Jean-Baptiste Lagrange

STUDENTS WORKING ON MODELS; AN ON-GOING EXPERIMENTATION IN MATHEMATICS AND CHEMISTRY

This paper focuses on modelling at upper secondary level. The objective is to give students an understanding of mathematical concepts and methods in close relationship to a domain of reality, as well as to give them insight into the contribution of models of different kinds. This has led to the development of a framework for modeling activities based on the Mathematical Working Spaces theory. The questions at stake concern the operationality of this framework. To what teaching situations can it lead? How do students work in these situations? We examine these questions through an on-going experimentation on models of acidbase transformations.

APPROACHES TO MODELING IN MATHEMATICS EDUCATION AND SCIENCE DIDACTICS

Beginning in the 2000s, interest in involving real-world contexts has grown in mathematics education. The ICMI 14 study (Bloom 2002) kicked off a lively stream of research and focused on mathematics as an important activity in society. This stream seems to us marked by two lines of force, problem solving and the modeling cycle as a theorization of the modeling activity. Many authors indeed characterize modeling activities as solving authentic problems, but the activities they propose focus more on the solution to a contextualized problem than on models. For example, in [START_REF] Blum | Mathematical modelling: Can it be taught and learnt[END_REF], the task involves a lighthouse of a given height and students have to find a value for the visibility distance that is valid specifically for that height. The authors identify steps in students' problem solving consistent with the modeling cycle, but these steps lead to the solution rather than to a model. Overall, the modeling cycle remains close to a classical resolution scheme where the problem and the solution are expressed at the extra-mathematical level and solving is done at the mathematical level. The cycle specifies steps and transitions and this allows, among other things, the interpretation of students' trajectories in their complexity. Nevertheless, the "real" and the "mathematical" remain two levels insufficiently intertwined to account for how working on a model articulates mathematics and real-world objects and phenomena [START_REF] Czocher | How does validating activity contribute to the modeling process[END_REF].

In experimental science didactics, the main concern is the relationship between an "empirical referent" [START_REF] Sanchez | Quelles relations entre modélisation et investigation scientifique[END_REF]) made up of objects and phenomena as they are perceived and spontaneously mobilized by the students, and a "scientific referent" consisting of theoretical elements. By appropriating a model, the students can relate Lagrange 1 -2 PME 45 -2022 these two referents and thus progress both in their perception of everyday objects and phenomena and in their understanding of scientific concepts. Nevertheless, the mathematical aspects of the model are generally not questioned as such, and science didactics privileges models where these aspects are minor for fear of complexity.

Thus, dominant approaches in mathematics education emphasize problem solving rather than working on models, and science didactics favors appropriation of a model while leaving aside mathematical aspects. Drawing on science didactics, we aim to engage students in explicit work on models, but we also want to include mathematics in this work. The experimentation we are carrying out starts from a laboratory technique taught in the chapter about acid-base reactions of the chemistry course in the non-vocational upper secondary stream in France. Students often describe this technique as "a cooking recipe", since mathematical methods are used, but not explained with reference to acid-base reaction models. The purpose of the experimentation is then to look for ways to make students study models both in their chemical and mathematical aspects and get a better understanding of these aspects.

APPROACH AND FRAMEWORK

Lagrange et al. (forthcoming) distinguish between modelling and mathematization of a domain. While mathematization is global in scope, modeling aims to account for certain aspects of the domain in order to understand it, even partially, and to act on it. A corollary is that there is not a single model: several models are as many ways to approach a reality. Modeling thus has a subjective and social dimension: all models can be useful, but each one must be discussed and confronted with others. In each model, the contribution of mathematics results from a specific mathematical work, in collaboration with experts in the field, in order to make the model more intelligible and facilitate its use. Thus, there is not a real model on one side and a mathematical model on the other side, but a plurality of models, each with a specific implication of mathematics into the same domain of reality. We consider students' activity in modelling as a work of appropriation of two or more models, and as a work of uncovering relationships between models, in order both to get better insight into the domain and to progress in the mathematical concepts used; for instance [START_REF] Lagrange | Connected working spaces: designing and evaluating modelling-based teaching situations[END_REF] proposed to consider four models of a suspension bridge for a high school teaching project, one based on a study of tensions, a second one on arithmetical relationships, a third one being a computer simulation, and finally a fourth model based on notions in real analysis (functions, integration, etc.).

Modelling implies collaboration between experts with different viewpoints (Lagrange et al. forthcoming) and that is why the above approach has led to organizing students' work in a "jigsaw classroom". The work starts from a question. For the present study, the question will be about how a model of the reaction justifies the laboratory technique. There are four phases: (1) Presentation of the question and work on prerequisite concepts (2) Expert groups: each group works on a model from a specific Lagrange PME 45 -2022 1 -3 viewpoint (3) Jigsaw groups: each group gathers experts from each expert group and progress in understanding the models (4) Whole class discussion and conclusion by the teacher.

Regarding the notion of work in educational settings, we refer to the theory of Mathematical Working Spaces (MWS). According to [START_REF] Kuzniak | Mathematical Working Spaces in schooling: an introduction[END_REF], a MWS is an abstract space that is organized to support mathematical work in an educational setting1 . The theory of MWS distinguishes three levels:

 A reference working space (WS) is a space in which somebody educated in a specific domain is expected to do the work in this domain.  A suitable WS helps manage the work for beginners in a teaching project.  A personal WS is particular to individuals. As Menares-Espinoza and Vivier (forthcoming) explain, beginners approach a new domain with their prior knowledge and cognitive processes. Teaching must design tasks to help students' personal WSs evolve towards the reference level and this requires designing suitable WSs. Here a reference WS is what allows for scientific thinking about the laboratory technique. Models on which this thinking can be based are described below. This paper focuses on suitable WSs, both a priori with reference to models of the reaction and a posteriori from student observation, leaving for further research a study of personal WSs.

The research question follows. RQ: What are the suitable WSs that provide a conceptual basis for a mathematical-chemical approach of acid-base reaction models? How do they predict students' behavior and cognitive processes? The laboratory technique we start with is titration, i.e. the use of a solution of known 1 -4 PME 45 -2022 concentration (titrant) to determine the unknown concentration of another solution (analyte). The titrant is added from a graduated buret to a known quantity of analyte.

MODELS, WORKING SPACES AND TASKS

In case of an acid-base reaction, the analyte is an acidic solution, characterized by a preponderance of oxonium ions and the titrant is a basic solution characterized by a preponderance of hydroxides ions. During the titration, the oxoniums and the hydroxides react and then the pH (minus the decimal logarithm of the concentration in oxonium) of the mixture increases and a table of values (volume added, pH) is obtained (Table 1 on the left). The experimental curve is a sigmoid whose inflection point (called neutralization point) corresponds to a volume added for which the mixture is neutral, i.e. has equal concentration in oxoniums and hydroxides corresponding to pH 7. The position of the neutralization point allows to know the quantity of hydroxides added and consequently the concentration in oxoniums of the analyte. A geometrical technique (method of tangents) is used to determine this position. It is based on the quasi-symmetry of the experimental curve with respect to the inflection point (Table 1 on the right). As said before, no theoretical justification in chemistry and mathematics is given to students. The underlying model of the reaction is the evolution of the pH based on empirical observation, increasing and almost symmetrical with regard to the neutralization point. This is Model 1. Titrant: 8 ml acid, oxinium concentration 0.0125 mole/liter. Analyte: base, hydroxide concentration 0.01 mole/liter.

Oxinium concentration for x ml base< 10:

pH for x ml base < 10:

f(x)= -log(10-x)+log(8+x)+2

Hydroxide concentration for x ml base> 10:

pH for x ml base> 10:

f(x)= log(x-10) -log(8+x) + 12
Model 2 is another model in which the titration curve is obtained by the calculation of a mathematical function (Table 2). It is based on a simplifying assumption: hydroxide and oxonium ions neutralize so that the mixture contains only one of the two types of ions. This assumption allows to obtain a function whose curve is coherent with the empirical curve, except in the vicinity of the neutralization point where the function tends towards infinity. This is a consequence of a phenomenon called partial dissociation of water, that contradicts the assumption: acidic solutions contain

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PME 45 -2022 1 -5 hydroxides and basic solutions contain oxoniums, but this is negligible except in the vicinity of the neutralization point. Because tangents are drawn at a distance from the neutralization point, the model is a basis for a mathematical work to justify the method. Mathematics also allows to question the simplifying assumption, and thus students working on the models should progress both in chemistry and mathematics.

