragkos | Towards predictive simulations of gaseous pool fires[END_REF]. The PRI reported in Table 7.1 are consistent with the highest resolution reported in the literature for similar simulations.

Flames D (m) HRR (kW ) D=¡ (-) D £ =¡ (-) No TRI £ i @h iAh Gi £ i @h iAa i @h i; h TiAI b @h TiA

Q emi = Q (-) Q abs = Q emi (-) F1D
ReS TRI £ i @ A G £ i @ Aa i @ ; TAI b @ TA Full TRI £ i G £ i a i I b
Table 7.2: Different TRI closures for filtered absorption and emission terms.

Comparison with available experimental data

This section presents the comparison between LES and the experimental data of [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF]. technique for thermal inertia [START_REF] Crauford | Experimental and numerical simulation of a buoyant fire[END_REF]. Consequently, the overall overestimation observed in Fig. 7.3 is not surprising.

The predicted puffing frequency of 1.67 Hz compares also reasonably well with the measured one of 1.96 Hz [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF]. quantitative agreement with that reported from experiments on medium-scale methane pool fires by [START_REF] Cox | Some stochastic properties of fire plumes[END_REF]. It should be pointed out that turbulent intensity for F4D reaches similar levels in both the CF and IF regions. Figure 7.4 shows also the ratio between the resolved temperature variance, h g T 0 P i ReS a h T P i h T i P and the total temperature variance.

It can be observed that more than 807 of temperature variance is on the whole resolved by the present LES for the three flames.

Effects of TRI on radiative loss

The radiant fraction can be re-expressed as R a emi

I Q abs = Q emi . emi a Q emi =
Q is the ratio of total emission to the HRR where Q abs = Q emi represents the part of total emission reabsorbed within the flame, with

I Q abs = Q emi
being the flame transparency (Consalvi and Nmira, 2019). emi increases significantly with the flame size from about 477 for F1D to more than 807 for F4D. On the other hand, Q abs = Q emi increases also with the flame size from 577 for F1D to 727 for F4D. Therefore, the moderate increase in R with the flame size reported in [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF].

The contributions of the resolved-scale, SGS and total fluctuations to R can be evaluated from the following decomposition:

@ R A F ullT RI a @ R A NoT RI C @ R A ReST RI @ R A NoT RI | {z } ReST RI C @ R A F ullT RI @ R A ReST RI | {z } SGST RI | {z }
T RI (7.1) Table 7.3 shows that @ R A NoT RI is around 0.04 for the three flames. This value is considerably lower than those of 0.19 measured by [START_REF] Fischer | The structure and radiation of an ethanol pool fire[END_REF] and 0.202 predicted by the Full TRI model for F1D (see Table 7.3). These differences highlight the importance of TRI in pool fires. The contribution of TRI to radiative loss is evaluated from T RI a @ R A F ullT RI @ R A NoT RI =@ R A F ullT RI ¢ IHH. Table 7.3 shows that TRI represents about 807 of the radiative loss. This high contribution agrees with the results reported in the literature for medium-scale non-luminous pool fires [START_REF] Consalvi | Influence of turbulence-radiation interactions in laboratory-scale methane pool fires[END_REF][START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF] and can be explained, on the one hand, by the rather low time-averaged temperature in pool fires and, on the other hand, by the high level of turbulence intensity due to the puffing process (see Fig. 7.4). In addition, this percentage increases slightly with the flame size (see Table 7.3). ReS R a @ R A ReST RI @ R A NoT RI =@ R A F ullT RI ¢ IHH and SGS R a @ R A F ullT RI @ R A ReST RI =@ R A F ullT RI ¢ IHH quantify the contributions of resolved and SGS fluctuations to R , respectively (see to radiative loss appears to be significant and increases with the pool size.

It represents about 187 for F1D and reaches 337 for F4D. SGS T RI a @ R A F ullT RI @ R A ReST RI = @ R A F ullT RI @ R A NoT RI ¢IHH quantifies the contribution of SGS fluctuation to TRI which increases from 237 for F1D to 407 for F4D. This illustrates further the increasing importance of the contribution of SGS TRI with the pool size.

7.1.7 Effects of SGS TRI on radiative heat flux 7.3). As a consequence, the temperature is overpredicted in the IF region (z £ > H:HV m • kW P=S ) with a maximum deviation of 82 K at z £ a H:P m • kW P=S (see Fig. 7.6(a)). This induces an overestimation of the velocity by about 107 (see Fig. 7.6(d)).

Conclusions

LES involving state of the art combustion and radiation sub-models have been exercised to evaluate and quantify the contributions of TRI in ethanol pool fires ranging from medium to large size. The following conclusions can be drawn:

1. The model reproduces well the experimental data in terms of temperature, species concentration, puffing frequency and radiant fraction.

2. TRI accounts for more than 807 of the radiative loss in non-luminous pool fires.

3. For LES that resolves more than 807 of the temperature variance, the 7.2 Influence of gas radiative property models

Experimental and computational details

Experimental details

The 1 m diameter methanol pool fire investigated experimentally by [START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF] is considered for simulations. The burner was located 30 cm R;S , measured by [START_REF] Klassen | Structure and radiation properties of pool fires[END_REF].

Computational details

Simulations are performed in a computational domain @x; y; zA of W ¢ W ¢ U m Q (see Fig. 7.7). In the present study, z represents the vertical direction. Consistently with the experimental configuration, the burner is located 30 cm above the floor and the fuel lip height is set equal to 1 cm. The filtered RTE (see Eq. 3.51) is solved by using the FVM [START_REF] Raithby | A finite-volume method for predicting a radiant heat transfer in enclosures with participating media[END_REF][START_REF] Lee | Finite volume method for radiation heat transfer[END_REF] [START_REF] Kim | A new angular discretization scheme of the finite volume method for 3-D radiative heat transfer in absorbing, emitting and anisotropically scattering media[END_REF]. In the FT n FVM, the polar angle, , is divided uniformly into an even number, n, while the azimuthal angle, , is uniformly divided into the numbers of the sequence of R, V, IP,...,Pn R, Pn, Pn, Pn R,..., V and R in each level of the polar angle [START_REF] Kim | A new angular discretization scheme of the finite volume method for 3-D radiative heat transfer in absorbing, emitting and anisotropically scattering media[END_REF]. This results in n ¢ @n C PA control angles.

Using a frozen field analysis is justified by the fact that meshes finer than 16

¢ 24 were found to have no influence on the radiative heat transfer inside the flame and to change only the vertical distributions of radiative flux outside the flame. Refining the angular mesh further than 48 ¢ 96 was found to have no effect on the vertical distributions of radiative flux outside the flame and this discretization will be used as reference.

All simulations were performed on the supercomputer Gaia of EDF R&D. This cluster is made of 1224 computer nodes and 42912 cores. Processor are Intel Xeon Gold 6140 CPU @ 2.30 GHz. The present LES were performed on 940 cores.

Quality of the LES

The ratio of resolved temperature variance, h g T 0 P i ReS a h T P i h T i P , to the total temperature variance, h g T 0 P i a h f T P i h T i P , is considered to assess the quality of the present LES. heights of the rms values of temperature fluctuations. Figure 7.11(a) shows that the measured axial temperature fluctuation is of the order of 290 K very close to the burner and decreases down to 247 K at about z a H:QH m. After this location, it increases with z to reach a peak of about 330 K at z a H:V m before decreasing as z is further increased. The numerical model captures well these trends. In particular, the peak location as well as the rates of increase and decrease are well reproduced. ford et al., 1985;[START_REF] Weckman | Processing effects in simultaneous velocity and temperature measurements in an intermittent combusting flow[END_REF][START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

Table 7.5 reports the total emission, Q emi a R V Q HHH emi dV , the total absorption, Q abs a R V Q HHH abs dV , the ratio of Q abs to Q emi that represents the part of emission reabsorbed within the flame and can be used to quantify the flame optical thickness [START_REF] Nmira | Turbulence-Radiation Interaction in Large Eddy Simulation of Non-Luminous Pool Fires[END_REF], the radiant fraction, R a @ Q emi Q abs A= Q, and R;S . These results show that about 59.87 of emission is reabsorbed within the flame. The predicted R is 0.269 and overestimates the experimental one of 0.22 ¦ 167 by about 227. The predicted R;S is 0.059, in closer agreement with the experimental data of 0.055 ¦ 217.

The vertical, radial and feedback distributions of radiative flux are plotted in Fig. 7.12(a)-(c [START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF].

16¢24 angular discretization. Predictions of the radiative flux in the downward direction outside the burner (see Fig. 7.12(b)) and the radiative feedback towards the fuel surface (see Fig. 7.12(c)) are in overall good agreement with the experimental data. Nevertheless, the model overpredicts the radiative flux at vicinity of the burner (see Fig. 7.12(b)) and underpredicts the radiative feedback by about IH7 (see Fig. 7.12(c)). On the other hand, the vertical distribution of radiative flux exhibits clearly ray effects, especially for H z I m (see Fig. 7.12(a)). Such effects are a major drawback of both DOM and FVM and are favoured in optically thin medium [START_REF] Coelho | Advances in the discrete ordinates and finite volume methods for the solution of radiative heat transfer problems in participating media[END_REF]. As a consequence, the present configuration is particularly sensitive to ray effetcts since radiation travels mainly a transparent medium before reaching the location x a P:HU m

where the vertical distribution is measured. The ray effects can be reduced by refining the angular mesh [START_REF] Coelho | Advances in the discrete ordinates and finite volume methods for the solution of radiative heat transfer problems in participating media[END_REF].

Radiation calculations were also performed by neglecting TRI with the 16

¢ 24 angular mesh. In these calculations, radiation is solved in decoupled manner by using time-averaged temperature and mole fractions of CO P and H P O obtained from the LES simulations. The total emission and the radiant fraction, obtained by neglecting TRI, were found to be Q emi;NoT RI a40 ¢IH Q kW and R;NoT RI = 0.051, respectively. A comparison with the predictions reported in Table 7.5 shows that TRI represents about 807 of the total emission and radiative loss. This high contribution agrees with the results reported in the literature for medium-scale non-luminous pool fires [START_REF] Nmira | Turbulence-Radiation Interaction in Large Eddy Simulation of Non-Luminous Pool Fires[END_REF][START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF] and can be explained, on the one hand, by the rather low time-averaged temperature in pool fires and, on the other hand, by the high level of turbulence intensity due to the puffing process.

Effects of angular mesh

Results in Table 7.5 and Fig. 7.12(a)-(c) show that using the uniform 48 ¢ 96 angular mesh instead of the uniform 16 ¢ 24 angular discretization does not modify the predictions of ¢ 96 angular mesh, but considers 2400 control angles instead of 4608.

Model Angular mesh

Q emi (kW ) Q abs (kW ) Q abs = Q emi (-) R (-) R;S (

Effects of radiative property models

Table 7.7 shows that the WSGG-NG overestimates significantly both the total emission and the total absorption and, by a less extent, the ratio Q abs = Q emi , the radiant fraction, R , and the normalized radiative feedback towards the fuel surface, R;S . In particular, these three latter quantities are moderately overestimated by about 8.07, 7.03 and 2.107, respectively. The WSGG-G-CB and WSGG-G-FB underestimate significantly both the total emission and the total absorption as well as Q abs = Q emi that is only of about 17.57 and 9.27

for the WSGG-G-CB and WSGG-G-FB, respectively, instead of 59.87 for the RCFSK. These results show that the grey models are not able to reproduce the radiative structure of the flame, predicting an optically-thin flame instead of an optically intermediate medium.

The WSGG-G-CB overpredicts R and R;S by 36.47 and 32.97, respectively.

