Introduction

L'étude de la convection naturelle en milieu confiné fait encore l'objet de nombreuses recherches, tant numériques qu'expérimentales. Dans ce type de problème, les différents modes de transfert de chaleur (convection, conduction, rayonnement) peuvent intervenir de manière couplée. Cependant, lorsque le transport radiatif est considéré, un problème particulier se pose lorsque le fluide absorbe et émet un rayonnement infrarouge. Il est alors nécessaire de prendre en compte une source de chaleur interne au milieu, résultant de la différence entre l'énergie rayonnante absorbée et émise par chaque élément de volume. De nombreuses études ont étudié ce phénomène dans une cavité différentiellement chauffée (Yücel, Acharya, and Williams [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF], Tan and Howell [START_REF] Tan | Combined radiation and natural convection in a twodimensional participating square medium[END_REF], Colomer et al. [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF], Colomer, Consul, and Oliva [START_REF] Colomer | Coupled radiation and natural convection: Different approaches of the SLW model for a non-gray gas mixture[END_REF], Soucasse et al. [START_REF] Soucasse | Numerical study of coupled molecular gas radiation and natural convection in a differentially heated cubical cavity[END_REF], Billaud, Saury, and Lemonnier [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF]). Le problème d'un apport de chaleur placé dans un environnement confiné a également retenu l'attention de nombreux chercheurs, soit avec une source ponctuelle (Tetsu, Itsuki, and Haruo [START_REF] Tetsu | Buoyant plume above a horizontal line heat source[END_REF], Urakawa, Morioka, and Kiyota [START_REF] Urakawa | Swaying motion of the buoyant plume above a horizontal line heat source[END_REF], Xin et al. [START_REF] Xin | Numerical simulations of natural convection around a line-source[END_REF], Hernandez [START_REF] Hernandez | Natural convection in thermal plumes emerging from a single heat source[END_REF], etc.) ou avec obstacles solides de taille finie (Kuznetsov and Sheremet [START_REF] Kuznetsov | Two-dimensional problem of natural convection in a rectangular domain with local heating and heat-conducting boundaries of finite thickness[END_REF], Kuznetsov and Sheremet [START_REF] Kuznetsov | Conjugate heat transfer in a closed domain with a locally lumped heat-release source[END_REF], Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF], Kuznetsov, Maksimov, and Sheremet [START_REF] Kuznetsov | Natural convection in a closed parallelepiped with a local energy source[END_REF], Souayeh et al. [START_REF] Souayeh | Prediction of unsteady natural convection within a square cavity containing an obstacle at high Rayleigh number value[END_REF], Gibanov and Sheremet [START_REF] Gibanov | Effect of the buoyancy force on natural convection in a cubical cavity with a heat source of triangular cross-section[END_REF],Iyi, Hasan, and Penlington [START_REF] Iyi | Numerical simulation of 2D turbulent natural convection of humid air in a cavity filled with solid objects[END_REF], Rahmati and Tahery [START_REF] Rahmati | Numerical study of nanofluid natural convection in a square cavity with a hot obstacle using lattice Boltzmann method[END_REF] etc.), mais dans la plupart des cas, la source de chaleur est placée dans le fluide, sans aucun contact avec les murs. Ici, nous abordons le cas d'un obstacle chauffant placé au fond de la cavité. Cela constitue une nouvelle configuration intéressante.

Tout d'abord, concernant la convection naturelle à l'intérieur d'une cavité contenant un obstacle opaque, Kuznetsov and Sheremet [START_REF] Kuznetsov | Conjugate heat transfer in a closed domain with a locally lumped heat-release source[END_REF] ont étudié les effets du nombre de Grasshof (10 5 -10 7 ) sur le mouvement du fluide et ont conclu à l'influence de ce paramètre sur le champ thermique dans l'enceinte. Kuznetsov and Sheremet [START_REF] Kuznetsov | Two-dimensional problem of natural convection in a rectangular domain with local heating and heat-conducting boundaries of finite thickness[END_REF] ont étudié la même configuration, mais avec une plage différente du nombre de Grasshof (10 7 -10 9 ) et ont souligné que, lorsque ce nombre augmente, l'écoulement et le processus de transfert de chaleur sont stabilisés. Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF], Kuznetsov, Maksimov, and Sheremet [START_REF] Kuznetsov | Natural convection in a closed parallelepiped with a local energy source[END_REF], Souayeh et al. [START_REF] Souayeh | Prediction of unsteady natural convection within a square cavity containing an obstacle at high Rayleigh number value[END_REF], Gibanov and Sheremet [START_REF] Gibanov | Effect of the buoyancy force on natural convection in a cubical cavity with a heat source of triangular cross-section[END_REF], Iyi, Hasan, and Penlington [START_REF] Iyi | Numerical simulation of 2D turbulent natural convection of humid air in a cavity filled with solid objects[END_REF],Rahmati and Tahery [START_REF] Rahmati | Numerical study of nanofluid natural convection in a square cavity with a hot obstacle using lattice Boltzmann method[END_REF] ont étudié différents écoulements convectifs de manière numérique et expérimentale avec l'influence de la numéro de Rayleigh. Ils ont rapporté que l'augmentation de ce paramètre intensifie le mouvement du fluide et augmente le transfert de chaleur par processus convectif. De plus, l'étude sur la taille de l'obstacle chaud de Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF] a souligné que, lorsque la hauteur de l'obstacle est la moitié de la cavité, le transfert de chaleur convectif est le pire parmi les cas analysés. En outre, Bouafia and Daube [START_REF] Bouafia | Natural convection for large temperature gradients around a square solid body within a rectangular cavity[END_REF] ont examiné les comportements instables dans cette configuration à différents rapports d'aspect de l'enceinte et ont conclu que le mécanisme de l'instabilité était dû au cisaillement lorsque ce paramètre vaut 1 et 2. Pour un rapport d'aspect plus élevé [START_REF] Colomer | Coupled radiation and natural convection: Different approaches of the SLW model for a non-gray gas mixture[END_REF] , le principal mécanisme était les instabilités de flottabilité. Hernandez [START_REF] Hernandez | Natural convection in thermal plumes emerging from a single heat source[END_REF] vi a étudié une cavité carrée avec une source de chaleur attachée au sol. Ses résultats indiquent que le comportement instable était dû à la vitesse horizontale élevée juste au-dessus de l'obstacle.

Ensuite, le problème de la convection naturelle combinée au rayonnement de surface a été étudié par Sun, Chénier, and Lauriat [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF], Martyushev and Sheremet [START_REF] Martyushev | Surface radiation influence on the regimes of conjugate natural convection in an enclosure with local energy source[END_REF], Saravanan and Sivaraj [START_REF] Saravanan | Surface radiation effect on convection in a closed enclosure driven by a discrete heater[END_REF], Patil, Sharma, and Velusamy [START_REF] Patil | Conjugate laminar natural convection and surface radiation in enclosures: Effects of protrusion shape and position[END_REF], Miroshnichenko, Sheremet, and Chamkha [START_REF] Miroshnichenko | Turbulent natural convection combined with surface thermal radiation in a square cavity with local heater[END_REF]. Ils ont conclu que l'augmentation de l'émissivité renforce le mouvement du fluide près de la paroi, intensifie le transfert radiatif mais réduit le transport convectif. Sun, Chénier, and Lauriat [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF] ont également signalé que le rayonnement de surface ralentit la transition vers l'instabilité à l'intérieur de la cavité.

Jusqu'à présent, une revue détaillée de littérature a montré que la convection naturelle couplée au rayonnement volumique n'étaient pas encore prise en compte dans la configuration d'une cavité contenant un obstacle. De plus, le coefficient d'absorption qui varie avec le nombre d'ondes, la fraction molaire du composant absorbant et la température du milieu pose également un problème remarquable. Notre objectif est donc d'étudier ce couplage (à la fois en convection thermique pure et double convection) à l'intérieur d'une cavité cubique dont les parois horizontales sont adiabatiques et les parois verticales isothermes. Un obstacle cubique est situé au centre de plancher. Il est opaque et sa surface est uniformément maintenue à une température supérieure à celle des parois de la cavité. L'étude comprend, dans un premier temps, une analyse des phénomènes en gaz transparent (convection thermique et panache confiné en double diffusion: cas aidant et cas opposant). Une première approche des gaz participants est ensuite effectuée en supposant que le mélange absorbant est gris. La dernière approche -et la plus importante -est la simulation en gaz réel. Concrètement, on considère les mélanges air -H 2 O et air -CO 2 avec une concentration prescrite d'espèces absorbantes à la surface de l'obstacle et une concentration nulle sur les parois verticales de la cavité. Nous analysons les résultats par comparaison avec les cas transparents, mais aussi en examinant dans quelle mesure les observations faites avec le modèle des gaz gris restent pertinentes.

Ce manuscrit est divisé en six chapitres. Premièrement, le présent chapitre présente la motivation de la thèse, ses objectifs et une revue de bibliographie sur certaines recherches connexes des dernières décennies. Ensuite, dans le deuxième chapitre, nous fournissons plus de détails sur les modèles mathématiques ainsi que sur les méthodes numériques utilisées dans cette étude. Ensuite, le chapitre trois présentera Code Saturne, l'outil de simulation CFD utilisé tout au long de notre travail et la mise en oeuvre de notre propre modèle SLW dans le module radiatif intégré de ce code. Nous présentons également quelques tests de validation pour évaluer la précision de nos calculs dans des configurations de plus en plus complexes. Le quatrième chapitre contient les résultats et l'analyse de la convection thermique combinée au rayonnement dans un gaz gris ainsi que dans un mélange gazeux réel. Ensuite, le chapitre cinq se concentrera sur les effets du rayonnement dans de nombreuses situations typiques de convection de double diffusion, coopérante ou opposée, dans un mélange de gaz gris ou réels. Pour terminer, le chapitre de conclusion synthétisera les principaux résultats de cette étude et fournira des perspectives pour les travaux futurs. vii

Methodologie Modèle mathématique

Les hypothèses utilisées dans ce travail sont les suivantes :

• L'écoulement dans la cavité est tridimensionnel et laminaire.

• Le fluide est considéré comme newtonien et incompressible.

• Les surfaces actives (parois verticales de l'enceinte et surfaces extérieures de l'obstacle) sont noires par rapport au rayonnement tandis que les surfaces adiabatiques (plafond et plancher de l'enceinte) sont purement réfléchissantes.

• Les variations de température et de concentration à l'intérieur de la cavité sont suffisamment faibles pour permettre l'approximation de Boussinesq. Par conséquent, les variations des propriétés du fluide sont ignorées, sauf pour la densité dans l'expression de la pousée d'Archimède.

• La dissipation visqueuse et le travail de pression sont négligées.

• Les effets de Soret et de Dufour sont négligés.

Équations de la dynamique des fluides

Plusieurs équations de conservation régissent les mouvements d'écoulement et les processus de transfert dans l'enceinte. Elles expriment un équilibre local en termes de masse, de quantité de mouvement, d'énergie et de composition au sein du fluide.

• Équation de continuité

∇ • u = 0 (1)
• Équation de quantité de mouvement

ρ 0 ∂u ∂t + ρ 0 u • ∇u = -∇p + ρ 0 (β T (T 0 -T) + β C (C 0 -C))g + µ∇ 2 u (2)
Le terme source ρ 0 (β T (T 0 -T) + β C (C 0 -C))g représente la force de flottabilité qui met le fluide en mouvement (exprimée ici sous l'approximation de Boussinesq).

• Équation dénergie

ρ 0 C p ∂T ∂t + ρ 0 C p u • ∇T = λ∇ 2 T -∇ • q (3)
• Équation de conservation de la concentration

∂C ∂t + u • ∇C = D∇ 2 C (4)
viii Toutes les équations de conservation sont couplées : le champ dynamique influence le transport des quantités scalaires (T et C) qui, à leur tour, entraînent l'écoulement par les effets de la flottabilité. De plus, dans le cas du mélange binaire, la concentration a un effet direct sur le champ thermique puisqu'elle modifie les propriétés d'absorption-émission du milieu. Par conséquent, la source radiative dans le bilan énergétique est impactée.

Équation de transfert radiatif

La cavité est remplie d'un milieu gazeux semi-transparent à l'équilibre thermodynamique local, qui absorbe et émet le rayonnement en tout point de l'espace. La luminance I η (s, Ω) 1 représente le flux radiatif (par unité d'angle solide et par unité de nombre d'onde) qui se propage au point s dans la direction Ω au nombre d'onde η. Dans un milieu non diffusant, le changement local de luminance est décrit par l'équation de transfert radiatif. Ω • ∇I η (s, Ω) = -κ η (s)I η (s, Ω) + κ η (s)I bη (T(s))

La solution dépend de trois coordonnées de position, de deux variables de direction (deux angles polaires ou deux cosinus de direction) et du nombre d'ondes. La luminance totale peut être trouvée par intégration sur l'ensemble du spectre.

I(s, Ω) = ∞ 0 I η (s, Ω)dη (6)
Le terme -∇ • q qui apparaît dans l'équation d'énergie est la divergence totale du flux radiatif. Ce flux peut être calculé à partir de la luminance totale par l'expression :

q(s) = 4π 0 I(s, Ω)ΩdΩ = 4π 0 ∞ 0 I η (s, Ω)ΩdηdΩ (7)
Conditions aux limites Toutes les surfaces de l'obstacle2 sont portées à une température et une concentration constantes et uniformes : En ce qui concerne la cavité, les parois verticales sont uniformément maintenues à une température et une concentration constantes. Les parois horizontales sont adiabatiques, imperméables et supposées se comporter comme des surfaces entièrement réfléchissantes.

Modèle de rayonnement

Méthode des ordonnées discrètes

Dans les problèmes couplés impliquant le rayonnement, nous devons résoudre le problème du transfert radiatif en plus des équations de conservation. Historiquement, plusieurs méthodes ont été développées pour atteindre cet objectif. Dans la présente étude, nous avons utilisé la méthode des ordonnées discrètes pour nos calculs radiatifs en raison de son bon compromis entre la précision et le coût de calcul et de sa facilité d'implantation dans de nombreux codes CFD. En particulier, Code Saturne, un code CFD open-source développé par EDF, offre un module DOM ix radiatif déjà intégré.

Méthode SLW

Chaque fois que l'on considère le rayonnement gazeux, il faut tenir compte du comportement spectral réel des propriétés d'absorption des fluides. À cette fin, divers modèles de gaz ont été introduits avec différents niveaux de complexité, d'exigences de calcul et de précision. Ces modèles peuvent être classés en trois groupes principaux comme suit :

• Modèle raie par raie • Modèles de bandes

• Modèles globales

Dans le modèle raie par raie et les modèles de bande, les propriétés radiatives sont évaluées sur chaque raies ou sur un intervalle donnée de nombres d'onde. D'autre part, les modèles globaux déterminent les caractéristiques radiatives sur l'ensemble du spectre.

Mise en oeuvre du modèle de gaz SLW Le modèle SLW (un modèle global) implique un nombre fini de gaz gris (N g ) et un gaz transparent. Le coefficient d'absorption du gaz j th est calculé comme suit

κ j = N • Y • C j (8)
où C j est la section efficace d'absorption. Connaissant N et Y, le problème restant est de déterminer C j .

Calculs de la fonction de distribution du corps noir de la ligne d'absorption et des poids des gaz gris

Les poids associés à chaque gaz gris sont calculés à partir de la fonction de distribution globale du coefficient d'absorption pondéré par la fonction de Planck (ALBDF). Cette fonction est évaluée comme l'intégrale de la fonction de Planck calculée à une température de source T b sur les intervalles de nombres d'onde de telle que la section efficace d'absorption C η (φ g ) à un état thermodynamique du gaz φ g est inférieure à une valeur spécifiée de C.

Le poids du gaz gris j th a j correspond à la différence de l'ALBDF aux deux sections d'absorption supplémentaires qui définissent le j me intervalle [ Cj-1 , Cj ]

Code Saturne et Calcul du Rayonnement Code Saturne

Toutes les équations de notre problème ont été résolues en utilisant Code Saturne version 5.0.4 (Archambeau, Méchitoua, and Sakiz [START_REF] Archambeau | Code Saturne: A Finite Volume Code for the Computation of Turbulent Incompressible Flows, Industrial Applications[END_REF]), un logiciel open source de calcul CFD développé par EDF. Un module radiatif intégré est disponible, dans lequel nous avons implémenté nos propres données pour l'intégration directionnelle x et un module spécifique pour le rayonnement gazeux selon le modèle SLW.

Mécanique des fluides

Code Saturne utilise une méthode de volumes finis pour résoudre les équations régissant le mouvement des fluides et le transfert de chaleur et de masse. Pour l'équation de quantité de mouvement, le logiciel recourt à l'algorithme SIMPLEC. Différentes discrétisations dans l'espace et dans le temps sont disponibles.

Discrétisation temporelle

Le schéma temporel est implémenté dans Code Saturne est un schéma θ avec: θ = 1 pour un schéma d'Euler implicite de premier ordre, θ = 1 2 pour le schéma Crank-Nicolson de second ordre. [START_REF] Xin | Numerical simulations of natural convection around a line-source[END_REF] Il existe deux options pour définir le pas temporel: constant ou variable. Dans ce dernier cas, le code calcule automatiquement après chaque itération le pas de temps qui satisfait au critère CFL.

Discrétisation spatiale

Code Saturne propose différents schémas de premier ordre (Upwind) et de second ordre (Centré ou Second-Order-Linear-Upwind (SOLU)) pour la discrétisation spatiale et. Dans cette étude, nous avons sélectionné le schéma de second ordre centré.

Méthode des ordonnées discrètes

La méthode des ordonnées discrètes (DOM) a été utilisée pour résoudre l'équation de transfert radiatif correspondant à chaque gaz gris du modèle SLW. Cette méthode consiste à remplacer les intégrales angulaires par une sommation sur un ensemble de directions discrètes telles que:

4π 0 f (Ω)dΩ ≈ M ∑ m=1 ω m f (Ω m ) (10)
où M désigne le nombre de directions et ω m est le poids attribué à l'élément m. Par conséquent, la distribution du rayonnement incident et du flux radiatif est approximée par:

G P η = 4π 0 I P η (Ω)dΩ ≈ M ∑ m=1 ω m I P η,m (Ω) (11)
q P m = 4π 0

I P η (Ω)ΩdΩ ≈ M ∑ m=1 ω m I P η,m (Ω)Ω m (12)
où P désigne ici le centre d'un volume de contrôle. Il existe plusieurs façons de définir les ensembles de directions discrètes. Un aperçu plus général de ce problème se trouve dans [START_REF] Koch | Evaluation of quadrature schemes for the discrete ordinates method[END_REF]. Nous avons sélectionné ici la quadrature symétrique de niveau S N en utilisant l'ensemble de données amélioré xi suggéré par [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[END_REF].

Critères de convergence

Code Saturne n'a pas de critère intégré pour déterminer la convergence de la solution. Au lieu de cela, il est conseillé de surveiller l'évolution temporelle des variables considérées à différentes positions dans le champ d'écoulement pour décider si le calcul atteint un état stationnaire (EDF [START_REF] Edf | Code_Saturne version 5.0.0 practical user's guide[END_REF]). L'utilisateur peut arrêter le calcul chaque fois qu'il trouve les résultats acceptables ou le code s'exécutera jusqu'à atteindre le nombre maximal de pas de temps déclaré. De plus, les conservations du flux de chaleur et de masse peuvent être un critère pour tester l'atteinte d'un régime stationaire car les flux totaux qui arrivent aux parois actives de la cavité (à faible valeur de température et de concentration) doivent être les mêmes que ceux qui sortent des surfaces de l'obstacle (à des valeurs élevées de température et de concentration).

Validation du Code

Dans cette section, Code Saturne et son modèle radiatif amélioré sont validés en convection thermique pure et en convection de double diffusion couplée au rayonnement. les comparaisons avec les travaux de De Vahl Davis [START_REF] De | Natural convection of air in a square cavity: a bench mark numerical solution[END_REF], Colomer et al. [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF] et Fusegi and Hyun [START_REF] Fusegi | Laminar and transitional natural convection in an enclosure with complex and realistic conditions[END_REF], Yücel, Acharya, and Williams [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF] et Laouar-Meftah [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF], Billaud, Saury, and Lemonnier [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] et Soucasse, Rivière, and Soufiani [START_REF] Soucasse | Natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation at Rayleigh numbers up to 3× 109[END_REF], Sezai and Mohamad [START_REF] Sezai | Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients[END_REF], Cherifi [START_REF] Cherifi | Modelisasion de la convection de double diffusio en presence de rayonnement dans une cavite 3D en regime instationaire[END_REF], Sun, Chénier, and Lauriat [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF] Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF] donnent un bon accord entre les résultats.

Couplage convection naturelle-rayonnement

Un premier ensemble de résultats concerne la convection thermique pure. Dans cette configuration, le fluide est de composition homogène et l'écoulement n'est régi que par les gradients de température.

Dans cette partie, nous analysons tout d'abord l'effet du rayonnement sur la structure de l'écoulement et le transfert de chaleur de manière simple, en supposant que le milieu remplissant la cavité est gris et possède des propriétés radiatives uniformes. Différentes valeurs du coefficient d'absorption κ sont considérées et l'opacité du milieu est caractérisée par l'épaisseur optique τ = κL liée à la taille de la cavité, L. Tous les calculs ont été effectués à Ra = 5 • 10 6 , Pr = 0, 71 et θ 0 = 11, 1. Les parois chaude et froide sont noires et les parois adiabatiques sont purement réfléchissantes. De ce fait, il n'y a pas de couplage radiatif-convectif lorsque τ = 0 (milieu transparent). Ce cas limite sert de référence pour déterminer les effets du rayonnement sur l'écoulement.Nous avons réalisé des calculs avec plusieurs valeurs de l'opacité: τ = 0.1; 0.2; 0.5; 1; 2.

Ensuite, nous considérons la présence d'un composant absorbant-émettant (H 2 O) dilué à différentes concentrations dans un gaz transparent (air sec). Le spectre d'absorption réel de la vapeur d'eau doit être pris en compte pour permettre des simulations réalistes. À cette fin, et suite à la discussion présentée au chapitre 2, nous avons recours xii au modèle SLW associé à l'approche de "Rank correlattion". Concernant les conditions limites, les surfaces de l'obstacle sont fixées à T h = 580K 3 , les parois verticales sont uniformément maintenues à T c = 530K tandis que le plafond et le sol sont supposés parfaitement réfléchissants ( = 0) et adiabatiques.

Les comparaisons entre les résultats obtenus et la référence transparente ont mis en évidence les phénomènes suivant :

• Le rayonnement a tendance à accélérer de manière non uniforme les couches limites de la cavité et de l'obstacle intérieur. Il fait met en mouvement certaines parties du fluide qui étaient stagnantes dans le cas transparent. L'écoulement du panache et sa recirculation interfèrent et créent les modèles d'écoulement de cisaillement.

• Le rayonnement modifie partiellement le gradient thermique près des parois de la cavité : les valeurs du nombre de Nusselt convectif sont augmentées dans la moitié supérieure et diminuées dans la moitié inférieure. Cependant, à la surface horizontale de l'obstacle, le gradient thermique est renforcé.

• Le rayonnement modifie la stratification thermique à l'extérieur du panache et uniformise légèrement la température moyenne.

Couplage convection de double diffusion-rayonnement

En convection de double diffusion, il existe deux gradients qui pilotent l'écoumenent : le gradient thermique et le gradient de concentration. L'ampleur relative des effets de ces deux gradients est définie par le rapport de flottabilité de masse/thermique N : son signe caractérise la coopération (> 0) ou l'opposition (< 0) de la conduite induite.

Dans le cadre de ce travail, nous avons effectué des calculs en convection de double diffusion, y compris des cas où le rayonnement gazeux est pris en compte. Des prédictions sans rayonnement (fluide transparent) sont également fournies pour différents rapports de masse/flottabilité thermique et servent de valeurs de référence mettant en évidence l'influence du transfert radiatf sur la structure d'écoulement et le transfert de chaleur et de masse. Tous les calculs sont effectués à Ra = 5.10 6 , Le = 1, pour un recouvrement parfait des couches limites thermiques et de concentration, Pl = 4.43 • 10 -3 et θ 0 = 11.1. Concernant les conditions limites, une xiii forte concentration de l'espèce absorbante est appliquée sur toutes les surfaces de l'obstacle (C h ), et une concentration nulle (C l = 0) le long des parois verticales de la cavité. L'émissivité des surfaces limites (y compris l'obstacle) est fixée à l'unité, sauf le plafond et le plancher, qui sont considérés comme parfaitement réfléchissants. Pour chaque rapport de force, des calculs sont réalisés à différent opacités: τ =0.1; 0.2; 0.5; 1 et 2.

• Écoulement aidant

A N = 1, l'introduction de rayonnement gazeux n'affecte pas beaucoup le champ de concentration. Il accélère légèrement les couches limites mais réduit la vitesse maximale à l'intérieur du panache. Concernant le champ thermique, le rayonnement volumique tend à homogénéiser le milieu. Il diminue la température dans la moitié supérieure de la cavité et redistribue les isothèmes (passant d'une stratification presque verticale à une stratification horizontale). L'augmentation de l'opacité du milieu renforce ces effets. A N = 2, les mêmes tendances sont observées mais leur amplitude est réduite.

• Écoulement opposant

A N = -1, pour un milieu transparent, aucun écoulement ne se produit à l'intérieur de la cavité en raison de la symétrie parfaite des gradients thermique et de concentration. Mais, avec le rayonnement, cet équilibre est rompu et de nouveaux mouvements de fluides s'établissent. L'écoulement dans la partie inférieure de la cavité est dominé par le gradient de masse pour toutes les épaisseurs optiques considérées. Plus haut, le gradient thermique régit le flux. Pour des valeurs de 0 à 1, le rayonnement intensifie le panache thermique. Cependant, à τ = 1, l'augmentation de la concentration dans l'axe du panache limite le mouvement vertical et provoque son étalement. À τ = 2, ces changements se renforcent, entraînant la séparation entre le panache et une zone de fluide presque immobile au centre de la cavité. En ce qui concerne le champ thermique, la prise en compte du rayonnement gazeux a réduit la température dans les régions proches des parois verticales de l'obstacle. Audessus du niveau de l'obstacle, la température est généralement réduite par rapport au cas transparent. Cependant, elle augmente avec l'opacité, sauf une légère diminution au centre de la cavité lorsque τ = 2. En ce qui concerne le champ de concentration, les tendances d'altération sont les mêmes que pour le champ thermique à l'exception d'une légère augmentation de cette quantité près des parois verticales froides dans la partie inférieure de l'enceinte.

A N = -2, la domination du gradient de masse sur le thermique, le rayonnement gazeux affecte la dynamique, la thermique et le champ de concentration de la même manière que pour N = -1 mais avec une amplitude plus faible.

Un autre cas considéré est le couplage entre la convection de double diffusion et le rayonnement du mélange de gas réél (air -H 2 O et air -CO 2 ). L'obstacle étant à la fois source de chaleur et de polluant, selon le mélange appliqué, nous avons deux types d'écoulements : opposant et aidant. La vapeur d'eau, dont on sait qu'elle est xiv plus légère que l'air, crée un gradient de masse qui a la même direction que le gradient de température. Ces deux gradients provoquent donc des écoulements de même sens (cas aidant). Dans le cas du mélange air -CO 2 , nous avons des écoulements opposant provoqués par deux gradients agissant en directions opposés car la densité molaire de CO 2 est plus grande que celle de l'air sec. Les parois verticales de la cavité sont maintenues à T c = 530K, C l = 0 et pour l'obstacle, elles sont T h = 580K et C h 4 . Toutes les parois actives sont supposées noires alors que les parois adiabatiques sont totalement réfléchissantes. Les principales conclusions issues de résultats de simulations sont :

• Mélange air -H 2 O
Lorsque le rayonnement est pris en compte, aucun changement significatif n'est constaté par rapport au cas transparent, sauf dans le champ thermique. Le rayonnement tend à réduire la température du fluide dans la moitié supérieure de la cavité (où l'émission domine sur l'absorption). Cet effet augmente avec la fraction molaire de vapeur d'eau dans le mélange. Le rayonnement diminue également le transfert thermique total à l'intérieur de la cavité en raison de la diminution du transport convectif près des parois verticales et de l'atténuation du transfert radiatif par l'effet d'absorption. Le transfert de masse semble être inchangé en raison de la structure dynamique préservée.

• Mélange air -CO 2 Le rayonnement tend à ralentir les couches limites (parois verticales de la cavité, surfaces latérales de l'obstacle) dans la partie inférieure de l'enceinte. Plus haut, il renforce le gradient thermique, crée un panache qui remplace le flux descendant dans l'enceinte observé dans le cas transparent. En outre, le rayonnement réduit la température dans la région proche de la paroi verticale dans la partie basse de la cavité. En revanche, il augmente le niveau de température dans la partie supérieure. Il renforce le panache thermique qui amène plus de fluide fortement chargé vers les régions élevées de la cavité, augmentant ainsi la concentration à ces niveaux. La présence de rayonnement réduit le transfert thermique total (par convectif près des parois verticales et transport radiatif le long de la surface supérieure de l'obstacle). De plus, elle diminue légèrement le transfert de masse. a j weight of j th gray gas

xvii

List of Figures

c p specific heat J • kg -1 • K -1 C concentration mol • m -3 C h high concentration mol • m -3 C l low concentration mol • m -3 C j j th absorption cross section m 2 • mol -1 D mass diffusivity m 2 • s -1 g gravitional acceleration vector: 9.81 m • s -2 G incident radiation W • m -2 I radiative intensity W • m -2
= β C (C h -C l ) β T (T h -T c ) N g number of
T re f reference temperature T re f = (T h + T c )/2 K Ra Rayleigh number Ra = gβ T ∆TL 3 αν u, v, w velocity component m • s -1 U re f reference velocity U re f = α √ Ra L m • s -1 U x normalized velocity component U x = u U re f U y normalized velocity component U y = v U re f U z normalized velocity component U z = w U re f R ideal gas constant: 8.3144621 J • K -1 • mol -1 x, y, z Cartersian coordinate m X, Y, Z normalized coordinate X = x L , Y = y L , Z = z L x CO 2 , x H 2 O molar fraction of C0 2 and H 2 O in the mixture ∆C concentration difference mol • m -3 ∆T temperature difference K α thermal diffusivity m 2 • s -1 β C mass expansion coefficient m 3 • mol -1 β T thermal expansion coefficient K -1 emissivity κ absorption coefficient m -1 λ thermal conductivity W • m -1 • K -1 µ dynamic viscosity m 2 • s η wavenumber cm -1 ν kinematic viscosity Pa • s -1 ρ density kg • m -3 τ optical thickness Ω solid angle sr Ω m m th discrete direction σ Stefan Boltzmann constant: 5.670367 × 10 -8 W • m -2 • K -4
Chapter 1

Introduction

Context and Configuration

Natural convection flows are sensitive to the effects of volume radiation in the presence of infrared active gases. This is appearent in particular in mixtures where an absorbing component diffuses into a transparent gas. While several studies have investigated this phenomenon in differentially heated cavities, in single and double diffusion, the case of a heating and diffusing obstacle placed on the floor constitutes a new worthy configuration of interest. Very schematically, it mimics the combustion of an object in the center of a room. The release of heat and the injection of polluting infrared active gases generate a confined buoyant plume due to the combined effects of thermal and concentration gradients. Moreover, radiative absorption and emission within the fluid depends on the gas composition, strongly influences the temperature distribution and, in turn, alters the flow motion.