In accordance with the RQ, the experimentation aimed to build and evaluate suitable WSs thanks to which students could recognize Model 1 and 2 as the foundations of the titration technique, and compare the two models. These WSs are designed for the two group phases (expert groups and jigsaw groups). A presentation of the WSs and associated tasks follows, summarized in Table 3.

WSJ:

Idem Ec + symbolic calculations Task J: Compare curves (Tasks Ea, Eb,Ec). Show how f (Task Eb) models the pH as a function of the added volume. Justify the tangent method. WSEa is suitable to work on Model 1 and thus includes elements of chemistry (ions, pH, concentration, etc.) but also of mathematics (measurements, curve, tangents, ratios, etc.) We choose to have Ea students work on computer software simulating titrations. The purpose of the software is to help systematize and reflect on the method: students must enter the data of a given titration; they obtain a curve simulating the empirical curve, they can choose points to perform the tangent method and observe the accuracy of the method (Task Ea).

WSEb is the space for a complete mathematical study of the Model 2 function. The signs, the theoretical frame of reference as well as the use of a software for functions WSEc is the space for students to numerically compute concentrations along the titration, using the assumption of complete neutralization. The elements of chemistry are the same as in Model 1, but they must be systematically quantified; pH formulas must be used. From a mathematical point of view it is an arithmetic space. Task Ec requires students to have a good mastering of chemistry notions, as well as ratios and unit conversion. The calculation of the concentration must take into account the quantity of ions, but also the increase of the volume along the titration.

WSJ correspond to the jigsaw groups in Phase 3. This is the space for students to understand Model 2, make the connection to the assumption that the mixture contains only one type of ions and make sense of the tangent method at a symbolic level. Task J should lead students to perform a computation similar to Task EC, but at a symbolic level as in Table 2, and to justify that two parallel tangents, one on each branch, are nearly symmetrical with regard to the neutralization point.

IMPLEMENTATION AND ANALYSIS

In the context of the pandemic in 2020, the experimentation was carried out in the form of a "lock-down online jigsaw classroom", i.e. the six participants were physically separated, communicating on a platform that allowed either all students to be gathered together (Phases 1 and 4) or split into groups (three groups of two in Phases 2, and two groups of three in Phase 3). The students were of average level, some more proficient in mathematics, other more in experimental sciences. They had previously performed titrations on real solutions. The platform allowed the recording of exchanges and productions. This data was completed by a e-mail survey. In the analysis of the data we leave aside the aspects related to the online work, underlining only that even online, the jigsaw classroom kept its potential for collaborative work. This is the analysis of the group work phases (about one hour each). Table 4 presents extracts of the reports made by the three experts groups (Ea, Eb, Ec, Phase 2) and the two jigsaw groups (J1, J2, Phase 3). Ea appropriated the simulation software after having difficulty understanding menus and data needed. They were able to use the tangent method for several values and compare the accuracy. Eb's study of the function remained partial as the log decimal function was unfamiliar. As shown in their report they were also not comfortable with a piece-wise function. They noted an undefinitness, but did not mention infinite limits that would have shown inconsistency with the experimental curve. Pseudo-symmetry and use of parallel tangents for the position of the center were discussed. Ec was comfortable with the chemistry concepts and the various calculations, but had difficulty taking into account the variation in the volume of the solution. They made the connection with the titration curve. Overall, Ea and Ec's work can be seen as fitting in the suitable

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PME 45 -2022 1 -7

WSEa and WSEc after initial difficulties. This is not the case for Eb. The students were not comfortable with the function of Model 2, which is different from routine functions they had been trained with. They were able to draw curves and tangents thanks to the software. It was consistent with WSEc only for the use of software. Group Ea We used a software to enter the data, and we got curves for concentration and pH . We saw clearly the turning point that makes it go from acidic to basic. We used the tangent method to get the pH. Group Eb We had two functions, one for x<10 and the other for x>10, but it was the same function. The two functions are not defined in 10.

We drew the curves with GeoGebra. The pH curve, was symmetrical with regard to the neutralization point. We placed the points A and B according to the given abscissas then we traced the tangents and the mid line on the computer to find the neutralization point. Group Ec We calculated the quantity of oxoniums for each added volume. Then we applied the formula m/V after converting the volume into liters to get the concentrations, and the formula -log(Ca) for the pH values. We did similar calculations for the hydroxides after neutralization. We saw that values increased and that it looked like the empirical curve. Groups J J1 We observed that when we apply the formulas of group Eb, we obtain the data found by group C. We can therefore deduce that the curve representing pH are related to the function of group Eb. We calculated the derivatives to get the slopes of the tangents. When the difference between these is close to 0, we get more accurate results. J2 We had to develop a formula that actually calculates everything at once. We made calculations like Ec did but with a variable x and we got the function of group Eb for the acidic part. Both J groups observed that the curves obtained by the three experts groups were similar. Group Eb's remark (the two functions are not defined in 10) did not lead students to observe a discrepancy between models near neutralization. Group J1 concluded that the similarity of curves is sufficient evidence that f is a model of pH evolution. They started a study of the slope of the tangents with regard to the accuracy of the method, using the derivative of f with difficulty. Group J2 looked for a an analytic proof that f is a model and succeeded only for the acid part. The behaviors in both groups show students' partial appropriation of WSJ, the suitable working space that should provide mastery of Model 2. One shortcoming is that students' symbolic calculation skills taught in math class were poorly enacted. Another shortcoming is that they were not able to recognize a discrepancy between the models. In Phase 4, after the J groups reported on their work, the teacher emphasized the use symbolic computation and the discrepancy between models near Lagrange 1 -8 PME 45 -2022 neutralization, which he explained by the dissociation of water. Answers to postquestions by email showed that these points were partially understood by students.

Conclusion and perspectives for research

The suitable WSs prepared for this experimentation allowed the students some appropriation of the models both in their chemical and mathematical dimensions. A critical look highlights achievements and gaps. WSEa did not help students distinguish Model 1 from Model 2. WSEb was too demanding in symbolic calculation. WSEc seemed appropriate, with students Ec completing the task and contributing to the work in Phase 3. WSJ was affected by Ea and Eb's shortcomings.

Another implementation was then carried out in 2021 with a variation of the tasks. In Task Ea the simulation was a computer program that the students could read and interpret. In all tasks it was asked to get values of the pH (Tasks Ea and Ec) or of the function (Task Eb) very close to the neutralization point. The discrepancy between values obtained by Ea on one side and by Eb and Ec on the other side brought a discussion in groups J. The students did not reach a consensus. Some students emphasized the validity of the model underlying the computer program and its conformity with empirical observations, and others maintained that Model 2 was more reliable, stressing the inaccuracy of empirical observations compared to mathematics. These results may be somewhat surprising and unsatisfaying but they confirm the interest of this situation and provide insights for further experimentation.

  school calculus, with the exception of decimal logarithms and the piece-wise function which are unfamiliar. Task Eb is a study of properties of the function. It is classical in the form, but the function is unusual.

Table 1 :

 1 The laboratory technique. Empirical procedure and tangent method.

Table 2 :

 2

Mathematical function based on assumption: mixture contains only one type of ions

  There are three expert groups. Students in group Ea should become experts in Model 1, students in Eb should become experts of the mathematical component of Model 2 and students Ec should develop an expertise in quantifying the evolution of concentrations throughout the titration. WSEa, WSEb and WSEc are suitable WS, each representing the respective expertise targeted in each group and WSJ is the suitable WS pour the Jigsaw groups.

Table 3 :

 3 Suitable working spaces and tasks in the group works.

	WSEa:	Task Ea: Appropriate a simulation
	Acid, base, neutralization (visual),	software. Simulate for given data. Operate
	pH (reading), measure, proportion,	the tangent method for varied positions and
	curve (experimental), tangents (visual).	compare accuracy.
	WSEb:	Task Eb: Study the function f (Table 1);
	Functions (symbolic), curve, tangents	growth, limits. Trace the curve and tangents
	(software), decimal logarithms.	at abscissas 10-x and 10+x for varied values
		of x ; observe the mid-lines.
	WSEc:	Task Ec: Calculate hydroxides and oxynium
	Ions, concentration, neutralization, pH.	concentration for varied added volume.
	Volumes, ratios, formula.	Calculate pH for these values and draw curve.

Table 4 :

 4 Extracts of reports of students' group work.

Because of limited space, we do not insist on the three dimensions that structure a MWS: semiotic, instrumental and discursive. However they are important in the domain of modelling, ensuring that mathematics are not simply considered as a "language".

PME 45 -2022 1 -9 dans l'enseignement des sciences de la terre? Éducation & Didactique 2 (2), 97-122.