On the other hand, as observed in previous studies involving decoupled radiative heat transfer calculations [START_REF] Demarco | Assessment of Radiative Property Models in Non-Gray Sooting Media[END_REF][START_REF] Fraga | On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires[END_REF], the WSGG-G-FB provides better predictions of the radiative loss than the WSGG-G-CB. Nevertheless, discrepancies of the order of 18.27 are still observed for R;S . 447 and -22.057, 18.957 and -19.917, and 40.27 and -13.977, respectively. The effects of the radiative property models on axial Consistently with the results in Table 7.7, the WSGG-NG overestimates significantly both local emission and absorption terms, but predicts a mean radiative source term in much better agreement with the RCSFK. However, the WSGG-NG produces noticeable discrepancies on the mean temperature, especially in the region located between z a H and z a H:US m. Eventually, Fig. 7.9 shows that the gas radiative models have no effect on the prediction of the puffing frequency as well as on the first harmonic. Chapter 8

Conclusions

Fire plumes under oxygen-dilution quenching conditions

In this chapter, LES of methane fire plumes under oxygen-dilution quenching conditions is conducted. As discussed in the introduction, this configuration represents an idealization of under-ventilated fire scenarios and is particular relevant for fires in nuclear plants. The UMD turbulent line fire experimental data are used as reference to validate numerical simulations. The UMD turbulent line burner experimental facility provides a canonical line fire configuration for the study of a low-strain, buoyancy-driven, fully-turbulent 8.2. FLAMELET LIBRARY CALCULATION diffusion flame (Fig. 8.1) [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. The burner features a groundglass filled, stainless-steel fuel port, measuring 5 cm wide by 50 cm long, with 1.5 mm thick side walls. The burner is centrally located at the outlet of a surrounding oxidizer port, measuring 50 cm wide by 75 cm long, with 10 cm thick side walls. A thin, 5 mm tall, 5 cm wide strip of ceramic fibreboard, is positioned above the oxidizer port and surrounding the fuel port, so top of the board is 20 mm below the lip of the fuel port. This board is used as a flow blockage to reduce the oxidizer velocity near the flame base, forcing the onset of buoyancy-generated turbulence upstream towards the fuel port and reducing the tendency to form laminar structures at the base of the flame. The oxidizer port sits 15 mm below the fuel port, while the perimeter around the oxidizer port sits at the same elevation as the fuel port.

Experimental details

Methane gas is released with a flow rate of 1.00 ¦ 0.02 g • s I , leading to a total heat-release rate of roughly 50 kW under complete combustion. The oxidizer is provided at a fixed total flow rate of 85 ¦ 7 g • s I . The oxygen molefraction in the coflow, X O2 , varies between 217, corresponding to the baseline case in pure air, and 117, which corresponds to a case well-below the oxygen extinction limit. Two kinds of configuration were considered by including or not an oxygen anchor at the base of the flame. The objective of this oxygen anchor is to reduce the flame lift-off and guarantee a prolonged stable regime.

Global extinction occurs for X O2 a 0.151 when the flame is not anchored. This value of X O2 is reduced to 0.13 when the flame is anchored.

The flame structure is characterized by measurements of the mean and rms gas temperature using 12.7-m-diameter Type-S (Pt/107Rh) thermocouples. The radiative loss from the micro-thermocouples is neglected. The flame behaviour is also characterized through measurements of the radiant fraction. The radiant fraction is determined from the measurement of a radiative heat flux at 1 m distance from the flame combined with time-resolved infrared camera imaging and a multipoint radiation source model [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. [START_REF] Pitsch | FlameMaster: A C++ computer program for 0D combustion and 1D laminar flame calculations[END_REF]. Radiative loss was incorporated in the each flamelet library by using the methodology as described in the Chapter 2. In this study, we consider SLF and FPV models. The look-up tables were parametrized by (Z; ; X R ) for SLF and (Z; c; X R ) for FPV. As discussed in Chapter 2, two definitions of the reaction progress variable are considered in this numerical study, c I and c P (see Eq. 2.17). The FPV model using c I as progress variable will be referred to as FPV-c I hereafter whereas that using c P will be referred to as FPV-c P . The advantages and disadvantages of the two definitions for the simulation of fire problems, where combustion occurs generally at low strain rates, have been discussed in Chapter 2. Using c I provides a relevant mapping of the flamelets in the stable branch whereas flamelets in the unstable branch are poorly mapped. An opposite behaviour is observed when using c P . These observations suggest that FPV-c I may be more accurate when simulating well-ventilated fires whereas FPV-c P may be better suited when the combustion is governed by the unstable branch.

Flamelet library calculation

Let us analyse the physics of oxygen-diluted quenching at the level of the Sshaped curve. Figure 8.2 shows the S-shaped curves for different X O2 under adiabatic condition. It can be observed that the scalar dissipation rate at flame extinction and the corresponding extinction temperature decrease as X O2 is reduced. This behaviour results directly from the weakening of the flame as X O2 is reduced. As a consequence, the S-shaped curves are shifted towards lower values of st as X O2 is progressively reduced. In addition, for a given st , the temperature of the stable burning branch is reduced as X O2 decreases. An opposite trend is observed for the unstable burning branch.

In addition, the relationships between T st and c st for different X O2 are plotted in Fig. 8.3. that the T st -c st curves are significantly less sensitive to changes in c P;st than to changes in c I;st . This shows that, for a given value of T st , the mass fractions of CO P and H P O at stoichiometry remains relatively unchanged as X O2 decreases whereas those of CO and H P are significantly reduced. In this work, the Favre-filtered governing Eqs. 3.1 to 3.4 are solved by the solver proposed in the Chapter 4. The second-order Crank-Nicolson scheme is used for time advancement. The Superbee-TVD scheme [START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF] is applied for convection in the mixture fraction transport to limit numerical oscillation whereas a second-order CDS is used for diffusion. For the momentum equation both convective and diffusive terms are discretized by using a second-order CDS.

The subgrid contribution to the momentum stress and scalar flux is computed using DSM and DEDM models, respectively. For both SLF and FPV models, the SGS mixture fraction variance required to retrieve solutions from the Simulations are performed in a computational domain @x; y; zA of Q:S¢Q:S¢P:Q m Q . The burner platform is lifted up by 50 cm in order to mimic the experi- Results are compared with experiments and discussed to highlight the performance of SLF and FPV models. The FPV-c I is used to perform all simulations with X O2 ranging from 13% to 21%, while simulations using FPV-c P are carried out only for X O2 a 13%, 14% and 15% to emphasis the differences brought by the choice of the progress variable in simulating fires close to oxygen extinction limit. The reason why FPV-c P is not applied to cases with X O2 ! 16% is that this definition does not respect the monotonic evolution of the progress variable over the entire mixture fraction range. As a consequence, this definition does not ensure a good mapping in the stable branch for low strain rates as discussed previously in Chapter 2.

Comparison with experimental data

Predictions obtained with SFL and FPV-c I are compared to the available exper- imental data for X O2 = 21%, referred to as the baseline case hereafter. Figure 8.5(a) shows that the measured axial temperature increases rapidly near the flame base, up to a peak value close to 1200 K, then exhibits a double-peak behaviour before decreasing for z larger than 0.25 m. The predictions obtained with both SLF and FPV-c I models are also plotted and the comparison with the experimental data indicates a good agreement. Both SLF and FPV-c I models predict a higher temperature for the first peak. In addition, over-estimations of temperature by about 150 K for the two models are also noticed between z a H:Q m and z a H:S m. In this region, temperature computed from the SLF model is slightly higher than that from the FPV model by about 50 K at most. It can be also observed that the decreasing rate of temperature from z a H:S m is well reproduced by the two models. In this latter region, the temperature computed by the FPV model is higher than that computed by the SLF model by about 50 K. The baseline case corresponds to a well-ventilated fire plume in which there is no occurrence of flame extinction. As a consequence, it is not surprising that the overall fire structure is well reproduced by both SLF and FPV-c I models.

However, slight differences appear between the two models. As discussed by [START_REF] Pierce | Progress-variable approach for large-eddy simulation of turbulent combustion[END_REF], the scalar dissipation rate value associated with the flamelet selected by the progress variable interpolation in the FPV-c I model is not expected to be equal to the one computed from the flow fields. This may explain these differences.

Local extinction and flame lift-off

When the oxygen mole-fraction in the coflow is progressively reduced from the baseline case, the flame becomes subjected to local quenching, resulting first in flame lift-off, and finally in total extinction [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. Similar behaviours were observed in laminar-coflow diffusion flames [START_REF] Bibliography Takahashi | Extinguishment mechanisms of coflow diffusion flames in a cup-burner apparatus[END_REF][START_REF] Guo | A numerical study on the effects of CO P /N P /Ar addition to air on liftoff of a laminar CH R /air diffusion flame[END_REF]. In this region, the temperature increases rapidly to reach 1000 K and both SLF and FPV models predict similar axial temperature distributions (see Fig. In this region, the flame emission predicted by the SLF model is significantly higher that predicted by both FPV-c I and FPV-c P models. Above the lift-off regions similar trends are observed for the three models. branch. This suggests that the flame is close to total extinction. This is consistent with the experimental observations when an oxygen anchored is considered [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF]. However, the present numerical simulations do not consider the oxygen anchor, and, in this case, the experiments show that the fire is fully suppressed. This inconsistency between the FPV simulations and experiments is attributed to the fact that the transition from the unstable branch to the purely mixing branch is not included in the steady FPV models Whatever the reaction progress variable definition used, the predictions of mean and rms temperature by the FPV models are significantly lower than those obtained from SLF in the flame region. The differences between FPV-c I and FPV-c P can be explained at the light of the previous discussions concerning these two formulations at low stretch rate and for combustion dominated by the unstable branch. As FPV-c P provides a better mapping of the unstable branch at low strain rate than FPV-c I , the mean and rms temperature distributions obtained with FPV-c P are expected to be more accurate. Interestingly, Fig.

8.9 shows that the FPV-c I predicts an evident re-igition process at z a H:S m that is not observed with the FPV-c P model.

Snapshots of the radiative emission term for X O2 a 13% are plotted in Fig. 8.10. The SLF model predict a substantially higher emission term because it fails to capture the flame lift-off. Consistently with the discussion made previously on temperature, the flame emission is lower for FPV-c P than for FPV-c I . processes were observed for these oxygen mole fractions [START_REF] White | Radiative emissions measurements from a buoyant, turbulent line flame under oxidizer-dilution quenching conditions[END_REF].

Global radiant fraction

As X O2 is further decreased the SLF model does not capture the change in slope observed experimentally. Similar predictions were obtained by [START_REF] Xu | Simulations of the coupling between combustion and radiation 178 BIBLIOGRAPHY Li MA BIBLIOGRAPHY in a turbulent line fire using an unsteady flamelet model[END_REF] by using an unsteady non-adiabatic flamelet formulation and the non-grey WSGG radiation model. The FPV models provide solutions in close agreement with the SLF model as long as local extinction processes are not observed. For X O2 H:IS, the solutions provided by the FPV models start to deviate from that obtained by the SLF model, showing that these models capture local extinction/re-ignition processes. However, as expected from the previous discussions, none of the FPV models are able to capture the rapid global extinction process observed for the non-anchored case. The rate of decrease in R predicted by FPV-c P is higher than predicted by FPV-c I and is more consistent with the experimental data. In particular, the FPV-c P solutions fit the experimental data obtained in the oxygen anchored case. The better ability of the FPV-c P formulation is expected from the previous discussions where FPV-c P was found to provide a better mapping of the unstable branch than FPV-c I at low strain rate.

158 CHAPTER 8. FIRE PLUMES UNDER OXYGEN-DILUTION QUENCHING CONDITIONS Li MA 8.6. CONCLUSIONS

Conclusions

Through this study, the following conclusions can be drawn:

1. As long as the fire plume combustion is complete, both SLF and FPV formulations are capable of correctly predicting the fire plume structure as well as the radiative loss.