The configuration under consideration in this thesis is illustrated in figure 1.1. It deals with a cubical cavity of size L = 0.25m having adiabatic horizontal walls, while the vertical ones are maintained at a constant and uniform temperature (T c ). A small solid cube of size l = 0.05m is located at the center of the bottom wall. It creates an opaque obstacle whose surfaces are set uniformly at a higher temperature than the walls (T h > T c ). The enclosure is filled with either dry air or a binary mixture involving an absorbing component, depending on the case study. In pure thermal convection, the medium is homogeneous in composition and the temperature difference between the obstacle and the cavity walls generates a natural convection flow. In doublediffusive cases, a concentration gradient is prescribed within the enclosure: it creates additional buoyancy forces that may either cooperate or oppose to those of thermal origin, depending on the molar weight of the injected species. The highest value C h of this component is uniformly fixed at the obstacle surfaces and the lowest concentration C l is prescribed along the active (cold) walls of the enclosure.

Objective of the thesis

Our goal is to study the natural convection flows generated in the configuration described above in the presence of volume radiation within the gas. These flows are, by nature, complex, especially in double diffusion where the cooperating/opposing configurations can generate original structures, and this even in the absence of radiative participation.

Transparent medium

The study will therefore include, firstly, an analysis of the phenomena in a transparent gas (confined double diffusion plume: co-operating and opposite case). This first approach can be carried out using a dimensionless formulation: this will highlight the characteristic parameters of the configuration (in particular, the definition of the Rayleigh number). The results will concern the dynamic, thermal and mass fields in laminar regime and at steady state, for different values of the mass-to-thermal buoyancy ratio. Steady state solutions are mainly considered, but unsteady behaviors can be reached in some typical cases. This series of calculations will provide reference results when dealing with cases involving radiative effects.

Gray gas

A first approach to participating gases can be carried out by assuming that the absorbing mixture is gray. This amounts to working with a hypothetical medium whose opacity can be freely varied, which is advantageous for a parametric study. It is also possible, at this stage, to maintain a formulation in non-dimensional quantities. The idea is to reproduce some simulations, already carried out without radiation, with a gas whose reference optical thickness varies from 0 (transparent case) to a few units (2, in general). This approach is expected to reveal major trends in the influence of radiation on the flow behavior and on heat and mass transfers.

Real gases

The last -and the most important -part is the simulation of real gas flows. Concretely, we will consider air -H 2 O and air -CO 2 mixtures with a prescribed concentration of absorbing species at the surfaces of the obstacle and a zero concentration on the vertical walls of the cavity. Cases with water vapor will give rise to 1.3. Bibliography review 3 cooperating flows, those with CO 2 to opposing flows. Concentrations will be varied in situations where thermal diffusion dominates, mass diffusion dominates, or both phenomenons are equivalent.

We will analyse the results by comparison with the cases involving a transparent gas.

Bibliography review 1.3.1 Natural convection with the consideration of radiation

The first work about the coupling of natural convection and radiation was performed by Goody [START_REF] Goody | The influence of radiative transfer on cellular convection[END_REF] with the gray gas assumption in the framework of Rayleigh-Bernard configuration in the stellar atmospheres. The authors concluded that the fluid radiation delays the trigger of the instability by decreasing the thermal stratification and damping the temperature fluctuations. Bdéoui and Soufiani [START_REF] Bdéoui | The onset of Rayleigh-Bénard instability in molecular radiating gases[END_REF] have extended this problem to the real gas mixture using a rigorous linearisation of the radiative source term and the same conclusion was found.

In recent decades, many researchers have given more attention to the configuration of the differentially heated enclosure. They considered the effects of surface radiation (transparent gas) as well as volume radiation from a participating fluid.

Concerning surface radiation, the coupling of convection with radiative transport is indirectly created through the boundary conditions of a prescribed flux on the bounding surfaces. In this framework, Behnia, Reizes, and De Vahl Davis [START_REF] Behnia | Combined radiation and natural convection in a rectangular cavity with a transparent wall and containing a non-participating fluid[END_REF] have investigated the coupled process within a rectangular cavity. The horizontal walls were assumed to be adiabatic, a vertical wall was opaque and maintained at high temperature (150 • C) while the other was semi-transparent and was exchanging with the outer environment (kept at 20 • C) by convection and radiation. The results obtained in the range of Rayleigh number [10 4 -3 • 10 5 ] showed that surface radiation accelerates the fluid motion and this effect increases with this parameter.

Wang, Xin, and Le Quéré [START_REF] Wang | Étude numérique du couplage de la convection naturelle avec le rayonnement de surfaces en cavité carrée remplie d'air[END_REF] considered the natural convection-radiation coupling inside a rectangular cavity filled with air. All the walls had the same emissivity. The results showed that the radiation of adiabatic wall lowers the average temperature of top wall while increasing the temperature of the bottom wall. This reduces the thermal stratification of the fluid inside the cavity and the influence of the radiation was still found significant at low temperature and with weak emissivities. It was also observed that the radiative coupling lower the critical Rayleigh number beyond which the unsteady solutions occur.

Regarding the volume radiation effects within the fluid, Lauriat [START_REF] Lauriat | Combined radiation-convection in gray fluids enclosed in vertical cavities[END_REF] has investigated the coupled transfer in a vertical tall cavity (whose height to width ratio was varied between 5 and 20) filled with a gray gas. The computations were performed at different opacities and Rayleigh numbers. The predictions of the radiant field was achieved using the P 1 method. The results showed that, in convective regime, the radiation of gray gas increases the velocities in boundary layers and decreases the vertical thermal stratification.

Chapter 1. Introduction

The same observations were reported by Yücel, Acharya, and Williams [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF] while considering a square cavity whose walls are black and with the optical thickness varying between 0.2 and 5. The Rayleigh number was 5 • 10 6 . In addition, the authors concluded that volume radiation warms up the center of the cavity and, through that, alters the thermal field.

Similarly, Tan and Howell [START_REF] Tan | Combined radiation and natural convection in a twodimensional participating square medium[END_REF] have addressed the effects of volume radiation on natural convection inside a 2D cavity at different Ra in the range of [10 3 -10 5 ] and Pr = 0.72. The sensitivity to the parameters that characterize radiation (wall emissivity, optical thickness, Planck number and albedo) has been analyzed and it was found that all of them, except the albedo, significantly affect the heat transfer.

Other works by Han and Baek [START_REF] Han | The effects of radiation on natural convection in a rectangular enclosure divided by two partitions[END_REF] and Lari et al. [START_REF] Lari | Combined heat transfer of radiation and natural convection in a square cavity containing participating gases[END_REF] in the same framework provided similar conclusions.

Colomer et al. [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF] have extended the calculation of coupled natural convection and radiation of a gray gas from a bi-dimensional enclosure to a cubical cavity. The results showed that the radiation increases the 3-D effects at the intermediate optical thickness. In addition, with a constant Rayleigh number, the heat flux at the hot wall decreases as the optical thickness increases.

Later, Colomer, Consul, and Oliva [START_REF] Colomer | Coupled radiation and natural convection: Different approaches of the SLW model for a non-gray gas mixture[END_REF] as well as Lari et al. [START_REF] Lari | Numerical study of non-gray radiation and natural convection using the full-spectrum k-distribution method[END_REF] have studied the coupled natural convection and radiation using different approaches for modeling the real gas mixtures. The authors concluded that the use of the gray gas approximation overestimates the radiative transfer and the fluid circulation of real gas mixtures.

Regarding real gas mixtures, Soucasse et al. [START_REF] Soucasse | Numerical study of coupled molecular gas radiation and natural convection in a differentially heated cubical cavity[END_REF] have studied natural convection in a cubical cavity filled with an air/H 2 O/CO 2 mixture. The results showed that radiative transfer homogenizes the thermal field and accelerates the vertical boundary layer. This steady state problem was later extended to the weakly turbulent and unsteady regime in Soucasse et al. [START_REF] Soucasse | Transitional regimes of natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation[END_REF]. The authors reported a transition to the unsteadiness at Ra = 3 • 10 8 and, beyond this value, volume radiation intensifies the turbulent fluctuations and decreases the thermal stratification in the center of the cavity.

Recently, Billaud, Saury, and Lemonnier [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] have considered the case of a differentially heated cubical enclosure filled with humid air and have varied the cavity size. They used the Discrete Ordinates Method associated with the SLW model for the calculation of radiative source term within the fluid. All the simulations were performed at Ra = 10 6 and Pr = 0.71. The results showed that the thermal field and velocities field depend on the cavity length even when keeping the Ra-value constant. More generally, radiation was found to accelerate the global circulation in the enclosure to limit the stagnant core region of the cavity.

Beside the numerical works, there exist a few experimental studies by Fusegi and Farouk [START_REF] Fusegi | A computational and experimental study of natural convection and surface/gas radiation interactions in a square cavity[END_REF] and Clergent [START_REF] Clergent | Influence du rayonnement thermique sur les écoulements de convection Naturelle en Espace Confiné[END_REF], for instance, for the coupled natural convection and radiation in a differentially heated cavity. In order to prevent the condensation of water vapor, the gas used in these experiments is usually carbon dioxide or ammonia. An interferometry method has been used for the measurement of temperature. Because of the difficulty of setting the boundary conditions and fluid characteristics as precisely as in the numerical calculations, the comparisons between numerical and experimental results are delicate.

Combined double diffusive convection and radiation

Several studies dealing with the coupling of double diffusive convection and the radiation have been performed. Borjini et al. [START_REF] Borjini | Effect of radiative heat transfer on the three-dimensional Boyancy flow in cubic enclosure heated from the side[END_REF] have investigated the case of a square cavity filled with a gray gas whose absorption coefficient was assumed independent of the concentration of the absorbing component. The calculations have been performed at Ra = 10 5 , Le = 2 and Pr = 13.6. Different optical thicknesses have been considered. The results showed that the radiation alters the flow structure inside the cavity.

Rafieivand [START_REF] Rafieivand | Etude numérique de la convection de double diffusion en présence de rayonnement en cavité rectangulaire/par Mehrdad Rafieivand[END_REF] and Mezrhab et al. [START_REF] Mezrhab | Numerical study of double-diffusion convection coupled to radiation in a square cavity filled with a participating grey gas[END_REF] were the first to report results in a more realistic case for a differentially heated square cavity filled with a binary gas mixture, still considered as a gray gas, whose radiative properties were depending on the local concentration of the absorbing component. The calculations were performed at Ra = 5 • 10 6 , Pr = 0.71 and different values of the mass-to-thermal buoyancy ratio. They showed that radiation could either alter or eliminate the vertical stratification of density, which may drive the oscillatory behavior in unsteady state.

Meftah et al. [START_REF] Meftah | Coupled radiation and double diffusive convection in nongray air-CO2 and air-H2O mixtures in cooperating situations[END_REF], Laouar-Meftah et al. [START_REF] Laouar-Meftah | Gas radiation effects on opposing double-diffusive convection in a non-gray air-H2O mixture[END_REF] extended the previous study to real gas mixtures such as air -H 2 O and air -CO 2 at different mole fraction. They used the SLW model to determine the fluid radiative properties as function of the local thermodynamic state. Their results show that radiation breaks the centro-symmetry of the thermal field, concentration field and flow structure compared to the transparent cases. It was also pointed out that the fluid is accelerated in both the vertical and horizontal boundary layers when thermal and concentration gradients cooperate. On the other hand, gas radiation has little influence on mass transfer, but due to the homogenization of the temperature field, the vertical thermal stratification decreases and thus reduces the convective transfer. Moreover, the radiative transfer between the two active walls also decreases because of the absorption by the participating gas. For these two reasons, the global heat transfer is lowered.

Ibrahim and Lemonnier [START_REF] Ibrahim | Numerical study of coupled double-diffusive natural convection and radiation in a square cavity filled with a N2-CO2 mixture[END_REF] studied the transient processes in 2D-configurations filled with a N 2 -CO 2 mixture with the Rayleigh number up to 1.5 • 10 9 . They reported that radiation, for cooperating flows, stabilizes the fluid motion and slightly accelerates the transition to the steady state. Conversely, for opposing flows, it delays the achievement of the steady state and may even promote the development of thermalsolutal instabilities.

The work of Cherifi et al. [START_REF] Cherifi | Interaction of radiation with double-diffusive natural convection in a three-dimensional cubic cavity filled with a non-gray gas mixture in cooperating cases[END_REF] is an extension to a 3-D configuration of the study by Laouar-Meftah et al. [START_REF] Laouar-Meftah | Gas radiation effects on opposing double-diffusive convection in a non-gray air-H2O mixture[END_REF]. The authors investigated only the cooperating flow. The results showed that radiation slightly affects the fluid motion near the transverse walls. The radiation also breaks the symmetry of thermal and concentration field as well as flow structure compared to a transparent medium. The total heat transfer is reduced while the influence of radiation on mass transfer is not sensible.

Enclosures with an obstacle or a heat source

First of all, concerning the natural convection inside a cavity containing and opaque obstacle within it, Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF] published a study based on numerical calculations and experiments inside a square cavity with an obstacle located on its floor. Regarding the experimental approach, the authors used the PIV (Particle Image Velocimetry) method for the detection of flow structures (velocity field, stream function and velocity vector distribution) and an interferometry technique to evaluate heat transfer, and especially the local and mean Nusselt numbers. The investigation were performed for Rayleigh numbers ranging from 3 • 10 4 to 3.5 • 10 5 . The results showed that the Nusselt number increases with this parameter. Besides it was found that among the three considered heights of the hot source: 0, 0.25 and 0.5 (compared to the cavity size), when the size of the heated obstacle reaches one half of the cavity, the convective heat transfer is the worst while the height of one fourth the cavity length gives the best performance.

Gibanov and Sheremet [START_REF] Gibanov | Effect of the buoyancy force on natural convection in a cubical cavity with a heat source of triangular cross-section[END_REF] have investigated the natural convection in a 3-D enclosure with a heat source of triangular cross section on its floor. The authors performed the numerical simulations for the Rayleigh number in the interval [10 4 -10 6 ]. They observed that the increase in Rayleigh number leads to the decrease of the thermal boundary layer while intensifying the convective flow.

Mousa [START_REF] Mousa | Modeling of laminar buoyancy convection in a square cavity containing an obstacle[END_REF] has modeled the natural convection inside a differentially heated square cavity containing an adiabatic obstacle. The calculations have been run for Pr = 0.71 and the Rayleigh number ranging in [10 2 : 10 7 ]. The results have been considered with respect to the aspect ratio between the obstacle and cavity size. It is reported that as the aspect ratio increases, the heat transfer rate decreases at Rayleigh number in [10 2 : 10 4 ], augments for the Ra values in [10 5 : 10 6 ] and seems to be maintained at Ra = 10 7 .

Raji et al. [START_REF] Raji | Effect of the subdivision of an obstacle on the natural convection heat transfer in a square cavity[END_REF] have considered the effect of the subdivision of an obstacle on the natural convection in a square cavity. The investigation was performed for different values of Rayleigh number in [10 3 -10 8 ] with different numbers of sub-obstacles. The results showed that the increase in the number of blocks reduced the heat transfer and fluid motion.

Kuznetsov and Sheremet [START_REF] Kuznetsov | Two-dimensional problem of natural convection in a rectangular domain with local heating and heat-conducting boundaries of finite thickness[END_REF] have investigated the natural convection inside a rectangular cavity with a local heating on a vertical wall. The calculations were performed at Pr = 0.71, Gr = 10 7 -10 9 . The results showed that the heat transfer was increased with greater Grashof numbers. The authors have also considered the problem of conjugate heat transfer in a closed domain with a locally lumped heatrelease source at Pr = 0.71 and Gr = 10 5 -10 7 . The results reported in the work by Kuznetsov and Sheremet [START_REF] Kuznetsov | Conjugate heat transfer in a closed domain with a locally lumped heat-release source[END_REF] pointed out that as the Grashof number increases, the structure of central vortex, which drives the formation of the temperature profile inside the room, changes in the manner that its center shifts toward the right part of calculation domain. An experimental research on convective heat transfer has been conducted by Kuznetsov, Maksimov, and Sheremet [START_REF] Kuznetsov | Natural convection in a closed parallelepiped with a local energy source[END_REF] in the configuration of a closed parallelepiped containing a local energy source. [START_REF] Kuznetsov | On the possibility of controlling thermal conditions of a typical element of electronic equipment with a local heat source via natural convection[END_REF] have simulated a typical element of electronic equipment by a 3-D gas filled cavity surrounded by thick solid walls that contains a local heat source located on its floor. This configuration has one vertical wall with variable thermal physical properties, which is in contact with outer environment. The others are insulated. The authors have investigated the convective process inside the cavity with respect to the variation in intensity of the heat source and to the environment conditions. It was shown a destabilizing role of the heat source on the flow structure and a significant effect of external conditions on the hydrodynamic and heat transfer in the system. Souayeh et al. [START_REF] Souayeh | Prediction of unsteady natural convection within a square cavity containing an obstacle at high Rayleigh number value[END_REF] have studied the unsteady natural convection within a square cavity containing an obstacle at Rayleigh number in the range [5 • 10 5 -10 7 ]. The authors observed the slight decrease in the extreme values of the stream function, which was explained by the appearance of a small are of re-circulation occurring at the horizontal wall of the obstacle.

Kuznetsov and Sheremet

Bouafia and Daube [START_REF] Bouafia | Natural convection for large temperature gradients around a square solid body within a rectangular cavity[END_REF] have studied the natural convection for large temperature gradients within a rectangular cavity with an inner square solid body. The effects of the aspect ratio as well as of the Rayleigh numbers have been investigated. The results showed that, for any considered values of the aspect ratio, the steady flow can be obtained at a low enough Rayleigh number; for sufficiently large values of Rayleigh number, a periodical flow always appears, but the transition to unsteadiness occurs in different manners depending on the aspect ratio.

Hernandez [START_REF] Hernandez | Natural convection in thermal plumes emerging from a single heat source[END_REF] has studied the natural convection generated by a heat source placed at the center of the bottom of a rectangular cavity. The computation were carried out at different values of Rayleigh number : 10 4 , 5 • 10 4 , 10 5 . The author concluded that the unsteadiness of the flow in high aspect ratio cavity at high Rayleigh number and low Prandlt number come from the shear instability of the interaction between ascending and descending fluid layers.

Concerning the coupled natural convection and surface radiation, Sun, Chénier, and Lauriat [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF] have studied the coupling of natural convection and surface radiation inside a square cavity with an obstacle at its center. The results showed that surface radiation stabilizes the fluid motion inside the cavity. The range of Rayleigh number in which the transition between steady and oscillatory flows appears was shifted from Ra c1 = 2 • 10 5 and 1.7 • 10 5 < Ra c2 < 1.75 • 10 5 for pure natural convection to Ra c1 = 3.15 • 10 5 and 2.85 • 10 5 < Ra c2 < 2.9 • 10 5 . The authors have also investigated the effects of the inner body size and pointed out that at the aspect ratio between the obstacle and the cavity of A = 0.8, the conduction dominates the heat transfer process in the enclosure.

Patil, Sharma, and Velusamy [START_REF] Patil | Conjugate laminar natural convection and surface radiation in enclosures: Effects of protrusion shape and position[END_REF] have investigated the combined natural convection and surface radiation in an enclosure containing a protrusion. The studies on the protrusion shape and position and surface emissivity have been performed for Rayleigh numbers in range of 10 3 -10 6 with surface emissivity values varying in [0; 1]. The results showed that the surface radiation did not much alter the velocity field but it changes noticeably the wall temperature: it partially increases this quantity at the bottom wall and decreases this parameter at the top temperature. [START_REF] Martyushev | Surface radiation influence on the regimes of conjugate natural convection in an enclosure with local energy source[END_REF] have investigated the effect of surface radiation on the natural convection in an enclosure with a local energy source. In their works, different sizes of the hot obstacle and different positions have been considered. The Chapter 1. Introduction calculations have been performed at Ra = 10 6 and Pr = 0.7 while the emissivities of the active surfaces are set to 0 ≤ < 1. The results pointed out that the increase in the emissivity leads to intensify the radiative transfer but reduce the convective Nusselt number. Besides, the increase in the length of the energy source induces the slow down of the arrival at a steady state.

Martyushev and Sheremet

Very recently, Ying Wang [START_REF] Wang | Simulations numériques de panaches thermiques dans une cavité confinée en présence de couplage convection-rayonnement volumique[END_REF] has considered a confined thermal plume inside an air-filled cubical cavity containing a line heat source. The simulations have been performed at Ra in the interval [10 6 -10 9 ]. The impacts of gas radiation on the flow have been taken into account using a gray gas assumption and the SLW model (for predicting the radiative properties of an air -H 2 O mixture). The results have reported that volume radiation stabilizes the plume, delays the transition to the instability. It also homogenizes the thermal field.

Thesis organization

Chapter 2

Methodology

In this chapter, we present the mathematical model of the double diffusive convection and the method for solving the radiative transfer equation. We recall that the studied configuration is illustrated in figure 1.1. It deals with a cubical cavity containing an obstacle (diffusion source of heat and pollutant) located on its floor. The enclosure is filled with dry air or a binary mixture involving an absorbing-emitting component.

Mathematical model

Main assumptions

• The flow in the cavity is three-dimensional, laminar.

• The fluid is considered as Newtonian and incompressible.

• The active surfaces (vertical walls of the enclosure and outer surfaces of the obstacle) are black with respect to radiation while the adiabatic surfaces (ceiling and floor of enclosure) are purely reflective.

• The variations in temperature and concentration within the cavity are weak enough to allow the Boussinesq approximation. Consequently, the variations of the fluid properties are ignored, except for density in the buoyancy force expression, which is written as:

ρ(T, C) = ρ 0 [1 -β T (T -T 0 ) -β C (C -C 0 )] (2.1)
Here, ρ 0 is the density of the mixture in an average state (T 0 , C 0 ) and β T , β C denote, respectively, the thermal and concentration expansion coefficients:

β T = - 1 ρ ( ∂ρ ∂T ) P,C (2.2
)

β C = - 1 ρ ( ∂ρ ∂C ) P,T (2.3)
• The viscous dissipation and pressure work are negligible.

• Soret and Dufour effects are negligible.

Fluid Dynamics Equations

Several conservation equations govern the flow motions and the transfer processes in the enclosure. They express a local balance in mass, momentum, energy and composition within the fluid:

• Continuity equation Since density variations are neglected, the total mass conservation reads:

∇ • u = 0 (2.4)
• Momentum equation

ρ 0 ∂u ∂t + ρ 0 u • ∇u = -∇p + ρ 0 (β T (T 0 -T) + β C (C 0 -C))g + µ∇ 2 u (2.5)
The source term ρ 0 (β

T (T 0 -T) + β C (C 0 -C))
g accounts for the buoyancy force that sets the fluid into motion (here expressed under the Boussinesq approximation).

• Energy equation

ρ 0 C p ∂T ∂t + ρ 0 C p u • ∇T = λ∇ 2 T -∇ • q (2.6)
The divergence term -∇ • q is the internal radiative source resulting from the difference between the absorbed and emitted radiant energy in each elementary volume of fluid.

• Concentration equation

In this study, we only consider binary mixtures. A component (which absorbs and emits radiation) diffuses into a transparent gas. Its concentration obeys a conservation equation that, under the Boussinesq approximation, is expressed as :

∂C ∂t + u • ∇C = D∇ 2 C (2.7)
All the conservation equations are coupled: the dynamic field influences the transport of scalar quantities (T and C) which, in turn, drive the flow through the buoyancy effects. Moreover, the concentration has a direct effect on the thermal field since it changes the absorption-emission properties of the medium. Therefore, the radiative source in the energy balance is impacted.

Radiative transfer equation

The cavity is filled with a semi-transparent gaseous medium at local thermodynamic equilibrium, which absorbs and emits radiation at any point in space. The spectral radiation intensity I η (s, Ω) 1 represents the radiant flux (per unit solid angle and per unit wavenumber) that propagates at point s = (x, y, z) in the direction Ω at the 1 s is the position vector 2.1. Mathematical model 11 wavenumber η. In a non-scattering medium, the local change of intensity is described by the radiative transfer equation:

Ω • ∇I η (s, Ω) = -κ η (s)I η (s, Ω) + κ η (s)I bη (T(s)) (2.8)
where κ η (s) is the local spectral absorption coefficient defined as κ η (s) = N(s) X(s) C η (φ(s)). In this expression, C η (φ(s)) is the spectral absorption cross section, which depends on the local thermodynamic state, φ(T, P, C), X(s) the mole fraction and N(s) the molar density of the absorbing species (Denison and Webb [START_REF] Denison | An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer[END_REF]). The radiative intensity depends on three position coordinates, two direction variables (either two polar angles or two direction cosines) and the wavenumber. The total intensity can be found by integration over the whole spectrum as:

I(s, Ω) = ∞ 0 I η (s, Ω)dη (2.9)
The term -∇ • q which appears in the energy equation is the total divergence of the radiative flux. This flux can be calculated from the total intensity by the expression:

q(s) = 4π 0 I(s, Ω)ΩdΩ = 4π 0 ∞ 0 I η (s, Ω)ΩdηdΩ (2.10)
and, as a result:

-∇ • q(s) = ∞ 0 4π 0 (Ω • ∇I η (s, Ω))dΩdη = ∞ 0 4π 0 κ η (s)I η (s, Ω) -κ η (s)I bη (T(s)) dΩdη = ∞ 0 4π 0 κ η (s)I η (s, Ω)dΩdη -4π ∞ 0 κ η (s)I bη (T(s))dη (2.11)

Boundary conditions

Conservation equations

All the surfaces of the obstacle2 are set at constant and uniform temperature and concentration:

T = T h C = C h (2.12)
Regarding the cavity:

• Vertical walls are uniformly maintained at constant temperature and concentration:

T = T c C = C l (2.13)
• Horizontal walls are adiabatic, impermeable and assumed to behave as fully reflective surfaces:

∂T ∂z = 0 at z = 0, L (2.14) 12
Chapter 2. Methodology

∂C ∂z = 0 at z = 0, L (2.15)
Zero velocities are applied to all the walls of the cavity and all the surfaces of the obstacle.

Radiative Transfer Equation

To solve the radiative transfer equation, we need to input the intensity coming from the bounding walls towards the fluid. This intensity is prescribed for all direction Ω pointing inward to the cavity such that Ω • n > 0 where n is the local unit vector3 on the boundary limit. Assuming gray diffuse surfaces, the boundary condition reads:

I η (s, Ω (Ω•n>0) ) = I bη (T(s)) + (1 -)q inc η (2.16)
where:

q inc η = Ω•n<0 I η (s, Ω)|Ω • n|dΩ (2.17)
In this study, the surfaces of the obstacle and the vertical walls of the cavity are black ( = 1) and the horizontal walls of the enclosure are purely reflective ( = 0).

Heat and Mass Transfer

To investigate the thermal and mass wall fluxes, we calculate the Nusselt and Sherwood numbers 4 . Their local values are defined as follows:

• The local convective Nusselt number refers to the ratio of convective to conductive heat transfer at a boundary in a fluid:

Nu C = L ∆T | ∂T ∂x | x=0,L 5
(2.18)

• The local radiative Nusselt number represents the ratio of radiative to conductive heat transfer at a bounding wall of the enclosure:

Nu R = L λ∆T |q net r,x | x=0,L (2.19)
where q net r,x = (σT ( s) 4 q inc (s)) at x = 0, L

• The local total Nusselt number adds up the contribution of convective and radiative transfer:

Nu T = Nu C + Nu R (2.20)
• The local Sherwood number refers to the ratio of convective mass transfer to diffusive mass transport:

Sh = L ∆C | ∂C ∂x | x=0,L (2.21)
Integrating these quantities over the bounding walls yields the following mean values:

• Mean convective Nusselt number:

Nu C = L A • ∆T A | ∂T ∂x | dA (2.22)
where A is area of the considered surface.

• Mean radiative Nusselt number:

Nu R = L A • ∆T A | q net r | dA (2.23)
• Mean total Nusselt number:

Nu T = Nu C + Nu R (2.24)
• Mean Sherwood number:

Sh = L A • ∆C A | ∂C ∂x | dA (2.25)

Gas Radiation Model

Resolution Methods of Radiative Transfer Equation

In coupled problems involving radiation transport, we have to solve the radiative transfer problem in addition to the conservation equations. Historically, several methods have been developed to achieve this goal. We just present here a short overview of the most popular approaches, and we refer the reader to the main textbooks in this domain for a complete survey (Modest [59], Lewis and Miller [60] for instance):

• Multiflux models. This method was first introduced in the pioneering works of Schuster [START_REF] Schuster | Radiation through a foggy atmosphere[END_REF] and Schwarzschild [START_REF] Schwarzschild | On the equilibrium of the sun's atmosphere[END_REF]. It is based on a division of the angular space, where the radiation intensity is considered as uniform in each discrete solid angles. The most popular approach remains the two-flux method for 1-D problems, in which the direction space is splitted into only two solid angles (one in each coordinate direction) over which the radiant intensities are assumed constant.