Louis Le Moigne

Dr David Gajan

Dr Jennifer Lesage De La Haye

Dr Jean Raynaud
email: jean.raynaud@univ-lyon1.fr

Emmanuel Lacôte
email: emmanuel.lacote@univ-lyon1.fr

Tommaso Posenato

Dr L Lemoigne

Catalyst-Free Transfer Hydrogenation from Amine-Borane Small Oligomers

Keywords: boron, dehydrogenation, hydrogen transfer, amineboranes, oligomers

come

Introduction

Hydrogenations are ubiquitous in organic synthesis. However, efforts have been made in the last decades to circumvent the use of gaseous dihydrogen, which requires pressurized containers to enhance concentration. This has led to the development of transfer hydrogenation methodologies, [1] as well as reagents able to serve as dihydrogen source, such as isopropanol or borazane (also called ammonia-borane, H3N•BH3). The latter has elicited considerable interest because of its high 19.6 wt% hydrogen content, more than 13 wt% of which retrievable at a reasonable energetic cost. [START_REF] Rossin | The dehydrogenation of borazane, with or without help from transitionmetal catalysts has been extensively investigated, see for instance[END_REF] Ammoniaborane and amine-boranes can dehydrogenate thermally, in the presence of a transition metal catalyst or with a frustrated Lewis pair (FLP). [START_REF] Rossin | The dehydrogenation of borazane, with or without help from transitionmetal catalysts has been extensively investigated, see for instance[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF][5][6][7] The dihydrogen or hydride generated can be transferred to various aldehydes, ketones, imines, pyridines, or olefins. [8,9] With the highest dihydrogen content, simple borazane is thus especially attractive. [START_REF] Rossin | The dehydrogenation of borazane, with or without help from transitionmetal catalysts has been extensively investigated, see for instance[END_REF] However, it is a solid with a tendency to generate unwanted by-products and soiled dihydrogen (traces of borane and ammonia as well as aminoborane primary products usually contaminate the produced H2). [START_REF] Rossin | The dehydrogenation of borazane, with or without help from transitionmetal catalysts has been extensively investigated, see for instance[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF] The Lewis pair can also fall out during the transfer, leading to the binding of the reduced substrate to the borane moiety, which requires a subsequent hydrolysis step. A lot of studies have examined substituted amine-boranes. [START_REF] Rossin | The dehydrogenation of borazane, with or without help from transitionmetal catalysts has been extensively investigated, see for instance[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF][START_REF] Similarly | the dehydrogenation of amine-boranes, with or without help from transition-metal catalysts has been extensively investigated[END_REF][5][6][7]10] This addressed part of the limitations. As part of our program devoted to boron polymers [11] we have investigated polyboramines, i. e. polymers containing hydrogen-rich amine-boranes in the repeat units. [11a,b] We observed that polyboramines retained the ability to transfer hydrogen to imines and ketones, without catalyst. In addition, the products were recovered directly without any hydrolysis step. [11a] In order for the direct hydrogen transfer from amine-boranes to proceed, the substrate must generate a complex with the amineborane, which induces the double transfer of the hydrogens. This requires the disruption of the dihydrogen bond (DHB, Figure 1) networks that are specific to amine-boranes, and this has an energetic cost. [2e, 12] The enhanced reactivity of the polymers is likely a consequence of entropy, which hampers strongly ordered chain/chain interactions in polymers, relative to molecular amine-boranes, and facilitates the approach of the small molecule substrates to the polymers in comparison. In the present publication, we explore the intermediate structural space between simple molecules and long-chain polymers, that of oligomers of amine-boranes. We introduce two oligomer structures O1-2 (Schemes 1, 2 and S1) and examine their thermal reactivity, as well as with a variety of transfer hydrogenations substrates (aldehydes, ketones, imines), but also electron-deficient alkenes and alkynes. Our focus aims at understanding how the chain-lengths influence the reactivities of repetitive amine-boranes. For comparison purposes, we also included dimers 2a-d in the panel studied.

Results and Discussion

Synthesis

Amine-boranes 2a-c were synthesized first. They feature two amine-borane Lewis pairs derived from phenyldiboronic acid, with different groups at the amine end of the amine-borane (Scheme 1). 2a and 2c are derived from ethylenediamine with two different protecting groups at the -amino end. 2b is derived from benzylamine. 2d is also derived from benzylamine, but grafted on a more flexible aliphatic 1,2-diboronic core. All substrates were prepared from the corresponding ammonium chlorides (Scheme 1, see SI pp 5-15 for their preparations). Lewis pairs 2a-b were obtained by reducing phenyl-1,4bisboronic acid 1a with LiAlH4 in the presence of the ammonium chlorhydrates. The inorganic aluminium and lithium salts were easily filtered off. In the case of 2c, the lithium borohydride derivative was prepared and isolated first. The direct reaction led to degradation in that case, presumably because LiAlH4 reacted with the Boc group rather than converting the boronic acid to the trihydridoborate intermediate needed to generate the amineborane. All three bis-amine-boranes were obtained in quasiquantitative yields and analytically pure. Although minute amounts of residual salts could still be present we did not purify the compounds further. 2d was obtained directly from bis-boronate ester 1b. It exhibits a 11 B signal at -9 ppm, as expected for the amine-borane.

However, in this case the reduction generates pinacol salts which proved very soluble in THF. This made their elimination extremely difficult. As a result, 2d is contaminated by much more salts. We nonetheless kept it to get a general idea of its reactivity, but decided not to use 1b to generate oligomers as the presence of salts would perturb too much the stoichiometry required for the AA/BB type polycondensation (see below). With the synthesis of the bis-amine-boranes in hands, we selected an AA/BB polycondensation strategy. An end-capping agent was added to stop the polymerization at the oligomer level. We prepared two oligomers derived from bis-boronic acid 1a. The flexibility of the oligomers was controlled via the diamine partner (Scheme 2). 1,4-dianiline is a rigid diamine and butyl-1,4-diamine is flexible. Benzylamine was used as capping agent in both cases. In a typical experiment, a mixture of the bisboronic acid and the bis-ammonium hydrochloride in THF was reacted with LiAlH4 in the presence of benzylamine hydrochloride, in a procedure similar to that used for preparing 2a-b. A degree of polymerization (DP) between 3 and 4 was targeted using a non-stoechiometric amount of bisboronic acid and diamine monomers. Oligomer O1 (resp. O2) was obtained in 53% (resp. 74%) yield. Using 1 H NMR we determined the numbers of repetition units as being 1-2 for O1 and 2-4 for O2 (see SI, pp 13 & 45). The theoretical values were not reached likely because of chain discrimination during filtration. The longer chains of O1 may have precipitated in THF and therefore been lost in the salt filtration step. This is also in line with the observed yields. Nonetheless, the formation of the amine-boranes was evidenced by 11 B NMR (broad singlet from in the -9 to -11 ppm), and by 1 H NMR (broad singlet in the 2.3-2.5 ppm region), as well as by IR (B-H ~ 2300 cm -1 , see the SI, pp 60-64 for the complete analyses).

Scheme 2. Preparation of the amine-borane oligomers (with benzylamine as end-capping agent).

Thermal dehydrogenation

We first focused on the thermal dehydrogenation profiles of the materials obtained in order to assess their inherent ability to release molecular dihydrogen, i. e. without catalysis. Differential scanning calorimetry (DSC) indicated that neat amine-boranes 2a-c liberate a first equivalent of H2 at respectively 84, 111, and 112 °C. In all cases the processes initially generating aminoboranes were exothermal (see SI, pp 55-60). [11a,b] O1 and O2 liberated their first equivalent of H2 at 120 °C and 102 °C respectively, also as an exothermal process. However, in the case of O2 the exotherm was immediately preceded by an endothermal event (see SI, page 57). This might be attributed to a transition to higher chain mobility and/or the formation of the intermolecular dihydrogen-bonding network necessary for the thermal dehydrogenation, which is more pronounced in the oligomers relative to the small molecules. The (exothermal) DSC profiles of the first dehydrogenation show that oligomers O1-2 behave like their molecular counterparts. An endothermic dehydrogenation profile was observed for the corresponding polyamine-boranes. [11a] In the polymer, the architecture reorganization needed to establish the interactions required for the dehydrogenation is likely both more pronounced than in the oligomers. It is therefore shifted at slightly higher temperatures, and finally overlaps with (and offsets) the exothermic dehydrogenation. The overall process is therefore endothermic. Therefore, the thermal dehydrogenation profile of oligomers O1 and O2 bridges the gap between pure molecules and polymers. The second dehydrogenation to the fully dehydrogenated iminoboranes was more difficult to observe because it is spread over a wider temperature range. The baseline is not very stable and the DSC chromatograms are thus more difficult to integrate. Nonetheless, for 2a-c exothermal dehydrogenations were observed at respectively 137, 179 and 154 °C, and endothermal ones were detected for O1 and O2 at respectively 164 and 142 °C, that once again could result from two concomitant processes, like in the polymers.