2. Consistently with experimental observations, the FPV results show that significant local extinction appears when X O2 is reduced below 16%. In such conditions, the use of FPV models improves significantly the predictions of the SLF model. Nevertheless, the present steady FPV models do not capture the value of X O2 at which the global extinction occurs.

3. The use of a progress variable defined by the sum of CO P and H P O mass fractions provides a better mapping of the flamelets along the unstable branch at low strain rate and, in turn, a better agreement with the experimental data than that defined by the sum of CO P , H P O, H P and CO mass fractions.

Li MA CHAPTER 8. FIRE PLUMES UNDER OXYGEN-DILUTION QUENCHING CONDITIONS

Chapter 9

Conclusions and perspectives

Conclusions

The objective of this thesis was to develop a LES-based fire simulator including state-of-the-art submodels for combustion and radiation in view to simulate non-sooting well-ventilated and under-ventilated purely buoyant fire plumes. In addition, the simulations of these fire plumes was used to address specific modelling issues and the main conclusions from these studies are the following:

• Laminar instabilities that form periodically at the base of purely buoyant fire plumes and grow to generate energy containing structures that govern the mixing process are initially non-dissipative and, as such, cannot be handle by conventional subgrid-scale models of turbulence that are dissipative in nature. As a consequence, a great attention has to be paid to have a sufficiently refined grid to resolve these instabilities. In addition, numerical simulations revealed that burner boundary geometries affect the formation and growth of these instabilities and the fire plume dynamics. As a consequence, the experimental burner boundary conditions have to be reproduced scrupulously in the simulations for relevant validations of numerical models.

• Concerning the gas radiative property models, the grey WSGG models have to be avoided since they fail to reproduce the radiative structure of the fire plume. In the opposite way, the non-grey WSGG model leads to

Li MA CHAPTER 9. CONCLUSIONS AND PERSPECTIVES 161 9.2. PERSPECTIVES predictions of radiative outputs within 10% -15% of the RCFSK and can be a reasonable alternative for non-sooting pool fires.

• Neglecting subgrid-scale turbulence-radiation interaction affects significantly the predictions of the fire plume structure and radiative heat flux even for medium-scale pool fires. These effects become of increasing importance as the size of the pool fire increases.

Perspectives

This work is the part of a development program of a LES-based fire simulator to model fire spread in nuclear plants. It was a first step in these developments and, as such, is limited to the modelling a non-luminous fire plumes in non-confined environments although under-ventilated configurations were considered. As such, the perspectives of the work and the future developments are exhaustive. Among them, future works will include obviously soot production processes and related heat transfer, the coupling between gas phase and condensed phase through the development of thermal degradation models, the inclusion of solid inert or flammable walls and the associated conjugated heat transfer,... As a consequence, in order to avoid an exhaustive description of such new problems, the perspective will deal with improvements of the current fire model and the related physics:

• As discussed previously, laminar instabilities that form close to the edge of burner and plays an important role in fire dynamics cannot be handle by the current subgrid-scale turbulence models. Some attempts have been made to develop subgrid-scale models, which could reproduce the effects of such mixing related to the growth of these instabilities on the fire plume dynamics, without having to resolve the details of their dynamics [START_REF] Tieszen | Toward the development of a LES-SGS closure model for buoyant plumes[END_REF]. However, no conclusive models have been reported

to date and this task constitutes an important fundamental problem in the modelling of the fire plume dynamics.

• The non-adiabatic flamelet/presumed FDF models results in look-up tables where the thermochemical state variables are typically stored as a function of the independent parameters. In the models developed in the present study, the independent variables are the filtered mixture fraction, 162 CHAPTER 9. CONCLUSIONS AND PERSPECTIVES Li MA 9.2. PERSPECTIVES mixture fraction variance, scalar dissipation rate or progress variable and the enthalpy defect. This results in 4-entry look-up tables that need to be interpolated at each iteration of the computation sequence to retrieve the filtered variables. Currently, the thermochemical state variables are stored in structured manner as a function of the independent parameters and 4D linear interpolations are performed to retrieve the desired variables. The size of the chemistry database will increase in the future with the development of new applications, introducing severe limitations on the resolution that can be afforded in multi-dimensional look-up tables.

Several methods have been proposed to address this problem, including the use of orthogonal polynomials [START_REF] Turanyi | Parameterization of reaction mechanisms using orthogonal polynomials[END_REF], in situ adaptive tabulation (ISAT) [START_REF] Pope | Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation[END_REF], artificial neural networks (Ihme et al., 2008), high-dimensional model representation [START_REF] Rabitz | General foundations of high-dimensional model representations[END_REF], self-similarity properties [START_REF] Ribert | Tabulation of complex chemistry based on self-similar behavior of laminar premixed flames[END_REF], adaptive Cartesian meshes [START_REF] Xia | Consistent properties reconstruction on adaptive cartesian meshes for complex fluids computations[END_REF] and the kd-trees [START_REF] Shunn | Large-eddy simulation of combustion sysytems with convective heat-loss[END_REF].

Li MA CHAPTER 9. CONCLUSIONS AND PERSPECTIVES

  (c) the algebraic model. . . . . . . . . . . . . . . . . . . . . . . 6.18 Radial distribution of mean temperature at different heights for the 45 kW fire plume. . . . . . . . . . . . . . . . . . . . . . . . 6.19 A schematic diagram of the three burner boundary conditions for the: (a) baseline case, (b) NoLip case, and (c) NoLipNoEnt case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20 Radial evolution of resolved part of temperature variance at different heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Axial profile of mean temperature. The experimental data are taken from Refs.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.28 Axial profiles of mean molar fractions for: (a) methanol, (b) carbon dioxide, (c) oxygen, (d) carbon monoxide, (e) water vapor and (f) hydrogen. The experimental data are taken from Ref. (Hamins and Lock, 2016). . . . . . . . . . . . . . . . . . . . . . 111 6.29 Heat fluxes: (a) Vertical distribution of radiative flux at a distance x a H:T m from the pool axis, (b) radial distribution of the radiative heat flux in the downward direction outside the burner, and (c) radiative heat feedback towards the fuel surface

  Uniform velocity vectors are plotted to illustrate the flow pattern. . . . . . . . . 113 6.31 Snapshots of temperature at different times for the NoLip case. The time between two snapshots is 0.06 s. Uniform velocity vectors are plotted to illustrate the flow pattern. . . . . . . . . 114 6.32 Snapshots of temperature at different times for the NoLipNoEnt case. The time between two snapshots is 0.06 s. Uniform velocity vectors are plotted to illustrate the flow pattern. . . . . . . 115 6.33 Effects of pool boundary conditions on axial and radial profiles at different heights: (a)-(d) rms of temperature and (e)-(h) axial velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.34 Effects of pool boundary conditions on radial profiles at different heights: (a)-(d) rms of temperature and (e)-(h) axial velocity. of pool boundary conditions on heat feedback to the pool surface: (a) radiative flux measured at the fuel surface and (b) total flux at 13 mm elevation above the fuel surface. The experimental data for radiative and total heat fluxes are taken from

7. 1

 1 Radial profiles of temperature at different heights above the pool basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 Radial profiles of mole fractions of CO, CO P and H P O at different heights above the pool basis. . . . . . . . . . . . . . . . . . . . 126 7.3 Radial profiles of temperature fluctuations at different heights above the pool basis. . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4 Centreline evolution of the turbulence intensity (left y-axis) and resolved-part of temperature variance (right y-axis) as a function the normalized height. . . . . . . . . . . . . . . . . . . . . . . . 127 7.5 Effects of SGS TRI on the axial distribution of net radiative heat flux at a normalized distance of r £ a r= Q P=S a H:IV m • kW P=S from the fire plume axis as a function of the normalized height. 130 7.6 Effects of SGS TRI on the: (a)-(c) axial mean temperature and (d)-(f) axial mean velocity as a function of the normalized height.131 7.7 A schematic diagram of the computational domain and the locations of the radiative heat flux distributions. . . . . . . . . . 133 7.8 Radial evolution of resolved-part of temperature variance at different heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.9 Predicted power spectrum Density from axial velocity along the flame axis at z a H:T m . . . . . . . . . . . . . . . . . . . . . . 136 7.10 Axial and radial profiles of mean temperature. The experimental data are taken from Ref. (Sung et al., 2020). . . . . . . . . . . 137 7.11 Radial profiles of rms values of temperature fluctuations at different heights. The experimental data are taken from Ref. (Sung et al., 2020). . . . . . . . . . . . . . . . . . . . . . . . . . . . of angular meshes on radiative heat fluxes. The location of the distributions are described in Fig. 7.7. (a) and (d) Vertical distribution at a distance r a P:HU m from the pool axis, (b) and (e) radial distribution outside the burner along the plane of the burner rim. The radial distance is measured from the pool centre, and (c) and (f) feedback towards the fuel surface. The experimental data in the diagrams (a) and (b) are taken from Sung et al. (2020) whereas those in diagram (c) are taken from Klassen and Gore (1992). All the simulations were obtained with the RCFSK. The diagrams (a), (b) and (c) investigate the effects of the uniform meshes whereas the diagrams (d), (e), and (f) those of the FT n FVM angular discretizations. . . . . . . . 140 7.13 Effects of gas radiative property models on raditive heat flux. The location of the distributions are described in Fig. 7.7. (a) Vertical distribution of radiative flux at a distance x a P:HU m from the pool axis, (b) radial distribution outside the burner along the plane of the burner rim. The radial distance is measured from the pool centre, and (c) radiative feedback towards the fuel surface. These radiative fluxes are computed in frozen field analysis using the FVM 48 ¢ 96 angular discretization. . . 143 7.14 Effects of radial property models on the axial distributions of: (a) emission term, (b) absorption term, (c) divergence of the radiative flux, (d) mean temperature, and (e) rms values of temperature fluctuations. These radiative fluxes are computed in frozen field analysis by using the FVM 48 ¢ 96 angular discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.1 Schematic of the experimental configuration (left); Plan view of UMD line burner (right). These schemes are taken from White et al. (2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.2 Adiabatic S-shaped curves for different X O2 . . . . . . . . . . . 150 8.3 Temperature as a function of the reaction progress variable at stoichiometric conditions at different X O2 : (a) c I definition and (b) c P definition. . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.4 Illustration of the numerical configuration. . . . . . . . . . . . profiles of: (a) mean temperature and (b) rms values of temperature fluctuations. . . . . . . . . . . . . . . . . . . . . . 8.6 Radial profiles of: (a)-(d) mean temperature and (e)-(f) rms values of temperature fluctuations at different heights. . . . . . 8.7 Axial profiles of: (a) mean temperature and (b) rms values of temperature fluctuations for X O2 a15%. . . . . . . . . . . . . . 8.8 Snapshots of instantaneous volumetric emissions predicted by: (a) SLF, (b) FPV-c I and (c) FPV-c P models for X O2 a 15%. . . 8.9 Axial profiles of: (a) mean temperature and (b) rms values of temperature fluctuations for X O2 a 13%. . . . . . . . . . . . . . 8.10 Snapshots of instantaneous volumetric emissions predicted by: (a) SLF, (b) FPV-c I and (c) FPV-c P models for X O2 a 13%. . . 8.11 Global radiant fraction as a function of oxygen mole-fraction in the oxidizer co-flow, computed from the SLF and FPV models. parameters for the MMS (Shunn et al., 2012). . . . 4.2 Observed order for the 2D-AD MMS for CFL = 0.5. . . . . . . 4.3 Observed order for the 2D-ODF MMS for CFL = 0.2. . . . . . 5.1 Computation details for the previous LES of the O'Hern helium plume O'Hern et al. (2005). . . . . . . . . . . . . . . . . . . . . 6.1 Resolution parameters for the LES calculations. . . . . . . . . . 6.2 Puffing frequency computed by VTE, STE and algebraic closure models for the 57.5 kW fire plume on the different grids. . . . . 6.3 Radiant fraction and optical thickness computed by VTE and STE for the 57.5 kW fire plume on the different grids. . . . . . 7.1 Flame configurations. emi and Q abs = Q emi are computed with the Full TRI model. . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Different TRI closures for filtered absorption and emission terms.