The RTE is thus reduced to two differential equations. Recently, Dombrovsky, Randrianalisoa, and Baillis [START_REF] Dombrovsky | Modified two-flux approximation for identification of radiative properties of absorbing and scattering media from directional-hemispherical measurements[END_REF], [START_REF] Dombrovsky | Combined two-flux approximation and Monte Carlo model for identification of radiative properties of highly scattering dispersed materials[END_REF] have applied this approach for the identification of the radiative properties of absorbing and scattering media and radiative properties of highly scattering dispersed materials in combination with a Monte Carlo method. This two-flux approximation was in the past extended Chapter 2. Methodology to four-flux (Vargas [65], Maheu, Letoulouzan, and Gouesbet [START_REF] Maheu | Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters[END_REF], Maheu and Gouesbet [START_REF] Maheu | Four-flux models to solve the scattering transfer equation: special cases[END_REF]) and six-flux models (Brucato et al. [START_REF] Brucato | Estimating radiant fields in flat heterogeneous photoreactors by the six-flux model[END_REF], Puma and Brucato [START_REF] Puma | Dimensionless analysis of slurry photocatalytic reactors using two-flux and six-flux radiation absorption-scattering models[END_REF]) models for solving multi-dimensional problems.

• Spherical Harmonics Approximation : P N The P N method was first proposed by Jeans [START_REF] Jeans | The Equations of Radiative Transfer of Energy[END_REF] in the domain of the astrophysics and was further developed in the field of neutronics. It consists in developing the radiative intensity on a basis of orthogonal functions truncated at the order N: the spherical harmonics associated with the Legendre polynomials P N (Shen [START_REF] Shen | Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials[END_REF], Weisstein [START_REF] Weisstein | Legendre polynomial[END_REF]). High order P N approximations for radiative transfer in arbitrary geometries were introduced by Bayazitoilu and Higenyi [START_REF] Bayazitoilu | Higher-order differential equations of radiative transfer: P3 approximation[END_REF], Mengüç and Viskanta [START_REF] Mengüç | Radiative transfer in axisymmetric, finite cylindrical enclosures[END_REF]. Practice shows that order 1 can give good results in certain configurations and that moving to higher orders increases considerably the computational cost for a moderate gain in accuracy. This method (P 1 or differential approximation, Modest [START_REF] Modest | Radiative heat transfer[END_REF]) has the advantage of the simplicity and the compatibility with standard methods for the solution of the energy equation. But in the optical thin limit, errors appear while treating the radiative flux coming from the bounding surfaces. A modification has been proposed by Olfe [START_REF] Olfe | A modification of the differential approximation for radiative transfer[END_REF] to eliminate this error. Radiation coming from walls is calculated separately with an accurate method and the P 1 -approximation only applies to radiation originating from medium emission. Modest [START_REF] Modest | Modified differential approximation for radiative transfer in general three-dimensional media[END_REF] has extended this modified version to three-dimensional and linear-anisotropically scattering media with reflecting boundaries.

• Discrete Ordinate Method : DOM The method was introduced by Chandrasekhar [START_REF] Chandrasekhar | Radiative Transfer Dover Publications Inc[END_REF] in the field of astrophysics. Then, Lee [START_REF] Lee | The discrete Sn approximation to transport theory[END_REF], Lathrop [START_REF] Lathrop | Use of discrete-ordinates methods for solution of photon transport problems[END_REF] and Carlson [START_REF] Carlson | Transport theory-the method of discrete ordinates[END_REF] used the DOM to solve neutron transport problem. After that, Fiveland [START_REF] Fiveland | Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures[END_REF], Fiveland [START_REF] Fiveland | Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media[END_REF], Fiveland [START_REF] Fiveland | Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method[END_REF], Truelove [START_REF] Truelove | Discrete-ordinate solutions of the radiation transport equation[END_REF], Truelove [START_REF] Truelove | Three-dimensional radiation in absorbing-emitting-scattering media using the discrete-ordinates approximation[END_REF] have adapted the method to the solution of radiative heat transfer. Jamaluddin and Smith [START_REF] Jamaluddin | Predicting radiative transfer in axisymmetric cylindrical enclosures using the discrete ordinates method[END_REF] applied the DOM for the heat transfer problem in an axisymmetric cylindrical enclosures, while Kim and Baek [START_REF] Kim | Analysis of combined conductive and radiative heat transfer in a two-dimensional rectangular enclosure using the discrete ordinates method[END_REF] used this approach in analysis of combined conductive and radiative transfer in a two-dimensional rectangular enclosure. Colomer et al. [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF] have investigated the combined radiation and natural convection in a three dimensional cavity working with the DOM. Overall, the principle of the discrete-ordinate method is to replace the angular integrals by a numerical quadrature formula:

4π 0 f (Ω)dΩ ≈ M ∑ m=1 ω m f (Ω m ) (2.26)
The selections of set of directions and weights is in general constrained by the need of preserving the symmetry of radiant propagation. Different quadratures have been introduced by Lee [78], Lathrop and Carlson [START_REF] Lathrop | Discrete ordinates angular quadrature of the neutron transport equation[END_REF], Truelove [START_REF] Truelove | Discrete-ordinate solutions of the radiation transport equation[END_REF], Fiveland [START_REF] Fiveland | Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures[END_REF], Thurgood [START_REF] Thurgood | A critical evaluation of the discrete ordinates method using HEART and T (N) quadrature[END_REF], Koch et al. [START_REF] Koch | Discrete ordinates quadrature schemes for multidimensional radiative transfer[END_REF] and Balsara [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[END_REF], among others.

The solutions from the DOM are affected by two types of inaccuracies: false scattering and ray effect, which are caused by the spatial and angular discretization errors respectively. The false scattering is similar to the 'numerical diffusion' in CFD calculations. It is related to the interpolation schemes that are involved by the method. The first order upwind scheme (STEP) creates false diffusion (however, this problem can be reduced by using a fine spatial discretization), while the second order DIAMOND scheme may produce negative intensities and thus fluctuations in radiative flux. A solution for this problem is to use high order bounded schemes such as the CLAM scheme (Coelho [START_REF] Coelho | The role of ray effects and false scattering on the accuracy of the standard and modified discrete ordinates methods[END_REF]). In order to reduce the 'ray effect', Ramankutty and Crosbie [START_REF] Ramankutty | Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures[END_REF], Ramankutty and Crosbie [START_REF] Ramankutty | Modified discrete-ordinates solution of radiative transfer in three-dimensional rectangular enclosures[END_REF] introduced a modified version of the DOM for treating separately the radiation coming from surfaces and medium in two and three dimensional problem respectively. In a recent past, this approach has also been considered for the radiative transfer problems with irregular geometries by Amiri, Mansouri, and Coelho [START_REF] Amiri | Application of the modified discrete ordinates method with the concept of blocked-off region to irregular geometries[END_REF].

Besides, for treating complex configurations, different structured and unstructured grid procedures such as block off, embedded boundaries, body-fitted structure, body-fitted unstructured, multi-block, local grid refinement have been proposed and then reported by Coelho [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]. Recently, different spatial schemes in discrete ordinates method using 3D unstructured mesh have been compared by Joseph et al. [START_REF] Joseph | Comparison of three spatial differencing schemes in discrete ordinates method using three-dimensional unstructured meshes[END_REF], while, the block-off and embedded boundary procedure have been used to mesh the irregular enclosures with Cartesian grid by Aghanajafi and Abjadpour [START_REF] Aghanajafi | Discrete ordinates method applied to radiative transfer equation in complex geometries meshed by structured and unstructured grids[END_REF]. In addition, Le Hardy et al. [START_REF] Hardy | Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method[END_REF] have developped specific numerical algorithms for handling specular reflection when solving 3D radiative transfer equation using DOM.

• Finite Volume Method : FVM Raithby and Chui [START_REF] Raithby | A finite-volume method for predicting a radiant heat transfer in enclosures with participating media[END_REF] first introduced this method for predicting the radiant heat transfer in enclosures with participating media. Other researches in the same frame work were introduced by Chui, Raithby, and Hughes [START_REF] Chui | Prediction of radiative transfer in cylindrical enclosures with the finite volume method[END_REF] for radiative problems in cylindrical enclosures. In this method, the RTE is integrated over the space and the solid angle like in Discrete ordinates method.

The main difference between these two methods regards the angular discretization (Coelho [95]). In the DOM, integrals over solid angles are replaced by the quadratures while, in the FVM, the RTE is integrated over a solid angle, often referred to as a control angle, ∆Ω which arises from the discretization of the entire spherical solid angle. Hunter and Guo [START_REF] Hunter | Comparison of the discrete-ordinates method and the finite-volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates[END_REF] have compared these two methods over the problems of radiative transfer problem in cylindrical geometries and concluded that with the same grid size and number of discrete direction, the DOM is more efficient than the FVM (less memory used and faster calculation).

• Zonal Method : ZM Zonal method was first introduced in radiative heat transfer by Hottel and Cohen [START_REF] Hottel | Radiant heat exchange in a gas-filled enclosure: Allowance for nonuniformity of gas temperature[END_REF]. In this method, the surface and the volume of an enclosure are divided into a number of zones, each assumed to have a uniform distribution of temperature and radiative properties. The direct exchange areas (factors) between the surface and volume elements are evaluated and the total exchange areas are determined using matrix inversion techniques (Viskanta and Mengüç [START_REF] Viskanta | Radiation heat transfer in combustion systems[END_REF]). Since this method was first introduced for an absorbing, emitting and non-scattering gray gas with constant absorption coefficient, Hottel and Sarofim [START_REF] Hottel | Radiative Transfer[END_REF] improved it to relax this last restriction. Recently, Ebrahimi et al. [START_REF] Ebrahimi | Zonal modeling of radiative heat transfer in industrial furnaces using simplified model for exchange area calculation[END_REF] have used the zonal method to calculate radiative heat transfer in Chapter 2. Methodology industrial furnaces using a simplified numerical integration to evaluate the exchange areas. The advantage of Zonal Method is its simplicity for the adaptation of different sets of boundary conditions but it is difficult to apply it to complex geometries. Moreover, the calculation of the direct exchange area may require a high computational cost.

• Monte Carlo Method : MCM The method was initially developed in the context of nuclear transport. Its first application to the thermal radiation problems is due to Fleck Jr [START_REF] Fleck | The calculation of nonlinear radiation transport by a Monte Carlo method[END_REF] and Howell and Perlmutter [START_REF] Howell | Monte Carlo solution of thermal transfer through radiant media between gray walls[END_REF]. In this approach, the method consists in simulating a finite number of photon histories through the use of random number generator (Lewis and Miller [START_REF] Lewis | Computational methods of neutron transport[END_REF]). In its standard form, photo bundles are traced in a forward direction but, in case radiation comes to a small area, it may become inefficient. Collins et al. [START_REF] Collins | Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres[END_REF] have introduced a new approach called 'Backward Monte Carlo' based on the review of Case [START_REF] Case | Transfer problems and the reciprocity principle[END_REF]. Modest [START_REF] Modest | Backward Monte Carlo simulations in radiative heat transfer[END_REF] recently published his research about Backward Monte Carlo in a scattering media and show that this backward method become inefficient when the scattering coefficient increases. The Monte Carlo method can easily handle anisotropically scattering media, complex geometries and spectral aspects. Its accuracy regularly serves as reference for other calculation tools. Its main drawback is the large computational time needed to achieve the results.

Recently, Fournier et al. [START_REF] Fournier | Radiative, conductive and convective heat-transfers in a single Monte Carlo algorithm[END_REF] have presented the problem of combined heat transfer using a single Monte Carlo algorithm. It is then applied in complex geometry problems by Ibarrart et al. [START_REF] Ibarrart | Combined conductive-convective-radiative heat transfer in complex geometry using the Monte Carlo method[END_REF] and Caliot et al. [START_REF] Caliot | Combined conductive-radiative heat transfer analysis in complex geometry using the Monte Carlo method[END_REF].

• Discrete Transfer Method: DTM This method was first introduced in the work by Lockwood and Shah [START_REF] Lockwood | A new radiation solution method for incorporation in general combustion prediction procedures[END_REF]. It is similar to the ray tracing method in choosing a set of directions along which the propagation of radiation is computed. The ray from each point of a surface is traced in a given direction through the medium until it meets another surface. Henson and Malalasekera [START_REF] Henson | Comparison of the discrete transfer and Monte Carlo methods for radiative heat transfer in three-dimensional nonhomogeneous scattering media[END_REF] have compared the DTM and Monte Carlo for radiative heat transfer in three-dimensional, non homogeneous, scattering media and pointed out the good agreement between the two methods. Selçuk and Kayakol [START_REF] Selçuk | Evaluation of discrete ordinates method for radiative transfer in rectangular furnaces[END_REF] have compared the DTM and DOM for radiative transfer calculation in rectangular furnaces and the results showed that S 4approximation and DT64 (64 rays per wall node) give good predictions of flux density and radiative energy source term compared to the exact solutions, but DOM consumes about 3 orders of magnitude less CPU times than the DTM. This method, however, carries the advantage of easily treating the irregular geometries but, like the DOM, it may suffer from 'ray effect', an error linked to the angular discretization. Cumber [START_REF] Cumber | Improvements to the discrete transfer method of calculating radiative heat transfer[END_REF] and Cumber [START_REF] Cumber | Application of adaptive quadrature to fire radiation modeling[END_REF] have proposed some modifications for this method while Coelho and Carvalho [START_REF] Coelho | A conservative formulation of the discrete transfer method[END_REF] have developed a conservative formulation of DTM. Heugang, Kamdem Tagne, and Pelap [START_REF] Heugang | A Discrete Transfer Method for Radiative Transfer through Anisotropically Scattering Media[END_REF] have performed the calculations of radiative heat transfer through anisotropically scattering media and showed that DTM can correctly deal with this problem, but a finer angular discretization is necessary when the scattering anisotropy is strong.

Some years ago, Feldheim and Lybaert [START_REF] Feldheim | Solution of radiative heat transfer problems with the discrete transfer method applied to triangular meshes[END_REF] have developped a DTM approach for the radiative transfer equation in a gray medium on unstructured triangular meshes. It was validated and found to perform well on pure radiative as well as combined heat transfer problems.

Coelho et al. [START_REF] Coelho | Modelling of radiative heat transfer in enclosures with obstacles[END_REF] solved the radiation problem in 2-D enclosures with an obstacle using DTM, DOM, FVM, MC, ZM. They found that DOM and FVM are the most efficient in terms combined accuracy and computational cost.

In the present study, we have used the Discrete Ordinate Method for our radiative calculations because of its good compromise between the accuracy and computational cost and its easy implantation in many CFD codes. In particular, Code Saturne, an open-source CFD code developed by EDF (Archambeau, Méchitoua, and Sakiz [START_REF] Archambeau | Code Saturne: A finite volume code for the computation of turbulent incompressible flows-Industrial applications[END_REF]), offers an already integrated radiative DOM module. More details about the Discrete Ordinate Method are provided in the next chapter along with the description of its implementation in the Code Saturne code.

Gas models

Whenever gas radiation is considered, the actual spectral behavior of the fluid absorption must be accounted for. To that end, various gas models have been introduced with different levels of complexity, computational requirements and accuracy. These models can be classified into three main groups as:

• Line by Line model

• Band models • Global models

Line by Line model

The Line by line model is considered as the most accurate. It is constructed by discretizing the absorption spectrum into discrete values (up to one million) so that the full spectral dynamics of the gas mixture can be recovered. As a result, the radiative transfer equation must be solved as many times as there are discrete κ-values. It therefore requires a huge computational time. This is why it remains in practice not affordable in coupled problems (where radiation is calculated iteratively at each time step), but this method serves as a reference for assessing the accuracy of simplified models.

Band models

In the band models, the spectral domain is divided into intervals of a given size and the radiative properties of the gas are evaluated over each of them. There are mainly two groups of band models depending on the width of the spectral band: the narrow band and wide band models.

The Narrow Band Models are constructed by selecting a band size, which is narrow enough to keep the Planck function constant. There still are two main approaches: Statistical Narrow Band (SNB) and k-distribution. The SNB model was first introduced by Goody [START_REF] Goody | A statistical model for water-vapour absorption[END_REF] and Godson [START_REF] W L Godson | The computation of infrared transmission by atmospheric water vapor[END_REF]. In this method, the radiative properties of the gas (transmissivity or emissivity) are calculated by adopting a statistical model to describe the distribution of line intensities, widths and spacing. Therefore, this approach is not compatible with some methods of solution of the RTE, such as P 1 or DOM, which requires explicit values of the absorption coefficients (Taine and Soufiani [START_REF] Taine | Gas IR Radiative Properties: From Spectroscopic Data to Approximate Models[END_REF]). The k-distribution model was first reported in Kondratyev [127]. It is based on the observation that the absorption coefficients κ attains the same value k over a narrow spectral range. These identical quantities are, therefore, reordered as an increasing function of k with respect to the reordered artificial wave number g 6 to reduce the repetition of computations with the same values of κ (Modest [START_REF] Modest | Radiative heat transfer[END_REF]). To deal with nonhomogeneous gases, the correlated-k distributions has been introduced by Goody et al. [START_REF] Goody | The correlated-k method for radiation calculations in nonhomogeneous atmospheres[END_REF] Lacis and Oinas [START_REF] Lacis | A description of the correlated k distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres[END_REF] and Fu and Liou [START_REF] Fu | On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres[END_REF]. The term 'correlated' indicates a hypothesis that the heterogeneities of the medium (temperature, in particular) are treated by assuming that the spectra between different state are correlated. The k-distribution differs from the SNB by the fact that it directly provides a (reordered) absorption coefficient representation and, therefore, can be used with any arbitrary RTE solver (including P 1 , DOM, FVM,...).

The Wide Band Models consider the bands whose spectral range is adjusted to the width of the physical absorption bands of the component. In principle, wide band correlations are found by integrating narrow band results across an entire band Modest [START_REF] Modest | Radiative heat transfer[END_REF]. The most popular model is the exponential wide band, which was initially presented by Edwards [START_REF] Edwards | Molecular gas band radiation[END_REF] and its applications to radiative transfer problems have been discussed by Ströhle and Coelho [132]. This method is less accurate than the narrow band ones, but it allows significant reductions in calculation time.

Global models

In the line by line model and the band models, one considered the radiative properties over each line of the spectral representation or a specified interval of wavenumbers. On the other hand, the global models find the radiative characteristic over the entire gas spectrum.

The simplest model of this group is the gray gas model in which the absorption coefficient is assumed to be constant over the whole spectrum. With only one value of local absorption coefficient κ a , the radiative transfer equation can be directly expressed in total quantities and no further spectral integration is needed.

Another model of this group is the weighted-sum-of-gray-gases (WSGG). This model was first introduced by Hottel and Sarofim [START_REF] Hottel | Radiative Transfer[END_REF] in the frame of the zonal method. Its principle is to replace the continuous spectral absorption coefficient by a finite set of values (each of them being related to a gray gas) with their associated weights. Modest [START_REF] Modest | The weighted-sum-of-gray-gases model for arbitrary solution methods in radiative transfer[END_REF] applied this method for the solution of the radiative transfer equation with the assumption of spatially constant absorption coefficients for all gray gases, but letting their weights vary with temperature. The radiative parameters (absorption coefficients and their associated weights) were originally identified by making the total emissivity ≈ ∑ n j=1 a j • (1e -κ j L ) fit the experimental data.

Based on the same idea as the WSSG model, Denison and Webb [START_REF] Denison | An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer[END_REF] have presented the Spectral Line-Based Weighted-sum-of-gray-gases models (SLW). The weights of the gray gases are now determined by using the global distribution function of the absorption coefficient weighted by the Planck function. This distribution function was calculated directly from high resolution spectral databases. Similar to SLW, the Absorption Distribution Function model (ADF) (Pierrot et al. [START_REF] Pierrot | A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases[END_REF]) and Full Spectrum K-distribution model (FSK) (Modest and Zhang [START_REF] Modest | The Full-Spectrum Correlated-k Distribution for Thermal Radiation From Molecular Gas-Particulate Mixtures[END_REF], Modest and Mehta [START_REF] Modest | Full spectrum k-distribution correlations for CO 2 from the CDSD-1000 spectroscopic databank[END_REF]) have been reported. These three methods have differences in the way of calculating the gas radiative properties, but their relationship has been brought out in the article of Solovjov and Webb [START_REF] Solovjov | Global Spectral Methods in Gas Radiation: the Exact Limit of the SLW Model and its Relationship to the ADF and FSK Methods[END_REF]. Recently, Solovjov, Lemonnier, and Webb [START_REF] V P Solovjov | The SLW-1 model for efficient prediction of radiative transfer in high temperature gases[END_REF] introduced the SLW-1 model such that the calculations is now performed by using only one gray (optimized) gas and one transparent component.

Goutiere, Liu, and Charette [START_REF] V Goutiere | An assessment of real-gas modelling in 2D enclosures[END_REF] and Goutière, Charette, and Kiss [START_REF] V Goutière | Comparative performance of nongray gas modeling techniques[END_REF] through their works about the comparisons of the different gas models, showed that the SLW provides the best compromise between the accuracy and the computational cost. The SLW model is therefore, chosen for the calculations of radiative properties of the gas mixtures in all the simulations presented in this thesis.

Recently, the SLW model in non uniform media has been developped using different novel approaches: the Rank Correlated (RC) ( [START_REF] V P Solovjov | The rank correlated SLW model of gas radiation in nonuniform media[END_REF]), Scaled (SC) (Solovjov et al. [START_REF] V P Solovjov | The Scaled SLW model of gas radiation in non-uniform media based on Planck-weighted moments of gas absorption cross-section[END_REF]) and Locally Correlated (LC) (Solovjov et al. [START_REF] V P Solovjov | Locally Correlated SLW Model for Prediction of Gas Radiation in Non-Uniform Media[END_REF]) models.

The SLW gas model

In this section, the SLW model will be described in more details for isothermal, homogeneous as well as non-isothermal, non-uniform media.

Calculation of the absorption coefficients

The SLW model involves a set of finite number (N g ) gray gases and one clear (transparent) gas. The absorption coefficient of j th gas is calculated as:

κ j = N • X • C j (2.27)
where C j is the absorption cross section, N is molar density and X the mole fraction.

Knowing N and X, the remaining problem is to determine C j .

The wavenumber range under consideration prescribes an overall absorption cross section interval [C min ,C max ]. The N g gray gases discretize this range into several intervals ∆ j such that, for the j th gray gas:

∆ j = η : Cj-1 < C η (φ g ) < Cj : gray gas intervals ∆ 0 = η : C η (φ g ) ≤ C0 = C min : clear gas intervals (2.28)
where Cj-1 and Cj are discrete values of C defined over [C min ,C max ]. They are termed the supplemental cross sections.

The value of C j can be arbitrarily chosen in the interval [ Cj-1 , Cj ] for calculating the absorption coefficient κ j . In all our simulations, and following the classical approach, the supplemental values were defined as: Cj = C min (C max /C min ) j/N g (2.29)

C j = Cj-1 Cj (2.30) Chapter 2.

Methodology

The resulting set of local absorption coefficients, was therefore obtained as:

κ j = NXC j = NX Cj-1 Cj : with j=1,...,N g κ 0 = 0 for the clear gas (2.31)

Calculations of Absorption line black body distribution function and the weights of gray gases

The weights associated with each gray gas are calculated from the global distribution function of the absorption coefficient weighted by the Planck function: it is named the absorption line black body distribution function (ALBDF) in the seminal work by Denison and Webb [START_REF] Denison | k-distributions and Weighted-Sum-of-Gray-Gases-A hybrid model[END_REF]. This function is evaluated as the integral of the Planck function calculated at a source temperature T b over the wavenumber intervals such that the absorption cross section C η (φ g ) at a gas thermodynamic state φ g 7 is below a prescribed value of C, namely:

F(C, φ g , T b ) = 1 E b (T b ) η:C η (φ g )<C E bη (T b )dη = π σT 4 b η:C η (φ g )<C I bη (T b )dη (2.32)
where E bη (T b ) is the Planck spectral emissive power emitted by a blackbody at temperature T b and E b is the total blackbody emissive power given by the Stefan-Boltzmann Law.

The ALBDF is determined by performing integrations over the whole spectrum at high resolution (line by line) at different pressures, temperatures and compositions. The resulting data are made available for the main participating species (H 2 O, CO 2 , CO) either as mathematical correlations or in look-up tables. The most recent contribution is reported in Pearson et al. [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] based on HITEMP-2010 spectral database. The main features of two methods are shortly introduced below:

Mathematical correlations

• For air -H 2 O mixtures:

F w (C, T g , T b , X w ) = 1 2 tanh[P w (T g , T b , ξ -ξ p )] + 1 2 (2.33)
7 φ g = (X, P, T g ) where T g is the gas temperature, Y the molar fraction, and P the total pressure.

Gas Radiation Model

21

where :

P w (T g , T b , ξ -ξ p ) = 3 ∑ l=0 3 ∑ m=0 3 ∑ n=0 b lmn T g 2500 n T b 2500 m (ξ -ξ p ) l ξ p = 3 ∑ l=0 3 ∑ m=0 3 ∑ n=0 u lmn T g T b 2500 2 n ξ m ψ l+1 w ψ w = 1 10 ln(100p e ) p e = (1 + 8.17X w )p ξ = ln(C) with 1 • 10 -4 ≤ C ≤ 60 m 2 mol
Here, F w denotes the approximation of the ALBDF when considered participating species is water vapor and P w is a temporary variable.

• For air -CO 2 mixtures:

F c (C, T g , T b ) = 1 2 tanh[P c (T g , T b , ξ -ξ p )] + 1 2 (2.34)
where :

P c (T g , T b , ξ -ξ p ) = 3 ∑ l=0 3 ∑ m=0 3 ∑ n=0 d lmn T g 2500 n T b 2500 m (ξ -ξ p ) l ξ p = 3 ∑ l=0 3 ∑ m=0 3 ∑ n=0 v lmn T g T b 2500 2 n ξ m ψ l+1 c ψ w = 1 10 ln(100p) ξ = ln(C) with 1 • 10 -4 ≤ C ≤ 600 m 2 mol
Here, F c denotes the approximation of the ALBDF when considered participating species is CO 2 and P c is a temporary variable.

The parameters b lmn , u lmn , d lmn , v lmn are tabulated in the work by Pearson et al. [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].

Look-up table

The tabulated approach consists in storing the ALBDF values obtained discrete value of the cross sections C, temperature T, pressure P and composition X. In more details, these tabulations are made: every 100K for T between 300 and 3000K; for 70 values of C between 10 -4 and 10 3 m 2 /mol; for 10 values of P in the range of [0.1 -50] atm and in the case of air -H 2 O mixture, for 9 discrete values of molar fraction between 0 and 1. All this tabulated data were assembled by V.P. Solovjov and are available at http://albdf.byu.edu. Then, through the interpolations over P, X (for air -H 2 O mixture only), C, T b and T g , the value of F at local conditions (φ g = (X, P, T g )) and for a given black body temperature T b are determined.

Chapter 2. Methodology

The weight of the j th gray gas a j corresponds to the difference in the ALBDF at the two supplemental absorption cross sections that define the j th interval [ Cj-1 , Cj ]:

a j = F( Cj , φ g , T b ) -F( Cj-1 , φ g , T b ) : for j = 1,...,N g a 0 = F( C0 , φ g , Tb) (2.35)
where T b = T g , the gas temperature, in a uniform medium and T b = T w at the boundaries (T w , is the wall temperature). The determination of these quantities for nonuniform gases will be described in the following section.

Implementation of SLW model in the radiative transfer equation

We consider the spectral integration of the monochromatic RTE over the wavenumber ranges corresponding to the j th gray gas:

∆ j Ω • ∇I η (s)dη = ∆ j (-κ j I η (s) + a j κ j I bη )dη (2.36) ∆ j = [a i,j (s); b i,j (s)
] is the wavenumber intervals also defined in (2.28). Applying the Leibnitz formula to the left hand side term in the equation (2.36), yields:

Ω • ∇ ∆ j (s) I η (s, Ω)dη = ∆ j (s) Ω • ∇I η (s, Ω)dη + ∑ i I η [s, η = b i,j (s), Ω]Ω • ∇b i,j (s) -I η [s, η = a i,j (s), Ω]Ω • ∇a i,j (s)
Leibnitz terms (2.37) For an isothermal, homogenous medium, the Leibnitz terms in the equation (2.37) are null because all the wavenumber intervals ∆ j are identical (therefore, Ω • ∇a i,j (s) = 0 and Ω • ∇b i,j (s) = 0).

For non-isothermal and/or non-homogeneous media, the Leibnitz terms are no longer null and vary with the position s. Their evaluation at each point of the domain and for each gray gas would require a huge calculation effort.

A first option is to simply neglect these terms when solving the RTE. But it leads to significant errors in predicting the radiative quantities when large temperature and concentration gradients are present (like in combustion problems, for instance).