Transfer hydrogenation

We next investigated the transfer hydrogenation of the amineboranes to various substrates: aldehydes, ketones, imines, alkenes, and alkynes (Table 1). All yields and conversions were determined by 1 H and 11 B NMR spectroscopy, using tetrachloroethane as internal standard. The amine-boranes were first reacted with aldehydes at room temperature in THF-d 8 (Table 1, Entries 1-2). This led to almost quantitative conversions, except for 2a-b, and good yields of the desired alcohols. The putative aminoborane by-products were insoluble and could be easily filtered off. However, imines obtained from the reaction between benzaldehyde and the benzylamine from amine-boranes 2b and O2 were isolated in 46% and 23% yield, respectively. We suppose that unwanted side-products form when the transfer hydrogenation is slow, and the imine is easily formed and stable. The amine-boranes were then mixed with acetophenone and benzophenone (Entries 3-4). The conversions were lower than for the aldehydes, except for dimer 2c and oligomer O1 that led to good yields of the reduced products (~80%). The selectivity was 100% in all cases, with exclusively the alcohol formed, thus suggesting that no decomposition occurred. [b] (74) [E] 25 [b] (94) [E] 16 [c] (99) [E] 28 [d] (99) [E] 5 [e] (99) [E] 8 13 [f] (84) [E/Z = 2.2:1] 5 [g] (47) [E/Z = 1.5:1] 45 [h] (99) [E/Z = 1:1] 11 [b] (81) [E/Z = 2.7:1] 32 [h] (99) [E/Z = 1.9:1] 9 [d] (81) [E/Z = 2:1] 9

12 (45) 0 (32) 23 ( 54) 1 (42) 28 ( 53) 5 (31) 10 14 [i] (68) 3 [j] (96) 23 [j] (99) 0 [k] (93) 41 [l] (99) 5 [m] (72)

[a] Conditions: Substrate (0.08 mmol), amine-borane (0.08 mmol of Lewis pairs), THF-d The reduction of diphenylimine and sulfonylimine proceeded in high conversions (Entries 5-6), especially with the more electrophilic one, the sulfonylimine, albeit the more rigid O2 was slightly less reactive toward diphenylimine. Some side-products were still observed. The alcohols coming from the reduction of the aldehyde part of the starting imines (benzaldehyde) were observed when 2a (7% for entry 5; 11% for entry 6), 2c (15% for entry 5; traces for entry 6) or 2d (4% for entry 5 only) were used.

Similarly, as in entries 1-2 the imines derived from the decomplexation of the corresponding amine from the borane and subsequent condensation were also observed (see SI, pp 16-25 for details). In some cases, these by-products were actually the major ones, as for the reduction of diphenylimine with 2d (41%), or that of the sulfonylimine with 2b (77%) or 2d (58%). Also, the reaction of the sulfonylimine with O2 delivered 65% of 4-(benzylideneamino)aniline. In this case it is the diamine (1,4-dianiline) at the core of the oligomer which is transferred, rather than one positioned at the end of a chain. Despite these shortcomings, the isolated amine yields and selectivities are the highest for amine-boranes 2c and particularly O1which are consistently more reactive all throughout entries 1-6.

Overall, given that 2b and 2d roughly behaves similarly, it shows that the transfer hydrogenation is not highly sensitive to the presence of the salt residues.To summarize the results in entries 1-6, the amine-borane dimers and oligomers proved quite efficient as high yields could be accessed in most cases using only one equivalent of amine-borane relative to the substrate. This is to be compared with H3N•BH3 which often requires excesses of amine-borane and/or a metal catalyst to assist the hydrogen transfer, and as a fluffy solid is practically difficult to deal with. This is in line with the existing literature body on hydrogenations with small-molecules substituted amine-boranes.

To better understand the role of the oligomer/polymer environment we wished to get a deeper insight into the reactivity. Two distinct pathways have been proposed to account for the direct transfer of dihydrogen from the amine-boranes to different substrates (Figure 2). [8h] A first path proceeds via a concerted double H transfer with a six-center transition state (Figure 2a). It is generally followed in the hydrogenations of imines and ketones/aldehydes. [8d,e,f] A second pathway, which is often followed for olefins and less reactive carbonyls, involves a sequential transfer with an initial hydroboration step (hydride from the boron end), followed by a protodeboronation (proton from the amine end) one (Figure 2b). [13] The mechanism for these polarized substrates is likely a concerted double-hydrogen-transfer reaction with a sixmembered transition state (Figure 2a). [8d,e,f] Oligomer O1 (and to a lesser extent O2) as well as diamine-borane 2c (and to a lesser extent 2a) perform better than the other reagents. In the case of 2c we believe this is due to the likely formation of a chelate via an additional H-bond (Figure 2c) that enhances the polarization of the reagent. This additional H-bond can also be established in the case of 2a, but with less effect as it is now an amine, not a carbamate. The reactivity enhancement in the oligomers likely arises from similar synergistic effects favouring necessary six-membered intermediates. O2 is less reactive than O1 because it is less flexible (and different Lewis pairs), which would be consistent with O1 readily able to develop favorable transfer hydrogenation intermediates. Considering the previous results, we chose to study the more challenging alkenes and alkynes, whose hydrogenation likely proceed via a stepwise transfer when the alkenes are highly polarized (Figure 2b). [8g, 14-16] To gauge the reactivity of our amine-borane oligomers, we selected two electron-poor yet symmetrical alkynes, so as not to facilitate the hydrogenation by tweaking the electronic configuration of the unsaturations. We also picked dimethyl fumarate and dimethyl maleatethe corresponding electron-poor alkenes. Both alkynes led to high conversions (Entries 7-8) with all amine-boranes. Yet, only average hydrogenation yields were observed. The maximum yields were again observed with amine-borane 2c (25% and 45%, respectively) and oligomer O1 (28% and 32%, respectively). The major isomer for the diethyl diester was the alkene E (fumarate) although the stereoselectivities remain poor. Oddly, the dimethyl diester led to only dimethyl fumarate. Given that the yields from the dimethyl diester (Entry 7) are somewhat lower than those of entry 8, it is possible that this enhanced selectivity is due to selective degradation. The main by-product (which actually is the main product in most cases) is the result of the hydro-amination: the amine part of the amine-borane reacts with the substrate. In the case of 2c, the nucleophilic attack of the chain-end amine (H2N-CH2-CH2-NH-Boc) led first to the E isomer. After 4-6 hours, the kinetic product was gradually converted into the Z isomer, which seems to be thermodynamically favoured, although the final ratio between both isomers depends on the amine-borane used (see SI, pp 16-25). The reduction of dimethyl fumarate (E isomer) (entry 9) and dimethyl maleate (Z isomer) (entry 10) were also attempted. In the case of the former, a similar trend as for entries 7-8 was observed, albeit with much reduced conversions. Again, 2c and O1 led to the best yields. In the case of the latter substrate, the conversions were much higher in all cases. This is likely due to the equilibrated isomerization of the maleate into the fumarate during reaction. This equilibration (reminiscent of the presence of the trans hydrogenation product in entry 8) coupled to the observation of hydroamination products when starting from the alkynes suggest that this inversion may be due to a reversible amine transfer, with a concerted attack of nucleophilic amine (Michael addition). Nonetheless, in the case of the alkynes, a stepwise mechanism, leading to a stereochemical loss either at the initial hydroboration step, or at the protodeboronation one, cannot be excluded.

As before, we believe oligomer O1 and amine-borane 2c perform better because of the possible cooperative interactions they can develop, either because they are multivalent oligomers, or because of the additional H-bond.

Comparison between thermal dehydrogenation and transfer hydrogenation

Amine-boranes 2c and O1 were heated in a Schlenk tube for 5h at about 160 °C and the spent material was analyzed in DSC.