u

  Abbrievation CDS Central difference scheme CF Continuous flame CFL Courant-Friedrichs-Lewy number DEDM Dynamic linear eddy diffusivity model DGGM Dynamic generalized gradient model DLGGM Dynamic full linear generalized gradient model DOM Discrete ordinates method DSM Dynamic Smagorinsky model EDC Eddy dissipation concept EDM Linear eddy diffusivity model FDF Filtered density function FPV Flamelet/Progress variable FS Full-spectrum FSCK Full-spectrum correlated-k-distribution model FVM Finite volume method HRR Heat release rate IF Intermittent flame LES Large-eddy simulation MMS Method of manufactured solution PDF probability density function PIV Particle image velocimetry PLIF Planar laser induced fluorescence PRI Plume resolution index PSD Power spectrum density PSR Perfectly stirred reactor RANS Reynolds averaged Navier-Stokes equations RCFSK Rank-correlated full-spectrum k-distribution model RHT Radiative heat transfer RTE Radiative transport equation SGS Subgrid-scale

  Figure 1.1: Flame structure of pool fire.

  by considering buoyant, turbulent line flames under oxidizer-dilution quenching conditions. This study highlighted the extinction mechanisms in oxygen diluted atmospheres, showing that flame extinction is preceded by a lift-off and occurs 4 fractions of 0.151 for methane and 0.138 for propane. When

Figure 2 . 3 :

 23 Figure 2.3: Profiles of temperature as a function of the reaction progress variable at stoichiometric conditions: (a) c I definition and (b) c P definition. Fuel: CH R ; Oxidizer: pure air.

Figure 2 .

 2 Figure2.5 shows the evolution of c I and c P as a function of the mixture fraction for different flamelets covering both the stable and unstable branches in the case of CH R -Air flame. It is interesting to investigate the profiles of c I and c P at small st that characterizes usually combustion in fire scenarios. It can be observed that c I provides a better mapping in the stable branch than c P since some profiles of c P overlap in this region. In the opposite way, c P provides a better mapping in the unstable branch. These observations suggest that c I is

  simulationLES is based on a separation of scales. This separation is commonly introduced by a filtering operation which decomposes the velocity and scalar fields into a resolved (filtered) part and an unresolved (subgrid-scale) part. In addition, for variable density flow, the resolved quantities are density-weighted (Favre filtered) as defined by " a . The Favre-filtered Navier-Stokes equations in low Mach number formulation supplemented with transport equations for the filtered enthalpy, h and mixture fraction, Z, are:

  and sgs u i u k are calculated by Eqs. (3.15), (3.7), and (3.5), respectively.

  53) Eqs. (3.52) and (3.53) are expressed in the case of the SLF model. Similar equations can be obtained for the FPV model. to the simulations of turbulent buoyant fire plumes and, more generally, flows encountered in fire safety applications are based on low Mach number variable-density Navier-Stokes solvers (FireFoam n.d.; McGrat-

  and Q Y are used to construct the manufactured solutions.The following iterative procedure is implemented into Code_Saturne, a finitevolume code developed at EDF, employing cell-centred collocated grids[START_REF] Archambeau | Code_Saturne: a Finite Volume Code for the Computation of Turbulent Incompressible Flows[END_REF] Code_Saturne n.d.). In the following, the superscripts n and n C I denote the known value at n¡t and the value to be computed at @n C IA¡t, respectively. The superscript k refers to the solution at each sub-iteration within one time step. At each time step, the scalar equation is first resolved to get the new scalar and density fields. Then a pressurevelocity step is performed by resolving the momentum and Poisson equations with the computed density. At each time step, sub-iterations were performed to achieve second-order accuracy and numerical stability. The second-order Crank-Nicolson scheme is employed for time advancement. For the scalar, a second-order Superbee-TVD scheme[START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF] is used for the convective term whereas a second-order central difference scheme is used for diffusion. For the momentum equation both convective and diffusive terms are discretized by using a second-order central difference scheme. The sub-iteration procedure is described below: of mass flux, pressure, source terms, and density:

  2. Solve the scalar transport equation to get the provisional scalar value, Y nCI;kCI . 3. Get nCI;kCI from Eq. (4.4) with Y nCI;kCI . 4. Update the scalar field to ensure the conservativity: Y nCI;kCI a nCI;k Y nCI;kCI nCI;kCI(4.10) 5. Solve the momentum equation to get the predicted velocity, u nCI;kCI . 6. A constant coefficient Poisson equation for pressure is obtained by introducing the continuity equation into the momentum equation and solved to ensure mass conservation. This equation is written as:

  (4.11), (4.12) and (4.13) yields the Poisson equation for the pressure increment:

Figures

  Figures 4.1(a) and (b) investigate the spatial and temporal convergences, re-

  Figures 4.1(a) and (b) investigate the spatial and temporal convergences, respectively. Grid resolutions consisting of n x =100, 200, 400, 800, 1600, 3200,

  Figures 4.1

  (c) and(d) show the time evolution of the L P -error for the scalar and velocity fields for different spatial resolutions and a time-step of 0.00125 (n t = 800).These figures confirm the convergence of the algorithm. Figure1edepicts the L P -error as a function of n x for a constant Fourier number, F o a D¡t=¡x P Li MA CHAPTER 4. NUMERICAL METHODS 4.2. ALGORITHM VERIFICATION BY MMS of 6. This figure confirms the second-order accuracy of the scheme for this problem.

Figure 4

 4 Figure 4.1: 1D-D MMS. (a) L P -error as a function of the spatial resolution for ¡t a H:HHIPS. (b) L P -error as a function of the temporal resolution for n x a TRHH, (c) Time evolution of the L P -error for density, (d) Time evolution of the L P -error for velocity, (e) L P -error as a function of the spatial resolution for a constant F o a T.

Figures 4

 4 Figures 4.2(b), (c),(d) show that the L P -error remains on the whole smooth over the simulation. In addition, these figures confirm that the second-order accuracy of the algorithm is maintained over the entire simulation.

  Figure 4.3: 2D-ODF MMS. (a) L P -error as a function of the spatial resolution for a constant CFL of 0.2. (b), (c) and (d) Time evolution of the L P -error for scalar, density and velocity, respectively.

  N x ¢ N y a IPV ¢ IPV, PST ¢ PST, SIP ¢ SIP and IHPR ¢ IHPR cells, respectively. The corresponding time steps are ¡t a I ¢ IH Q , S ¢ IH R , P:S ¢ IH R and I:PS ¢ IH R s, respectively. A no-slip boundary condition is applied at the top while a wall condition is considered at the bottom. Periodic conditions are imposed at the left and right sides.

Figure 4

 4 Figure 4.4: Contour plot of at t a H:I s (left), t a H:R s (middle) and t a H:US s (right) computed on mesh IPV ¢ IPV. The lines correspond to = 0.2, 0.5 and 0.8 levels. Red colour corresponds to the heavy fluid ( a I kg • m Q ) and blue colour to the light one ( a H:I kg • m Q ).

Figure 4 .

 4 Figure4.4 shows the density fields at t a H:I s, t a H:R s, and t a H:US s, simulated on the mesh IPV ¢IPV. They compare well with the density fields in Fig.4.5 reported by[START_REF] Mullyadzhanov | Verification of a low Mach variable-density Navier-Stokes solver for turbulent combustion[END_REF]. In addition, density fields at t a H:US s for the three finer meshes are also displayed in Fig.4.6. The density

Figure 4 Figure 4

 44 Figure 4.6: Density fields at t a H:US s computed on mesh PST ¢ PST (left), SIP ¢ SIP (middle), IHPR ¢ IHPR (right).

Figure 4

 4 Figure 4.8: Comparison with results in literatures for mesh IPV ¢ IPV (left), and for mesh SIP ¢ SIP (right).

  1: Computation details for the previous LES of the O'Hern helium plume[START_REF] O'hern | Experimental study of a turbulent buoyant helium plume[END_REF].to modelling approaches based on a LES description of turbulence.These modelling studies used either standard or dynamic Samgorinsky (SM or DSM) and standard or dynamic linear eddy diffusivity (EDM or DEDM) models to close the SGS stress tensor and scalar flux, respectively.[START_REF] Desjardin | Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume[END_REF] elucidated the mechanisms responsible for puffing. They showed that vortices are generated at the base of the plume near the outer edge where vorticity generation from buoyancy is the greatest. These vortices are then quickly destabilized leading to the formation of "finger-like" instabilities that contribute to the breakdown of the large-scale vortices and enhance the local mixing. More generally, all the reported LES were found to be able to capture the large-scale dynamics. However, the mixing rate was systematically underestimated, leading to overpredictions of mean and fluctuation of the helium concentration near the plume centreline. These discrepancies were attributed to a lack of resolution of the buoyancy-induced turbulence mixing.The present simulations considered the DSM as SGS stress model. The EDM model assumes a linear alignment with the large-scale scalar gradient which is inconsistent with the physics of turbulent convection and cannot correctly reflect the local geometric property of the SGS scalar flux vector at each time step[START_REF] Peng | On a subgrid-scale heat flux model for large eddy simulation of turbulent thermal flow[END_REF][START_REF] Higgins | The heat flux and the temperature gradient in the lower atmosphere[END_REF][START_REF] Wang | A general dynamic linear tensor-diffusivity subgrid-scale heat flux model for large-eddy simulation of turbulent thermal flows[END_REF] Wang 58 CHAPTER 5. HELIUM PLUMELi MA 5.3. COMPUTATIONAL DETAILS et al., 2007a). To correct this behaviour, two other approaches are also considered in the present work. The first is based on tensorial forms of the diffusivity for the SGS scalar flux modelling[START_REF] Wang | New dynamic subgrid-scale heat flux models for large-eddy simulation of thermal convection based on the general gradient diffusion hypothesis[END_REF], leading to the dynamic generalized gradient model (DGGM) and the dynamic full linear generalized gradient model (DLGGM). The second consists in adding a gradient model (based on Taylor expansion of the filtering operation) to either the DEDM or

Figure 5

 5 Figure 5.1 compares the experimental centreline axial velocity at z=d a H:S, where d is the source diameter, with that computed with the DGGM on the finer grid. Model predictions in terms of amplitude are in satisfactory agreement with the experimental data. In addition, a clear puffing frequency is evidenced for both the model and experiments. The model predicts a total of 8 cycles over the 6 s sequence whereas only 7 cycles are observed experimentally. A Fast Fourier Transform of the numerical signal confirms that the model overestimates the puffing frequency, predicting I:SH Hz as compared to I:QU ¦ H:I Hz for the experiments. However, a better agreement is obtained with the frequency of 1.45 Hz measured in test 4 (see Ref. (O'Hern et al., 2005)). The predicted value of I:S Hz was found to be insensitive to both SGS

Figs

  these quantities, only the simulations with the DEDM model and the coarse grid (I:T cm) are shown since their predictions were found to be insensitive to the different SGS scalar flux closures and grid refinements.[START_REF] Tieszen | Toward the development of a LES-SGS closure model for buoyant plumes[END_REF] reported the same observation for the mean radial velocity.

Figures 5 .

 5 Figures 5.2 to 5.4 show that model predictions are in good agreement with the experimental data, being on the whole within the experimental uncertainty.