Another approach is to select the wavenumber intervals such that they do not depend on the locations: ∆ j (s) = ∆ j = const. This idea is related to the assumption of an "ideal spectrum behavior" proposed in Denison and Webb [START_REF] Denison | k-distributions and Weighted-Sum-of-Gray-Gases-A hybrid model[END_REF]. The main hypothesis can be summarized as follows: consider a reference state φ re f = (P re f , T g = T re f , X re f ) and a local state φ loc = (P loc , T g = T loc , X loc ). The interval over which C η (φ re f ) remains below a fixed value C re f is the same as the interval over which the cross section C η (φ loc ), at local state, remains below a value C loc when the black body temperature is the same. In other words:

η : C η (φ re f ) < C re f = η : C η (φ loc ) < C loc (2.38)
This equality assumes that the gas spectrum of the related species is correlated, in the sense that there exist some relation linking the spectra at different conditions. This approximation leads to the equality of the values of the ALBDF calculated for a fixed source temperature

T b =T re f F(C loc , φ loc , T b = T re f ) = F(C re f , φ re f , T b = T re f ) (2.39)
The cross section valid in the local state C loc can therefore be deduced from the reference value C re f by inverting the equation (2.39):

C loc = C(F re f , φ loc , T b = T re f )
where C(F, φ, T b ) is the distribution function of cross section with respect to F, that is the reciprocal function of F and where

F re f = F(C re f , φ re f , T b = T re f ).
The above assumption is referred to as the Reference Approach (RA). More recent implementations of the SLW model include the Rank Correlated (RC) ( [START_REF] V P Solovjov | The rank correlated SLW model of gas radiation in nonuniform media[END_REF]), Locally Correlated (LC) (Solovjov et al. [START_REF] V P Solovjov | Locally Correlated SLW Model for Prediction of Gas Radiation in Non-Uniform Media[END_REF]) and Scaled approaches (SC) (Solovjov et al. [START_REF] V P Solovjov | The Scaled SLW model of gas radiation in non-uniform media based on Planck-weighted moments of gas absorption cross-section[END_REF]). We will here focus on the RA-SLW and RC-SLW taking advantage of the Chapter 2. Methodology explanations given by Solovjov et al. [START_REF] V P Solovjov | The rank correlated SLW model of gas radiation in nonuniform media[END_REF] and Solovjov, Webb, and André [START_REF] V P Solovjov | Radiative Properties of Gases[END_REF]:

Reference Approach

This approach involves different steps that may be summarized as follow:

1. Choose a reference state: φ re f = T re f , X re f , p re f 2. At reference state, chose the set of the reference supplemental cross sections Cre f j using (2.29).

3. Solve the following implicit equation to determine the local supplemental cross sections Cloc

j F( Cre f j , φ re f , T b = T re f ) = F( Cloc j , φ loc , T b = T re f ) (2.40)
4. Use equation (2.30) with the determined supplemental cross section for the calculation the local gray gas absorption coefficients:

κ j (s) = N loc X loc C loc j = N(s)X(s) Cloc j Cloc j-1
(2.41)

5. Calculate the local weights attributed to the j th gray gas:

a j (s) = a loc j = F( Cre f j , φ re f , T b = T loc ) -F( Cre f j-1 , φ re f , T b = T loc ) (2.42)
In case of boundaries emitting at the temperature T w , it is advised to set the black body temperature equal to these wall temperature (Solovjov, Webb, and André [START_REF] V P Solovjov | Radiative properties of gases[END_REF]) Rank Correlated Approach In this approach, Solovjov et al. [START_REF] V P Solovjov | The rank correlated SLW model of gas radiation in nonuniform media[END_REF] suggest a modification of the assumption of "ideal spectrum" or "correlated spectrum" discussed above. They consider a less restrictive assumption of rank correlation regarding the relationship between absorption spectra at different thermodynamic states. The main hypothesis underlying this approach can be summarized as follows. Consider a fixed value of the ALBDF at a fixed black body temperature T b : the wavenumber intervals ∆ 1 = η : C η (φ 1 ) < C(F, φ 1 , T b ) and ∆ 2 =η : C η (φ 2 ) < C(F, φ 2 , T b ) are the same. This expression is assumed to be correct for any two thermodynamic states, thus, ∆ 1 = ∆ 2 = constant (see figure 2.3) for a given value of F at a fixed T b . It ensures the elimination of the Leibnitz term in equation (2.37). It can be observed that we no longer need to specify any reference state, but only a black body source temperature T b . Moreover, the process involves a discretization of the ALBDF in the interval of [F min , F max ] rather than of the absorption cross section. The following algorithm can therefore be used: With a large number of gray gases N, even simple uniform subdivision is likely satisfactory for accurate results (Webb, Solovjov, and André [START_REF] Webb | An exploration of the influence of spectral model parameters on the accuracy of the rank correlated SLW model[END_REF]). However, for a small number of gray gases, a more efficient discretization can be achieved by using the Gauss-Legendre quadrature nodes and weights. Firstly, the positive abscissas x j > 0 and the corresponding weights w j , j = 1, 2, ..., n of Gauss-Legendre quadrature for integration over the interval [-1,1] are calculated. And then, we can determine the reference values of ALBDF F re f j and supplemental reference values Fre f j in interval [F min , F max ] (Solovjov, Webb, and André [START_REF] V P Solovjov | Radiative Properties of Gases[END_REF]) such that:

a j (T w ) = F( Cre f j , φ re f , T b = T w ) -F( Cre f j-1 , φ re f , T b = T w ) (2.43)
F re f j = F min + x j (F max -F min ) Fre f j = F min + (F max -F min ) j ∑ k=1 w k Fre f 0 = F min (2.44) with j = 1, 2, ..., N g 2.3. Conclusion 27
2. Use the inverse ALBDF to find the values of local absorption cross sections and also local supplemental absorption cross sections:

C loc j = C(F re f j , φ loc , T re f ) Cloc j = C( Fre f j , φ loc , T re f ) (2.45)
3. Calculate the local gray gas absorption coefficients:

κ loc j = N loc X loc C loc j κ loc 0 = 0 (2.46)
4. Calculate the local corresponding weights of local gray gas absorption coefficients with the help of the ALBDF at local cross sections Cloc j :

a loc j = F( Cloc j , φ loc , T loc ) -F( Cloc j-1 , φ loc , T loc ) : with j = 1,2,...,N g a loc 0 = F( Cloc 0 , φ loc , T loc ) (2.47)
Applying the newly calculated radiative properties to eq. (2.36) yields the radiative transfer equations for j th gray gas written as :

Ω • ∇I j (s) = -κ j I j (s) + a j κ j I b (s) (2.48)
where ∆ j Ω • ∇I η dη = Ω • ∇I j Consequently, the total intensity and radiative source term are given by the following expressions:

I(s, Ω) = N g ∑ j=1 I j (s, Ω) (2.49) ∇ • q = N g ∑ j=1 (4πa j κ j I b (s) -κ j G j (s)) (2.50)
where G j (s) = 4π 0 I j (s, Ω)dΩ is the incident radiation in the case of j th gray gas.

Conclusion

In this chapter, the mathematical basis of the study was presented. Besides, we have shortly described the methods of resolution for the RTE and the Discrete Ordinates Method (DOM) has been selected as the solver for our simulations. Finally, we have mentioned some existed gas models and focused on the formation of the SLW method with the center of interest was the Reference Approach and the Rank Correlated for the treatment of non-isothermal, non-homogeneous medium.

Chapter 3

Code Saturne and Radiative Calculation

Code Saturne

In this section, we introduce the resolution of the governing equations of our problem. For this purpose, we have used Code Saturne version 5.0.4 [START_REF] Archambeau | Code Saturne: A finite volume code for the computation of turbulent incompressible flows-Industrial applications[END_REF], an open source software for CFD calculation developed by EDF. A built-in radiative module is available, in which we have implemented our own data for directional integration and gas radiation modeling.

Computational Fluid Dynamic

Code Saturne uses a finite volume method to solve the governing equations of fluid motion and heat and mass transfer. For the momentum equations, Code Saturne uses an algorithm of type prediction-correction called SIMPLEC, which stands for Semi-Implicit Methods for Pressure Linked Equations Consistent. Different discretizations in space and in time are also available.

Time stepping

The time scheme is used in Code Saturne is a θscheme with: θ = 1 for an implicit first order Euler scheme, θ =1 2 for second order Crank-Nicolson scheme.

In our study, we have used the implicit first order backward Euler scheme 1 2 . Code Saturne provides two options for the temporal step: constant or variable (where the code automatically calculates the time step after each iteration that satisfy the CFL criterion).

Spatial discretization

In the finite volume approach, the equations are integrated over each cell. Using the Green Theorem, the volume integrations become surface integrations. We, therefore, need only to calculate the face gradients of each variables. Code Saturne proposes different schemes of first order (Upwind) and second order (Centered or Second-Order-Linear-Upwind (SOLU)) for spatial discretization. In this study, we have selected the centered second order. 30 Chapter 3. Code Saturne and Radiative Calculation

Discrete Ordinate Method in Code Saturne

A set of discrete directions with associated weights has to be specified. For each of these directions, the radiative transfer equation (2.8) is solved over the entire spatial domain. Integrating this equation on a control volume centered at the grid node P, yields :

CV Ω • ∇I P η (Ω)dV = - CV κ P η I P η (Ω)dV + CV κ P η I P bη dV (3.2)
Applying the Green theorem to the left hand side of this expression with the assumption that the variables on the right hand side remains constant within the control volume, gives:

A Ω • nI P η (Ω) = -Vκ P η I P η (Ω) + Vκ P η I P bη (3.3)
where A and V denote, respectively, the surface and the volume of the elementary cell while n is the outer unit vector normal to a cell face.

Approximating the integral over the boundaries of the control volume by a discrete sum, yields:

F ∑ f =1 Ω • n f I f η (Ω)A f = -Vκ P η I P η (Ω) + Vκ P η I P bη (3.4)
Here, f denotes a cell face with its area A f , F is the total number of cell faces of the

F ∑ f =1 (Ω•n f >0) Ω • n f A f + κ P η V I P η (Ω) = F ∑ f =1 (Ω•n f <0) |Ω • n f |I U η A f + Vκ P η I P bη (3.5)
and solving this equation makes the whole intensity field -attached to one given direction -available at all the grid points. To further obtain the discrete values of incident radiation and radiative flux, each nodal intensity must be integrated over all the directions in space (4π sr). The DOM, in this situation, replaces the angular integrals by a summation over a set of discrete directions such as:

4π 0 f (Ω)dΩ ≈ M ∑ m=1 ω m f (Ω m ) (3.6)
where M denotes the number of directions in the set and ω m is the weight attributed to the m th element. Consequently, the distribution of incident radiation and radiative flux are approximated by:

G P η = 4π 0 I P η (Ω)dΩ ≈ M ∑ m=1 ω m I P η (Ω m ) (3.7)
q P m = 4π 0 I P η (Ω)ΩdΩ ≈ M ∑ m=1 ω m I P η (Ω m )Ω m (3.8)
There are several ways to define the discrete direction sets. We will here focus on the quadrature available in Code Saturne and the ones we have introduced to improve this part of the code. A more general overview on this problem can be found in Koch and Becker [START_REF] Koch | Evaluation of quadrature schemes for the discrete ordinates method[END_REF].

• The first (and more classic) set is the Level Symmetric quadrature S N . It was introduced for radiative transfer by Lee [START_REF] Lee | The discrete Sn approximation to transport theory[END_REF]. Different sets of directions and weights were later introduced, for example, by Lathrop and Carlson [START_REF] Lathrop | Discrete ordinates angular quadrature of the neutron transport equation[END_REF] and Fiveland [START_REF] Fiveland | The selection of discrete ordinate quadrature sets for anisotropic scattering[END_REF]. The principle of this quadrature scheme is to apply strict rules of symmetry in order to equally treat all the directions of propagation. These rules prescribe that, if Ω m , characterized by the direction cosines value (µ m ,η m ,ξ m ) is in the quadrature:

i. all the direction (±µ m ,±η m ,±ξ m ) are also in the discrete ordinates set and have the same weight.

ii. all the directions coming from a permutation of (µ m ,η m ,ξ m ) are also in the quadrature and are assigned the same weight.

However, the customary choice of discrete ordinates has to preserve the zeroth, first and second moments of integration of the intensity over the whole direction range (Modest [59]). Fiveland [START_REF] Fiveland | Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media[END_REF] and Truelove [START_REF] Truelove | Discrete-ordinate solutions of the radiation transport equation[END_REF] have studied different direction sets and they pointed out the requirement of satisfying, in Chapter 3. Code Saturne and Radiative Calculation addition, the first moments over a half range of direction to avoid biasing the evaluation of wall fluxes.

• The T N quadrature set is due to Thurgood [START_REF] Thurgood | A critical evaluation of the discrete ordinates method using HEART and T (N) quadrature[END_REF]. The basic idea of this scheme is to divide the unit sphere into spherical triangles. For each triangle, an associated direction is defined from the center of the sphere to the center of the triangle region. The quadrature weight are calculated according to the spherical triangle areas. The order N refers to the number of subdivisions of the edges of the principal triangle.

• The LC -11 quadrature set. Based on the fundamentals of Sobolev [START_REF] Sobolev | On mechanical quadrature formulae on the surface of a sphere[END_REF], Lebedev [START_REF] V I Lebedev | Quadratures on a sphere[END_REF] proposed quadrature schemes which are rotational invariant to the group of regular polyhedrons and are capable to exactly integrate the spherical harmonics functions Y n (Ω) up to the order of N on the unit sphere (N = 11 for the LC-11 quadrature). Quadratures of that type are developed so that the spherical harmonics of order N = 11, 15 can be integrated exactly. All the weights are identical while the directions can be adjusted.

• The DCT -020 -2468 quadrature set is one of the DCT (Double Cyclic Triangles) quadrature schemes. It has been designed to provide as many degrees of freedom as possible for satisfying additional moment conditions (Koch et al. [START_REF] Koch | Discrete ordinates quadrature schemes for multidimensional radiative transfer[END_REF]), (Koch and Becker [START_REF] Koch | Evaluation of quadrature schemes for the discrete ordinates method[END_REF]). The naming convention of these quadrature schemes is DCTxyzabcd... where x, y, z denote respectively the number of non-degenerated tuples 3 , the number of single degenerated tuples and the number of double degenerated tuples; abcd... denotes the moment conditions that are satisfied by the quadrature.

Encountered Difficulties and Applied Modifications

The S N quadrature set implemented in the version 5.0.4 of Code Saturne comes from the selection of Lathrop and Carlson [START_REF] Lathrop | Discrete ordinates angular quadrature of the neutron transport equation[END_REF] and Fiveland [START_REF] Fiveland | Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures[END_REF]. We have assessed the accuracy of these implementations by comparisons in a test case where the exact (analytical) solution is available. We have also input in the code more recent values of the S 8 and S 12 sets based on the modifications proposed by Balsara [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[END_REF].

As a benchmark, we have considered the radiative equilibrium in a 2-D square cavity of size 1m × 1m (see Figure 3.2). The walls are black, one of them (hot wall) is heated up to 1000K while the others are uniformly set at 500K (cold walls). The medium inside the cavity is homogeneous and gray. The evaluation criterion used here is the distribution of the radiative net flux on the cold wall facing the hot wall. It is known to be the most sensitive result. We have also checked the integral value of this flux over the entire wall. The reference solution is taken from Crosbie and Schrenker [START_REF] Schrenker | Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation[END_REF] who have provided a semi-analytical solution to this problem. The calculations are run with Code Saturne and, for sake of comparison, with a homemade code using the discrete ordinates in 2-D geometries. Two different values of the gray absorption coefficient were considered κ = 0.25m -1 and κ = 1.0m -1 . The results and comparisons are illustrated in the figures below (3.3 and 3.4).

It appears that the S 8 quadrature implemented in Code Saturne leads to erroneous results (at least, in the version we have used). Indeed, none of the calculations returns the correct reference distribution, but this is a well-known bias of the DOM named "ray-effect". It is due to the discrete representation of the direction space, and these errors are attached to the choice of the quadrature set. However, it is known that the integrated flux (over the entire wall) is preserved when using the S N data: therefore, our results must match the exact (reference) value. This is true for our S N set (based on Balsara data), both for the homemade code or after implementation in Code Saturne (see table 3.1 and 3.2). Conversely, the built-in S 8 quadrature in Code Saturne returns erroneous values. This is probably due to some inaccuracies when inputting the related data in the code. Note that:

-The other quadrature (S 4 , S 6 , ...) implemented in Code Saturne give satisfactory results.

- These comparisons led us to choose the quadratures of Balsara [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[END_REF] in all our calculations (mainly S 8 ).

Implementation of SLW model

Dealing with a non-gray gas mixture requires a spectral model to account for the variation of the absorption coefficient in the medium. To that end, we have implanted the SLW model, which is described in the previous chapter, in the radiative module of Code Saturne. This implementation is presented in the diagram below: 2. The choice of the number of gray gases N g (user input) and the tabulated data of the ALBDF are then used for determining the local radiative properties: the absorption coefficient and the corresponding weights for each gray gas.

3. The radiative properties are transferred to the module solving the radiative transfer equation using the Discrete Ordinate method. The solution is repeated for each gray gas and the integrations are performed over all directions to get the incident radiation and radiant flux. Then, these quantities are summed up over all the gray gases and are assembled to get the radiative source (divergence of the total radiative flux).

4. This radiative source is then injected into the right hand side of the energy equation.

In more details, for implanting SLW model into the built-in radiative module of Code Saturne (which contains tens of source files and hundreds of subroutines), we firstly had to find out the data-flow between these files for determining the location where our own model can be inserted. Next, the extraction of the inputs (from the conservation equations) had to be performed. Besides, a new library was created to store all the new variables and information related to SLW model. Finally, the subroutines for calculating the radiative properties had to be added. The highest difficulty of this implantation was to insert our own data without breaking the coherence of the built-in module. More details on the implementation are given in appendix B .

Convergence criteria

Code Saturne has no criteria to determine the convergence of the solution in term of variable residuals. Instead, it is advised to monitor the time evolution of the considered variables at different positions in the flow field to decide whether the calculation reaches a steady state (EDF [START_REF] Edf | Code_Saturne version 5.0.0 practical user's guide[END_REF]). The user can stop the computation whenever he finds the results stable enough or the code will run until it attains the declared number of time-steps. Besides, the conservations of the heat and mass flux can be a criterion for the consideration of a steady solution where the total fluxes arriving at the cavity walls have to be the same as those coming from the obstacle surfaces.

In our study, the convergence of the solution is qualitatively monitored by observing the temporal evolutions of the variables (temperature, concentration, velocity components) at different points inside the cavity. Quantitatively, the convergence is evaluated by the balance of the heat exchange between the hot source and the cavity active walls, which is written as:

| ∑ f ∈F 1 Nu t f S f -∑ f ∈F 2 Nu t f S f | ∑ f ∈F 2 Nu t f S f ≤ 10 -3 (3.9)
where F 1 and F 2 are the active walls of the enclosure and the hot source respectively, S f being the surface area.

Code Validation

Differentially heated cavity

In this section, Code Saturne and its improved radiative model is validated by the simulations of both pure thermal convection and double diffusive convection coupled with radiation in a differentially heated cavity (see figure 3.6). Regarding 3-D cases, we have performed calculations in the configurations studied by Colomer et al. [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF] and Fusegi and Hyun [START_REF] Fusegi | Laminar and transitional natural convection in an enclosure with complex and realistic conditions[END_REF]. An uniform 81 3 grid has been used. The computations have been run at Pr = 0.71. A good agreement is observed between our results and these references (see table 3

Pure Thermal Convection coupled with radiation

We now present the comparisons between our computations and those of Yücel, Acharya, and Williams [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF] and Laouar-Meftah [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF] for coupled thermal convection and radiation in a gray gas inside a 2D differentially heated square cavity. In [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF], the authors have used a non-uniform 50 2 grid (but they did not specify which type) and performed the calculations at different values of the overall optical thickness τ = κ • L (L is the cavity size and κ the gray absorption coefficient). The Rayleigh number was fixed at 5 • 10 6 with Pr = 0.72 and the Planck number defined as Pl = λ 4σLT 3 0 is set to Pl = 0.02. The dimensionless temperature θ 0 defined as T 0 /∆T was 1.5 while the emissivities ( i ) of the bounding walls were set at 1. In our work, a non-uniform 81 2 grid like in reference [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF] but with a different function for nodal distribution has been used: it is a tangent hyperbolic function in [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF] while we used a cosine hyperbolic one. The results are listed in table 3.5. They show that the difference between our calculations and the reference does not exceed 4 %. This difference may come from the different interpolation schemes used in the DOM (Lathrop scheme in [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF], STEP scheme in our study) For coupling thermal convection and radiation in a real gas (that is, accounting for the real absorption spectrum of the medium), we have performed different validation tests in a 3-D differentially heated cavity. We have re-produced the works of Billaud, Saury, and Lemonnier [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] by considering a cubic enclosure filled with humid air (air -H 2 O mixture). Different case studies based on the radiative behavior of the bounding walls and the medium are performed. They are described in the table 3.6. Our calculations have been carried out using a non-uniform 91 3 grid like in reference [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF]. Our results are also compared with the data obtained by Soucasse, Rivière, and Soufiani [START_REF] Soucasse | Natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation at Rayleigh numbers up to 3× 109[END_REF] and presented in table 3.7. The comparisons show a fairly good agreement between our predictions and the two references.

Chapter

Double diffusive convection without radiation

We now simulate a double diffusive convection flow in a cubic enclosure with opposing temperature and concentration gradients. The calculations are performed at Ra = 10 7 , Le = 1, Pr = 0.71 and different mass-to-thermal buoyancy ratios. No radiation is included at this stage, neither from the fluid (transparent), nor from the surfaces. The results are compared to the works of Sezai and Mohamad [START_REF] Sezai | Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients[END_REF].

N

Code Validation

39

Calculations were performed over a grid of 80 3 uniform cells. The comparisons in table 3.8 shows a good agreement with the results of the reference [START_REF] Sezai | Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients[END_REF]: the maximum difference in Nusselt number is less than 0.5%.

Double diffusive convection coupled with radiation

We now address the coupling of double diffusive convection and radiation in a real gas whose radiative properties are evaluated by SLW model. The comparisons of the Nusselt and Sherwood numbers between our results and the work by Cherifi [START_REF] Cherifi | Modelisasion de la convection de double diffusio en presence de rayonnement dans une cavite 3D en regime instationaire[END_REF] are performed and reported below for air -CO 2 and air -H 2 O mixtures at different concentration x. A non-uniform grid generated from a cosine function was used in the calculations of the reference [START_REF] Cherifi | Modelisasion de la convection de double diffusio en presence de rayonnement dans une cavite 3D en regime instationaire[END_REF]. In our computations, the mesh is 81 × 81 × 81 with a nonuniform distribution along each edge whose density is given by a hyperbolic tangent function.

x CO 2 Our work Cherifi [START_REF] Cherifi | Modelisasion de la convection de double diffusio en presence de rayonnement dans une cavite 3D en regime instationaire[END_REF] In all cases, our results coincide with the reference within a tolerance of 1.5%.

Cavity with a hot obstacle located inside

Our purpose is now to test the ability of our simulation code to handle geometries involving obstacles. The first validation test considers a 2-D square cavity with a heat source located at the center of the bottom wall (see figure 3.7). The enclosure has the dimension of 0.05 × 0.05 m 2 . The heat source is 0.01m wide and 0.025m high. The enclosure is filled by dry air (transparent) and the Prandtl number is set to 0.71. The heat source is maintained at T h = 301.16K while two lateral walls of cavity are prescribed at T c = 291.16K. The remaining surfaces of the enclosure are adiabatic.

The tests are performed with different Rayleigh numbers, using a 100 3 uniform grid and the results are compared to the experiments and the simulations of Paroncini and Corvaro [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF]. Ra Reference [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF] (exp) Reference [START_REF] Paroncini | Natural convection in a square enclosure with a hot source[END_REF] (num) Our work The comparisons show a good agreement with the reference, the maximum difference being less than 1% when comparing the two numerical simulations. Difference are larger with respect to measured quantities (but this is also true in the reference work). Some thermal leakage through the plexiglas plates (imperfect insulation) may explain these variations. The second validation test considers a hot obstacle located at the center of a 2-D square cavity (see figure 3.8). Now, two vertical walls of the cavity are kept adiabatic while the horizontal one are maintained at a constant lower temperature compared to the isothermal heated obstacle. The enclosure is filled with dry air. The simulation is run at a Rayleigh number of Ra = 2 • 10 5 and a Prandlt number of Pr = 0.71. The comparisons with the results of Sun, Chénier, and Lauriat [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF] reported in table 3.14 point out that our code can efficiently handle this type of configuration, both for the flow description and thermal transport (the maximum difference is less than 0.5%).

Our work Reference [START_REF] Sun | Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body[END_REF]

Conclusion

In this section, we have described the Discrete Ordinates Method and the improvements we have brought to its implementation in Code Saturne. We have also detailed the introduction of the SLW model in this code.

Different validation tests in differentially heated cavity (thermal or double diffusive convection with/without radiation) as well as the configuration of the cavity with an obstacle located inside have been performed. The comparisons between our predictions and different references point out that:

• Code Saturne can accurately handle the calculation of thermal convection or double diffusive convection.

• The built-in radiative module of Code Saturne is reliable after the correction of some flaws that are present in version 5.0.4.

• Our implemented SLW model is able to predict the radiation effects with a satisfactory accuracy in configuration involving real gases.

Chapter 4

Coupling between Pure Thermal Convection and Radiation

A first set of results concerns pure thermal convection. In this configuration, the fluid is homogeneous in composition and the flow is only governed by the temperature gradients.

Introduction

We firstly present the cross section planes and the crosslines used to display the results. The dimensionless temperature, concentration and velocity fields are plotted in the median vertical plane of the cavity (Y = 0.5 or y = 0.125m) (see figure 4.1). We also consider the profiles of these quantities along different crosslines in the plane Y = 0.5:

• Z-lines: Z = 0.1, Z = 0.5, Z = 0.8

• X-lines: X = 0.2, X = 0.5, X = 0.8 (where α = λ ρ 0 C p is the thermal diffusivity).

Convergence on spatial grid

Before conducting original simulations, we have analyzed the convergence of results with respect to the spatial meshing, the angular discretization and the number of gray gases (spectral divisions) in the SLW model. We have run different tests, but we only present here those concerning a case of natural convection at Ra = 5 • 10 6 , Pr = 0.71, T 0 = 555K and ∆T = 50K with black active walls and purely reflective adiabatic walls. Three uniform grids of different size have been considered: 80 × 80 × 80 (80 3 ), 100 × 100 × 100 (100 3 ) and 120 × 120 × 120 (120 3 ). Mesh u max (m/s) w max (m/s) 80 3 0.191 0.339 100 3 0.191 0.342 120 3 0.191 0.342 Mesh Front wall Back wall Left wall Right wall 80 3 1.690 1.690 1.690 1.690 100 3 1.682 1.682 1.682 1.682 120 3 1.681 1.681 1.681 1.681 From the result displayed above, we consider that the simulations are converged with respect to the spatial meshing when using a 100 × 100 × 100 grid. Therefore, we have selected this mesh for our subsequent calculations.

Convergence on angular discretization

When radiation is present, the sensitivity of the results to the angular discretization needs to be assessed. To that end, we consider the coupling of natural convection and radiation in a gray gas at Ra = 5 • 10 6 , Pr = 0.71 and θ 0 1 = 11.1 with an optical thickness of τ = 0.5. Different orders of the level symmetric quadrature S N are tested: S 6 , S 8 , S 12 . The corresponding results in terms of temperature and velocity are displayed in figures. 4.3-4.4 and in table 4.2.

1 θ 0 is defined as T 0 T h -T c . This dimensionless parameter relates the absolute temperature T 0 (which governs the radiation problem) to the temperature difference (which drives the convection motion) The comparisons presented above confirm that, with a S 8 quadrature set only (80 directions), we can obtain the results with a good accuracy. This was confirmed for other optical thicknesses ranging from 0.1 to 2. Consequently, we have selected this discrete directions set for our following calculations.

Convergence on the number of gray gases for the SLW model

The tests are performed for the case of coupling natural convection and radiation in a real (non gray) gas mixture air -H 2 O at concentration x H 2 O = 0.20. For the radiative calculations, we have used the rank-correlated approach, as described in Chapter 2. The results are displayed below (figure 4.5 and table 4. The comparisons show that using only 5 gray gases in the SLW model is enough, in this type of problem, to achieve an acceptable level of accuracy within a very affordable computational cost. Concretely, the simulations are run on one core of Intel(R) Xeon(R) CPU E5-2620 @ 2GHz. With 11 gases, it needs 182 hours of CPU time while with 5 gases, it reduces to 103 hours (56 % compared to the 11 gas model) to reach the same physical simulation time.

Coupling with radiation in the gray gas assumption

In this part, we analyze the effect of radiation on the flow structure and heat transfer in a simple manner, by assuming that the medium filling the cavity is gray and has uniform radiative properties. This assumption allow us to study adimensionally the effect of gas radiation on the flow structure and heat transfer via the nondimensional optical thickness. This parameter is defined as τ = κL related to the cavity size, L and the absorption coefficient κ. Different fluid opacities may be considered by changing the optical thickness values.

All calculations were performed at Ra = 5 • 10 6 and Pr = 0.71 and θ 0 = 11.1. The emissivity of the hot and cold walls is 1 and of the adiabatic surface is 0.