The initial heating treatment was designed to extract the first equivalent of dihydrogen and generate the aminoborane, prior to the DSC analysis. This results in "isolating" the second dehydrogenation (see SI, pp 55-60). The spent material exhibits an endothermic process at about 150 °C. This corresponds to the second dehydrogenation in the previous DSC analysis. We used 11 B CP-MAS and direct-detection very high-field solidstate NMR spectroscopy to compare the products from the first thermal dehydrogenation and those isolated post-transfer hydrogenation (see SI p.54 for description of experiment). Both materials display very similar spectroscopic signatures (see Figure 3). Because these spectra were acquired on a 1 GHz spectrometer, and very high rotational speeds were used, the isotropic shift can almost be derived from these relatively narrow spectra. The chemical shifts are consistent with a mixture of aminoboranes (massif between 0-5ppm for unimeric form and ~25 ppm for di-hydrogen-bonded dimeric structures) and iminoboranes (massif between 15-20 ppm). [2-4, 11, 17] We based our assignment of the signals on that of the native amineboranes (see SI, page 54). We observed the signal corresponding to the amine-boranes 2c and O1 centered ~ -1 ppm, along with exactly two signals i) between 0-5 ppm corresponding to unimeric aminoboranes and ii) at ~25 ppm, corresponding to dimeric structures. Both signals arise from the first dehydrogenation. These signals grow simultaneously during the acquisition (see Figure 50 in the SI). The dehydrogenation observed during NMR acquisition is due to the 33.33 kHz spinning rate of the NMR rotor, which produces enough heat to promote partial release of the first H2 equivalent. Both analyses indicate the formation of polyaminoboranes, but also of a small proportion of iminoboranesi.e. coming from a second dihydrogen transfer. We conclude that the boron byproducts issued from the thermal and chemical dehydrogenation of the dimeric and oligomeric amine-boranes are similar. Solidstate NMR additionally suggests that more than one equivalent of H2 is utilized during transfer dehydrogenation with O1. The amount of iminoboranes obtained after the chemical reaction is significantly higher than in the case of the controlled thermal treatment.

The presence of signals arising from the transfer of the second equivalent of dihydrogen in our materials, led us to investigate whether this second equivalent of dihydrogen might be harnessed for the transfer hydrogenation without help from any external catalyst (Scheme 3). Borazane is able to accommodate this second transfer presumably via a solvent-stabilized form (S-H2B=NH2), [13] or cyclic structures such as cyclotriborazane or B-(cyclodiborazanyl)amino-borohydride. [8e] So we wished to learn whether our oligomers could also produce their H2 second equivalent in a synthetically productive way. Acetophenone was reacted with 2c (resp. O1, 0.5 equivalent of Lewis pair) in THF-d 8 in J-young tubes. Tetrachloroethane was used as internal standard. After 18 hours at room temperature, the conversion was 31% (resp. 59%). The reaction was left for one week. With amine-borane 2c, the conversion peaked at 36%, while it increased to 66% with oligomer O1. The selectivity was good, as no degradation or side-products were identified.

The isolated yields were identical to the conversions. The same conversions were obtained after only 18 h at 60 °C, albeit the isolated yields dropped when O1 was used. Benzophenone behaved identically, except that less degradation was observed at 60 °C. Gratifyingly, in the case of O1 both alcohols were obtained in more than 50% yield. This shows that O1 is able to engage the second equivalent of H2 in the transfer hydrogenation, but not 2c. This confirms the NMR spectroscopic of Figure 4. There is a clear improvement with the oligomers, further highlighting the potential of the oligomer and polymer structural space for hydrogen-rich amine-boranes. Interestingly this can be achieved in the absence of catalyst, but could also be advantageous for more challenging substrates and in the presence of a catalyst. [18]

Conclusion

In the present work we showed that amine-borane dimers and oligomers promote transfer hydrogenations with enhanced efficiency. We evidenced that this happened via cooperative dihydrogen-bond networks and likely favourable substratereducing agent dipolar interactions. Therefore, the flexible O1 oligomer is a better dihydrogen donor. A similar reactivity improvement was observed on shorter amine-boranes when an H-bond was adequately positioned to generate a cooperative effect, such as in 2c. Very interestingly, the oligomer is able to transfer more than one equivalent of dihydrogen. Furthermore it was shown that the oligomers bridge the gap between the DSC profiles of the thermal dehydrogenation of small molecules and poly(amine-borane)s. The differences likely arise from the increasing cost required to preorganize the materials for dehydrogenation. This illustrates the interest there is in exploring the poly(amine-borane) structural field to tune or change established reactivities. Further work will focus on the rehydrogenation/recycling of the identified corresponding polyaminoboranes and/or polyiminoboranes, ideally without catalyst assistance.

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Entry for the Table of Contents

Small hydrogen-rich amine-borane oligomers were prepared via capping agent-controlled AA/BB polycondensation. Solid-state NMR spectroscopy, Differential Scanning Calorimetry and reactivity in catalyst-free transfer hydrogenationsto aldehydes, imines, electron-poor alkenes and alkyneswere used to assess how the oligomers bridge the gap between molecular amine-boranes and poly(amine-borane)s.

Figure 1 .

 1 Figure 1. Dihydrogen bonds (DHB) of borazane.

Scheme 1 .

 1 Scheme 1. Synthetic route to the molecular amine-boranes 2a-d (Bpin = Boron pinacolato, see SI, pp 5-15 for details).

Figure 2 .

 2 Figure 2. Main mechanistic pathways for the amine-borane transfer hydrogenation: a) synchronous transfer; b) stepwise hydroboration/protodeboration; c) additional activation via intramolecular Hbond.

  Figure 3. 11 B solid-state NMR spectra of 2c and O1 after thermal treatment (blue) and transfer dehydrogenation (red).

Scheme 3 .

 3 Scheme 3. Reduction of (a) acetophenone and (b) benzophenone with a halfequivalent of amine-boranes 2c and O1.

Table 1 .

 1 Transfer hydrogenation from amine-borane dimers and oligomers to various substrates.

			Yield (conversion) (%) for the reduction by the amine-borane indicated [a]
	Entry	Substrate
			2a	2b	2c	2d	O1	O2
	1	Benzaldehyde	27 (70)	32 (90)	99 (99)	39 (99)	99 (99)	78 (99)
	2	Hexanal	97 (99)	29 (99)	90 (99)	39 (99)	99 (99)	97 (99)
	3	Acetophenone	22 (22)	32 (35)	82 (82)	31 (49)	85 (85)	37 (37)
	4	Benzophenone	8 (8)	27 (36)	73 (73)	36 (37)	83 (83)	34 (34)
	5	Diphenylimine	34 (88)	14 (87)	59 (96)	39 (92)	62 (87)	20 (59)
	6	Sulfonylimine	78 (99)	23 (99)	73 (99)	42 (99)	88 (99)	28 (99)
	7		0 (81)	12

Acknowledgements

We thank CNRS, Université Claude Bernard Lyon 1, CPE Lyon, ENS Lyon, CNES and ArianeGroup for funding of this work. The research was also supported by ANR (grant ANR2021-CE06-PRC-PolyBora). We also thank the TGIR-NMR and in particular the CRMN Lyon (UMR 5082) for NMR spectroscopy. We are grateful for our stimulating discussions with Dr. Franck Collas (Mettler Toledo) regarding DSC characterization of our samples.


Marc Gual Skopek

Marianna Braza

Yannick Hoarau

Frank Thiele

M Gual Skopek

Hybrid RANS-LES Modeling of a Strongly Detached Turbulent Flow around a Tandem Cylinders Configuration

published or not. The documents may come

Introduction

Within the ATAAC (Advanced Turbulence Simulations for Aerodynamic Application Challenges) EU program, it has been shown through its main testcases, that DDES (Delayed Detached Eddy Simulation) is a promising approach to capture the unsteady dynamics and the turbulence statistical content in strongly detached unsteady flows, by using reasonably fine grids, comparing to the grid sizes that would be needed by LES, for the present high-Reynolds number range flows around bodies, involving strong detachment. The present configuration highly interests aeroacoustics and aerodynamic noise control, generated by the two supports (tandem cylinders) of a generic configuration of a landing gear. The present test case has been the object of detailed experimental and numerical studies leaded by the NASA Langley Research Center, [START_REF] Jenkins | AIAA 2005-2812[END_REF][START_REF] Khorrami | Simulations of bluff body flow interaction for noise source modelling[END_REF], 2007, among other). The experiments were carried out by standard 2D PIV, at Reynolds number 166,000. In order to avoid natural transition development, the experiments applied two transition strips at two specific upstream locations. A first synthesis of numerical simulations carried out for this test case was carried out by Lockhard, regrouping an order of 13 contributions involving different modelling approaches, as well as previous simulations by [START_REF] Khorrami | Simulations of bluff body flow interaction for noise source modelling[END_REF], using URANS-SST. These simulations indicated that the majority of the approaches captured quite well the Strouhal number of the vortex shedding frequency around the first cylinder, (St=0.24) . Furthermore, as is seen in the ATAAC European program, the DES approaches better capture the complex vortex dynamics of the present flow, especially the formation of Kelvin-Helmholz vortices in the separated shear layers. In the experimental context, it was found that the shear layers formed downstream of the first cylinder wrap around the second cylinder and interact non-linearly with the complex turbulence background, producing predominant frequencies in the energy spectrum, in the range of acoustic noise. Moreover, the numerical studies reported by Lockard in the context of the workshop for airframe noise computation, evaluated the mean drag coefficient provided by the different simulations, that had shown quite a dispersion among the different studies, with a most probable mean value of order 0.484 around the first cylinder. The aim of the present paper is to evaluate the ability of the DDES methodology for an accurate prediction of the pressure fluctuations and frequencies responsible for the acoustic noise.