Figure 5 . 2 :

 52 Figure 5.2: Radial profile of density-weighted time-averaged axial velocity at different heights. The numerical simulation were performed with the DEDM on the coarser grid. The experimental data (O'Hern et al., 2005) are represented by the largest black circles with the experimental uncertainties delimited by the dotted lines.

Figure 5 .Figure 5 . 3 :Figure 5 . 4 :Figure 5 . 5 :

 5535455 Figure5.5 shows the radial profiles of rms values of axial velocity at different heights. The predictions reported in this figure were obtained using the coarser mesh (grid spacing of 1.6 cm). Simulations obtained with the DEDM are in good agreement with the experiments from the edges of the plume to x a ¦H:I

Figure 5 .

 5 Figure 5.7 shows the radial profiles of mean and rms values of helium mass fraction predicted by the DEDM, DGGM, and DLGGM models on the coarser grid, respectively. Mean helium mass fraction near the base of the plume (z H:P m), where the flow transits from a laminar to a turbulent state, is well predicted by the DEDM (see Fig. 5.7). For z > H:P m, as observed for the axial velocity fluctuations, the DEDM reproduces well the experimental mean helium mass fraction from the edge of the plume to about x a ¦H:I m whereas it under-predicts the mixing rate close to the centreline (see Fig.

Figure 5 . 7 :Figure 5 . 8 :Figure 5 . 9 :Figure 5 Figure 5 Figure 5 Figure 5

 5758595555 Figure 5.7: Radial profile of density-weighted time-averaged helium mass fraction at different heights on the coarser grid.

  2. Puffing frequency, mean axial velocity and mean and fluctuations of radial velocity are insensitive to grid refinements and SGS scalar flux models. This result is of significant importance for fire modelling since radial ve-air entrainment. In opposite way, fluctuations of axial velocity and mean and fluctuations of helium concentration are affected by both mesh refinement and SGS scalar flux models.3. The DGGM and DLGGM provide similar results and improve mixing rate predictions as compared to the DEDM, especially for z H:R m.Adding gradient model with the DEDM improves the DEDM whereas combining it to the DGGM has no noticeable effects on the predictions of the DGGM.4. Grid refinement improves more significantly helium mass fraction and its fluctuation than SGS scalar closures. Based on this observations, the DEDM model will be used in all the simulations reported hereafter.The first part of this chapter will be dedicated to the combustion model and, more particularly, to the modelling of subgrid mixture fraction variance and filtered scalar dissipation rate closures. This study will be made by considering the McCaffrey fire plumes[START_REF] Mccaffrey | Purely Buoyant Diffusion Flames Some Experimental Results[END_REF]. The second part will be devoted to the modelling of the well-documented methanol pool fire investigated experimentally by[START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF] for an exhaustive validation of the LES-based model. In addition, the effects of burner boundary conditions on the flame structure will be investigated.These two purely buoyant diffusion flames belong to the list of target flames identified by the workshop MacFP[START_REF] Brown | Proceedings of the first workshop organized by the IAFSS Work-ing Group on Measurement and Computation of Fire Phenomena (MaCFP)[END_REF] The 1st Measurement and Computation of Fire Phenomena (MaCFP) Workshop 2017) recently initiated by members of the fire community. The goals of this workshop are to develop and validate predictive models of fire plume, to identify well-defined target flames that are suitable for modelling, and to archive detailed data sets for these target flames.

6. 1 .

 1 SUBGRID VARIANCE MODELLING ture fraction variance prediction in the framework of the SLF/presumed FDF model. Three different models are studied: the first is based on the equilibrium between the production and dissipation of the variance at these subgrid scales with a dynamic estimation of the model constant whereas the two others are non-equilibrium models, namely the variance transport equation (VTE) and the second moment transport equation (STE). Both non-equilibrium models incorporate the same model for the scalar dissipation rate. The reader can refer to the section 3.3 for a detailed description.

Figure 6 . 1 :

 61 Figure 6.1: Radial distribution of time-averaged mixture fraction with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6 . 3 :

 63 Figure 6.3: Radial distribution of resolved mixture fraction variance with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.4: Radial distribution of total mixture fraction variance with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6 . 5 :

 65 Figure 6.5: Radial distribution of time-averaged temperature with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6 Figure 6 . 7 :

 667 Figure 6.6: Radial distribution of total rms temperature with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6 . 8 :

 68 Figure 6.8: Radial distribution of rms resolved axial velocity with VTE model (a)-(d) and STE model (e)-(h) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.9: Instantaneous snapshots of the density field, computed with the STE model on the different grids, in the centreline (x-z) plane over a sequence of 0.4 s for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.11: Radial distribution of total rms temperature with VTE and STE models on G4 ((a)-(d)) and G5 ((e)-(h)) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.12: Radial distribution of mean axial velocity with VTE and STE models on G4 ((a)-(d)) and G5 ((e)-(h)) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.13: Radial distribution of rms resolved axial velocity with VTE and STE models on G4 ((a)-(d)) and G5 ((e)-(h)) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.14: Radial distribution of rms resolved temperature with VTE and STE models on G4 ((a)-(d)) and G5 ((e)-(h)) at different heights for the 57.5 kW fire plume.

Figure 6

 6 Figure 6.15: Convergence of mean mixture fraction (index 1), subgrid mixture fraction variance (index 2) and total mixture fraction variance (index 3) with grid spacing along the plume axis for: (a) VTE, (b) STE and (c) the algebraic model.

6Figure 6

 6 Figure 6.16: Axial evolution of the time-averaged production and dissipation rates of the subgrid mixture fraction variance computed by VTE on the grid G5 for the 57.5 kW fire plume.

  Figures 6.17 shows the axial profiles of mean axial velocity (first line, index 1), mean temperature rise (second line, index 2), resolved temperature fluctuation (third line, index 3) and total temperature fluctuation (fourth line, index 4) as a function of the normalized height for the five plumes investigated experimentally by McCaffrey (1979). Model predictions obtained on G3 with VTE (first column, diagrams a), STE (second column, diagrams b) and the algebraic model (third column, diagrams c) are compared with experimental data. McCaffrey (1979) found that scaling the height and axial velocity by Q P=S and Q I=S , respectively, allows collapsing the experimental data for mean temperature rise above the ambient and mean axial velocity on a single curve independent of the HRR. Let us start to discuss the solutions obtained with VTE

Figure 6

 6 Figure 6.17: Axial distribution of axial velocity (index 1), mean temperature (index 2), resolved temperature fluctuation (index 3) and total temperature fluctuation (index 4) for the five fire plumes as a function of the normalized height with: (a) VTE, (b) STE and (c) the algebraic model.

Figure 6

 6 Figure 6.18: Radial distribution of mean temperature at different heights for the 45 kW fire plume.

Figure 6 .

 6 Figure 6.19: A schematic diagram of the three burner boundary conditions for the: (a) baseline case, (b) NoLip case, and (c) NoLipNoEnt case.

  Concerning the boundary conditions, typical outflow/inflow boundary conditions are used for open boundaries at the sides. At the inlet, a spatially uniform and steady inlet velocity is imposed to provide the specified HRR and both convective and diffusive mass and energy fluxes are accounted for. The inlet temperature is set to the boiling point of methanol, i.e. 338 K. At the domain exit, a convective condition is used. In the rest of the domain, the classical wall boundary condition is imposed.

Figure 6

 6 Figure 6.20: Radial evolution of resolved part of temperature variance at different heights.

Figures 6 .

 6 Figures 6.22 to 6.27 display the radial profiles of mean and rms temperature, axial velocity and radial velocity at different heights. Model predictions present an overall good agreement with the experimental data. In particular, the simulations capture well the radial spreading of the fire plume. The profiles of mean temperature are on the whole well reproduced by the model at all the heights (see Fig.6.22). However, in the lower part of the fire plume @z IH cmA, the predicted mean temperature exhibits M-shaped profiles that are not observed experimentally (seeFig. 6.22). This results in an underestimation of the

6

  .27). Nevertheless, the computed fluctuations exhibit well pronounced Mshaped profiles whereas the measured ones are flatter, which results in an underestimation of the fluctuations of radial velocity close to the centreline.

Figure 6 .Figure 6

 66 Figure6.28 shows the axial profiles of the molar fractions of CH Q OH, O P , CO P , H P O, CO and H P . The model reproduces quantitatively the experimental data

Figure 6

 6 Figure 6.23: Radial profiles of rms value of temperature at different heights.

Figure 6 .

 6 Figure6.24: Radial profiles of mean axial velocity at different heights. The experimental data are taken from Ref.[START_REF] Weckman | Experimental investigation of the turbulence structure of medium-scale methanol pool fires[END_REF].

Figure 6

 6 Figure 6.25: Radial profiles of rms value of axial velocity at different heights.

Figure 6 .

 6 Figure 6.29a shows that the model overestimates the radiative flux, especially for H z H:S m. A better agreement is observed downstream.

Figure 6 .Figure 6

 66 Figure 6.29(b) shows the radial distribution outside the burner along the plane of the burner rim. Predictions, obtained with the IT ¢ PR angular mesh, are in overall good agreement with the experimental data. Nevertheless, the model overpredicts the radiative flux at vicinity of the burner.Figure 6.29(c) shows both radiative and total heat feedback towards the fuel surface. The reported

Figure 6

 6 Figure 6.27: Radial profiles of rms value of radial velocity at different heights.

Figure 6

 6 Figure 6.28: Axial profiles of mean molar fractions for: (a) methanol, (b) carbon dioxide, (c) oxygen, (d) carbon monoxide, (e) water vapor and (f)hydrogen. The experimental data are taken from Ref.[START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF].

Figure 6

 6 Figure 6.29: Heat fluxes: (a) Vertical distribution of radiative flux at a distance

  Figure 6.30: Snapshots of temperature at different times for the baseline case.The time between two snapshots is 0.06 s. Uniform velocity vectors are plotted to illustrate the flow pattern.

Figure 6 .

 6 Figure 6.30(a) shows a coherent structure in the middle of the image. On the other hand, the flame is anchored on the top of the burner rim in accordance

  Figure 6.31: Snapshots of temperature at different times for the NoLip case. The time between two snapshots is 0.06 s. Uniform velocity vectors are plotted to illustrate the flow pattern.

  Figure 6.32: Snapshots of temperature at different times for the NoLipNoEnt case. The time between two snapshots is 0.06 s. Uniform velocity vectors are plotted to illustrate the flow pattern.

Figure 6

 6 Figure 6.33: Effects of pool boundary conditions on axial and radial profiles at different heights: (a)-(d) rms of temperature and (e)-(h) axial velocity.

Figure 6 .Figure 6

 66 Figure6.35 shows the influence of the fuel boundary conditions on both radiative and total heat feedback towards the fuel surface. Before started the discussion, it should be reminded that these predictions have been obtained by considering the fuel burning per unit area unchanged for the different boundary conditions. This figure shows that the heat feedback for the NoLip and the NoLipNoEnt cases are very similar. This is not surprising since these two boundary conditions lead to similar thermal and flow fields (seeFigs. 6.33 and

Figure 6

 6 Figure 6.35: Effects of pool boundary conditions on heat feedback to the pool surface: (a) radiative flux measured at the fuel surface and (b) total flux at 13

  of ethanol oxidation developed by Saxena and Williams (2007). The radiation model combines the RCFSK model for spectral gas radiation and the DOM with a S8 quadrature scheme as radiative transfer equation (RTE) solver. To analyse the effects of the flame optical thickness, two synthetic pool fires 122 CHAPTER 7. RADIATION IN FIRE PLUMES Li MA 7.1. ASSESSMENT OF SUBGRID-SCALE TURBULENCE-RADIATION INTERACTIONS are simulated by scaling up the pool diameter by 2 and 4 while maintaining the same fuel mass burning rate per unit area. A simple geometric scaling of the computational mesh is considered for all flames. The simulations relative to these flames are referred to as F2D and F4D, respectively. The flame configurations are summarized in Table 7.1. The fourth column provides the plume resolution index (PRI), defined as the ratio between the plume characteristic length scale, D £ a Q=@ I c p T I p gA P=S and the grid size, ¡. I , c p , and T I represent the density, heat capacity and temperature of ambient air, respec-

7. 1 . 3

 13 Analysis of TRITRI represents the contributions of the turbulent fluctuations to the mean absorption and emission terms. In the following, hi and 0 will denote the timeaveraged value and the fluctuation of , respectively. In LES, the contribution of turbulent fluctuations in temperature and mole fractions of the radiating species to TRI is decomposed into resolved-scale fluctuations, ReS TRI, and subgrid-scale fluctuations, SGS TRI. These latter requires modelling. Different closures for the filtered absorption and emission terms were investigated in order to quantify the effects of ReS TRI and SGS TRI on the radiative loss and are summarized in

  Figures 7.1 and 7.2 show that a satisfactory agreement is observed for the radial profiles of mean temperature and mole fractions of CO P , H P O and CO.