As the adiabatic walls are assumed to be purely reflective, there is no radiativeconvective coupling when τ = 0 (transparent medium). This limiting case serves as the reference for determining the radiation effects on the flow field. It is observed in figure 4.6 a nearly vertical stratification in absence of radiation on both sides of the cavity. But this distribution is broken when the radiative effects are introduced. The hot surfaces of the obstacle radiate toward the absorbing fluid between the heater and the cold wall of the enclosure (see figure 4.7 a). At medium and high levels (Z = 0.5, 0.8) (see figure 4.7 b,c), radiation cools down the fluid, since the gas in this region emits more than it absorbs. This is demonstrated by the negative values of the radiative source term (see figure 4.9 b,d,f). And the higher optical thickness, the stronger the radiation effects are. Overall, the radiative transport levels the temperature field in the cavity, at least in the range of opacities we have investigated. It is shown that, as the opacity increases, the fluid is more accelerated, inside the plume and next to the cavity surfaces. It is also observed the broadening of the plume and of the vertical boundary layers near the cavity walls when the radiation effects is considered (compared to the transparent cases). Besides, we observe that the plume at τ = 0.1 and τ = 0.2 keeps the same cone shape. However, as the τ = 0.5, the structure of the plume changes: it is compressed at medium altitude and takes the form of an hourglass. The reason is that, when the optical thickness increases, the temperature in the lower part of the cavity increases (see figure 4 Regarding the flow structure, there are three spirals, which are associated to low velocities and cannot be found in the vector field above, on each side of the cavity (see figure 4.11 a) when the medium is transparent. However, when the medium participates to radiation, the number of spirals decreases (see figure 4. 11 b,c,d). The profiles of vertical velocities at different Z-levels (figure 4.12) reveal that the fluid that was stagnant in the transparent case is now moving. As a consequence, the ascending plume and the descending movement along the cavity walls interfere and create a shear flow. Besides, we also observe the broadening of the obstacle vertical boundary layers (figure 4.12 a). As the optical thickness increases, this effect is accentuated and at τ = 0.5, these boundary layers broaden enough to reach the cavity lateral walls and even block the downward flow (slow down and redirect). In addition, the mass transport driven by the plume is increased, causing an acceleration of the return stream along the ceiling (mainly) and the floor (to a lesser extent) (see figure 4.13 a). For further analyzing the flow structure, we investigate the Q-criterion3 , which is used as a method to identify the swirl zones within the fluid (where Q > 0). Q < 0 stands for the regions where the deformation dominates over the rotation. Concerning the 3-D structures, we observe that the swirl takes place in the area between the plume flow and its re-circulation along the cavity walls. Looking closely at figure 4.14, which represents the iso-surfaces at Q = 0.024 , we denote an expansion of the surfaces as the opacity increases. This means that the trend of auto-rotation dominates over deformation, becoming a global effect rather than a local phenomenon as it can be seen in figure 4.14 a. This explains the disappearance of the spiral flows. We have also considered the distribution of negative values of Q plotted in the median plane (Y = 0.5) (see figure 4.15). The differences observed in the patterns of the Q = -0.02 iso-value lines for different fluid opacities result from the interference of the upward plume and downward boundary layers where, literally, the deformation dominates over rotation.

Steady flows

Unsteady flows

The two cases τ = 1 and τ = 2 are now considered. The case τ = 1 leads to an unsteady laminar flow (see figure 4.17 a), while the other produces turbulent results (see figure 4.17 b). The turbulent behavior can also be demonstrated by the observation of the spectrum analysis where we do not found any dominant frequency (see figure 4.18). We only investigate here the case at τ = 1. At point P 2 (see figure 4.20), the Fourier analysis yields a fundamental frequency f = 0.028 and its two harmonics f 2 = 2 f and f 3 = 3 f . On the other hand, at point P 1 (see figure 4.19), P 3 (see figure 4.21) and P 4 (see figure 4.22), it reduces to f 1 = 0.014. However, the frequency of 0.028 is present at all the considered positions, even when it is not the fundamental one. It may refer to a global phenomenon while other identified frequencies belong to the local fluctuations. The figures 4.23 show the flow lines at different instants over one oscillation period recorded at P 3 . It is observed the appearance and vanishing of the two small eddies right above the upper surface of the obstacle. This process is periodically repeated with a frequency that is exactly the fundamental one f 1 returned by the FFT. We, therefore, may conclude that this phenomenon drives the fluctuations of temperature considered at P 3 . A similar conclusion was reported by Souayeh et al. [START_REF] Souayeh | Prediction of unsteady natural convection within a square cavity containing an obstacle at high Rayleigh number value[END_REF] for the results in a square cavity with a hot obstacle located on its floor. In addition, Bouafia and Daube [START_REF] Bouafia | Natural convection for large temperature gradients around a square solid body within a rectangular cavity[END_REF] have pointed out that this type of unsteadiness is due to the shear instabilities, which occur in the zones of high velocity gradient within the primary flow.

(a) t 0 (b) t 0 + 0.15T 1 (c) t 0 + 0.35T 1 (d) t 0 + 0.43T 1 (e) t 0 + 0.50T 1 (f) t 0 + 0.68T 1 (g) t 0 + 0.83T 1 (h) t 0 + 0.93T 1 (i) t 0 + T 1

Chapter 4. Coupling between Pure Thermal Convection and Radiation

However, differing from these two references, in our results, it is also observed the deformation of two large flow cells on both sides of the plume and, in addition, the appearance and vanishing of the small vortices near the vertical walls of the cavity (below the large cells). This alteration may explain the local frequencies found in the signal recorded at P 1 , P 3 , P 4 . Figure 4.24 displays the Q distribution recorded at P 3 in the mid-depth plane (Y = 0.5) with six iso-values at different instants within one period. We here consider the evolution of the rate of shear strain which is represented by the negative values of Q. It is observed the formation and the deformation of the bubble zones where the rate of strain dominates over the vorticity. This demonstrates the shear instability which occurs due to the interference of a continuous fluid at two velocities. In addition, this process repeats periodically with exactly the fundamental frequency that is found from Fourier Transform process. For further discussing, we observe that in figure 4.25 there are not any core horizontally stratification of the fluid density (equivalently represented by temperature stratification), which following Le Quéré and Behnia [START_REF] Quéré | From onset of unsteadiness to chaos in a differentially heated square cavity[END_REF] is the structure that can sustain the internal gravity waves (with the fundamental frequency f is smaller than the cut-off Brunt-Väisälä one f BV ). We therefore conclude that the fluctuation propagating inside the cavity cannot be related to either the internal gravity waves or the traveling waves (where f > f BV ).

(a) t 0 (b) t 0 + 0.15T 1 (c) t 0 + 0.35T 1 (d) t 0 + 0.43T 1 (e) t 0 + 0.50T 1 (f) t 0 + 0.68T 1 1 (g) t 0 + 0.83T 1 (h) t 0 + 0.93T 1 (i) t 0 + T 1
As a conclusion, the radiation with an opacity at τ = 1 has changed the distribution of the temperature field in the cavity. This alters the dynamic field which experiences shear instabilities in the zones of high velocity gradient. It induces the deformations of fluid circulations in the areas right above the hot obstacle, next to the vertical walls of the enclosure and between the upward plume and downward boundary layers flow, which are repeated periodically. Furthermore, increasing the fluid opacity (τ = 2) leads to a turbulent flow.

Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture

We now move from a fictitious gas to a real air -H 2 O mixture. It involves an absorbing-emitting component (H 2 O) diluted at different concentrations into a transparent gas (dry air). The actual absorption spectrum of water vapor must be considered to allow realistic simulations. To that end, and following the discussion presented in Chapter 2, we resort to the SLW model associated to the rank correlated approach. Concerning the boundary conditions, the obstacle surfaces are considered as black ( = 1) and prescribed at T h = 580K5 , the vertical walls are also black ( = 1) and uniformly maintained at T c = 530K, while the ceiling and the floor are assumed perfectly reflective ( = 0) and adiabatic. In the range of Rayleigh numbers under consideration, the flow remains laminar and it always reaches a steady state. Unlike with the gray gas assumption, the radiative properties now depends on the local temperature inside the cavity. Therefore, the absorption coefficient is not uniform and thus we cannot use a single (gray) non-dimensional number τ to fully characterize each configuration.

In each case, the results are compared to the reference (transparent) values, which are generated using the same gas mixture, but without including any radiation effects in volume. b,d,f). The radiant flux coming from the hot surfaces of the obstacle heats up the medium between the heater and the vertical walls of the cavity. As a result, the nearly vertical thermal stratification in absence of radiation is broken. The air (outside the thermal plume), therefore, slightly becomes more uniform in temperature. As a whole, radiation does not significantly change the thermal profile along the centerline (Z = 0.5) (see figures 4.27 d,e,f) and in the lower half of the cavity (Z = 0.1) (see figures 4.27 g,h,i). But at a higher position (Z = 0.8) (see figures 4.27 a,b,c), a slight decrease of temperature in the re-circulation zone between the plume and the wall boundary layer can now be observed. In this region of low convective transport, the gas radiates toward the cold surfaces and the colder parts of the fluid (it emits more than it absorbs): this results in a negative radiative source within the fluid (see figures 4.28 a,c,e). Similar modifications in dynamic and thermal field were reported by Billaud, Saury, and Lemonnier [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] for a differentially heated cavity: gas radiation was found to accelerate the global circulation and to set into motion some parts of the fluid that were stagnant in the transparent case.

Velocity and thermal fields

Chapter 4. Coupling between Pure Thermal Convection and Radiation

Regarding the 3-D structure of the flow, figures 4.33 represent the distributions of the Q inside the cavity. As in the previous section (gray gas), we observe that the iso-surfaces Q = 0.02 extends when the fluid opacity -linked to the concentration of water vapor -increases. The spirals observed in the figures 4.32 now disappear gradually as the medium becomes more absorbing. The plots in the figure 4.34 clearly show that the convective Nusselt number along the centerline of any vertical wall of the enclosure is decreased close to the roof of the cavity and increased elsewhere. The reason is that, in the upper part, the fluid was cooled down (figures 4.28 a,c,e) before reaching the lateral walls. On the other hand, in the lower part, absorption dominates over the emission (figures 4.28 b,d,f), thus, the medium is heated up and the thermal gradient is increased. However, the increased part does not compensate the decreased one. Therefore, overall, the average convective Nusselt number over the vertical wall of the enclosure is reduced (see table 4

Heat Transfer

.7).

This decreasing trend in the local convective Nusselt number is also found when considering the vertical hot surfaces of the obstacle. This is illustrated in figure 4. [START_REF] Cherifi | Modelisasion de la convection de double diffusio en presence de rayonnement dans une cavite 3D en regime instationaire[END_REF]. The reason is that the thermal gradient is reduced when radiation is considered. Indeed, the fluid is warmed up (by absorption) along the floor of the enclosure before arriving at the vertical hot surfaces of the obstacle. Consequently, the average values of Nu C displayed in table 4.9 are decreased when the radiation is taken into account. However, it is observed that the convective transfer along the horizontal upper surface of the obstacle increases. When radiation is present, the fluid layer near the hot surfaces becomes cooler due to emission (it is evidenced by the negative values of the radiative source in the figures 4.28. Consequently, the thermal gradient in this region is enhanced, which induces the increase in local values of Nusselt number with the appearance of the radiation (figure 4.36). This explains the higher average convective Nusselt number obtained in table 4.9. The values of average convective Nusselt number and average total Nusselt number, which are defined as Nu = 1

A A Nu(s)dA, are presented in the tables below: Regarding the total heat transfer, tables 4.7, 4.8 and 4.9 reveal that, compared to the transparent cases, the average total Nusselt number along any black surfaces is decreased. This, along the vertical walls, is due to the drop off in the convective Nusselt number. In addition, the attenuation of radiative transfer by the absorbing medium also contributes to the reduction of total thermal transport, especially, when considering this quantity along the upper surfaces of the obstacle, where the convective transfer is accelerated.

x H 2 O (%)
x H 2 O (%)

Conclusions

In this chapter, we have presented our calculation of pure thermal convection coupled to gas radiation in a cavity (size L = 0.25m) containing a hot source (size l = 0.05m). Firstly, the sensitivities of the results with respect to spatial discretization, angular discretization and spectral divisions have been studied in order to select the most suitable computation settings for subsequent calculations. In more details, the simulations have been performed using a 100 3 uniform grid, the S 8 quadrature for the discrete direction set and the SLW model (associated with RC approach) with 5 gray gases.

Next, the computations on the thermal convection in the enclosure filled by gray gas (at different opacities) or real gas (SLW model) (at different molar fraction of water vapor) mixtures have been investigated. The comparisons between obtained results and the reference pointed out the influences of radiation effects:

• Radiation tends to non-uniformly accelerate the boundary layers along the cavity wall and the hot inner obstacle. It also makes moving some parts of the fluid that were stagnant in the transparent cases. The plume flow and its recirculation interfere and create shear flow patterns.

• Radiation partly modifies the thermal gradient near the bounding surfaces of the cavity: the convective Nusselt values are increased in the upper half and decreased in the lower half. However, at the horizontal surface of the obstacle, this thermal gradient is strengthened.

• Radiation modifies the nearly vertical thermal stratification outside the plume and slightly uniformizes the medium temperature.

• In real gases, radiation reduces the total thermal transfer, especially the convective part on the vertical walls and the radiative exchange between the upper surface of the obstacle and the ones of the cavity.

• All these effects increase when the medium becomes more absorbing (in the considered range of optical thicknesses and molar fraction).

• Besides, in the case of a gray gas, when the optical thickness of the medium is unity, radiation makes the flow depart from the steadiness and reach a unsteady regime. This mechanism is due to the shear instability created by the interference of ascending plume and the boundary layers flowing downward.

It is worth mentioning that, at τ = 2, the flow becomes totally turbulent at Ra = 5 • 10 6 .

Chapter 5

Double Diffusive Convection Coupled to Gas Radiation

In double diffusive convection, there are two gradients that govern the flow: the thermal and the concentration ones. The relative magnitude of the induced driving forces is defined by the mass-to-thermal buoyancy ratio N:

N = β C ∆C β T ∆T (5.1)
where

∆C = C h -C l , ∆T = T h -T c .
Its sign characterizes the cooperating (> 0) or opposing (< 0) effects of these two gradients.

In the frame of this work, we have performed some calculations in double diffusive convection including cases where gas radiation is accounted for. Predictions without radiation (transparent fluid) are also provided at different mass-to-thermal buoyancy ratios and serve as reference for highlighting the influence of radiant transport on the flow characteristics.

Gray gas model

All the calculations are performed at Ra = 5.10 6 , Le = 1 (allowing a perfect overlap of the thermal and concentration boundary layers), Pl = 4.43 • 10 -3 and θ 0 = 11.1. A high concentration of the absorbing species is prescribed on all the surfaces of the obstacle (C h ), and a null concentration (C l = 0) along the vertical walls of the cavity. The emissivity of the bounding surfaces (including the obstacle) are set to unity except the ceiling and the floor, which are considered as perfectly reflecting. In all subsequent computations, the absorption coefficient is made proportional to the local concentration. We will therefore consider that :

κ(C * ) = κ 0 C * C re f (5.2)
where C * is a dimensional value and κ 0 and

C re f = C h +C l 2
are the reference absorption coefficient and reference concentration, respectively. Using the dimensionless quantity C = (C * -C re f )/(C h -C l ) in (5.2) yields:

κ(C) = κ 0 • (2C + 1) (5.3)
All the results presented hereafter are normalized using the reference temperature We assume that the fluid supplied by the (hot) obstacle and removed by the (cold) cavity walls is lighter than the main component of the mixture. Differences in concentration then create a convective motion in the same direction as those induced by the thermal gradient (cooperative action) and the magnitude of these two effects (mass and thermal) are similar (N = 1 and Le = 1). At low value of τ (optically thin limit), a nearly vertical stratification is established between the plume and the boundary layers along the cavity walls. No obvious change in this pattern occurs below τ = 0.2 or even τ = 0.5. That was already observed in the case of combined radiation and pure thermal convection (homogeneous mixture). As the optical thickness goes beyond unity, we clearly recognize the transformation of the thermal field. Moreover, the temperature profiles along different horizontal crosslines (see figure 5.2) prove that the temperature at medium (Z = 0.5) and high levels (Z = 0.8) decreases significantly when radiation is taken into account. The vertical temperature distributions plotted in figure 5.3 confirm this trend. The hot fluid carried by the plume emits towards the cold walls and, therefore, its temperature decreases with respect to the transparent case. The amplitude of this phenomenon, which is also illustrated by figures 5.4-5.8, increases with the optical thickness, at least in the range of values we have investigated 1 . On the other hand, the lower part of the cavity (Z = 0.1) is mainly driven by the two differentially heated surfaces (the obstacle and the cold walls): it is less sensitive to gas radiation. The velocity vectors in figure 5.9 illustrate the impacts of radiation on the fluid motion. There are no significant changes in the flow structure, but a slight broadening of the plume is observed at τ = 2. The horizontal profiles of vertical velocity in the median plane display more clearly the difference. At low altitude (Z = 0.1) (see figure 5.10 a), the fluid near the obstacle and the vertical walls of the cavity is accelerated while, at intermediate and high altitudes (Z = 0.5; 0.8) (see figures 5.10 b,c), the only evident change is the drop of the maximum velocity (the peak of the profiles). This is due to the decrease of temperature in the plume, which alters the buoyancy source. Besides, the profile of horizontal velocity along the vertical crossline (X = 0.2) (see figure 5.11 a) proves that the boundary layers near the floor and the ceiling of the enclosure are also accelerated. The circulation near the bottom brings cooled fluid from the cavity wall to the region between the obstacle and the enclosure (near the floor). The fluid is further heated up by the radiation coming from the hot cube. At a higher opacity, the radiative effects dominate over the convective transfer and the fluid, in this region, becomes warmer (but not as much as in the thermal convection case). Simultaneously, the medium close to the ceiling emits more than it absorbs, and, thus, becomes cooler. These two tendencies explain the typical distribution of the thermal field mentioned above. Figure 5.12 represents the iso-surfaces of value 0.01 of the Q inside the cavity. It can be observed that, when the optical thickness is one or more, several of these isosurfaces disappear. This illustration gives a better view of the changes experienced by the flow structure depending on the fluid opacity. Concerning the concentration field, no significant differences are found in the calculations with and without the radiative effects (see figure 5.13). This observation 5.1. Gray gas model 91 still holds when considering the concentration profiles along different cross lines (see figures 5.14 and 5.15). The reason is that gas radiation does not directly influence the concentration field, but only through the dynamic field and, as presented above, radiation does not significantly alter the flow structure. Compared to the reference results (transparent case), the flow field in participating medium at different opacities has no evident change when we consider the velocity vectors (see figure 5.16). It is observed in all the plots a motionless volume of fluid between the ascending plume and the descending boundary layers along the cavity walls. However, some differences are found when considering the dynamic profiles along the horizontal cross lines: a slight decrease of the maximum velocity at medium and high levels (Z = 0.5 and Z = 0.8) and a weak acceleration of the fluid near the cavity walls and the hot body surfaces (Z = 0.1). It can also be seen in the representation of the Q-criterion (see figure 5.19) that the flow structure is preserved at all the optical thicknesses, except a stretching of the iso-surface at mid-height of the enclosure (see figure 5.19). The reasons for this unchanged flow structure is that the mass gradient governs the fluid motion. Therefore, radiation, which primarily acts on the temperature field, has a weak impact on the dynamic field. Although the dynamic is not significantly altered, radiation, as usual, plays an important role in redistributing the thermal energy. Due to emission, the fluid at medium (Z = 0. There is no obvious change in the concentration field when the radiation effects are taken into account. C is stratified nearly vertically along the plume boundaries (see figure 5.28). The almost unchanged structure of its distribution is confirmed when plotting different profiles along the horizontal and vertical crosslines at various positions in the cavity (see figures 5.29 and 5.30). Figure 5.31 represents the velocity field in the mid-depth plane (Y = 0.5) of the cavity. Because radiation has broken the symmetry between the thermal and concentration fields, the fluid is now set into motion. Investigation of the velocity profiles (see figures 5.32 a,b,c and 5.33 a) reveals that, in the lower part of the cavity (Z = 0.1), the flow is dominated by the mass gradient. It descends along the obstacle (at high concentration) and climbs along the cavity walls (at low concentration). These two motions are well separated. In addition, there exists at all opacities a thermal boundary sublayer along the hot source vertical walls. On the other hand, a thermally driven motion prevails in the upper part of the cavity (Z = 0.8). We recognize an ascending plume in the center and descending boundary layers along the lateral cold walls. However, at τ = 1, an increase in concentration on the plume axis tends to limit the vertical movement and causes the plume to spread. From τ = 2, the flow dynamic is very attenuated (at the limit, if the opacity were infinite, one would retrieve the motionless solution of the transparent case). At intermediate levels (Z = 0.5), the situation is more complex. At low opacities (τ = 0.1, 0.2), there simultaneously exists a thermally driven flow (above the hot obstacle) and a mass governed one (ascending boundary layers along the cavity walls). At τ = 0.5 and 1, the flow is totally driven by the thermal gradient (ascending plume and descending boundary layers along the cold walls). Moreover, the thermal plume expands as the opacity increases from 0.1 to 1 (see figures 5.31 b-e). At τ = 1, the intensification of the mass force slows down the plume and tends to separate it into two parts (see figures 5.31 e and 5.32 b). At τ = 2, this separation is complete and the hot fluid is pushed toward the side walls. Near these surfaces, the fluid temperature is increased but it drops at the center of the cavity. This alters the thermal buoyancy force and thus, can explain the typical pattern of the velocity field. However, temperature and concentration act together on the flow. We therefore consider the dimensionless buoyancy source term3 plotted at different levels (figures 5.32 j,k,l). At Z = 0.5 we observe that, for τ = 2, this quantity raises up near the vertical walls and then drops in the center, which more clearly demonstrates the formation of the fluid movements. The increase of the Boussinesq source can be explained by a locally dominant thermal effect induced by radiative absorption (see figure 5.38 b for positive radiative source term). when compared to the transparent case, it is still decreased for all considered τ). This change is more obvious inside the plume. There, the fluid absorbs more than it emits (see figures 5.34 b, 5.35 b, 5.36 b and 5.37 b), in addition, the plume is spread up. These two processes warm up the medium. However, at τ = 2, this tendency is reversed. At Z = 0.8, temperature is decreased compared to the solution at τ = 1 (see figure 5.32 f). This is due to the damping of the plume motion. At Z = 0.5, thermal level is almost risen up except a drop off in the center. Here, the typical upward circulation brings more hot fluid toward side walls but the amount sent to the center of the enclosure is cut down(see figures 5.31 f and 5.32 b). The flow structure is also illustrated when considering the Q-criterion (see figure 5.42). New iso-surfaces of Q = 0.002 (for instance) appear and are transformed as the medium becomes more absorbing.

T re f = T h +T c 2 , the reference concentration C re f = C h +C l

Mass-to-Thermal Buoyancy ratio N = -2

In this section, we assume that N = -2. We therefore consider the case where the mass driven force dominates over the thermal one (|N| > 1). These two forces still act in opposite directions (N < 0). When the medium is transparent (see figure 5.43 a), the convective motion is fully driven by the concentration gradient. Near the hot obstacle, the fluid at high concentration is heavier and, therefore, flows downward. Close to the cavity vertical walls, the cold fluid at low concentration is pushed upward because the opposite trend induced by temperature is too weak. These two motions create a global clockwise circulation in the left part of the enclosure (and counterclockwise in the right). As the opacity increases up to τ = 0.5, the flow structure changes. In higher parts of the cavity (Z = 0.8), for τ between 0.1 and 0.2, the dynamic structure is preserved (ascending boundary layers near the vertical walls and descending flow in the center) except a slight slow down (see figure 5.44 c). From τ = 0.5, the fluid motion is increasingly dominated by the thermal gradient. And at τ = 1; 2, we clearly observe an ascending plume in the center and the descending movements near the cavity lateral walls. These two motions interfere and create a shear flow pattern. It can also be noted an acceleration of the fluid circulation near the ceiling (see figure 5.45 a). At Z = 0.5, the tendency is the same as at Z = 0.8 except that, at τ = 1, in the region close to the cold walls, there are upward motions instead of downward flows (see figure 5.44 b). This can be explained by the formation of new thermally driven flows above the obstacle. However, in the lower part of the cavity, there is no significant change except a slight slow down of the fluid near the bottom of the enclosure (see figure 5.45 a). The flow, in this region, is governed by the mass gradient for all the considered optical thicknesses. As the medium opacity increases, the transformations of the thermal patterns from nearly horizontal lines to inclined and vertical ones are clearly displayed and still denote the development of a thermal plume inside the cavity (see figure 5.46). In addition, the temperature profiles at different positions show that, at low level (Z = 0.1), the medium is less homogeneous in temperature: the fluid is more cooled down near the cavity walls and more warmed up around the obstacle (see figure 5.47 a). The fluid at intermediate and high altitude (Z = 0.5; 0.8) also gets warmer with the increase of the opacity (see figures 5.47 b,c). On the other hand, radiation has little effect in the upper part of the cavity (see figures 5. 49-5.53). However, at intermediate altitude, the radiative absorption dominates over the emission with the increase of the optical thickness (see figures 5.49 b-5.53 b): here, the fluid is warmed up (see figure 5.48 b for positive radiative source term). This increases the thermal gradient, which counters, then dominates over the concentration gradient and, finally, generates the thermal plume. This flow brings the fluid at high temperature to the layers near the ceiling and the vertical walls of the enclosure, which explain the increase of thermal level in these regions. There are no significant changes in the concentration field as the optical thickness remains below τ = 0.5 (see figures 5.54 and 5.55). At low level (Z = 0.1) the concentration increases near the obstacle surfaces and decreases near the vertical walls (see figure 5.55 a). This alteration is strengthened with the increases in opacity. However, an opposite trend is observed in the upper half of the cavity: the concentration is augmented near the cold surfaces as the medium becomes more absorbing (see figure 5.55 b,c). These changes come from the transformation in the flow structure.

The alterations in flow structures are also illustrated in figure 5.57, which displays the iso-surfaces of the Q-criterion. The formation and expansion of the new iso-surfaces are easily observed, showing the stronger impact of the radiative effects with the increase of opacity.

Synthesis

In this chapter, we analyze the influences of the radiation of a gas mixture on the double diffusive convection, using the gray gas assumption. The calculations were performed at different mass-to-thermal buoyancy ratios (N = -2; -1; 1; 2) (which describes the flow characteristics: cooperating or opposing and mass driven or thermally governed) and different values of the optical thickness defined as τ = κ 0 × L (τ = 0.1; 0.2; 0.5; 1; 2). The comparisons between the obtained results and the reference solutions (transparent case) reveal the transformations inside the enclosure which are summarized as:

• Cooperating flow At N = 1, introducing gas radiation does not significantly affect the concentration field. It slightly accelerates the boundary layers but reduces the maximum velocity inside the plume. Volume radiation tends to thermally homogenize the medium. It decreases the temperature in the upper half of the cavity and redistributes the iso thermal patterns (from nearly vertical stratification into horizontal one). Increasing the opacity of the medium strengthens these effects.

At N = 2, the same modifications take place but with weaker magnitudes.

• Opposing flow

At N = -1, for a transparent medium, no flow occurs in the cavity due to the perfect symmetry of the thermal and concentration gradients. But with radiation, this balance is broken and new fluid motions are established. The movement in the lower part of the cavity is dominated by the mass gradient for all the considered optical thicknesses but, at medium and high levels, the thermal gradient governs the flow. For τ between 0 and 1, radiation intensifies the thermal plume. However at τ = 1, the increase of concentration in the axis of the plume limits the vertical motion and causes it to spread. At τ = 2, these changes are strengthened, resulting in the plume separation with a zone of nearly motionless fluid in the center of the cavity. In addition, gas radiation reduces the temperature in the regions near the obstacle vertical surfaces. Above the source, temperature is generally reduced compared to transparent case, it increases with the opacity except in the center of the enclosure at τ = 2. The alteration of the concentration field are the same as for the thermal one except a slight increase of this quantity near the vertical cold walls in the lower part of the cavity.

At N = -2, gas radiation affects the dynamic, the thermal and the concentration field in the same manner as for N = -1 with the changes in optical thicknesses (in the participating mediums). However, the temperature and concentration fields are reduces at the intermediate and high levels when comparing to the transparent case.

Real gas mixtures

This section deals with the coupling of double diffusive convection and radiation in real gas mixtures (dry air and an absorbing-emitting component, which can be viewed as a pollutant). The physical properties of the fluid depend on its compositions and are calculated using the expressions detailed in appendix A. In this study, we consider either air -H 2 O or air -CO 2 mixtures. Depending on the added component, this leads to two types of flow: opposing or cooperating. Water vapor, which is lighter than air, creates a mass driven force acting in the same direction as the thermal one (cooperating case). On the other hand, the air -CO 2 mixture yields opposing flows caused by two counter-direction forces as the molar mass of CO 2 is larger than that of dry air. Regarding the boundary conditions, we recall that the cavity vertical walls are maintained at T c = 530K and C l = 0 while the obstacle surfaces are set to T h = 580K and a given concentration of pollutant C h . All the active walls are black and the adiabatic ones are purely reflective.

Air -H 2 O mixture

As presented above, this mixture generates a cooperating flow in the cavity. The characteristic non-dimensional numbers related to this configuration are listed in table 5.1. All the values are comparable to those of the gray case ( §5.1). However, the Lewis number is now about 0.8 and the mass boundary layers are then expected to be slightly thicker than the thermal ones. The average mole fraction of pollutant (here H 2 O) is defined as:

x = C re f * R * T re f P (5.4)
where R is universal gas constant and P is the pressure (here 1 atm). We recall that C re f = C h +C l 2 , or, when C l = 0 (as it is assumed here), C h /2. The x-parameter is given three different values 5%, 10% and 20%. It leads to three different types of flow: temperature driven (N < 1), balanced thermal and mass effects (N ≈ 1) and mass driven (N > 1). The distribution of the radiative source term (see figure 5.61) shows that radiation affects the fluid in the upper half of the cavity, inside the plume and in a limited region around the hot vertical surfaces. In particular, there is no significant radiative effect near the vertical bounding walls in the lower half of the cavity (which differs from the homogeneous cases). This explains the unchanged thermal distribution in this region. Cinematic profiles along horizontal and vertical crosslines (see figure 5.66 and 5.67) show that, in the transparent cases, as the proportion of water vapor increases, the magnitude of the velocity also augments. However, introducing gas radiation does not impact the flow characteristics. At low molar fraction of water vapor, although the thermally driven force dominates over the mass one, the change in the temperature is too weak to bring any significant alteration to the momentum source and, in turn, to the dynamical structure. As x H 2 O rises up to 0.10 and then 0.20, the domination of the thermally induced force decreases and that of mass origin now prevails. Consequently, gas radiation has weaker and weaker impact on the dynamic structure in the enclosure. In contrast to the homogeneous case, the convective Nusselt number slightly increases along the vertical hot surfaces when radiation is taken into account (see figure 5.69). In this region, the thermal gradient augments due to emission of the gas layers close to the hot surfaces (which reduces the fluid temperature). On the horizontal hot surface, the convective Nusselt number augments when gas radiation is considered (see figure 5.70) since the nearby fluid is cooled down by emission (as evidenced by negative values of radiative source in figures 5.61 a,c,e). As a result, the thermal gradient is enhanced, inducing a higher value of the convective Nusselt number.