In the present study, the DDES approach is considered, involving OES modelling [START_REF] Braza | Turbulence Properties in the cylinder wake at high Reynolds number[END_REF], in the URANS part, accounting for improvement of the nearregion flow physics in respect of non-equilibrium turbulence. In [START_REF] Bourguet | [END_REF], the DES-OES modelling was successfully applied for the simulation of the strongly detached flow around an airfoil beyond stall. The present DDES-OES, as the previous one, aims at providing a smooth passage from the URANS towards the LES region of flow detachment and at keeping the statistical region extent quite significant around the body.

Test-case Description

Experimental set-up

The experiments were conducted at the NASA Langley Basic Aerodynamic Research Tunnel (BART), which is a subsonic, atmospheric wind tunnel. The free stream velocity was set to 56 m/s and in a second experiment to 44 m/s. The diameter of the cylinders in the second experiment was made higher to accommodate more pressure orifices, to be able to have a more precise pressure distribution around the cylinder. Therefore both velocities correspond to a Reynolds number based on the cylinder diameter of Re = 166,000. The free stream turbulence level at these conditions is about 0.09%. Figure 1 shows schematically the configuration with the Tandem Cylinders placed in the wind tunnel, as well as a sketch of the computational used. An oil flow visualisation is presented in fig. 2. Transition strips were taped on the cylinder between  = 50°-60° and  = 300-310°. Experiments were performed to demonstrate the effectiveness of these strips, where the cp-curve of a single cylinder with transition strips was compared to the cp-curve of a high Reynolds test-case at Re = 8,4 x10 6 of a single cylinder (Roshko, 1961). The distance between the cylinders is L = 3:7D (Figure 5.1 (a)) and the length of each cylinder is 16D. The transition strips were placed on the first cylinder, as the second cylinders boundary is already fully turbulent from the stagnation point. This is because of the turbulence caused by the flow separation at the first cylinder. The Mach number of the experiment is M = 0.128, corresponding to an incompressible flow.

Numerical method and turbulence modelling

The simulations have been performed with the NSMB solver, (Navier-Stokes Multi Block), based on structured grids architecture. IMFT is part of the NSMB consortium (Vos et al, 1998), contributing with turbulence modelling development for strongly detached unsteady aerodynamic flows. For the present simulations, central difference scheme has been applied, among the variety of spacial discretisation schemes available in the solver, after performing detailed tests [START_REF] Barbut | Analyse physique par simulation numérique d'écoulements turbulents instationnaires autour de surfaces portantes fixes ou en mouvement, à nombres de Reynolds et de Mach élevés[END_REF][START_REF] Gual-Skopek | Hybrid RANS-LES Modelling on a strongly detached turbulent flow[END_REF]. Dual time stepping is used for the temporal discretisation.

The computational grid used is the one provided by NTS -St Petersbourg, by the partners M. Strelets and M. Shur in the ATAAC European program, shown in figure 3. The mesh refinement can be appreciated in Figure 3 (lower). The closest point to the wall at the second cylinder is at a distance of y=D = 3.0 x10 -5 . The influence on the aerodynamic forces and on capturing the predominant frequencies is discussed. The turbulence modelling is the DDES approach, in which the RANS turbulence length scale is improved by means of the Organised Eddy Simulation [START_REF] Braza | Turbulence Properties in the cylinder wake at high Reynolds number[END_REF]. It is recalled that distinction of coherent and chaotic turbulent structures is performed in OES by dual spectrum splitting, solving the ensemble-averaged Navier-Stokes equations, where the turbulence stresses are modelled by modified turbulence scales modelling. This has been be derived from DRSM (Differential Reynolds Stress Transport Modelling) and takes into account modification of the turbulence stresses in respect of non-equilibrium turbulence. The URANS turbulence length scale in the k transport equation of DDES is chosen as the l OES length scale, in the same way as reported in Bourguet et al. The turbulence viscosity has been chosen as the minimum between the eddy-viscosity provided by the k--SST [START_REF] Menter | [END_REF] and the k--OES models.

Results

In figure 4, comparison of two DDES models is performed, using the more economic time step, t=0.01. The results obtained show a rich turbulence statistical content, as well as formation of alternating shear layers and the wrap around mechanism towards the second cylinder. It can be seen that the DDES-OES modelling provides finer shear layers, governed by the shedding of Kelvin-Helmholtz vortices, responsible for the acoustic noise (figure 4, right) and the contouring mechanism of the shear-layer vortices around the second cylinder, in accordance with the experiments. The 3D instantaneous visualisation of the flow field is shown in figures 5 and 6, where the complex vortex dynamics of finerscale chaotic structures interacting with the coherent structures is illustrated. Figure 8 shows the averaged velocity in the rear axis, sensitive to the flow detachment, where it is shown that a reasonably good agreement with the experiment is achieved. Figure 9 (up) shows the averaged Cp coefficient corresponding to the coarser time step, t=0.01 in comparison with the experiments. Concerning the first cylinder, the averaged maximum pressure values are lower than in the experiment, where a better agreement is obtained for the second cylinder. Figure 9 (down) shows the averaged pressure coefficient corresponding to the finest time step, t=0.0005, where the pressure peaks values are quite closer to the experimental ones. The computations need to be continued to achieve fully convergent statistics in the pressure plateau regions. The influence of the time step on the mean drag coefficient has been studied. The averaged drag coefficient C D is found 0.628 with t=0.01 and 0.500 for t=0.005.

Comparing to numerical median value evaluated from a number of thirteen different numerical simulations reported by Lockard, (CD=0.484), it is shown that the finer time step provides a much closer drag coefficient to this value. The experimental drag is not available in the present experiments.

The turbulence spectra show the principal instability modes, among which the von Kármán mode, as well as formation of a spectral 'bump' corresponding to the Kelvin-Helmholtz eddies. Of course, the chaotic turbulence effect provides the fact that these eddies are governed by a random vortex smearing around a predominant K-H mode. The turbulence spectra presented in fig. 10, corresponding to the coarser and the finer time steps, illustrate this interaction effect of the random turbulence on the predominant frequencies. Due to these facts and to the complex vortex flow system, the predominant frequencies are subjected to a modulation in time, that produces the aforementioned frequency bumps. As shown in the numerical flow visualizations (figures 5 to 7), the apparently chaotic flow structure is governed by a temporal coherence, interacting non-linearly with the random turbulence background.

A way to investigate the predominant frequencies modulation in time is the application of wavelet analysis to selected monitoring signals in the separated shear layers. The Morlet's wavelet analysis has been applied (for details of signal processing, see Marple, 1987, Grossman and[START_REF] Grossmann | Decomposition of Hardy functions into square integrable wavelets of constant shape[END_REF].

Conclusions

A numerical study has been carried out to analyse the complex vortex pattern around two cylinders in tandem at Reynolds number 166,000, by means of Hybrid Turbulence modelling approaches. The Delayed Detached Eddy Simulation DDES-SST and DDES-OES models have been applied and comparison of results with the experiment is carried out. Furthermore, the influence of the The statistical content of the DDES-OES is more detailed and the shear layer interfaces are finer and contain a number of Kelvin-Helmholtz vortices. Moreover, the spectral analysis shows that the flow is mainly governed by the von Kármán mode. A reasonably good agreement with the experiments carried out in the NASA-Langley Research Center is shown, especially concerning the mean streamlines structure, velocities and predominant frequencies. The influence of the time step on the flow parameters shows an improvement of the pressure coefficient and of the mean drag (compared with numerical results in the literature), with the finer time step. The computations with the finer time step are continued to provide statistics including more than a decade of periods of the main vortex shedding. The interaction between the Von Kármán mode and the shear layer frequencies bump is quantified by means of advanced signal processing, especially using wavelet analysis. The present study shows the ability of the DDES to simulate the present complex flow dynamics and predominant modes, associated with landing gear aerodynamic noise.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Schematic representation of the tandem cylinders cross section and of the computational domain size

Fig. 3 .