  Figure 7.3 displays radial profiles of temperature fluctu- ations. This figure shows that model predictions follow the same trends as the experiments. Fischer et al. (1987) measured temperatures with 75 m 124 CHAPTER 7. RADIATION IN FIRE PLUMES Li MA 7.1. ASSESSMENT OF SUBGRID-SCALE TURBULENCE-RADIATION INTERACTIONS thermocouples with bead diameters of 100 m, which can lead to a significant underestimation of temperature fluctuations despite the use of compensation

Figure

  Figure 7.1: Radial profiles of temperature at different heights above the pool basis.

Figure 7 . 3 :

 73 Figure 7.3: Radial profiles of temperature fluctuations at different heights above the pool basis.

Figure 7 . 4 :

 74 Figure 7.4: Centreline evolution of the turbulence intensity (left y-axis) and resolved-part of temperature variance (right y-axis) as a function the normalized height.

Figure 7 .

 7 Figure 7.5 displays the axial distribution of the net radiative flux at a normalized distance of r £ a r=Q P=S a H:IV m • kW P=S from the fire plume axis. This normalized distance corresponds to r a I m for F1D. Figure7.5 shows that the net radiative flux increases on the whole with the pool size. Whatever the pool fire considered, the radiative flux is underestimated along the entire distribution when SGS TRI is disregarded and this underestimation becomes increasingly important as the pool size is enhanced. In particular, neglecting SGS TRI leads to an underestimation of the peaks of net radiative flux that by about12.27, 25.67 and 29.27 for F1D, F2D and F4D, respectively.

Figure 7 .

 7 Figure7.6 displays the temperature and the axial velocity along the fire plume axis. This figure shows clearly that disregarding the SGS TRI affects the fire plume structures and the effects are enhanced with the pool size. For F1D, the radiative loss are underestimated by 187 when SGS TRI is not considered (see Table7.3). As a consequence, the temperature is overpredicted in the IF

  Figure 7.6(a) and (d) shows that considering SGS TRI improves the predictions for F1D. For F2D, neglecting the SGS TRI leads to an underestimation of the radiative loss by about 227.

  Figure 7.5: Effects of SGS TRI on the axial distribution of net radiative heat flux at a normalized distance of r £ a r= Q P=S a H:IV m • kW P=S from the fire plume axis as a function of the normalized height.

  Figure 7.6: Effects of SGS TRI on the: (a)-(c) axial mean temperature and (d)-(f) axial mean velocity as a function of the normalized height.

  kW . Fast Fourier power spectrum of the time-varying flame height revealed a puffing frequency of 1.37 Hz and a first harmonic of 2.75 Hz. Temperature measurements were performed using 50 m diameter Type S thermocouples and were corrected for thermal inertia effects. The compensation technique was based on an instantaneous time constant computed from the thermocouple bead geometry and an estimation of the flow conditions for the convective heat transfer. Vertical distribution, at a distance of 2.07 m from the pool axis, and radial distribution, along the plane of the burner rim outside the pool fires, of radiative flux were measured with wide-view angle, water-cooled, Gardontype total heat flux gauges (see Fig.7.7). Sung et al. (2020) also deduced from their data the total radiative feedback, dA (see Fig. 7.7). This quantity represents the incident radiative flux on the fuel surface integrated over the burner surface, A b . It was normalized by the HRR to provide R;S a Q R;S = Q. This latter set of data is completed by the local radiative feedback, q HH

  Typical outflow and entrainment boundary conditions are used for open boundaries at the outlet and sides, respectively. The fuel inlet temperature is set to the boiling temperature of methanol, i.e. 338 K. The inlet velocity is imposed to provide the specified HRR and both convective and diffusive mass and energy fluxes are accounted for. In the rest domain, the classical wall boundary condition is imposed.

Figure 7

 7 Figure 7.7: A schematic diagram of the computational domain and the locations of the radiative heat flux distributions.

  Figure 7.8: Radial evolution of resolved-part of temperature variance at different heights.

Figure 7 Figure 7 . 9 :

 779 Figure 7.9(a) shows the power spectrum density (PSD) computed with a fast Fourier transform of the times series of the axial velocity along the axis at a height of 0.6 m. The model predicts a puffing frequency of 1.25 Hz and a first harmonic of 2.50 Hz in good accordance with the experimental ones of 1.37 Hz and 2.75 Hz, respectively.Figure 7.10(a) shows the axial distribution of mean

Figure 7

 7 Figure 7.10: Axial and radial profiles of mean temperature. The experimental data are taken from Ref.[START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF].

  Figure 7.11 shows also that the experimental data exhibit pronounced M-shaped radial profiles close to the burner and transit progressively towards flatter profiles as z increases (see fig. 7.11(b)-(f)). The model captures also well these trends and provides predictions in overall good agreement with the experimental data. Nevertheless, it overestimates on the whole the temperature fluctuations. These discrepancies may be attributed, at least partially, to the measurements since previous experimental studies suggested that temperature fluctuations may be underestimated when measured Li MA CHAPTER 7. RADIATION IN FIRE PLUMES 137 7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS by 50 m thermocouples despite the use of compensation techniques (Crau-

Figure 7

 7 Figure 7.11: Radial profiles of rms values of temperature fluctuations at different heights. The experimental data are taken from Ref.[START_REF] Sung | The Thermal Character of a 1 m Methanol Pool Fire[END_REF].

Figure 7

 7 Figure 7.12: Effects of angular meshes on radiative heat fluxes. The location of the distributions are described in Fig. 7.7. (a) and (d) Vertical distribution at a distance r a P:HU m from the pool axis, (b) and (e) radial distribution outside the burner along the plane of the burner rim. The radial distance is measured from the pool centre, and (c) and (f) feedback towards the fuel surface. The experimental data in the diagrams (a) and (b) are taken from Sung et al. (2020) whereas those in diagram (c) are taken from Klassen and Gore (1992). All the simulations were obtained with the RCFSK. The diagrams (a), (b) and (c) investigate the effects of the uniform meshes whereas the diagrams (d), (e), and (f) those of the FT n FVM angular discretizations.

Q

  abs , R and R;S as well as the radiative feedback towards the fuel surface (see Fig. 7.12(c)). In addition, this has almost imper-140 CHAPTER 7. RADIATION IN FIRE PLUMES Li MA 7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS ceptible effects on the radial distribution of radiative flux along the plane of the burner rim (see Fig. 7.12(b)). In an opposite way, it improves significantly the vertical distribution of radiative flux by eliminating the ray effects (see Fig.7.12(a)).Table7.6 and Fig.7.12(d)-(f) investigate the effects of the FT n angular discretizations on both integrated and local radiative outputs. In agreement with the previous discussion, the results show clearly that the angular meshes affect only the vertical distribution of radiative flux outside the flame.

Figure 7 .

 7 Figure 7.12(d) shows that significant ray effects are observed for FT IP and FT IT .The FT IT and 16 ¢ 24 angular meshes have the same polar discretization and differ from the azimuthal discretization. These two angular meshes provide almost identical vertical distributions, showing that the discretization of the polar angle is mainly responsible for these ray effects. This can be explained by the statistically axisymmetric nature of the present fire plume. The FT PR angular discretization mitigates the ray effects and predicts the peak of vertical flux within 77 of the 48 ¢ 96 angular mesh. However, relative errors as large as about 167 are still observed at z a I:P m. Acceptable predictions of the vertical distribution of radiative flux are observed for the FT QP angular discretization, which involves 1088 control angles. The FT RV angular discretiza- tion provides the same vertical distribution of radiative flux as the uniform 48

Figure 7 .

 7 Figure 7.13 illustrates the effects of the radiative property models on the radiative flux distributions. Predictions with the WSGG-NG agree well with the RCFSK. On the other hand, the WSGG-G-CB model overpredicts notably

Figure 7

 7 Figure 7.13: Effects of gas radiative property models on raditive heat flux. The location of the distributions are described in Fig. 7.7. (a) Vertical distribution of radiative flux at a distance x a P:HU m from the pool axis, (b) radial distribution outside the burner along the plane of the burner rim. The radial distance is measured from the pool centre, and (c) radiative feedback towards the fuel surface. These radiative fluxes are computed in frozen field analysis using the FVM 48 ¢ 96 angular discretization.

Figure 7

 7 Figure 7.14: Effects of radial property models on the axial distributions of: (a) emission term, (b) absorption term, (c) divergence of the radiative flux, (d) mean temperature, and (e) rms values of temperature fluctuations. These radiative fluxes are computed in frozen field analysis by using the FVM 48 ¢ 96 angular discretization.

  Figure 7.14(d) shows that the peak value of mean temperature is lower by about 150 K than that computed with the RCFSK. A consequence of this underestimation of the temperature is that the buoyant forces are underestimated, producing in turn 144 CHAPTER 7. RADIATION IN FIRE PLUMES Li MA 7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS an underestimation of the temperature fluctuations as observed in Fig. 7.14(e). The WSGG-G-FB model provides a divergence of the radiative flux as well as mean and rms temperature comparable to those predicted by the WSGG-NG model. Nevertheless, it has to keep in mind that the WSGG-NG outperforms significantly the WSGG-G-FB in the prediction of the radiative flux (see Fig.

  LES of 1 m methanol pool fires have been exercised in order to investigate the effects of gas radiative property models on the radiative outputs and flame structure. The baseline radiation model involves the RCFSK coupled to the FVM with 16 ¢ 24 control angles. Three other gas radiative property models, namely the WSGG-NG, the WSGG-G-CB and the WSGG-G-FB, are also considered. In addition, solutions obtained with other angular meshes, including the uniform 48 ¢ 96 angular discretization and the FT n scheme with n up to 48, are assessed. The following conclusions can be drawn: 1. The present LES-based fire model involving the non-adiabatic/presumed FDF and RCFSK radiation model provides high fidelity predictions in terms of puffing frequency and first harmonic, mean and fluctuating temperatures, radiative feedback to the pool surface and radiative loss to the surrounding. 2. All radiative outputs at the exception of vertical distributions of radiative flux outside the flame can be accurately predicted with the FT IP angular discretization. On the other hand, finer angular meshes, considering at least 32 segments to discretize the polar angle, are required to compute accurately the vertical distributions of radiative flux outside the flame. 3. Whatever the definition of the mean path length, the grey WSGG models fail to reproduce the radiative structure of the fire plume, predicting an optically thin flame instead of an optically intermediate medium. The WSGG-G-FB provides a better description of the radiative loss than the WSGG-G-CB. However, it leads to relative errors in the range 15-207 on the radiative fluxes.

Figure 8 . 1 :

 81 Figure 8.1: Schematic of the experimental configuration (left); Plan view of UMD line burner (right). These schemes are taken from White et al. (2015).