The values of averaged convective Nusselt numbers and average total Nusselt numbers 4 , which are defined as Nu = Regarding the transparent cases, tables 5.2, 5.3 and 5.4 show that, as the proportion of water vapor increases, the mean convective Nusselt number is risen up along all active surfaces. This comes from the speed up of the fluid motion as mentioned above. When gas radiation takes place, although the dynamic structure remains unchanged, the convective thermal transport is now altered. At the vertical cold walls, this quantity is dramatically reduced (see figure 5.68) and keeps almost the same value for all the considered molar fractions (the reason is the stronger and stronger radiative emission in the upper part of the cavity as the quantity of water vapor increases, which lessens the thermal gradient). At the hot surfaces, the convective transport is pulled up (by an increase of the thermal gradient due to emission). Besides, radiative transfer is attenuated by the absorption of the medium. These effects induce the drop of the average total Nusselt numbers.

Regarding mass transfer, the average Sherwood number, which is defined as Sh = 1

A A Sh(s)dA is insensitive to radiation (see tables 5.5, 5.6 and 5.7), since radiative transport has no significant impact on the flow structure (see figures 5. [START_REF] Maheu | Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters[END_REF] The case with 5% concentration of CO 2 is not studied here. It was found to generate a non-symmetric flow (in presence of radiation) in a symmetric configuration. Its characteristics deserve a more focused study. In the transparent case, we receive an unsteady signal in the cavity, which is illustrated by the time evolution of the temperature at the center point ((X, Y, Z) = (0.5, 0.5, 0.5)) (see figure 5.71a). However when radiation is introduced, the flow converges again towards a steady state. In this configuration, the temperature diffuses faster than the pollutant (Le > 1). Therefore, close to the vertical surfaces of the obstacle, the fluid is initially heated more quickly than loaded in CO 2 . As a result, in the transparent case at t = 1, the buoyancy force of mass origin is less dominant than the thermal one in these regions (the large thermally driven flow cells with the small mass governed ones in the corners are presented in the figure 5.72 a). As time elapses, the mass driven motion dominates but only weakly (|N| = 1.102) (for instance, see figure 5.72 c at t = 666). The fluid circulations driven by these two effects unpredictably interact and create the fluctuations inside the cavity. Simultaneously, in the upper part of the enclosure, the increase of concentration in the plume axis also restraints the vertical motion and this contributes to the observed oscillation. Several authors have found a similar flow pattern in differentially heated cavities (N = -1.102 and Le = 1.282), namely a large thermal recirculation in the center, and mass driven cells in the corners ( [START_REF] Chang | Unsteady thermosolutal opposing convection of liquidwater mixture in a square cavity-II. Flow structure and fluctuation analysis[END_REF], [START_REF] Nishimura | Oscillatory double-diffusive convection in a rectangular enclosure with combined horizontal temperature and concentration gradients[END_REF], [START_REF] Ibrahim | Couplage de la convection naturelle et du rayonnement dans les mélanges gazeux absorbants-émettants[END_REF]). Such configurations are likely to yield oscillatory solutions attributed to the thermosolutal instability. This phenomenon is induced by a local and abrupt variation in fluid density caused by a uniform "bubble" bursting in temperature or concentration (see figure 5.74). Compared to their works, we observe the similar patterns in our configuration: large thermally induced flow cells above the obstacle and the mass driven ones in the lower part of the cavity. On the other hand, the spectrum analysis of the temperature signal in temporal range [400 : 3000] shows that there are many frequencies which govern the fluctuations in the cavity (see figure 5.73). They come from the periods which are comparable to the time scale of thermal diffusion through the temperature boundary layer thickness (δ t ), δ t α . This parameter corresponds to the dense zone of the temperature contours ( [START_REF] Nishimura | Oscillatory double-diffusive convection in a rectangular enclosure with combined horizontal temperature and concentration gradients[END_REF]) (see figure 5.74). For instance, the thermal boundary layer thickness determined at t = 775 is δ t = 0.066(m) (mean value of boundary layer thicknesses determined from two cavity vertical walls which are represented in the figure 5.75). The frequency calculated from this value is comparable to the most significant one observed in the figure 5.73 b: f = 0.00615. However, the plot of the thermal field at different instants in figure 5.74 do not show any periodic behavior. We, therefore, conclude that the flow inside the cavity is turbulent with a dominant frequency, which could be related to the thermosolutal instability rather than other mechanisms. . Indeed, the presence of radiation increases temperature (by absorption). Therefore, the thermal gradient is strengthened and weakens the domination of mass effect, resulting in the slow down of the fluid circulation near the bottom (see figure 5.80 a). At intermediate altitude (Z = 0.5), in the transparent medium, although we can see a complex flow with a weak thermal plume in the center and upward boundary layers near the cold walls, mass gradient still governs the fluid motions. As radiation is taken into account, the impact of thermal gradient increases and exceeds the mass one (due to radiative absorption). The changes are obvious (compared to the velocity magnitude in this region): the significant deceleration of the fluid layers next to the cold walls and a stronger thermal plume (see 5.79 b). At high level (Z = 0.8), this tendency is more pronounced (see 5.79 c). We also observe the acceleration of the boundary layer at the ceiling, while it was nearly an unmoving region in the transparent medium (see figure 5.80 a). When radiation is considered, the mass driven flow is totally replaced by the thermal one (upward plume and downward bounding layers). However, these changes are not significant when comparing to the velocity magnitude in the lower part of the enclosure. The Q-criterion illustrated in figure 5.84 shows that the presence of radiation does not significantly modify the dynamic structure in the cavity. Table 5.9 represents the mean convective and total Nusselt numbers on the active walls of the cavity and of the obstacle. When radiation is accounted for, the mean convective Nusselt number decreases at the cavity walls. Conversely, it slightly increases on the hot source surfaces. We recall that the fluid near the cold walls is cooled down by emission, resulting in a decrease of the thermal gradient. On the hot source horizontal surface, the average convective quantity is slightly augmented (radiative emission lowers the temperature of the nearby fluid). Besides, the radiative transfer is attenuated by absorption of the medium. Concerning the average total Nusselt number, this quantity is lessened on all the active surfaces. Introducing gas radiation slightly decreases the mass transfer inside the cavity (see table 5.10). Indeed, although the thermal plume is reinforced in the upper part of the cavity, which can raise up the concentration transport, it does not compensate for the reduction in this quantity due to the slow down of the mass driven flow in the lower part of the enclosure.

Mixture at x CO

Heat and Mass Transfer

Synthesis

This section presents the study of the double diffusive convection coupled to the radiation of a real gas mixture in a cavity hosting a hot obstacle. The composition involves either H 2 O or CO 2 which respectively induces cooperating or opposing flows in the enclosure. The computations are carried out at different average mole fractions of the absorbing species. The comparisons between the obtained results and the reference solutions (transparent case) reveal the impacts of gas radiation of the flow structure and heat and mass transfer, which are summarized as below:

• Air -H 2 O mixture
The presence of radiation does not influence the dynamic structure inside the cavity at all the considered average mole fractions. Consequently, the concentration field is found unchanged compared to the transparent case. However, gas absorption lessens the thermal level at the intermediate and high level of the enclosure.

Regarding the transport processes, radiation slightly reduces the convective transfer on the bounding walls of the cavity. But it marginally increases this quantitiy on the vertical surfaces of the obstacle and more significantly on the hot upper face. Besides, radiative transfer is attenuated by absorption through the medium, especially between the upper surface of the obstacle and the cavity walls. In turn, the total heat transfer is lessened. Because the dynamic behavior is not much altered when radiation is accounted for, the mass transfer in the enclosure remains nearly unchanged.

• Air -CO 2 mixture In this configuration, at x CO 2 = 0.10, in transparent medium, the turbulent flow occurs with a domninant frequency which maybe due to the thermosolutal instability in a transparent medium. However, when the gas radiation, the fluid motion is stabilized. Radiation increases the thermal gradient which balance the effect of the concentration one in the plume region.

At x CO 2 = 0.20, gas radiation tends to slow down the mass driven flow in the lower part of the cavity but intensifies the thermal plume in the upper part.

In addition, it also alters the thermal field: a slight increase of temperature is observed at intermediate and high levels and around the obstacle but a drop is found near the region close to the lower part of the vertical walls. The modification tendency in the concentration field is found similar for the thermal field except a slight augmentation of this quantity near the floor. Besides, introducing radiation reduces the total thermal transfer: the convective part near the vertical walls and the radiative transport along the obstacle upper surface. It also slightly decreases the mass transfer.

Conclusion

In this chapter, we analyze the influences of radiation of a gas mixture on the double diffusive convection, using the gray gas assumption and real gas model. The comparisons between the obtained results and the reference solutions (transparent case) reveal the transformations inside the enclosure which are summarized as:

• Cooperating flow Introducing gas radiation does not significantly affect the concentration field. It slightly accelerates the boundary layers but reduces the maximum velocity inside the plume. Volume radiation tends to thermally homogenize the medium. It decreases the temperature in the upper half of the cavity.

Radiation slightly reduces the convective Nusselt number on the bounding walls of the cavity. But it marginally increases these quantities on the vertical surfaces of the obstacle and significantly augments those on the hot upper face. Besides, radiative transfer is attenuated by absorption of the medium, especially between the upper surface of the obstacle and ones of the cavity. In turn, the total heat transfer is lessened. Because the dynamic behavior is not much altered when radiation is accounted for, the mass transfer in the enclosure remains nearly unchanged.

All these effects are strengthened when the medium becomes more absorbing (in the considered range of optical thicknesses and molar fraction).

• Opposing flow

Generally, the movement in the lower part of the cavity is dominated by the mass gradient but at medium and high levels, the thermal one governs the flow. Radiation also intensifies the thermal plume. However, the increase of concentration in the axis of the plume limits the vertical motion and causes it to spread. In addition, gas radiation reduces the temperature in the regions near the obstacle vertical surfaces. Above the source, temperature is generally reduced compared to transparent case. The alteration of the concentration field are the same as for the thermal one except a slight increase of this quantity near the vertical cold walls in the lower part of the cavity. These effects are reinforced with the optical thickness.

Particularly, in the air -CO 2 mixture, at x CO 2 = 0.10, radiation plays the role of stabilization of the unsteadiness found in the cavity when medium is considered transparent (due to the thermosolutal instability rather than other mechanisms).

Introducing radiation reduces the total thermal transfer: the part by convective process near the vertical walls and the radiative transport along the obstacle upper surface. In addition, it slightly decreases the mass transfer.

Chapter 6

General Conclusion

In this thesis, we have numerically studied the influence of the radiation of gray gas and real gas mixture on the pure thermal as well as double diffusive natural convection. The considered configuration is a 3D cavity containing a small cubical obstacle located on its floor. This object is prescribed at a high temperature and high concentration in species while the vertical walls of the enclosure are set at low temperature and low concentration. The other horizontal surfaces of the cavity and the floor of the obstacle are assumed adiabatic and impermeable. In addition, the active walls are considered black while the rest are perfectly reflective. The enclosure is filled with either a gray gas or a binary real gas mixture (air -H 2 O or air -CO 2 ).

Due to the moderate variation in temperature and concentration, the convective fluid motion inside the cavity is simulated using the Boussinesq approximation. The fluid motion as well as the mass and thermal transport are simultated by Code Saturne v5.0.4. In addition, the radiative transfer equation is solved using the built-in Discrete Ordinates module of this code. However, some improvements in the generation of direction sets and a new SLW model for real gas mixtures have been added. The compatibility and ability of the Code Saturne and our own model have been validated with the previous results given in the references.

Different calculations on the coupled of thermal or double diffusive convection with radiation of gray gas medium or real gas mixtures have been performed. Concerning the thermal convention cases, different gray gases at various opacities and an air -H 2 0 mixture at three different molar fractions (5%, 10% and 20%) of the water vapor have been accounted for. While for double diffusive situations, depending on the mass to thermal buoyancy ratio (N), the fluid motions inside the cavity can be classed into opposing or cooperating flows. Like in the thermal convection, the calculation on the gray gas at several optical thicknesses and on real gas mixtures (air -H 2 O and air -CO 2 ) at different average mole fractions have been studied. The main influences of gas radiation on natural convection are summarized as:

Coupling thermal convection and radiation

Generally, in the steady results using either gray gas assumption or real gas model, introducing radiation non-uniformly accelerates the boundary layers of the enclosure and of the hot obstacle. It also makes moving some parts of the motionless fluid observed in the transparent cases. The thermal plume and its recirculation interfere and create shear flow patterns. In addition, radiation reduces the temperature in the upper half of the cavity, while, in the lower part, it lessens this quantity inside the plume but slightly increases around. The heat transfer is also impacted by radiation.

At the cold walls, the convective Nusselt numbers are risen up in the upper part but dropped off in the lower half. At the obstacle horizontal surface, this quantity is significantly increased. In turn, the total thermal transfer is reduced, especially the part by convective process on the vertical walls and by the radiative attenuation due to absorption along the obstacle upper surface. These impacts are enhanced with the optical thickness. Particularly, in gray gas simulations, when the optical thickness of the medium is unity, radiation yeilds a periodic flow. Its mechanism is due to the formation and vanishing of small vortices right above the hot upper surface of the obstacle and of the fluid flow cells in the regions next to the vertical walls of the enclosure. Then, at τ = 2, the circulation becomes turbulent at Ra = 5 • 10 6 .

Coupling double diffusive convection and radiation

Concerning the cooperating flows, in a gray gas mixture, the presence of volume radiation slightly accelerates the boundary layers but reduces the maximum velocity inside the plume. It tends to lower the temperature in the upper part of the cavity, and thus, homogenizes the thermal field but a higher mass-to-thermal buoyancy ratio has an opposite effect. In all considered cases, the concentration field remains insensitive to radiation. In a real gas mixture (air -H 2 O), the alteration of the thermal distribution is similar as for gray gas. However, these changes do not bring any notable modifications to the dynamic and the concentration fields. Consequently, the total heat exchange is reduced, while the mass transfer remains unchanged, compared to the transparent medium.

Regarding the cooperating flows in gray gas mixtures, firstly, at N = -1, radiation breaks the perfect symmetry between the thermal and the concentration fields. It sets the stagnant fluid (in transparent case) into motion with a mass driven movement in the lower part and a thermal one in the upper part of the cavity. With the increase of the optical thickness, radiation intensifies the thermal plume. However, the increase of concentration in the axis of the plume limits the vertical motion and causes it to spread. In addition, gas radiation reduces the temperature in the regions near the obstacle lateral surfaces. Above the source, the temperature is generally reduced compared to the transparent case. The concentration field is modified in the same manner as the thermal one, except a slight increase of this quantity near the vertical cold walls in the lower part of the cavity. At N = -2, gas radiation still affects the dynamic, the thermal and the concentration field, but with a weaker amplitude because of the domination of the mass gradient. In the air -CO 2 mixture, at x CO 2 = 0.10, accounting for radiation stabilizes the flow, which, in the transparent case, was turbulent (with a dominant frequency which maybe due to the thermosolutal instability). At x CO 2 = 0.20, the modifications tendencies are similar as in a gray gas mixture when mass forces dominate: a slight reinforcement of thermal plume but a slow down of mass driven motion. Concerning the heat and mass transfer, considering gas radiation slightly lessens these quantities.

Perspective

The performed studies reveal the effects of a gas radiation on flow structures and on heat and mass transfer. In perspective, the following points maybe considered:

• Gas mixture with more than two components: In real context of a combustion, the diffusing gas mixture contains many components that can absorb and emit radiation. Therefore, it is necessary to study simultaneously the effects of a multi-component gas mixture to the flow structure and heat transfer. This would involve a modification of the gas radiation model to include more than one absorbing species, through the multiplication method, for instance Solovjov and Webb [START_REF] Vladimir | An efficient method for modeling radiative transfer in multicomponent gas mixtures with soot[END_REF].

• Turbulence regime. With larger temperature gradients or in larger enclosures, the Rayleigh number increases, turning the flow to transition or fully turbulent behaviors. The influences of gas radiation may switch on the flow characteristics (threshold values of Ra, for instance) and the turbulent quantities (intensities, frequencies, correlations) are remarkable problems. The question of a proper modeling of the turbulent-radiation interation may also arise.

• Ambient environment. These computations can be applied for the configuration containing a lower value of the reference temperature, of the temperature difference and of the average mole fraction of pollutant, which simulates the realistic conditions in the building environment. At room temperature, the concentration of radiant species are low (H 2 O, especially) but may have a significant role over large distances (several meters).

• Non-Boussinesq simulations. For highly anisothermal and heterogeneous gas mixtures, the thermo-physical properties may significantly vary from one point to another. A non-Boussinesq model is then necessary to correctly simulate the flow behavior. A low Mach model could be implemented.

• Varying the cavity parameters. Cavity size, ratio between the enclosure and the obstacle are key parameters, as well as the surface emissivities.

• Experimental approach. For evaluating the results of the numerical researches, the realistic way is the comparison with experiments. The results from the experiments can help validating the numerical studies. It however remains a challenge to build a device where gas concentration may be prescibed at the boundaries.

• Parallel calculation. The radiative transfer equation can be independently solved for each gray gas and each discrete direction and then, the values of radiative source term (at each cell) and of the incident flux (at each boundary control surface) are summed up. Therefore, these calculations can be simultaneously carried out with the help of parallel computing libraries (OpenMP, Cuda,...). A pioneering work in that field was performed by Cadet [START_REF] Cadet | Étude du couplage convection-rayonnement en cavité différentiellement chauffée à haut nombre de Rayleigh en ambiances habitables[END_REF]. where S = 1 2 (∇v + (∇v) T ) is rate of strain tensor and Ω = The quantity Q is defined as second invariant of ∇v (Hunt, Wray, and Moin [START_REF] Hunt | Eddies, streams, and convergence zones in turbulent flows[END_REF]) and written as:

Q = 1 2 ((∇ • v) 2 -tr((∇v) 2 )) (C.5)
For an incompressible flow, ∇ • v = 0. Thus:

Q = - 1 2 tr((∇v) 2 ) = 1 2 ( Ω 2 -S 2 ) (C.6)
• If Q > 0, the vorticity magnitude is greater than rate of shear strain. This characterizes the presence of rotation (Jeong and Hussain [START_REF] Jeong | On the identification of a vortex[END_REF]).

• If Q < 0, the shear strain rate dominates over the vorticity magnitude and this denotes stretching pattern.

The Q values are then calculated using the filter Gradient Of Unstructured DataSet in Paraview.

ÉTUDE NUMÉRIQUE DE LA CONVECTION NATURELLE COUPLÉE AU RAYONNEMENT GAZEUX DANS UN CAVITÉ CONTENANT UN OBSTACLE ACTIF

Notre objectif est d'étudier numériquement des écoulements de convection naturelle en milieu confiné, le fluide étant un mélange gazeux incluant des composants absorbants (CO 2 , H 2 O). On considère pour cela une cavité cubique avec une source localisée sur le plancher chauffant le fluide et diffusant un polluant participant au rayonnement. Nos calculs sont réalisés avec le code CFD Code Saturne, dans lequel nous avons implanté nos propres données pour la méthode des ordonnées discrètes (nouvelles quadratures) et pour modéliser le rayonnement des gaz (méthode SLW dans l'approche « rank-correlated »). En convection naturelle thermique pure les résultats montrent que le rayonnement du gaz modifie légèrement la structure de l'écoulement et la distribution de température. Il réduit les échanges convectifs entre le fluide et les parois de l'enceinte ainsi que l'échange radiatif entre la surface supérieure de l'obstacle et celles de la cavité. En double-diffusion, dans le cas aidant, le rayonnement du gaz tend à homogénéiser le champ thermique, accélère légèrement les couches limites pariétales, mais réduit la vitesse maximale à l'intérieur du panache. Par contre, il affecte peu le champ de concentration. Dans le cas opposant, le rayonnement intensifie le panache thermique qui se développe au-dessus de l'obstacle. Il réduit la température dans les régions proches des surfaces verticales de l'obstacle. Le champ de concentration montre les mêmes tendances d'altération que le champ thermique.

Mots clés:

Analyse numérique, Chaleur-Convection, Gaz-Écoulement, Couche limite, Modélisation CFD, Rayonnement thermique, Transfert de chaleur.

NUMERICAL STUDY OF NATURAL CONVECTION COUPLED TO GAS RADIATION IN A CAVITY CONTAINING AN ACTIVE OBSTACLE

Our objective is to study numerically natural convection flows in an enclosure, the fluid being a gaseous mixture including absorbent components (CO 2 , H 2 O). For this purpose, we consider a cubic cavity with a source located on the floor, heating the fluid and diffusing a pollutant participating to radiation. Our calculations are performed with the CFD software Code Saturne, in which we have implemented our own data for the discrete ordinates method (new quadratures) and for modelling gas radiation (SLW method in the rank-correlated approach). In pure thermal natural convection, the results show that gas radiation slightly changes the flow structure and the temperature distribution. It reduces the convective exchanges between the fluid and the walls of the enclosure as well as the radiative exchange between the upper surface of the obstacle and the cavity boundaries. In double diffusion, in the aiding case, gas radiation tends to homogenize the thermal field and slightly accelerates the parietal boundary layers, but reduces the maximum velocity within the plume. On the other hand, it has little effect on the concentration field. In the opposing case, the radiation intensifies the thermal plume above the obstacle. It reduces the temperature in regions close to the vertical surfaces of the obstacle. The concentration field shows the same modification trends as the thermal field.

Keywords: Numerical analysis, Heat-Convection, Gas flow, Boundary layer, Computational fluid dynamics, Heat-Radiation and absorption, Heat-Transmission.

= 2 •

 2 10 5 , Pr = 0.71) . . . . . . . . . . . . . . 4.1 Median plane (Y = 0.5) and crosslines used for the results display . . . 4.2 Temperature T-T re f T h -T c distribution in the median plane (Y = 0.5) with different mesh sizes: transparent medium. . . . . . . . . . . . . . . . . . 4.3 Temperature T-T re f T h -T c distribution in the median plane (Y = 0.5) for the gray medium at τ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Profiles of temperature T-T re f T h -T c and vertical velocities w U re f at different crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . 4.5 Profiles of temperature and vertical velocities at different crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Temperature T-T re f T h -T c distribution in the median plane (Y = 0.5) for different fluid opacities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Temperature T-T re f T h -T c profiles at different Z-crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Temperature T-T re f T h -T c profiles at different X-crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Distribution of radiative source term in the median plane (Y = 0.5) at different optical thicknesses. Sources are normalized by 4σT 4 re f /L. . . . 4.10 Velocity field on the median plane (Y = 0.5) for different fluid opacities. Velocities are normalized by U re f . . . . . . . . . . . . . . . . . . . . 4.11 Flow lines 5 on the median plane (Y = 0.5) for different fluid opacities. 4.12 Profiles of vertical velocities w U re f at different Z-crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 4.13 Profiles of horizontal velocities u U re f at different X-crosslines in the median plane (Y = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Representation of the Q-criterion at Q = 0.02 in the cavity. Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Isovalue lines at Q = -0.02 in the median plane (Y = 0.5). Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Considered points for tracking the time evolution of temperature. . . . 4.17 Time evolution of temperature T-T re f T h -T c at the center point of the cavity P 3 : (X, Y, Z) = (0.5, 0.5, 0.5) at different optical thicknesses. . . . . . . . 4.18 Power spectrum of the temperature signal at the point P 3 : (X, Y, Z) = (0.5, 0.5, 0.5) at τ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Time evolution and power spectrum of the temperature signal at the point P 1 : (X, Y, Z) = (0.2, 0.5, 0.2) in temporal range [5000:5800] at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Time evolution and power spectrum of the temperature signal at the point P 2 : (X, Y, Z) = (0.5, 0.5, 0.3) in temporal range [5000:5800] at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Time evolution and power spectrum of the temperature signal at the point P 3 : (X, Y, Z) = (0.5, 0.5, 0.5) in temporal range [5000:5800] at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 Time evolution and power spectrum of the temperature signal at the point P 4 : (X, Y, Z) = (0.8, 0.5, 0.8) in temporal range [5000:5800] at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 Flow lines in the median plane (Y = 0.5) at different instants over one period (T 1 = 72) at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Negative values of Q-criterion in the median plane (Y = 0.5) at different instants over one period (T 1 = 72) at τ = 1 (Black: 0; Red: -0.1; Blue: -0.2; Orange: -0.4; Purple: -1; Green: -2). Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.25 Temperature field T-T re f T h -T c at an instant in the median plane (Y = 0.5). . . 4.26 2D-contours of temperature T-T re f T h -T c in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor. . . . . . . . . . . . 4.27 Temperature T-T re f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5) at different concentration of water vapor. . . . . . . . . 4.28 Distribution of radiative source term in the median plane (Y = 0.5) at different mole fractions of water vapor. Sources are normalized by 4σT 4 re f /L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.29 Profiles of vertical velocities w U re f at different Z-crosslines in the median plane (Y = 0.5) at different concentration of water vapor. . . . . . 4.30 Profiles of horizontal velocities u U re f at different X-crosslines in the median plane (Y = 0.5) at different concentration of water vapor. . . . . . 4.31 Velocity field in the median plane (Y = 0.5) at different mole fractions of water vapor. Velocities are normalized by U re f . . . . . . . . . . . . . 4.32 Flow lines in the median plane (Y = 0.5) at different mole fractions of water vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Representation of Q-criterion at Q = 0.02 in the cavity at different mole fractions of water vapor. Values are normalized by U 2 re f /L 2 . . . . 4.34 Local convective Nusselt number along the vertical centerline of any lateral wall of the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 4.35 Local convective Nusselt number along vertical center line of each lateral surfaces of the obstacle. . . . . . . . . . . . . . . . . . . . . . . . 4.36 Local convective Nusselt number along horizontal centerline of the obstacle top surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Thermal field T-T re f T h -T c in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating cases, N = 1. . . . . . . . . . . . . 5.2 Temperature T-T re f T h -T c profiles along Z-crosslines in the median plane (Y = 0.5): cooperating flow, N = 1 . . . . . . . . . . . . . . . . . . . . . 5.3 Temperature T-T re f T h -T c profiles along different X-crosslines in the median plane (Y = 0.5): cooperating flow, N = 1 . . . . . . . . . . . . . . . . . . 5.4 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 0.1. Sources are normalized by 4σT 4 re f /L. 5.5 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 0.2. Sources are normalized by 4σT 4 re f /L. 5.6 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 0.5. Sources are normalized by 4σT 4 re f /L. 5.7 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 1. Sources are normalized by 4σT 4 re f /L. . 5.8 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 2. Sources are normalized by 4σT 4 re f /L. . 5.9 Vector field in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 1. Velocities are normalized by U re f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 1. . . . . . . . . . . . . . . 5.11 Horizontal velocity u U re f profiles along different X-crosslines in the median plane (Y = 0.5): cooperating case, N = 1. . . . . . . . . . . . . . . 5.12 Iso-surface of the Q-criterion at the value 0.01 in the cavity. Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Concentration field C-Cre f C h -C l in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 1. . . . . . . . . . . 5.14 Concentration C-Cre f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 1. . . . . . . . . . . . . . . 5.15 Concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5): cooperating case, N = 1. . . . . . . . . . . . . . . 5.16 Velocity vectors in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 2. Velocities are normalized by U re f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . 5.18 Horizontal velocity u U re f profiles along different X-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . 5.19 Iso-surface of the Q-criterion at the value 0.01 in the cavity. Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Thermal field in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 2. . . . . . . . . . . . . . . . . . . xx 5.21 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 0.1. Sources are normalized by 4σT 4 re f /L. 5.22 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 0.2. Sources are normalized by 4σT 4 re f /L. 5.23 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 0.5. Sources are normalized by 4σT 4 re f /L. 5.24 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 1. Sources are normalized by 4σT 4 re f /L. . 5.25 Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 2. Sources are normalized by 4σT 4 re f /L. . 5.26 Temperature T-Tre f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . . . . 5.27 Temperature T-Tre f T h -T c profiles along different X-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . 5.28 Concentration field in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 2. . . . . . . . . . . . . 5.29 Concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . 5.30 Concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5): cooperating case, N = 2. . . . . . . . . . . . . . . 5.31 Velocity vectors in the median plane (Y = 0.5) of the cavity for different fluid opacities: opposing case, N = -1. Velocities are normalized by U re f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.32 Vertical velocity w U re f , temperature T-T re f T h -T c , concentration C-C re f C h -C l and dimensionless buoyancy source term profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -1. . . . . . . . . . 5.33 Horizontal velocity u U re f , temperature T-T re f T h -T c and concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5): opposing case, N = -1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.34 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.1. Sources are normalized by 4σT 4 re f /L. 5.35 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.2. Sources are normalized by 4σT 4 re f /L. 5.36 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.5. Sources are normalized by 4σT 4 re f /L. 5.37 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 1. Sources are normalized by 4σT 4 re f /L. . 5.38 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 2. Sources are normalized by 4σT 4 re f /L. . 5.39 Thermal field T-T re f T h -T c in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -1. . . . . . . . . . . . . . . . . . . . . . 5.40 Distribution of concentration C-C re f C h -C l in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -1. . . . . . . . . . . . . . 5.41 Vertical velocities w U re f across the mid-height plane (Z = 0.5) for different fluid opacities: opposing case, N = -1. . . . . . . . . . . . . . . xxi 5.42 Iso-surface of the Q-criterion at the value 0.002 in the cavity for different fluid opacities: opposing case, N = -1. Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.43 Velocity vectors in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -2. Velocities are normalized by U re f . . . . . 5.44 Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -2. . . . . . . . . . . . . . . . 5.45 Horizontal velocity u U re f profiles along different X-crosslines in the median plane (Y = 0.5): opposing case, N = -2. . . . . . . . . . . . . . . . 5.46 Thermal field T-T re f T h -T c in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -2. . . . . . . . . . . . . . . . . . . . . . 5.47 Temperature T-T re f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -2. . . . . . . . . . . . . . . . . . . 5.48 Temperature T-T re f T h -T c profiles along different X-crosslines in the median plane (Y = 0.5): opposing flow, N = -2. . . . . . . . . . . . . . . . . . . 5.49 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 0.1. Sources are normalized by 4σT 4 re f /L. 5.50 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 0.2. Sources are normalized by 4σT 4 re f /L. 5.51 Distribution of radiative source in the median plane (Y = 0.5): opposing case, N = -2, τ = 0.5. Sources are normalized by 4σT 4 re f /L. . . . . 5.52 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 1. Sources are normalized by 4σT 4 re f /L. . 5.53 Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 2. Sources are normalized by 4σT 4 re f /L. . 5.54 Concentration C-C re f C h -C l field in the median plane (Y = 0.5) for different fluid opacities, N = -2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.55 Concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -2. . . . . . . . . . . . . . . . 5.56 Concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5): opposing case, N = -2. . . . . . . . . . . . . . . . 5.57 Iso-surface of the Q-criterion at the value 0.002 in the cavity for different fluid opacities: opposing case, N = -2. VValues are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.58 Thermal field T-T re f T h -T c in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor. . . . . . . . . . . . . . . . . . . 5.59 Temperature T-T re f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor. . . . 5.60 Temperature T-T re f T h -T c profiles along different X-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor. . . 5.61 Distribution of radiative source term in the median plane (Y = 0.5) at different mole fractions of water vapor. Sources are normalized by 4σT 4 re f /L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.62 Concentration C-C re f C h -C l field in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor. . . . . . . . . . . . . . . . . . 5.63 Concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor.xxii 5.64 Concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.65 Velocity vector in the median plane of the cavity (Y = 0.5) at different mole fraction of water vapor. Velocities are normalized by U re f . . . . . 5.66 Vertical velocities w U re f profiles at different Z-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor. . . 5.67 Horizontal velocities u U re f profiles at different X-crosslines in the median plane (Y=0.5) at different average mole fractions of water vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.68 Local convective Nusselt number along a vertical centerline of any lateral wall of the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.69 Local convective Nusselt number along a vertical center line of each lateral surfaces of the obstacle. . . . . . . . . . . . . . . . . . . . . . . . 5.70 Local convective Nusselt number along an horizontal centerline of the obstacle top surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.71 Time evolution of temperature at the center point of the cavity (X, Y, Z) = (0.5, 0.5, 0.5) at x CO 2 =0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.72 Flow lines on the median plane of the cavity (Y = 0.5) at different instants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.73 Time evolution and power spectrum of the temperature signal at the point (X, Y, Z) = (0.5, 0.5, 0.5) in temporal range [400 : 3000] at x CO 2 = 0.10: transparent medium. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.74 Thermal field T-T re f T h -T c of the cavity (Y = 0.5) at different instants. . . . . 5.75 Temperature T-T re f T h -T c profile at Z = 0.5 in the median plane at t = 775 . . 5.76 Temperature T-T re f T h -T c profile at Z = 0.1 in the range X = [0.35 : 0.65] at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.77 Distribution of radiative source term in the median plane (Y = 0.5) at x CO 2 = 0.10. Sources are normalized by 4σT 4 re f /L . . . . . . . . . . . . . 5.78 Velocity vector in the median plane of the cavity (Y = 0.5) at x CO 2 = 0.20. Velocities are normalized by U re f . . . . . . . . . . . . . . . . . . . 5.79 Vertical velocity w U re f , temperature T-T re f T h -T c and concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5) at x CO 2 = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.80 Horizontal velocity u U re f , temperature T-T re f T h -T c and concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5) at x CO 2 = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.81 Thermal field T-T re f T h -T c in the median plane of the cavity (Y = 0.5) at x CO 2 = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.82 Distribution of radiative source term in the median plane (Y = 0.5) at x CO 2 = 0.20. Sources are normalized by 4σT 4 re f /L . . . . . . . . . . . . . 5.83 Concentration field C-C re f C h -C l in the median plane of the cavity (Y = 0.5) at x CO 2 = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.84 Iso-surface of the Q-criterion at the value 0.01 in the cavity at x CO2 = 0.20. Values are normalized by U 2 re f /L 2 . . . . . . . . . . . . . . . . . . xxiiiList of Tables