 3 Fig. 3. Computational grid provided by NTS -St Petersbourg, Russia, for the ATAAC EU program partners

Fig. 4 .

 4 Fig. 4. Instantaneous spanwise vorticity cut (middle section), left: DDES-k--SST, right: DDES-k--OES

Fig. 7 .

 7 Fig. 7. View of the instantaneous velocity field (up) and of the averaged streamlines field (down) in comparison with the experiment of Jenkins et al.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Averaged velocity profiles in the gap region and at the rear flow

Fig. 7 .

 7 Fig. 7. Wall pressure spectra at the first and second cylinder, as in sketch, by means of t=0.01 (up) and t=0.0005 (down), first cylinder

Fig. 8 .

 8 Fig. 8. Wavelet analysis on a pressure signal in the upper shear layer at (x/D, y/D) = (1.2,0.5). Up: signal versus time, middle, Morlet wavelets and dimensionless frequency versus time, down: dimensionless frequency versus time by means of FFT

  has been studied on the same grid, provided by NTS -St Petersbourg. The results have shown that the flow structure between the two cylinders and past the second cylinder are well produced by the present methods.

Acknowledgement

This work is part of the ATAAC research program, "Advanced Turbulence Simulations for Aerodynamic Application Challenges", coordinated by DLR-Germany (D. Schwamborn). The authors are grateful for the use of the French supercomputing facilities of CINES, IDRIS, CEA, this last in the context of the PRICE EU initiative. The authors thank F. Déliancourt, Master's degree student in IMFT and to Dr. T. Deloze, post-doc IMFT for their valuable contributions.


Monica Siroux
email: monica.siroux@bourges.univ-orleans.fr

Eric Tang-Kwor

Simone Matteï
email: mattei@univ-paris12.fr

A Periodic Technique for Emissivity Measurements of Insulating Materials at Moderate Temperature

A periodic radiometric technique for total emissivity of insulating materials at moderate temperature is presented. The radiometric technique involves a mutifrequency modulation of the front side temperature of the sample. This method needs two measurements : one on the studied sample, the other one on a modulated thin reference. The infrared output signal of the other side (back side) as well as the front side temperature of the sample are measured. The measured signals are fitted with a temperature modulated conduction model, providing the emissivity without surface temperature measurement. As an experimental validation 25 measurements are carried out on a PVC sample 5 mm thick coated with an aluminium paint. The results are consistent with the results obtained by two other methods : the monofrequency radiometric method and the integrating directional measurements.

ABSTRACT 1. INTRODUCTION

The direct emissivity measurement method is built on the emissivity definition

) T ( L ) T ( L 0 = ε
, where L(T) is the radiance emitted by the sample at temperature T and L 0 (T) stands for the blackbody radiance. Instead of a blackbody, a well known material at the same temperature T can be used [START_REF] Hervé | Mesure de l'émissivité thermique Techniques de l'Ingénieur[END_REF]. This method requires measurement of the emissive power, and, any radiometer facing a sample measures both emitted and reflected flux. For a surface temperature below 600 K the emissive power cannot be measured in isolation unless the ambient reflected flux is strongly reduced by using a complex apparatus [2]. A means of emissivity measurement at moderate temperature has been proposed by S. Matteï and D.

Especel [START_REF] Matteï | Une méthode de mesure de l'émissivité thermique des matériaux opaques à température ambiante[END_REF]. This technique consists in modulating emitted flux by using a modulated temperature sample, the emitted and reflected fluxes can then be separated using a Fourier analysis. This method has been performed at room temperature on Duralumin and Aluminium Paint [START_REF] Matteï | Une méthode de mesure de l'émissivité thermique des matériaux opaques à température ambiante[END_REF], [START_REF] Especel | Total Emissivity Measurements without use of an Absolute Reference Infrared[END_REF]. The statistical precision of the results was found to be better than 0.01.

Moreover, the direct emissivity method requires knowledge of the surface temperature, and, for insulating specimen this temperature is not easy to determine because of the temperature profile within the medium. To circumvent this difficulty, one can paint a part of the studied surface and alternatively view the coated and the plain surfaces. One must then assume that the paint does not modify the surface temperature. Another solution suitable to diffuse surfaces consists in covering the sample with reflecting hemisphere [START_REF] Ono | Evaluation of the Effective Emissivity of Reference Sources for the Radiometric Emissivity Measurements[END_REF].

This paper presents a periodic emissivity measurement technique suitable for insulating materials at moderate temperature.

PERIODIC EMISSIVITY MEASUREMENT

Consider an opaque strip shaped sample (Figure 1.) whose front side temperature is in the form :

) t f 2 cos( T T ) t ( T 0 FS 0 ϕ + π + = (1)
Assuming an infrared detector facing the back side of the sample, the measured flux is the sum of an emitted flux and a reflected one. At temperature modulation frequency f 0 the reflected flux is not modulated, so the f 0 component of the infrared output signal Fourier

Transform can be written :

) f ( T . . ). f ( K ) f ( U ~0 4 BS s 0 0 BS σ ε = (2)
with :

) f ( K ~0
the complex frequency response of the infrared detection system, For low temperature amplitudes and provided that front side and back side mean temperature are equal

BS FS T T =
, the equation ( 2) reduces to :

) f ( H . . ). f ( K ) f ( R ~0 s s 0 0 s σ ε = (3)
where

) f ( T ) f ( U ) f ( R ~0 4 FS 0 BS 0 =
is the experimental complex ratio of the measured signals and

) f ( T ) f ( T ) f ( H ~0 FS 0 BS 0 s =
is the sample thermal transfer function, ratio of the back side to front side temperatures [6].

The modulus of ) f ( H ~s
is the temperature attenuation, and the phase of ) f ( H ~s the temperature phase shift between the two sides of the strip. Using a modulated temperature one-dimensional heat conduction model [6], we can obtain the following formula :

) e . q ( ch Bi e . q ) e . q ( sh Bi e . q H ~s + = (4)
e is the sample thickness. Bi is the Biot number given by Bi h e k = .

, where k is the sample conductivity and h is the heat transfer coefficient on the sample back side. This coefficient is the sum of a convective coefficient convection h and a radiative coefficient radiation h .

q is given by

2 / 1 ) a f 2 j ( q π =
, where a is the sample diffusivity.

A sample is considered as thermally thin if 1 ) f ( H ~0 s = , and thermally thick if 1 ) f ( H ~0 s < .

Thermally thin sample emissivity measurement

In order to obtain the sample emissivity two measurements are necessary, one on the sample, the other one on a modulated reference. If this reference is thermally thin

( 1 ) f ( H ~0 ref = )
, the emissivity sample is given by :

) f ( H ~1 . ) f ( R ~) f ( R . 0 s 0 ref 0 s ref s ε = ε (5) where ) f ( R ~0 is the modulus of ) f ( R ~0 and ) f ( H ~0 s is the modulus of ) f ( H ~0 s .
We consider a sample like thermally thin ( 1 ) f ( H ~0 s = ) if the temperature attenuation correction is smaller than the expected emissivity uncertainty. In this condition, the thermally thin sample emissivity can be reached by :

) f ( R ~) f ( R . 0 ref 0 s ref s ε = ε (6)

Thermally thick sample emissivity measurement

If the equation ( 6) cannot be used, then, two approaches are possible to determine the sample emissivity. The first one (the monofrequency method) consists in leaving out the sample thermal transfer function value ) f ( H ~0 s using a thick reference. The second one (the mutifrequency method) consists in calculating the sample transfer function using a mutifrequency modulation temperature and identifying parameters.

The monofrequency emissivity measurement method

The chosen reference is the sample coated with a known thin reference (Figure 2.).

The emissivity is then given by : We can show that, if the sample emissivity and the reference emissivity are very closed, or, if the Biot number is small then

) f ( H ~) f ( H . ) f ( R ~) f ( R . 0 s 0 ref s 0 ref 0 s ref s + ε = ε (7)
) f ( H ) f ( H ~0 s 0 ref s = +
, and the sample emissivity can be obtained using equation (6).

Generally, the emissivity relative systematic error, using this method is given by [6] :

) f ( H ~) f ( H 1 0 ref s 0 s s s + - = ε ε Δ (8)
Figure 3. shows the error versus emissivity for several materials (ceramic, Pyrex, PVC) 5 mm thick obtained for 5 . 12 f 0 = mHz and for

radiation convection h h h + =
, where

5 h convection = W.m -2 .K -1 and s radiation . 6 h ε = W.m -2 .K -1 .
We can note that for the ceramic sample this error is less than 0.5 %, but, for PVC and Pyrex sample, this error cannot be neglected. Thus the monofrequency emissivity method is not suitable for all materials. A more general method, involving a mutifrequency temperature modulation has been carried out.