  Figure 8.3(a) shows that, as X O2 decreases, the curves are shifted towards lower values of c I;st and the stoichiometric temperature span becomes smaller. On the other hand, comparing Fig. 8.3(b) with Fig. 8.3(a) evidences

Figure 8 . 3 :

 83 Figure 8.3: Temperature as a function of the reaction progress variable at stoichiometric conditions at different X O2 : (a) c I definition and (b) c P definition.

  mental configuration, as shown in Fig.8.4. It should be pointed out that no oxygen anchor is considered in the simulations. The mesh is uniformly refined in the region of H:IS ¢H:T¢H:S m Q centered around the burner with a minimal grid spacing equal to 5 mm. Outside this region, the mesh is progressively stretched in all the direction.

Figure 8 . 4 :

 84 Figure 8.4: Illustration of the numerical configuration.

  Figure 8.5: Axial profiles of: (a) mean temperature and (b) rms values of temperature fluctuations.

Figure 8 .

 8 Figure 8.6 displays the radial profiles of mean and rms temperature. The two model predictions present an overall good agreement with experiments except at z a H:PS and z a H:S m where the rms values of temperature fluctuations are overestimated by about 100 K (see Fig. 8.6(e) and (f)).

LiFigure 8

 8 Figure 8.6: Radial profiles of: (a)-(d) mean temperature and (e)-(f) rms values of temperature fluctuations at different heights.

Figure 8 .

 8 Figure8.7 shows the axial profiles of mean and rms temperature predicted by the SLF, FPV-c I and FPV-c P models for X O2 a 15%. The two FPV models reduce significantly the first peak of temperature by about 400 K at z H:PS m

8. 7

 7 (a)). On the other hand, the computed axial rms values of temperature fluctuations are almost insensitive to the combustion model. Snapshots of the radiative emission term predicted by the SLF, FPV-c I and FPV-c P models are shown in Fig. 8.8. This figure exhibits clear differences close to the burner which are manifestations of the flame lift-off and local extinction.

Figure 8 . 7 :

 87 Figure 8.7: Axial profiles of: (a) mean temperature and (b) rms values of temperature fluctuations for X O2 a15%.

Figure 8 .

 8 Figure 8.9 shows mean and rms temperature obtained for the simulation of the flame with X O2 a 13%. Simulations made with the FPV models show that almost the whole flame region is represented by flamelets in the unstable

  Figure 8.8: Snapshots of instantaneous volumetric emissions predicted by: (a) SLF, (b) FPV-c I and (c) FPV-c P models for X O2 a 15%.

Figure 8 .

 8 Figure 8.11 shows that the radiant fraction, R , as a function of X O2 . The two sets of experimental data corresponding to oxygen anchored and nonanchored conditions are plotted in this figure. In both cases, the radiant

Figure 8 .

 8 Figure 8.10: Snapshots of instantaneous volumetric emissions predicted by: (a) SLF, (b) FPV-c I and (c) FPV-c P models for X O2 a 13%.

Figure 8 .Figure 8

 88 Figure8.11 shows also the evolution of R predicted by SLF FPV-c I and FPVc P models. The SLF predicts well the decrease in R with X O2 up to X O2 a 0.15. This result is expected since no noticeable local extinction/re-ignition

  These developments and implementations have been made in Code_Saturne of EDF.The first step was the development and verification of a new second-order low-Mach number variable-density flow solver that was an indispensable prerequisite to describe accurately the strong unsteady processes related to the purely buoyant nature of the flow. Secondly, a significant effort was put on the selection of variable-density subgrid-scale (SGS) stress tensor and scalar flux models. In particular, the performance of a dynamic Smagorinsky model for SGS stress tensor and advanced SGS scalar flux models, including the dynamic eddy diffusivity model, dynamic tensorial diffusivity models and gradient models were assessed in LES of the 1 m helium plume. This study revealed that the dynamic Smagorinsky model and the dynamic eddy diffusivity model are good candidate for the simulations of purely buoyant fire plumes and were applied in the remaining simulations of thesis. A significant effort was also put on the modelling of subgrid-scale turbulence-chemistry interaction and turbulence radiation interaction. Both non-adiabatic steady laminar flamelet and flamelet/ progress-variable models coupled to a presumed filtered density function(FDF) were developed and implemented, with the flamelet progress variable being 9.1. CONCLUSIONS specifically developed for the modelling of under-oxygenated fires characterized by significant local extinction/re-ignition processes. A special attention was paid on the modelling of the SGS scalar variance and filtered scalar dissipation, that are important quantities for the formulation of flamelet approaches.Investigation of specific models in LES of the McCaffrey fire plume showed the limitations of equilibrium model and that the non-equilibrium model based the second moment transport equation is a good candidate to deal with this problem. Finally, the rank correlated full spectrum k was developed and implemented as gas radiative property model. Other simpler gas radiative property models, involving grey and non-grey formulation of the weighted-sum-of-grey gases, were also implemented.The model was exhaustively validated by simulating several well-documented fire plumes of the literature. It was found that the model is able to reproduce with high fidelity the dynamic and radiative structures of non-luminous fire plumes in both ventilated and under-ventilated configurations. In particular, the flamelet progress variable allows addressing oxygen-diluted oxidizer conditions close to extinction where significant local ignition/re-ignition processes are observed.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial profiles of mean radial velocity at different heights. The experimental data are taken from Ref. (Weckman and Strong, 1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.27 Radial profiles of rms value of radial velocity at different heights.

			LIST OF FIGURES
	6.26
	xii	LIST OF FIGURES	Li MA

  R -Air flame. It is interesting to investigate the profiles of c I and c P at small st that characterizes usually combustion in fire scenarios. It can be observed that c I provides a better mapping in the stable branch than c P since some profiles of c P overlap in this region. In the opposite way, c P provides a better mapping in the unstable branch. These observations suggest that c I is expected to be a better choice if the combustion is dominated by the stable

	Li MA branch whereas c P may become more attractive in cases where the unstable flamelet prevails. 0.0 0.2 0.4 0.6 0.8 1.0 Z [ -] 500 1000 1500 2000 T [K] Increasing χ st Increasing χ st Stable branch Unstable branch Figure 2.4: Profiles of Temperature as a function of mixture fraction in both stable and unstable branches with respect to st . Fuel: CH R ; Oxidizer: pure air. 0.0 0.2 0.4 0.6 0.8 1.0 Z [ -] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c 1 [ Increasing χ st Increasing χ st (a) Stable branch Unstable branch 0.0 0.2 0.4 0.6 0.8 1.0 Z [ -] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c 2 [ Increasing χ st Increasing χ st (b) Stable branch Unstable branch Figure 2.5: Profiles of reaction progress variable: (a) c I and (b) c P as a function of mixture fraction in both stable and unstable branches with respect to st . Fuel: CH R ; Oxidizer: pure air. 2.2.5 Extended non-adiabatic flamelet model Thermal radiation is an important phenomenon in reactive systems, especially for buoyant diffusion flames. Heat is released to the surroundings leading to additional interaction at the level of the flame structure. A model incorporating the effects of radiative loss on the flamelet is developed. The SLF or FPV models rely on the underlying assumption that the thermophysical state of 26 CHAPTER 2. MATHEMATICAL FORMULATION Li MA 2.2. COMBUSTION MODELLING a particular flamelet relaxes to the steady-state solution on a sufficiently fast time-scale. However, radiative heat-loss processes evolve on time-scales that are CHAPTER 2. MATHEMATICAL FORMULATION 2.2. COMBUSTION MODELLING slow compared to other processes in typical combustion applications

-]

-]

table ,

 ,

	such that each "	appearing in the left-hand side of Eqs. (3.1) to (3.4) is
	replaced by the filtered density computed as:
	8 > <	"	a " slf @ Z; V Z ; ; XR A; for SLF
	> :	"	a "

fpv @ Z; V Z ; c; XR A; for FPV (3.39) Li MA CHAPTER 3. LARGE-EDDY SIMULATION 3.4. SUBGRID SCALAR VARIANCE AND DISSIPATION

Table 4

 4

	46	CHAPTER 4. NUMERICAL METHODS	Li MA

.1: Calculation parameters for the MMS

[START_REF] Shunn | Verification of variable-density flow solvers using manufactured solutions[END_REF]

.

  H xexp@ k P tAA, and k I , k P , and w H are constant parameters given in Table4.1. This manufactured solution satisfies the continuity and mo-

	I u@x; tA a Pk P exp@ k I tA C I Y @x; tA H H I @x; tA	! I u u P C I	C	(4.21) @k I =k P IA@arctan u =RA w H exp@ k P tA !
	where u a exp@w mentum equations with	Q a H,	Q

u a H. On the other hand, a non-zero source term,

  Dirichlet boundary conduction is used at x a I, an outflow boundary condition is applied at x a Q, and periodic boundary conditions are imposed at y a ¦ I P . The calculations are conducted with seven meshes (n x ¢n y ) of PHH ¢SH; RHH ¢IHH; VHH ¢PHH; ITHH ¢ RHH; QPHH ¢ VHH, TRHH ¢ ITHH, and IPVHH ¢ QPHH cells. For each mesh, the time step is selected to keep the convective CFL based on u f constant and equal to 0.5. Figure4.2(a) shows the L P -error convergence, indicating the second-order accuracy of the proposed scheme. This conclusion is confirmed in Table4.2 which reports the error data for the scalar, velocity components, and pressure.

y I=P, and H t I. This set of equations satisfies the continuity equation with Q a H. Non-zero source terms appear in the x-, y-momentum, and scalar transport equations. The density front advects diagonally across the domain according to the relative values of u f and v f . The steepness of the initial density profile in the x-direction is governed by the value of b, while a and k dictate the x-amplitude and wavenumber of the sinusoidal perturbation, respectively. The rate at which the front broadens with time is determined by the inverse time scale, !.

  of the sinusoidal waves is a H:HHI and the wave numbers are ! k a 4,14, 23, 28, 33, 42, 51, 59, respectively, following the test case of[START_REF] Desjardins | High order conservative finite difference scheme for variable density low Mach number Li MA BIBLIOGRAPHY 167 BIBLIOGRAPHY turbulent flows[END_REF], A mixture fraction scalar field is introduced as Finally, the density is computed from the mixture fraction using the same equation of state as previously used in the MMS verification step. The two fluids have identical kinematic viscosity a H:HHI m P • s I and kinematic diffusivity D a S ¢ IH R m P • s I . The value for the gravity acceleration is g a W m • s P so that the Reynolds number is Re a

			4.2. ALGORITHM VERIFICATION BY MMS
		where the amplitude Y @x; yA a	I P	I C tnh	y int @xA y P	!!	(4.25)
		where the interface thickness is a H:HHHP.
		p n x n y Y	u	p
		32
		64	1.986 2.058 2.054
		128	1.762 2.008 2.012
		256	1.854 2.002 2.002
		512	1.956 2.001 2.002
		1024 1.989 2.000 2.001
		2048 1.998 2.000 2.001
		Table 4.3: Observed order for the 2D-ODF MMS for CFL = 0.2.
	interface. The heavy fluid ( H a I kg • m Q ) is above the light fluid ( I a H:I kg • m Q ). The mean interface is located at y a H in a domain size of x ¢ y a H:S; H:S ¢ H:S; H:S m P . The exact location of the interface is given by
		y int @xA a	U X kaI	os @! k xA	(4.24)
	52	CHAPTER 4. NUMERICAL METHODS	Li MA

q gL y L x = a QHHH. In this case, a Superbee TVD convection scheme

[START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method[END_REF]

is used for the scalar transport equation in order to ensure the positivity and to prevent undershoot and overshoot of the density. Simulations have been performed on four different meshes composed of

Table 5

 5

	Li MA	CHAPTER 5. HELIUM PLUME	57

.1 summarizes the grids, the SGS stress models, and the SGS scalar sub-models considered in these studies. This literature survey is limited