Figure 1 . 1 :

 11 Figure 1.1: Problem configuration

Figure 2 . 1 :

 21 Figure 2.1: SLW calculation diagram for j th gray gas (from Solovjov, Webb, and André [146])

Figure 2 . 2 :

 22 Figure 2.2: Schematic representation of the Reference Approach (from Solovjov, Webb, and André [147])

Figure 2 . 3 :

 23 Figure 2.3: Schematic representation of the Rank Correlated (from Solovjov et al. [141])

  control volume and I f η (Ω) is the mean monochromatic radiation intensity in a cell face f along the direction Ω. An interpolation is needed to relate I f η (Ω) to the unknown I P η (Ω). Different schemes may be applied but the implemented radiative module of Code Saturne resorts to the first order step scheme. It approximates the face value I f η (Ω) by the center value of the upstream control volume. For instance, for a face f 1 where the outer unit normal vector n 1 is such as Ω • n 1 < 0, the scheme prescribes I f 1 η (Ω) = I U η (Ω). Conversely, for the face f 2 where Ω • n 2 > 0, the I f 2 η (Ω) is set to I P η (Ω) (see figure 3.1).

Figure 3 . 1 :

 31 Figure 3.1: Presentation of step scheme for calculation of intensity at one face

  The correct S 8 set (based of Fiveland data) returns results close to what is obtained with the Balsara set.

Figure 3 . 2 :

 32 Figure 3.2: 2D -Square Cavity

Figure 3 . 5 :

 35 Figure 3.5: Implantation of SLW model in the Code Saturne

Figure 3 . 6 :

 36 Figure 3.6: Differentially heated cavity

Figure 3 . 7 :

 37 Figure 3.7: Calculation domain (Paroncini and Corvaro [13])

Figure 3 . 8 :

 38 Figure 3.8: Domain of calculation (Ra = 2 • 10 5 , Pr = 0.71)

(a)

  Plane under consideration (Y = 0.5) (b) Crosslines in the plane Y = 0.5

Figure 4 . 1 :

 41 Figure 4.1: Median plane (Y = 0.5) and crosslines used for the results display

Figure 4 . 2 :

 42 Figure 4.2: Temperature T-T re f T h -T c distribution in the median plane (Y = 0.5) with different mesh sizes: transparent medium.

8 - 5 Figure 4 . 4 :

 8544 Figure 4.4: Profiles of temperature T-T re f T h -T c and vertical velocities w U re f at different

Chapter

8 - 8 - 5 Figure 4 . 5 :

 88545 Figure 4.5: Profiles of temperature and vertical velocities at different crosslines in the median plane (Y = 0.5).

N

(a ) 5 Figure 4 . 6 :

 )546 Figure 4.6: Temperature T-T re f T h -T c distribution in the median plane (Y = 0.5) for different fluid opacities.

Figure 4 .

 4 Figure 4.6 displays the thermal fields in the median plane (Y = 0.5) for different case studies ranging from a transparent medium to an optical thickness of τ = 0.5.All these configurations lead to a steady state solution. Further increasing τ may yield periodic flows, as will be described in the next paragraph (4.2.2).

4. 2 . 51 -

 251 Coupling with radiation in the gray gas assumption

Figure 4 . 7 :

 47 Figure 4.7: Temperature T-T re f T h -T c profiles at different Z-crosslines in the median plane (Y = 0.5).

Figure 4 . 8 :

 48 Figure 4.8: Temperature T-T re f T h -T c profiles at different X-crosslines in the median plane (Y = 0.5).

  τ = 0.1: Absorption dominates (c) τ = 0.2: Emission dominates (d) τ = 0.2: Absorption dominates (e) τ = 0.5: Emission dominates (f) τ = 0.5: Absorption dominates

Figure 4 . 9 :

 49 Figure 4.9: Distribution of radiative source term in the median plane (Y = 0.5) at different optical thicknesses. Sources are normalized by 4σT 4 re f /L.

(a) 5 Figure 4 . 10 :

 5410 Figure 4.10: Velocity field on the median plane (Y = 0.5) for different fluid opacities.Velocities are normalized by U re f .

Figure 4 .

 4 Figure 4.10 represents the velocity vectors in the mid-depth plane (Y = 0.5) for different values of the optical thickness.It is shown that, as the opacity increases, the fluid is more accelerated, inside the plume and next to the cavity surfaces. It is also observed the broadening of the plume and of the vertical boundary layers near the cavity walls when the radiation effects is considered (compared to the transparent cases). Besides, we observe that the plume at τ = 0.1 and τ = 0.2 keeps the same cone shape. However, as the τ = 0.5, the structure of the plume changes: it is compressed at medium altitude and takes the form of an hourglass. The reason is that, when the optical thickness increases, the temperature in the lower part of the cavity increases (see figure4.8 a) due to the absorption effect (see figure4.9 f). As a result, the fluid in this region is pushed up, and then collides with the downward boundary layers near the cavity vertical wall.

  Figure 4.10 represents the velocity vectors in the mid-depth plane (Y = 0.5) for different values of the optical thickness.It is shown that, as the opacity increases, the fluid is more accelerated, inside the plume and next to the cavity surfaces. It is also observed the broadening of the plume and of the vertical boundary layers near the cavity walls when the radiation effects is considered (compared to the transparent cases). Besides, we observe that the plume at τ = 0.1 and τ = 0.2 keeps the same cone shape. However, as the τ = 0.5, the structure of the plume changes: it is compressed at medium altitude and takes the form of an hourglass. The reason is that, when the optical thickness increases, the temperature in the lower part of the cavity increases (see figure4.8 a) due to the absorption effect (see figure4.9 f). As a result, the fluid in this region is pushed up, and then collides with the downward boundary layers near the cavity vertical wall.

  Figure 4.10 represents the velocity vectors in the mid-depth plane (Y = 0.5) for different values of the optical thickness.It is shown that, as the opacity increases, the fluid is more accelerated, inside the plume and next to the cavity surfaces. It is also observed the broadening of the plume and of the vertical boundary layers near the cavity walls when the radiation effects is considered (compared to the transparent cases). Besides, we observe that the plume at τ = 0.1 and τ = 0.2 keeps the same cone shape. However, as the τ = 0.5, the structure of the plume changes: it is compressed at medium altitude and takes the form of an hourglass. The reason is that, when the optical thickness increases, the temperature in the lower part of the cavity increases (see figure4.8 a) due to the absorption effect (see figure4.9 f). As a result, the fluid in this region is pushed up, and then collides with the downward boundary layers near the cavity vertical wall.

5 Figure 4 . 11 :

 5411 Figure 4.11: Flow lines 2 on the median plane (Y = 0.5) for different fluid opacities.

8 Figure 4 . 12 :

 8412 Figure 4.12: Profiles of vertical velocities w U re f at different Z-crosslines in the median plane (Y = 0.5).

Figure 4 . 13 :

 413 Figure 4.13: Profiles of horizontal velocities u U re f at different X-crosslines in the median plane (Y = 0.5).

(a) 5 Figure 4 . 14 :

 5414 Figure 4.14: Representation of the Q-criterion at Q = 0.02 in the cavity. Values are normalized by U 2 re f /L 2 .

5 Figure 4 . 15 :

 5415 Figure 4.15: Isovalue lines at Q = -0.02 in the median plane (Y = 0.5). Values are normalized by U 2 re f /L 2 .

Figure 4 . 16 :

 416 Figure 4.16: Considered points for tracking the time evolution of temperature.

Figure 4 . 17 :

 417 Figure 4.17: Time evolution of temperature T-T re f T h -T c at the center point of the cavity P 3 : (X, Y, Z) = (0.5, 0.5, 0.5) at different optical thicknesses.

Figure 4 . 18 :

 418 Figure 4.18: Power spectrum of the temperature signal at the point P 3 :(X, Y, Z) = (0.5, 0.5, 0.5) at τ = 2

( a )

 a Time evolution of temperature (b) Power spectrum

Figure 4 . 19 :

 419 Figure 4.19: Time evolution and power spectrum of the temperature signal at the point P 1 : (X, Y, Z) = (0.2, 0.5, 0.2) in temporal range [5000:5800] at τ = 1.

Figure 4 . 20 :

 420 Figure 4.20: Time evolution and power spectrum of the temperature signal at the point P 2 : (X, Y, Z) = (0.5, 0.5, 0.3) in temporal range [5000:5800] at τ = 1.

( a )

 a Time evolution of temperature (b) Power spectrum

Figure 4 . 21 :

 421 Figure 4.21: Time evolution and power spectrum of the temperature signal at the point P 3 : (X, Y, Z) = (0.5, 0.5, 0.5) in temporal range [5000:5800] at τ = 1.

Figure 4 . 22 :

 422 Figure 4.22: Time evolution and power spectrum of the temperature signal at the point P 4 : (X, Y, Z) = (0.8, 0.5, 0.8) in temporal range [5000:5800] at τ = 1.

Figure 4 . 23 :

 423 Figure 4.23: Flow lines in the median plane (Y = 0.5) at different instants over one period (T 1 = 72) at τ = 1.

Figure 4 . 24 :

 424 Figure 4.24: Negative values of Q-criterion in the median plane (Y = 0.5) at different instants over one period (T 1 = 72) at τ = 1 (Black: 0; Red: -0.1; Blue: -0.2; Orange: -0.4; Purple: -1; Green: -2). Values are normalized by U 2 re f /L 2 .

4. 3 .

 3 Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture 63

Figure 4 . 25 :

 425 Figure 4.25: Temperature field T-T re fT h -T c at an instant in the median plane (Y = 0.5).

Figures 4 .

 4 Figures 4.26 display the thermal field in the median plane (Y = 0.5) in both cases of transparent and participating media at different concentration of water vapor. The most sensitive effect of radiation (at these concentrations) is a slight broadening of the temperature contours in the lower half of the cavity (Figures 4.26 b,d,f). The radiant flux coming from the hot surfaces of the obstacle heats up the medium between the heater and the vertical walls of the cavity. As a result, the nearly vertical thermal stratification in absence of radiation is broken. The air (outside the thermal plume), therefore, slightly becomes more uniform in temperature.

4. 3 .

 3 Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture 65 (a) x H 2 O = 0.05: Transparent medium (b) x H 2 O = 0.05: Participating medium (c) x H 2 O = 0.10: Transparent medium (d) x H 2 O = 0.10 :Participating medium (e) x H 2 O = 0.20: Transparent medium (f) x H 2 O = 0.20 :Participating medium

Figure 4 . 26 :

 426 Figure 4.26: 2D-contours of temperature T-T re f T h -T c in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor.

  (a) x H 2 O = 0.05 : Z = 0.8 (b) x H 2 O = 0.10 : Z = 0.8 (c) x H 2 O = 0.20 : Z = 0.8 (d) x H 2 O = 0.05 : Z = 0.5 (e) x H 2 O = 0.10 : Z = 0.5 (f) x H 2 O = 0.20 : Z = 0.5 (g) x H 2 O = 0.05 : Z = 0.1 (h) x H 2 O = 0.10 : Z = 0.1 (i) x H 2 O = 0.20 : Z = 0.1

Figure 4 . 27 :

 427 Figure 4.27: Temperature T-T re f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5) at different concentration of water vapor.

4. 3 .

 3 Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture 67

(

  a) x H 2 O = 0.05: Emission dominates (b) x H 2 O = 0.05: Absorption dominates (c) x H 2 O = 0.10: Emission dominates (d) x H 2 O = 0.10: Absorption dominates (e) x H 2 O = 0.20: Emission dominates (f) x H 2 O = 0.20: Absorption dominates

Figure 4 . 28 :

 428 Figure 4.28: Distribution of radiative source term in the median plane (Y = 0.5) at different mole fractions of water vapor. Sources are normalized by 4σT 4 re f /L.

4. 3 . 1 Figure 4 . 29 :

 31429 Figure 4.29: Profiles of vertical velocities w U re f at different Z-crosslines in the median plane (Y = 0.5) at different concentration of water vapor.

(a) x H 2 O

 2 = 0.05 : X=0.2 (b) x H 2 O = 0.10 : X=0.2 (c) x H 2 O = 0.20 : X=0.2

Figure 4 . 30 :

 430 Figure 4.30: Profiles of horizontal velocities u U re f at different X-crosslines in the median plane (Y = 0.5) at different concentration of water vapor.

Figure 4 .

 4 Figure 4.31 represents the velocity vectors in the mid-depth plane (Y = 0.5) for three concentrations of the absorbing component. They display the typical patterns explaining the formation of the plume. The fluid is accelerated along the lateral surfaces of the hot body, goes up and then combines above the top surface of the obstacle.Here, the fluid between the plume and its surrounding is pushed upward by the buoyancy force created by the temperature difference. The hot fluid moves along the ceiling of the enclosure and then flows down near the cold walls. Besides, it is clearly observed that the plume broadens and the boundary layers near vertical walls get thicker when the radiation effects are present (compared to the transparent cases).

4. 3 .Figure 4 . 31 :

 3431 Figure 4.31: Velocity field in the median plane (Y = 0.5) at different mole fractions of water vapor. Velocities are normalized by U re f .

(a) x H 2 O

 2 = 0.05: Transparent medium (b) x H 2 O = 0.05: Participating medium (c) x H 2 O = 0.10: Transparent medium (d) x H 2 O = 0.10: Participating medium (e) x H 2 O = 0.20: Transparent medium (f) x H 2 O = 0.20: Participating medium

Figure 4 . 32 :

 432 Figure 4.32: Flow lines in the median plane (Y = 0.5) at different mole fractions of water vapor.

4. 3 .

 3 Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture 73

  (a) x H 2 O = 0.05: Transparent medium (b) x H 2 O = 0.05: Participating medium (c) x H 2 O = 0.10: Transparent medium (d) x H 2 O = 0.10: Participating medium (e) x H 2 O = 0.20: Transparent medium (f) x H 2 O = 0.20: Participating medium

Figure 4 . 33 :

 433 Figure 4.33: Representation of Q-criterion at Q = 0.02 in the cavity at different mole fractions of water vapor. Values are normalized by U 2 re f /L 2 .

  (a) x H 2 0 = 0.05 (b) x H 2 0 = 0.10 (c) x H 2 0 = 0.20

Figure 4 . 34 :

 434 Figure 4.34: Local convective Nusselt number along the vertical centerline of any lateral wall of the cavity.

4. 3 .

 3 Coupling between Pure Thermal Convection and Radiation in a Real Gas mixture 75 (a) x H 2 0 = 0.05 (b) x H 2 0 = 0.10 (c) x H 2 0 = 0.20

Figure 4 . 35 :

 435 Figure 4.35: Local convective Nusselt number along vertical center line of each lateral surfaces of the obstacle.

(a) x H 2 0

 2 = 0.05 (b) x H 2 0 = 0.10 (c) x H 2 0 = 0.20

Figure 4 . 36 :

 436 Figure 4.36: Local convective Nusselt number along horizontal centerline of the obstacle top surface.

2 the 2 α √ Ra . 5 . 1 . 1

 22511 reference length L, the reference velocity U re f = α √ Ra L and the reference time L Mass-to-Thermal Buoyancy ratio N = 1

= 2 Figure 5 . 1 :

 251 Figure 5.1: Thermal field T-T re f T h -T c in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating cases, N = 1.

Figures 5 .

 5 Figures 5.1 a-f describe the thermal field in the median plane of the cavity (Y = 0.5) for different fluid opacities ranging from a transparent medium (τ = 0) to τ = 2. At low value of τ (optically thin limit), a nearly vertical stratification is established between the plume and the boundary layers along the cavity walls. No obvious change in this pattern occurs below τ = 0.2 or even τ = 0.5. That was already observed in the case of combined radiation and pure thermal convection (homogeneous mixture). As the optical thickness goes beyond unity, we clearly recognize

8 Figure 5 . 2 : 1 0Figure 5 . 3 :Figure 5 . 4 :Figure 5 . 5 :

 8521535455 Figure 5.2: Temperature T-T re f T h -T c profiles along Z-crosslines in the median plane (Y = 0.5): cooperating flow, N = 1

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 0.5. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 8 :

 58 Figure 5.8: Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 1, τ = 2. Sources are normalized by 4σT 4 re f /L.

Figure 5 .

 5 Figure 5.4, 5.5, 5.6, 5.7, 5.8 display the values of the radiative source term in the median plane (Y = 0.5) of the cavity. We recall that negative values correspond to the regions where the fluid emits more than it absorbs and conversely for positive values. The observed distributions are representative of double diffusive convection because of the concentration-dependent absorption coefficient. There are regions where κ is very low (even 0), and where radiation, consequently, has little effects on the fluid. This phenomenon distinguishes the combined radiation -double diffusive convection from radiation and pure thermal convection.

2 Figure 5 . 9 :

 259 Figure 5.9: Vector field in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 1. Velocities are normalized by U re f .

8 Figure 5 . 10 : 88 Chapter 5 . 2 Figure 5 . 11 :

 85108852511 Figure 5.10: Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 1.

(a) 2 Figure 5 . 12 :

 2512 Figure 5.12: Iso-surface of the Q-criterion at the value 0.01 in the cavity. Values are normalized by U 2 re f /L 2 .

5 (e) τ = 1 (f) τ = 2 Figure 5 . 13 :

 512513 Figure 5.13: Concentration field C-Cre f C h -C l in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 1.

8 Figure 5 . 14 : 5 Figure 5 . 15 :

 85145515 Figure 5.14: Concentration C-Cre f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 1.

2 Figure 5 . 16 :

 2516 Figure 5.16: Velocity vectors in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 2. Velocities are normalized by U re f .

8 Figure 5 . 17 : 2 (a) X = 0. 2 Figure 5 . 18 :

 851722518 Figure 5.17: Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2.

= 2 Figure 5 . 19 : 5 (e) τ = 1 (f) τ = 2 Figure 5 . 20 :

 2519512520 Figure 5.19: Iso-surface of the Q-criterion at the value 0.01 in the cavity. Values are normalized by U 2 re f /L 2 .

  5) and high (Z = 0.8) positions (see figures 5.21 a, 5.22 a, 5.23 a, 5.24 a and 5.25 a) reduces its temperature level (see figure 5.26 b,c) while the boundary layers around the obstacle at low altitude (Z = 0.1) are warmed up (see figure 5.26 a) by absorption (see figures 5.21 b, 5.22 b, 5.23 b, 5.24 b and 5.25 b). These two effects create the typical structure of the thermal field (see figure 5.20) already found when N = 1.

Figure 5 . 21 :

 521 Figure 5.21: Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 0.1. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 22 :Figure 5 . 23 :

 522523 Figure 5.22: Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 0.2. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 24 : 5 Figure 5 . 27 :

 5245527 Figure 5.24: Distribution of radiative source term in the median plane (Y = 0.5): cooperating case, N = 2, τ = 1. Sources are normalized by 4σT 4 re f /L.

= 2 Figure 5 . 28 :

 2528 Figure 5.28: Concentration field in the median plane of the cavity (Y = 0.5) for different fluid opacities: cooperating case, N = 2.

8 Figure 5 . 29 : 2 (b) X = 0. 5 Figure 5 . 30 :

 852925530 Figure 5.29: Concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2.

2 Figure 5 . 31 :

 2531 Figure 5.31: Velocity vectors in the median plane (Y = 0.5) of the cavity for different fluid opacities: opposing case, N = -1. Velocities are normalized by U re f .

Figure 5 . 34 :

 534 Figure 5.34: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.1. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 35 :

 535 Figure 5.35: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.2. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 36 :

 536 Figure 5.36: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 0.5. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 37 :Figure 5 . 38 :

 537538 Figure 5.37: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -1, τ = 1. Sources are normalized by 4σT 4 re f /L.

= 2 Figure 5 . 39 :

 2539 Figure 5.39: Thermal field T-T re f T h -T c in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -1.

5 (e) τ = 1 (f) τ = 2 Figure 5 . 40 :

 512540 Figure 5.40: Distribution of concentration C-C re f C h -C l in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -1.

= 2 Figure 5 . 41 :

 2541 Figure 5.41: Vertical velocities w U re f across the mid-height plane (Z = 0.5) for different fluid opacities: opposing case, N = -1.

Figure 5 .

 5 Figure 5.41 displays the vertical velocity distribution across the mid-height horizontal plane (Z = 0.5). It more clearly illustrate the flow structure, and especially the redistribution of the vertical movements that ensures the mass conservation. Close to its origin, the plume displays a quasi square cross section but turned by 45 • with respect to the obstacle geometry. This pattern is generated by the fluid input due to the ascending boundary layers along the obstacle verticle surfaces and by the interactions with recirculation flows in the corners of the cavtiy.

= 2 Figure 5 . 42 :

 2542 Figure 5.42: Iso-surface of the Q-criterion at the value 0.002 in the cavity for different fluid opacities: opposing case, N = -1. Values are normalized by U 2 re f /L 2 .

(a) 2 Figure 5 . 43 :

 2543 Figure 5.43: Velocity vectors in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -2. Velocities are normalized by U re f .

2 (c) Z = 0. 8 Figure 5 . 44 : 2 Figure 5 . 45 :

 285442545 Figure 5.44: Vertical velocity w U re f profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -2.

= 2 Figure 5 . 46 :

 2546 Figure 5.46: Thermal field T-T re f T h -T c in the median plane (Y = 0.5) for different fluid opacities: opposing case, N = -2.

8 Figure 5 . 47 :

 8547 Figure 5.47: Temperature T-T re f T h -T c profiles along different Z-crosslines in the median plane (Y = 0.5): opposing case, N = -2.

Figure 5 . 50 :

 550 Figure 5.50: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 0.2. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 51 :

 551 Figure 5.51: Distribution of radiative source in the median plane (Y = 0.5): opposing case, N = -2, τ = 0.5. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 52 :

 552 Figure 5.52: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 1. Sources are normalized by 4σT 4 re f /L.

Figure 5 . 53 :

 553 Figure 5.53: Distribution of radiative source term in the median plane (Y = 0.5): opposing case, N = -2, τ = 2. Sources are normalized by 4σT 4 re f /L.

= 2 Figure 5 . 54 :

 2554 Figure 5.54: Concentration C-C re f C h -C l field in the median plane (Y = 0.5) for different fluid opacities, N = -2.

5 (e) τ = 1 (f) τ = 2 Figure 5 . 57 :

 512557 Figure 5.57: Iso-surface of the Q-criterion at the value 0.002 in the cavity for different fluid opacities: opposing case, N = -2. VValues are normalized by U 2 re f /L 2 .

5. 2 . 1 . 1 Figure 5 . 58 :

 211558 Figure 5.58: Thermal field T-T re f T h -T c in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor.

Figure 5 . 1 Figure 5 . 59 :

 51559 Figure 5.58 represents the thermal field in the median plane (Y = 0.5) in both cases of transparent and participating media at different average mole fractions of water vapor. Gas radiation does not induce any significant change, except a slight

(a) x H 2 O 8 ( 8 ( 5 Figure 5 . 60 :

 2885560 Figure 5.60: Temperature T-T re f T h -T c profiles along different X-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor.

(Figure 5 . 61 :

 561 Figure 5.61: Distribution of radiative source term in the median plane (Y = 0.5) at different mole fractions of water vapor. Sources are normalized by 4σT 4 re f /L.

(a) x H 2 O

 2 = 0.05: Transparent medium (b) x H 2 O = 0.05: Real gas (c) x H 2 O = 0.10: Transparent medium (d) x H 2 O = 0.10: Real gas (e) x H 2 O = 0.20: Transparent medium (f) x H 2 O = 0.20: Real gas

Figure 5 .

 5 Figure 5.62: Concentration C-C re f C h -C l field in the median plane of the cavity (Y = 0.5) at different mole fractions of water vapor.

Figures 5 .

 5 Figures 5.62 displays the concentration distribution in the median plane (Y = 0.5) of the cavity. This field essentially remains insensitive to gas radiation.

(a) x H 2 O 8 ( 8 ( 5 Figure 5 . 64 :

 2885564 Figure 5.64: Concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor.

Figure 5 . 65 :

 565 Figure 5.65: Velocity vector in the median plane of the cavity (Y = 0.5) at different mole fraction of water vapor. Velocities are normalized by U re f .

Figure 5 .

 5 Figure 5.65 represents the velocity field in the median plane (Y = 0.5) of the cavity. A global circulation is established within the enclosure, combining a climbing plume above the hot obstacle and descending boundary layers along the vertical walls. Including gas radiation does not significantly change these distributions (compared to the transparent cases).

1 Figure 5 . 66 :

 1566 Figure 5.66: Vertical velocities w U re f profiles at different Z-crosslines in the median plane (Y = 0.5) at different average mole fractions of water vapor.

(a) x H 2 O 2 Figure 5 . 67 : 5 . 2 . 1 . 2 20 Figure 5 . 68 :

 22567521220568 Figure 5.67: Horizontal velocities u U re f profiles at different X-crosslines in the median plane (Y=0.5) at different average mole fractions of water vapor.

Figure 5 .

 5 Figure 5.68 displays the distribution of the convective Nusselt number along the center line of any vertical wall of the cavity. It is observed that gas radiation reduces this parameter. Indeed, although the dynamic structure remains unchanged at all the considered mole fractions, the temperature of the fluid close to the vertical boundaries is reduced by emission. It decreases the thermal gradient and this effect gets stronger as the quantity of water vapor increases.

( 20 Figure 5 . 69 :

 20569 Figure 5.69: Local convective Nusselt number along a vertical center line of each lateral surfaces of the obstacle.

20 Figure 5 . 70 :

 20570 Figure 5.70: Local convective Nusselt number along an horizontal centerline of the obstacle top surface.

10 Figure 5 . 71 :

 10571 Figure 5.71: Time evolution of temperature at the center point of the cavity (X, Y, Z) = (0.5, 0.5, 0.5) at x CO 2 =0.10

( a ) 1 ( b ) 1 ( c ) 666 Figure 5 . 72 :

 a1b1c666572 Figure 5.72: Flow lines on the median plane of the cavity (Y = 0.5) at different instants.