The multifrequency emissivity measurement method

The multifrequency method requires two measurements for several modulation frequency i f (Figure 4.) : one on the sample, the other one on a modulated thin reference ( ~( ) H f ref i = 1 ), leading to two experimental ratios ~( ) R f s i and ~( ) R f ref i . For each frequency f i we obtain :

) f ( H . ) f ( R ~) f ( R ~i s ref s i ref i s ε ε = (9)
The measured signals

) f ( R ) f ( R ~i ref i s are fitted with the theoretical values ) f ( H . ) ( i s ref s ε ε
calculated from a modulated one-dimensional heat conduction model, yielding the sample emissivity s ε . Moreover, this method let us to determine the sample diffusivity a and the Biot number Bi [START_REF] Siroux | Une nouvelle méthode calorimétrique périodique de mesure de l'émissivité des matériaux opaques à température ambiante[END_REF].

EXPERIMENTAL STUDY

Experimental set-up

The experimental set-up (Figure 5.) allows us to carry out multifrequency radiometric measurements at moderate temperature. The apparatus is composed of a sample holder (1), an infrared detector (2) and a data acquisition system [START_REF] Matteï | Une méthode de mesure de l'émissivité thermique des matériaux opaques à température ambiante[END_REF]. Each sample ( 4) is stock on a modulated sample holder fitted with a thermoelectric device (1b) squeezed between two metallic plates (1a and 1c). A thermocouple [START_REF] Vriendt | La transmission de la chaleur -La conduction[END_REF], embedded in the metallic plate which holds the sample, allows us to determine the sample's front side temperature. A cooling water system sinks the heat created by the thermoelectric devices Joule's effect.

The infrared detector is a thermopile with a KRS5 window (spectral range 1 -50 µm).

The field of view of the thermopile is about 120° (Figure 6.). Using this infrared detector, the measured emissivity depends on the conical angular response. The detector is motorised [START_REF] Laurent | Conductivité thermique des solides Techniques de l'Ingénieurs[END_REF] and can be successively placed in front of the different samples.

A PC data based acquisition system (3) controls the excitation voltage of the thermoelectric device and measures both sample temperature T t FS ( ) and infrared detection system output U t BS ( ) .

This apparatus can be placed in a vacuum chamber [START_REF] Tang-Kwor | Réalisation d'un corps noir destiné aux mesures d'émissivité à température ambiante IIIème Colloque Interuniversitaire Franco-Québécois[END_REF] and using a vacuum pump system, the pressure can be controlled. It is then possible to access either emissivity at low temperature or total hemispherical emissivity via Biot number, which only depends on the radiative coefficient [START_REF] Siroux | Une nouvelle méthode calorimétrique périodique de mesure de l'émissivité des matériaux opaques à température ambiante[END_REF].

Specimen

The multifrequency measurements have been carried out on a 5 mm thick PVC slab coated with an aluminium paint (Rust Oleum 7315) whose emissivity can be measured by other methods. The used reference is a black paint Nextel Velvet Coating 811-21 [START_REF] Lohrengel | Wärmeleitfähigkeit, Gesamtemissionsgrade und spektrale Emissionsgrade der Beschichtung Nextel-Velvet-Coating 811-21[END_REF], [START_REF] Hameury | Rapport d'essais du Laboratoire National d'Essais[END_REF] directly coated on the metallic plate (1a) of the sample holder.

Excitation voltage

The frequency range is [0.78 mHz , 12.5 mHz]. The reasons for choosing this frequency range are as follows :

below 0.78 mHz the heat transfer function doesn't change any more, -above 12.5 mHz, the infrared signal becomes too weak to obtain accurate measurements.

The excitation voltage is in the form :

∑ = - π + = 5 1 i 0 1 i i mean ) t f 2 2 cos( . V V ) t ( V (10)
Where mean V is the mean excitation voltage, and 0 f is equal to 0.78 mHz. i V is chosen in order that the ratio signal/noise is the same over all the spectrum.

Table 1. shows the modulation amplitudes values V i and the corresponding temperature amplitudes values T FS,i versus the modulation frequency i f . It can be seen that front side temperature is around 12 °C peak to peak. In this case we can consider that the thermal conductivity, the thermal diffusivity and the emissivity sample are constant [START_REF] Matteï | Une méthode de mesure de l'émissivité thermique des matériaux opaques à température ambiante[END_REF], [START_REF] Laurent | Conductivité thermique des solides Techniques de l'Ingénieurs[END_REF].

Parameters identification

The three parameters are identified using the minimisation between measured data

) f ( R ) f ( R ~i ref i s and the calculated ones ) f ( H . ) ( i s ref s ε ε
. The chosen estimator take into account the modulus and the phase of data [6]. should also be noted that the modulus of

) f ( R ) f ( R ~i ref i s
is very sensible to the emissivity ratio and so for a large frequency domain.

RESULTS AND DISCUSSION

25 multifrequency measurements have been carried out on the 5 mm thick PVC slab coated with the aluminium paint. for the five frequencies, as well as the statistical accuracies. These accuracies are estimated by measuring the amount of noise in the FFT spectra [6].

Emissivity verification using the monofrequency method

Emissivity value obtained by multifrequency method has been compared with value obtained by using monofrequency method. The monofrequency obtained ratio is

003 . 0 721 . 0 ref s ± = ε ε
. We note (Table 3.) that the difference between the two values is around 0.8 %.

Emissivity verification using integrating directional measurements

Integrated directional emissivity measurements have been carried out on aluminium paint and on the reference back paint (Figure 8.). The used apparatus is fitted with a pyroelectric detector (spectral range 1 -50 µm), a chopper, a look-in amplifier, a KRS5 lens and a diaphragm (diameter 1.8 mm). The measurements have been made from = θ 0° to 60°, with a step of 10° (Table 4.).

Integrating the directional emissivity measurements we can determine the ratio :

0 = θ
) and ) ( c n θ is the detector angular response. Using the [START_REF] Maillet | Les erreurs sur la diffusivité thermique mesurée par méthode flash, confrontation théorie-expérience[END_REF] formula for the aluminium paint and for the reference back paint, we obtain . 707 . 0 ref s

= ε ε

This value is in good agreement with the identified one (the difference being 1.4 %).

Diffusivity verification

The identified diffusivity value (Table 3.) is consistent with the value of the literature [START_REF] Maillet | Les erreurs sur la diffusivité thermique mesurée par méthode flash, confrontation théorie-expérience[END_REF], [START_REF] Vriendt | La transmission de la chaleur -La conduction[END_REF]. Moreover, the repeatability error is very small. Thus, the mutifrequency method can also be used to diffusivity determination.

CONCLUSION

The multifrequency method makes it possible to measure emissivity of insulating materials at moderate temperature without surface temperature measurement.

This technique entails the use of an emissivity reference. It has been shown that it is possible to use a modulated blackbody [6]. This blackbody is a diffusive, highly emissive modulated temperature surface associated with an highly reflective hemisphere [START_REF] Tang-Kwor | Réalisation d'un corps noir destiné aux mesures d'émissivité à température ambiante IIIème Colloque Interuniversitaire Franco-Québécois[END_REF].

On the other hand, the measured factor, noted ε is a total emissivity which depends on the detection system field of view. The detection optic could be modified to obtain a directional emissivity.

Furthermore, this technique could be applied at highest or lowest temperature.

  the complex back side temperature.Because of the back side temperature measurement is difficult, the front side temperature only is measured, leading to the complex ) f ( T ~0 FS (Figure1.).

  modulus of the thermal transfer function of the sample and the same coated sample.

Figure 7 .

 7 shows the reduced sensitivity of the modulus and phase of )

  Bi can be identified by fitting the measured values to the model. For each identified parameter, the mean, the standard deviation and the repeatability uncertainty are calculated (

Table 2 .

 2 presents the modulus and phase of the infrared

	signal	f U ~i BS (	)	and of the front side temperature	f T ~i 4 ( FS	)

Table 3 .

 3 ).

	In order to verify the	ε	s ε	ref	value we have carried out monofrequency emissivity
	measurements and directional emissivity measurements.

 [START_REF] Maillet | Les erreurs sur la diffusivité thermique mesurée par méthode flash, confrontation théorie-expérience[END_REF]where n ε is the total normal emissivity (obtained for