  • kW P=S ), at the transition 20 ¢ 66 24 ¢ 24 ¢ 80 30 ¢ 30 ¢ 100 40 ¢ 40 ¢ 133 60 ¢ 60 ¢ 200 Table 6.1: Resolution parameters for the LES calculations. between the CF and the intermittent flame regions (IF) and in the IF region (H:HV < z £ H:P m • kW P=S ), respectively (McCaffrey

	6.1. SUBGRID VARIANCE MODELLING
	Grids		G1	G2	G3	G4	G5
	Number of cells
	in 20 ¢ H:Q ¢ H:Q ¢ I:H m 3 ¡ (cm)	1.5	1.25	1.00	0.75	0.50
	=¡	1.04	1.25	1.56	2.08	3.12
	l d =¡ ¡= k	0.95 17.8	1.14 14.8	1.42 11.9	1.90 8.9	2.85 5.9
	Li MA		CHAPTER 6. FIRE PLUME DYNAMICS	73

  of grid spacing on VTE and STE Let us start by investigating the effects of the turbulent resolution scale ¡ on VTE and STE. Radial profiles of time-averaged mixture fraction, hZi, SGS variance, hV Z i, resolved-scale variance, hZ V;ReS i, and total variance, Z V;tot , at different heights of 0.1, 0.2, 0.4 and 0.6 m, computed with the VTE and STE models on the different grids, are presented in Figs. 6.1 to 6.4, respectively.

  resulting in lower values close

	Li MA 0.00 0.02 0.04 0.06 0.08 Zv, tot [-] 0.00 0.02 0.04 0.06 Zv, tot [-] 0.00 0.01 0.03 0.04 CHAPTER 6. FIRE PLUME DYNAMICS 6.1. SUBGRID VARIANCE MODELLING -0.2 0.0 0.2 (a) z = 0.1m G1 G2 G3 G4 G5 -0.2 0.0 0.00 0.02 0.04 0.06 0.08 (e) -0.2 0.0 0.2 (b) z = 0.2m -0.2 0.0 0.00 0.02 0.04 0.06 (f) -0.2 0.0 0.2 0.02 Zv,	0.2 0.2

tot

[-]

Table 6 .

 6 3: Radiant fraction and optical thickness computed by VTE and STE for the 57.5 kW fire plume on the different grids.

	Li MA	CHAPTER 6. FIRE PLUME DYNAMICS	91

  Q • s I . The correspond-

	6.2. EFFECTS OF BURNER'S GEOMETRY
	ing heat release rate (HRR,	Q) is 22.45 kW . Weckman and Strong reported
	measurements of mean and rms velocity and temperature. Velocities were mea-
	sured by using a two-component laser Doppler anemometer with uncertainties
	of ¦ 57 on mean values and ¦ 157 on rms values. In addition, power spec-
	tral density (PSD) computed from time series of radial and axial velocities
	revealed a puffing frequency of 2.8 Hz. Temperatures were measured with 50
	m diameter bare-wire Pt-Pt/107 Rh thermocouples with bead diameters in
	the range of 75-100 m. Temperature measurements were corrected to account
	Li MA	CHAPTER 6. FIRE PLUME DYNAMICS	99

for the thermal inertia of the thermocouple bead. The compensation technique was based on an instantaneous time constant computed from the thermocouple geometry and an estimation of the flow conditions for the convective heat

  mesh is uniformly refined in the region of H:R ¢H:R¢H:I m Q centred around the burner, with a minimal grid spacing equal to 2.5 mm. Outside this region, ¡x and ¡y are stretched progressively towards the sides of the computational domain. In the vertical direction, ¡z is stretched from z a H:I m to z a H:P m to reach 5 mm.

An uniform ¡z a S mm is then applied up to z a H:T m. Above z a H:T m, ¡z is stretched progressively. The flamelet library was generated using the 100 CHAPTER 6. FIRE PLUME DYNAMICS Li MA 6.2. EFFECTS OF BURNER'S GEOMETRY

  . Experimental data shows that the axial temperature increases with z £ up to z £ % H:HP m • kW P=S , reaches a plateau of about 1300 K between z £ % H:HP m • kW P=S and z £ % H:HR m • kW P=S , and decreases as z £ is further increased. The model reproduces well the temperature peak as well as the experimental profile for z £ ! H:HQ m • kW P=S . For lower z £ , the axial temperature is underpredicted although it is in overall within the measurement uncertainty of[START_REF] Li | The structure of a moderate-scale methanol pool fire[END_REF].

		1400 1600 1800		Num. Weckman and Strong Hamins and Lock
		1200
	T [K]	800 1000
		600
		400
		200
		0.00 0	0.05	0.10 z/Q 2/5 [m. kW 2/5 ] 0.15	0.20	0.25
	Figure 6.21: Axial profile of mean temperature. The experimental data are
	taken from Refs.

  Table7.2. The filtered absorption and emission terms were evaluated from: i) h T i, hx CO2 i and hx H2O i by ignoring all the fluctuations (No TRI), ii) T , xCO2 and xH2O by considering only the resolved fluctuations (ReS TRI) and then ignoring the SGS TRI, and iii) Eqs 3.52 and 3.53 based on presumed FDF that allows taking into account both ReS and SGS fluctuations (Full TRI). £ i and £ i a i I b are included in the 4D look-up table parametrized by Z, V Z , and XR . Consequently, the increase in CPU time is marginal as compared to the ReS TRI case.Radiation is solved in coupled manner for both ReS TRI and Full TRI cases and in decoupled manner for the No TRI case by using h T i, hx CO2 i and hx H2O i

	obtained from the Full TRI simulations.
	case	Absorption	Emission

Table 7

 7

		.3 (Full TRI line) is explained by a balance
	between an enhancement in emi and a reduction of	I	Q abs =	Q emi	.
	Li MA	CHAPTER 7. RADIATION IN FIRE PLUMES	127

Table 128

 128 ReS R decreases as the pool size is enhanced from about 607 for F1D to 497 for F4D. On the other hand, the contribution of SGS TRI

	CHAPTER 7. RADIATION IN FIRE PLUMES	Li MA

  It was found necessary to respect these experimental features to reproduce adequately the pool dynamics. A non-uniform structured mesh with 8.01 million of cells is considered. The mesh is refined over the fuel surface with a cell size (¡x, ¡y, ¡z) of 8.15 ¢ 8.15 ¢ 5 mm Q . Outside the pan, ¡x and ¡y are stretched progressively towards the sides. In the vertical direction, ¡z = 5 mm is applied between the fuel surface (z a H) and the burner rim (z a I cm). ¡z is then stretched from z a H:HI m to z a H:QPT m to reach 1.58 cm. A uniform ¡z = 1.58 cm is then applied up to z a I:V m. Above z a I:V m, ¡z is stretched progressively. The flamelet library was generated using the full chemical kinetic

		7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS
	scheme developed by Li et al. (2007).
	132	CHAPTER 7. RADIATION IN FIRE PLUMES	Li MA

  with a uniform angular mesh of 384 (N ¢ N = 16 ¢ 24) control angles. The subscripts and represent the polar angular meshes over the statistically stationary state and, then, the resulting instantaneous radiative intensities are averaged to get the mean absorption term and radiative fluxes. The other angular meshes considered in the present study are, on the one hand, a three-time finer uniform angular mesh with 4608 (N ¢ N = 48 ¢ 96) control angles and, on the other hand, the non-uniform FT n angular discretizations

	7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS
	and azimutal angles, respectively. The selection of this angular discretization
	was based on the study of Jensen et al. (2007) who suggested that, for uniform
	angular discretization, approximately 50 angles inside and 350 angles outside
	the fire are needed to adequately resolve radiation.
	A frozen field analysis will be considered to investigate the angular mesh ef-
	fects on the radiative outputs. In this analysis, instantaneous fields of filtered
	absorption coefficient (Eq. (3.52)) and emission term (Eq. (3.53)), computed
	with the uniform 16 ¢ 24 angular mesh, are saved during the statistically sta-
	tionary state. These fields are used to solve instantaneous RTE (Eq. (3.51))
	with the other
	Li MA	CHAPTER 7. RADIATION IN FIRE PLUMES	133

Table 7

 7

		.4 provides the CPU times for all the radiative models in
	one time step. A careful examination shows that CPU times for the different
	angular meshes and the different radiative property models increase almost
	linearly with number of control angles and the number of RTEs to be solved,
	respectively.
	134	CHAPTER 7. RADIATION IN FIRE PLUMES	Li MA

  In this section, the LES results obtained by the RCFSK/FVM (16 ¢ 24) radiation model are compared with the experimental data.

	7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS
	7.2.3 Comparison with available experimental data
	Li MA	CHAPTER 7. RADIATION IN FIRE PLUMES	135

  ). Let us start to analyse the results obtained with the

		7.2. INFLUENCE OF GAS RADIATIVE PROPERTY MODELS
	Model Angular mesh	Q emi (kW )	Q abs (kW )	Q abs =	Q emi (-) R (-) R;S (-)
	RCFSK	16 ¢ 24	172	102.6	0.598	0.269	0.059
	RCFSK	48 ¢ 96	172	102.6	0.598	0.269	0.059
	Table 7.5: Integrated flame radiative properties. Measured radiant fractions,
	R , and fractional radiative heat feedback towards the fuel surface, R;S , were
	0.22 ¦ 167 and 0.055 ¦ 217, respectively
	138	CHAPTER 7. RADIATION IN FIRE PLUMES		Li MA

Table 7 .

 7 6: Effects of FT n FVM angular discretization on the integrated flame radiative properties. Values in parenthesis represent the relative errors in 7 of the current model as compared to the RCFSK/FVM 48 ¢ 96. Results with FT n angular discretization higher than n a IT are not reported since these angular meshes do not introduce any discrepancies on these quantities.

	Li MA	CHAPTER 7. RADIATION IN FIRE PLUMES	141

Table 7 .

 7 7: Effects of the gas radiative property models on the integrated flame radiative properties. Values in parenthesis represent the relative errors in 7 of the current model as compared to the RCFSK/FVM RV ¢ WT.

  The peak of mean tempera-INFLUENCE OF GAS RADIATIVE PROPERTY MODELS ture is slightly shifted towards the fuel surface and its value is underpredicted by about 60 K as compared to the RCFSK. The WSGG-NG model leads also

	7.2.
	Li MA	CHAPTER 7. RADIATION IN FIRE PLUMES	143

  Decreasing X O 2 Decreasing X O 2 Decreasing X O 2 Decreasing X O 2 Decreasing X O 2 Figure 8.2: Adiabatic S-shaped curves for different X O2

	8.2. FLAMELET LIBRARY CALCULATION
				2200
				2000		Decreasing X O 2
				1800
				1400 1600 st [K] T
				1200
				1000
				10 -3 800	10 -2	10 -1 χ st [s -1 ] 10 0	10 1	10 2
		2200
		2000
		1800
	T st [K]	1400 1600		Decreasing X O2 Decreasing X O2 Decreasing X O2 Decreasing X O2 Decreasing X O2 Decreasing X O2
		1200
		1000
		0.05 800	0.10	0.15 c 1, st [ -] 0.20	0.25	0.30
	Li MA CHAPTER 8. FIRE PLUMES UNDER OXYGEN-DILUTION	149
					QUENCHING CONDITIONS

  .2. FLAMELET LIBRARY CALCULATION model is retained as radiative property model. The Filtered RTE is solved by using FVM with a non-uniform FT12 angular discretization.

filtered flamelet look-up table is computed by the STE model. The RCFSK 150 CHAPTER 8. FIRE PLUMES UNDER OXYGEN-DILUTION QUENCHING CONDITIONS Li MA 8

belle expérience au sein du département MFEE d'EDF R&D, en collaboration