Figure 5 . 73 :

 573 Figure 5.73: Time evolution and power spectrum of the temperature signal at the point (X, Y, Z) = (0.5, 0.5, 0.5) in temporal range [400 : 3000] at x CO 2 = 0.10: transparent medium.

Figure 5 .

 5 Figure 5.75: Temperature T-T re f T h -T c profile at Z = 0.5 in the median plane at t = 775

Figure 5 . 1 5. 2 .Figure 5 . 77 : 5 . 2 . 2 . 2 Figure 5 . 78 :Figure 5 .

 51257752225785 Figure 5.76: Temperature T-T re f T h -T c profile at Z = 0.1 in the range X = [0.35 : 0.65] at t = 1

Figure 5 . 79 : 152 Chapter 5 .

 5791525 Figure 5.79: Vertical velocity w U re f , temperature T-T re f T h -T c and concentration C-C re f C h -C l profiles along different Z-crosslines in the median plane (Y = 0.5) at x CO 2 = 0.20

( 5 Figure 5 . 80 :Figure 5 . 81 :Figure 5 . 82 :Figure 5 . 83 : 154 Chapter 5 .

 55805815825831545 Figure 5.80: Horizontal velocity u U re f , temperature T-T re f T h -T c and concentration C-C re f C h -C l profiles along different X-crosslines in the median plane (Y = 0.5) at x CO 2 = 0.20

Figure 5 . 84 :

 584 Figure 5.84: Iso-surface of the Q-criterion at the value 0.01 in the cavity at x CO2 = 0.20. Values are normalized by U 2 re f /L 2

  if (molecule == 1){ YY[0] = 0; YY[1] = 0.05; YY[2] = 0.1; YY[3] = 0.2; YY[4] = 0.3; YY[5] = 0.4; YY[6] = 0.6; YY[7] = 0.8; YY[8] = 1.0; j = locate(YY,8,Y); i = j*55664+m*1988+l*71+k;Fint[0]=Fdata1[i]+(Fdata2[i]-Fdata1[i])*(P-PP[P1])/(PP[P1+1]-PP[P1]); Fint[1]=Fdata1[i+55664]+(Fdata2[i+55664]-Fdata1[i+55664])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[2]=Fdata1[i+1]+(Fdata2[i+1]-Fdata1[i+1])*(P-PP[P1])/(PP[P1+1]-PP → [P1]); Fint[3]=Fdata1[i+55665]+(Fdata2[i+55665]-Fdata1[i+55665])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[4]=Fdata1[i+71]+(Fdata2[i+71]-Fdata1[i+71])*(P-PP[P1])/(PP[P1 → +1]-PP[P1]); Fint[5]=Fdata1[i+55735]+(Fdata2[i+55735]-Fdata1[i+55735])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[6]=Fdata1[i+72]+(Fdata2[i+72]-Fdata1[i+72])*(P-PP[P1])/(PP[P1 → +1]-PP[P1]); Fint[7]=Fdata1[i+55736]+(Fdata2[i+55736]-Fdata1[i+55736])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[8]=Fdata1[i+1988]+(Fdata2[i+1988]-Fdata1[i+1988])*(P-PP[P1])/(PP → [P1+1]-PP[P1]); Fint[9]=Fdata1[i+57652]+(Fdata2[i+57652]-Fdata1[i+57652])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[10]=Fdata1[i+1989]+(Fdata2[i+1989]-Fdata1[i+1989])*(P-PP[P1])/( → PP[P1+1]-PP[P1]); Fint[11]=Fdata1[i+57653]+(Fdata2[i+57653]-Fdata1[i+57653])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[12]=Fdata1[i+2059]+(Fdata2[i+2059]-Fdata1[i+2059])*(P-PP[P1])/( → PP[P1+1]-PP[P1]); Fint[13]=Fdata1[i+57723]+(Fdata2[i+57723]-Fdata1[i+57723])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[14]=Fdata1[i+2060]+(Fdata2[i+2060]-Fdata1[i+2060])*(P-PP[P1])/( → PP[P1+1]-PP[P1]); Fint[15]=Fdata1[i+57724]+(Fdata2[i+57724]-Fdata1[i+57724])*(P-PP[P1]) → /(PP[P1+1]-PP[P1]); Fint[0]=Fint[0]+(Fint[1]-Fint[0])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[1]=Fint[2]+(Fint[3]-Fint[2])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[2]=Fint[4]+(Fint[5]-Fint[4])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[3]=Fint[6]+(Fint[7]-Fint[6])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[4]=Fint[8]+(Fint[9]-Fint[8])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[5]=Fint[10]+(Fint[11]-Fint[10])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[6]=Fint[12]+(Fint[13]-Fint[12])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[7]=Fint[14]+(Fint[15]-Fint[14])*(Y-YY[j])/(YY[j+1]-YY[j]); Fint[0]=Fint[0]+(Fint[1]-Fint[0])*(Cabs-C[k])/(C[k+1]-C[k]);

  1.1 Problem configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 SLW calculation diagram for j th gray gas (from Solovjov, Webb, and André [146]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Schematic representation of the Reference Approach (from Solovjov, Webb, and André [147]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schematic representation of the Rank Correlated (from Solovjov et al.

[START_REF] V P Solovjov | The rank correlated SLW model of gas radiation in nonuniform media[END_REF]

) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Presentation of step scheme for calculation of intensity at one face . . . 3.2 2D -Square Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 2D -Radiant flux at the cold wall, opposed to the hot wall with κ = 0.25 3.4 2D -Radiant flux at the cold wall, opposed to the hot wall with κ = 1 . 3.5 Implantation of SLW model in the Code Saturne . . . . . . . . . . . . . 3.6 Differentially heated cavity . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Calculation domain (Paroncini and Corvaro [13]) . . . . . . . . . . . . . 3.8 Domain of calculation (Ra

  = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Radiative boundary conditions and radiative properties of the medium 38 3.7 Mean convective Nusselt number on the hot wall (air -H 2 O mixture, Ra = 10 6 , Pr = 0.707) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 Mean Nusselt number on the hot wall (Ra = 10 7 , Le = 1 and Pr = 0.71) 38 3.9 Mean convective Nusselt number on the hot wall for an air -CO 2 mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.10 Mean Sherwood number on the hot wall for an air -CO 2 mixture . . . 39 3.11 Mean convective Nusselt number on the hot wall for an air -H 2 O mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.12 Mean Sherwood number on the hot wall for an air -H 2 O mixture . . . 39 3.13 Mean Nusselt number on the lateral wall of the heated obstacle . . . . 40 3.14 Mean Nusselt number on different walls (Ra = 2 • 10 5 , Pr = 0.71) . . . 41 Average convective and total Nusselt numbers on any obstacle vertical surface: double diffusive convection and radiation in an air -H 2 O mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.4 Average convective and total Nusselt numbers on the obstacle horizontal surface: double diffusive convection and radiation in an air -H 2 O mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xxiv 5.5 Averaged Sherwood number on any cavity lateral wall: double diffusive convection and radiation in an air -H 2 O mixture. . . . . . . . . . 5.6 Averaged Sherwood number on any obstacle vertical wall: double diffusive convection and radiation in an air -H 2 O mixture. . . . . . . 5.7 Averaged Sherwood number on the obstacle horizontal wall: double diffusive convection and radiation in an air -H 2 O mixture. . . . . . . 5.8 Configuration parameters: air -CO 2 mixture . . . . . . . . . . . . . . . 5.9 Average convective and total Nusselt number on active walls of the cavity and of the obstacle for double diffusive convection and radiation in the air -CO 2 mixture at x CO 2 = 0.20 . . . . . . . . . . . . . . . . 5.10 Averaged Sherwood number on active walls of the cavity and of the obstalce for double diffusive convection in the air -CO 2 mixture at x CO 2 = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Changes in the S 8 quadrature . . . . . . . . . . . . . . . . . . . . . . . .

	3.1 Integrated wall flux, κ = 0.25, quadrature S 8 . . . . . . . . . . . . . . . . 34 3.2 Integrated wall flux, κ = 1, quadrature S 8 . . . . . . . . . . . . . . . . . 34 3.3 Mean Nusselt number on the hot wall . . . . . . . . . . . . . . . . . . . 37 3.4 Mean Nusselt Number on the hot wall (Pr = 0.71) . . . . . . . . . . . . 37 3.5 Mean Nusselt number on the hot wall (Pr = 0.71, Pl = 0.02, θ 0 = 1.5, 4.1 Maximum horizontal and vertical velocities for different mesh sizes. . 44 4.2 Mean Nusselt numbers at the bounding surfaces of the enclosure. . . . 45 4.3 Mean Nusselt numbers at the bounding surfaces of the hot source. . . 45 4.4 Mean convective and radiative Nusselt numbers over the vertical sur-faces of the enclosure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Mean convective and radiative Nusselt numbers at a lateral wall of the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.6 Description of the cases of calculation. . . . . . . . . . . . . . . . . . . . 64 4.7 Average convective and total Nusselt number along any vertical wall of the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.8 Average convective and total Nusselt number along any vertical wall of the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.9 Average convective and total Nusselt number along horizontal upper wall of the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Configuration parameters: air -H 2 O mixture. . . . . . . . . . . . . . . 129 5.2 Average convective and total Nusselt numbers on any vertical cavity wall: double diffusive convection and radiation in an air -H 2 O mixture.142 List of Symbols 5.3 xxv A area of the considered surface m 2

i

  • sr -1

	l	obstacle size	m
	L	cavity size	m
	Le	Lewis number Le = α D
	n	unit normal vector
	N	mass to thermal buoyancy ratio N

Table 3 . 1 :

 31 Integrated wall flux, κ = 0.25, quadrature S 8

		0.26
		0.25
		0.24
		0.23
	Net flux (-)	0.19 0.2 0.21 0.22
		0.18
		0.17		Crosbie Our Code 2D S8 Balsara
		0.16
		0.15
		0	0.2	0.4	0.6	0.8	1
				Z (-)

Table 3 . 2 :

 32 Integrated wall flux, κ = 1, quadrature S 8

3.2.1.1 Pure Thermal Convection First

  , we have performed the calculation of natural convection at different Rayleigh numbers while keeping the Prandlt number at 0.71. There is no radiation (transparent medium, non-emitting walls). Our calculations are run using an uniform grid of 80 × 80 cells. Many reference results are available (Le Quéré[START_REF] Le | Accurate solutions to the square thermally driven cavity at high Rayleigh number[END_REF], Tric, Labrosse, and Betrouni[START_REF] Tric | A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF],...). Here, the solutions are compared to the data provided by De Vahl Davis[START_REF] De | Natural convection of air in a square cavity: a bench mark numerical solution[END_REF] and are presented below. A good agreement is found between our works and this reference.

	Ra Our work De Vahl Davis [29] (Relative Difference %))
	10 3	1.113	1.117 (0.36)
	10 4	2.235	2.238 (0.13)
	10 5	4.507	4.509 (0.04)
	10 6	8.816	8.817 (0.01)

Table 3 . 3 :

 33 Mean Nusselt number on the hot wall

  .4).

	Ra Our work Colomer[30]	Fusegi[31]
	10 4	2.059	2.030 (1.42%) 2.100 (1.95 %)
	10 5	4.365	4.334 (0.71 %) 4.361 (0.09%)
	10 6	8.717	8.862 (1.63%) 8.770 (0.60 %)

Table 3 . 4

 34

: Mean Nusselt Number on the hot wall (Pr = 0.71)

Table 3 . 5 :

 35 [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF]. Code Saturne and Radiative Calculation Mean Nusselt number on the hot wall (Pr = 0.71, Pl = 0.02, θ 0 = 1.5, i = 1)

	τ	S N	Convective Nusselt Our work Ref.[1] Ref.[32] Our work Ref.[1] Ref.[32] Total Nusselt
	0.2 S 4	36.01	37.40	37.40	46.50	46.11	46.05
	1	S 8	31.88	31.25	31.25	39.38	38.93	38.81
	5	S 4	24.58	23.64	23.57	31.47	31.76	31.59

Table 3 . 6 :

 36 Radiative boundary conditions and radiative properties of the medium

	Case Our work	Billaud[6]	Soucasse[33]
	A	8.64	8.65 (0.11 %) 8.64 (0.0 %)
	B	7.24	7.42 (2.42 %) 7.55 (4.10 %)
	C	7.93	8.10 (2.09 %) 8.47 (6.37 %)
	D	8.48	8.01 (5.86 %) 8.48 (0.0 %)

Table 3 .7: Mean

 3

convective Nusselt number on the hot wall (air -H 2 O mixture, Ra = 10 6 , Pr = 0.707)

Table 3 .

 3

8: Mean Nusselt number on the hot wall (Ra = 10 7 , Le = 1 and Pr = 0.71)

Table 3 . 9 :

 39 Mean convective Nusselt number on the hot wall for an air -CO 2 mixture

	x CO 2 Our work Cherifi [35] Relative difference (%)
	0.05	19.53	19.38	0.77
	0.11	21.65	21.44	0.98
	0.20	23.96	23.77	0.80

Table 3 . 10 :

 310 Mean Sherwood number on the hot wall for an air -CO 2 mixture

	x H 2 O Our work Cherifi [35] Relative difference (%)
	0.05	15.65	15.78	0.82
	0.11	16.51	16.75	1.43
	0.20	18.17	18.03	0.77

Table 3 .11:

 3 Mean convective Nusselt number on the hot wall for an air -H 2 O mixture

	x H 2 O Our work Cherifi [35] Relative difference (%)
	0.05	15.82	15.74	0.5
	0.11	16.85	16.79	0.35
	0.20	19.15	19.02	0.68

Table 3 . 12 :

 312 Mean Sherwood number on the hot wall for an air -H 2 O mixture

Table 3 . 13 :

 313 Mean Nusselt number on the lateral wall of the heated obstacle

Table 3 .14: Mean

 3

	Relative difference (%)

Nusselt number on different walls (Ra = 2 • 10 5 , Pr = 0.71)

Table 4 . 1 :

 41 Maximum horizontal and vertical velocities for different mesh sizes.

Table 4 . 2 :

 42 Mean Nusselt numbers at the bounding surfaces of the enclosure.

	Mesh Front wall Back wall Left wall Right wall Top wall
	80 3	38.664	38.664	38.664	38.664	14.343
	100 3	38.508	38.508	38.508	38.508	14.166
	120 3	38.492	38.492	38.492	38.492	14.132

Table 4 . 3 :

 43 Mean Nusselt numbers at the bounding surfaces of the hot source.

Table 4 . 4 :

 44 [START_REF] Colomer | Coupled radiation and natural convection: Different approaches of the SLW model for a non-gray gas mixture[END_REF]. Coupling between Pure Thermal Convection and Radiation Mean convective and radiative Nusselt numbers over the vertical surfaces of the enclosure.

	S N Convective Nusselt number Radiative Nusselt number
	S 6	0.976	11.073
	S 8	0.989	11.055
	S 12	0.995	11.053

Table 4 . 5 :

 45 Mean convective and radiative Nusselt numbers at a lateral wall of the obstacle.

	g	Convective Nusselt number Radiative Nusselt number
	5	36.59	211.34
	11	36.68	211.15
	Relative Difference (%)	0.24	0.09

  The characteristic parameters of the simulations are presented in the table below • 10 6 7.251 • 10 -1 4.394 • 10 -3 10 4.693 • 10 6 7.351 • 10 -1 4.392 • 10 -3 20 4.753 • 10 6 7.553 • 10 -1 4.389 • 10 -3

	64	Chapter 4. Coupling between Pure Thermal Convection and Radiation
	(Table 4.6):
		x (%)	Ra	Pr	Pl
		5	4.663

Table 4 . 6 :

 46 Description of the cases of calculation.

Table 4 .

 4

		Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	1.67	1.34	12.47	12.38
	10	1.67	1.27	12.48	12.33
	20	1.68	1.24	12.50	12.26

7: Average convective and total Nusselt number along any vertical wall of the cavity.

Table 4 . 8 :

 48 Average convective and total Nusselt number along any vertical wall of the obstacle.

		Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	38.28	37.09	252.07	250.25
	10	38.34	36.75	252.19	249.31
	20	38.57	36.59	252.59	247.93
	x H 2 O (%) Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	14.12	14.50	239.91	238.26
	10	14.17	14.90	240.03	236.92
	20	14.30	15.85	240.34	234.74

Table 4 . 9 :

 49 Average convective and total Nusselt number along horizontal upper wall of the obstacle.

Table 5 . 1 :

 51 Configuration parameters: air -H 2 O mixture.

Table 5 . 2 :

 52 Average convective and total Nusselt numbers on any vertical cavity wall: double diffusive convection and radiation in an air -H 2 O mixture.

	x(%) Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	41.77	42.07	255.56	255.29
	10	45.27	45.67	259.12	258.45
	20	50.93	51.55	264.95	263.64

Table 5 . 3 :

 53 Average convective and total Nusselt numbers on any obstacle vertical surface: double diffusive convection and radiation in an air -H 2 O mixture.

	x(%) Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	14.72	15.65	240.51	239.27
	10	15.32	17.19	241.18	238.88
	20	16.63	19.82	242.67	238.33

Table 5 . 4 :

 54 Average convective and total Nusselt numbers on the obstacle horizontal surface: double diffusive convection and radiation in an air -H 2 O mixture.

Table 5 . 5 :

 55 Averaged Sherwood number on any cavity lateral wall: double diffusive convection and radiation in an air -H 2 O mixture.

	x(%)	Sherwood number
		Transparent Real gas mixture
	5	41.144	41.204
	10	44.081	44.081
	20	48.850	48.812

Table 5 . 6 :

 56 Averaged Sherwood number on any obstacle vertical wall: double diffusive convection and radiation in an air -H 2 O mixture.

	x(%)	Sherwood number
		Transparent Real gas mixture
	5	13.769	13.731
	10	14.283	14.246
	20	15.239	15.198

Table 5 . 7 :

 57 Averaged Sherwood number on the obstacle horizontal wall: double diffusive convection and radiation in an air -H 2 O mixture. Air -CO 2 mixtureWe now turn to a case of the double diffusive convection coupled to gas radiation in an air -CO 2 mixture. As introduced in the beginning of the chapter, this combination generates opposite flows inside the cavity. The characteristic dimensionless numbers related to this configuration are listed in the table 5.8. The average mole fractions of the CO 2 are still defined as x =

	C re f * R * T re f P	and are given two different

5.2.2

Table 5 . 8 :

 58 Configuration parameters: air -CO 2 mixture

Table 5 . 9 :

 59 Average convective and total Nusselt number on active walls of the cavity and of the obstacle for double diffusive convection and radiation in the air -CO 2 mixture at x CO 2 = 0.20

Table 5 . 10 :

 510 Averaged Sherwood number on active walls of the cavity and of the obstalce for double diffusive convection in the air -CO 2 mixture at x CO 2 = 0.20

  // reference molar density cs_real_t C[START_REF] Shen | Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials[END_REF]; // discrete absorption cross section for tabulated → data cs_real_t PP[START_REF] Hernandez | Natural convection in thermal plumes emerging from a single heat source[END_REF]; // discrete pressure for tabulated data cs_real_t T[START_REF] Edf | Code_Saturne version 5.0.0 practical user's guide[END_REF]; // discrete temperature for tabulated data cs_real_t YY[START_REF] Xin | Numerical simulations of natural convection around a line-source[END_REF]; // discrete molar fraction for tabulated data cs_real_t Fdata1[500976]; // Storage of ALBDF

	int P1;	// Initialize discrete values of concentration, pressure, temperature
	cs_real_t Ru ; //universal gas constant → and molar fraction
	cs_real_t xaref ; // reference molar fraction void data_set(void);
	cs_real_t Tref ; // reference temperature
	cs_real_t P ; // Normalized pressure // Calculate ALBDF at a thermodynamic state of a binary air-absorbnant
	cs_real_t *ka; // reference absorption coeffiecient → component mixture
	cs_real_t *xa; // cell center molar fraction for first absorbant cs_real_t FCC(cs_real_t Cabs,
	→ component	cs_real_t Tg,
	cs_real_t *xa1;	cs_real_t Tb,
	cs_real_t kmin; // min absorption cross section cs_real_t Y,
	cs_real_t kmax; // max absorption cross section cs_real_t P,
	cs_real_t *xabd; // boundary molar fraction for first absorbant int P1,
	→ component	int molecule,
	cs_real_t *xabd1;	cs_real_t C[],
	cs_real_t *tempbd; // boundary temperature cs_real_t Fdata1[],
	cs_real_t Fdata2[]); // Calculate ALBDF for general case. cs_real_t FMIX(cs_real_t Cabs, cs_real_t Tg, cs_real_t Tb, cs_real_t Y1, cs_real_t Nref; cs_real_t Fdata2[500976]; cs_real_t Y2,
	cs_real_t Fdata3[55664];	cs_real_t P,
	cs_real_t Fdata4[55664];	int P1,
			cs_real_t C[]);
	cs_real_t xgs[100]; // parameter of gaussian quadrature
	cs_real_t wgs[100]; // Determine the absorption cross section equivalent to a ALBDF value
		→ at a thermaldynamic state
	// initialize required data for SLW model cs_real_t CFMIX(cs_real_t F,
	void setup_uservalue(void);	cs_real_t Tg,
			cs_real_t Tb,
			cs_real_t Y1,
			cs_real_t Y2,
	// locate index of a value of an array	cs_real_t P,
	int locate(cs_real_t xx[],	int P1,
		int n,	cs_real_t C[]);
		cs_real_t x);
		// SLW model based on Rank-Correlated associated to the lookup table
	// Generate parameter of gaussian quadrature → method
	void gauss(int Ngmax, void SLW_RC_LBL(cs_real_t tempk[],
		cs_real_t x[],	cs_real_t tempbd[],
		cs_real_t w[]);	int Ngmax,
			cs_real_t Tref,
	// Import necessary spectral data	cs_real_t Y1REF,
	void data_read(cs_real_t Fdata1[],	cs_real_t Y2REF,
		cs_real_t Fdata2[], cs_real_t Y1LOC[],
		int molecule, cs_real_t Y2LOC[],
		int P1);	cs_real_t Y1FLOC[],
			cs_real_t Y2FLOC[],
			unsigned long n_cells,

  1 2 (∇v -(∇v) T ) is known as vorticity tensor.

	Noting that:
						∂u	∂u	∂u	
						∂x	∂y	∂z
				∇v =	 	∂v ∂x ∂w	∂v ∂y ∂w	∂v ∂z ∂w	 	(C.2)
						∂x	∂y	∂z
	We further obtain:
				∂u ∂x	1 2 ( ∂u ∂y + ∂v ∂x ) 1 2 ( ∂u ∂z + ∂w ∂x )	
	S =	 	1 2 ( ∂u ∂y + ∂v ∂x ) 1 2 ( ∂u ∂z + ∂w ∂x ) 1 2 ( ∂v ∂z + ∂w ∂v ∂y ∂y )	1 2 ( ∂v ∂z + ∂w ∂y ) ∂z ∂w	 	(C.3)
				0		1 2 ( ∂u ∂y -∂v ∂x ) 1 2 ( ∂u ∂z -∂w ∂x )	
	Ω =	 	-1 2 ( ∂u ∂y -∂v ∂x ) -1 2 ( ∂u ∂z -∂w ∂x ) -1 2 ( ∂v 0 ∂z -∂w ∂y )	1 2 ( ∂v ∂z -∂w ∂y ) 0	 	(C.4)

s est la vecteur de position le long du chemin optique

excepté la surface en contact avec le sol de la cavité

• le rayonnement réduit le transfert thermique total, en particulier la partie convective sur les parois verticales et l'échange radiatif entre la surface supérieure de l'obstacle et les murs de la cavité.• Tous ces effets augmentent lorsque le milieu devient plus opaque (dans la gamme étudiée des épaisseurs optiques et de la fraction molaire).• En outre, dans le cas du gaz gris, lorsque l'épaisseur optique du milieu est unitaire, le rayonnement mène à un écoulement périodique. Ce mécanisme est dû à l'instabilité de cisaillement créée par l'interférence du panache ascendant et des couches limites s'écoulant vers le bas. Enfin, à τ = 2, l'écoulement devient totalement turbulent à Ra = 5 • 10 6 .

C h dépend de la fraction molaire de référence fixée dans le milieu.xv

This manuscript is divided into six main chapters. Firstly, the present chapter introduces the motivation of the thesis, its objectives and a bibliography review about some related researches in the recent decades. Then, in the second chapter, we provide more details about the mathematical models as well as the numerical methods used in this study. Chapter three will introduce Code Saturne, the CFD simulation tool used all along our work and the implementation of our own SLW model into the built-in radiative module of this code. We also present some validation tests for assessing the accuracy of our calculations in configurations with an increasing degree of complexity. The fourth chapter contains the results and analysis about the combined thermal convection and radiation in a gray gas as well as in a real gas mixture. Then, chapter five will focus on the radiation effects in many typical situations of double diffusive convection, either cooperating or opposing, in gray or real gas mixture. To end with, the concluding chapter will synthesize the main results of this study and provide perspectives for future works.

Except the surface in contact with the floor of the cavity

n is the vector pointing into the medium that it is the inner vector for the cavity walls and the outer vector at the surface of the obstacle.

This quantities are considered only at the active surfaces which consist of: the lateral walls of the cavity, the vertical and top walls of the obstacle.

This expression holds for the walls that are normal to the x-direction of the cavity. For the other bounding surfaces of the enclosure and the obstacle, it needs to be changed to adapt the normal vector and the positions of the walls.

This is the default scheme in Code Saturne and it was validated to be suitable for our calculations.

This choice can be questioned when unsteady behaviors are addressed. In that case, a specific convergence steady was conducted to ensure that the selected time step was appropriate.

These flow lines represent the trajectory of fluid particles uniformly placed in the plane Y = 0.5. Each of them is generated by integration of velocity values.

See appendix C for the definition of the Q-criterion.

This value was found to produce the best illustration of the modifications in the flow structure.

Except the bottom one, which is in contact with the floor of the cavity

If we increase the fluid opacity, we will ultimately find an opaque situation, where there are no more radiative effects.

T-Tre fT h -T c profiles along different Z-crosslines in the median plane (Y = 0.5): cooperating case, N = 2.

The nondimensional buoyancy source term in the momentum equation is ρ 0 gβ T ∆T(NC + T). When Le = 1, the conservation of species and the energy equation are identical and share the same boundary conditions. Therefore, the nondimensional T and C-fields are identical and, when N = -1, the buoyancy force goes to zero.

This quantity is defined as β T (T h -T c )(NC + T).

These parameters are defined in chapter 2

cs_real_t tempbd[], int Ngmax, cs_real_t Tref, cs_real_t Y1REF, cs_real_t Y2REF, cs_real_t Y1LOC[], cs_real_t Y2LOC[], cs_real_t Y1FLOC[], cs_real_t Y2FLOC[], unsigned long n_cells, unsigned long n_faces, cs_real_t kgi[], cs_real_t agi[], cs_real_t agbi[]){ // Declear local variables cs_real_t Cmin = 1e-4; cs_real_t Cmax = 600.0; cs_real_t CTREF[Ngmax]; cs_real_t FTREF[Ngmax]; cs_real_t FCREF[Ngmax]; cs_real_t *CTLOC;

Code Saturne S8 Fiveland Code Saturne S8 Balsara Figure 3.3: 2D -Radiant flux at the cold wall, opposed to the hot wall with κ = 0.25

Chapter 3. Code Saturne and Radiative Calculation κ = 0.25 Integration Relative difference (%) Crosbie 0.342532207 Code 2D 0.343785469 0.3658 Code Saturne Fiveland 0.315125699 8.001 Code Saturne Balsara 0.345137495 0.7606 Code Saturne S8 Fiveland Code Saturne S8 Balsara

Figure 3.4: 2D -Radiant flux at the cold wall, opposed to the hot wall with κ = 1 κ = 1 Integration Relative difference (%) Crosbie 0.215529801 Code 2D 0.21866138 1.4529 Code Saturne Fiveland 0.205275837 4.7575 Code Saturne Balsara 0.217682182 0.9986

T h -T c distribution in the median plane (Y = 0.5) for the gray medium at τ = 0.5.

Chapter 5. Double Diffusive Convection Coupled to Gas Radiation

Mass-to-Thermal Buoyancy ratio N = 2

In this case, the temperature and concentration gradients still cooperate (N > 0) but the mass driving force dominates over the thermal one (N > 1).

Mass-to-Thermal Buoyancy ratio N = -1

We now assume that the fluid supplied by the (hot) obstacle and removed by the (cold) cavity walls is heavier than the main component of the mixture. In the specific case where N = -1, the thermal and mass buoyancy forces have exactly the same magnitude (|N| = 1), but act in opposite directions (N < 0). Literally, the momentum source created by the temperature and concentration gradients cancel each other 2 (since Le = 1). As a result, the fluid remains motionless. However, this holds for a transparent medium only. When volume radiation is taken into account, the similarity of temperature and concentration distributions is broken. A flow structure is then established.

profiles along different X-crosslines in the median plane (Y = 0.5): opposing case, N = -1.

Introducing gas radiation dramatically changes the thermal field, due to the appearance of convective motions. In the lower part of the cavity (Z = 0.1), near the obstacle (figure 5.32 d), the temperature is reduced as the opacity increases (5.33 b). Indeed, although the mass driven flow is slightly accelerated (which can brings more fluid at high temperature from the obstacle to the cavity walls), the radiative emission always dominates over absorption. This effect becomes stronger with the optical thickness (see figures 5.34 a, 5.35 a, 5.36 a, 5.37 a and 5.38 a for negative values of radiative source term). It explains the cooling down of the medium. However, at intermediate and high levels (figure 5.32 e,f), an opposite trend is observed. The temperature augments as the fluid opacities varies from 0.1 to 1 (even though Appendix A

Calculation of the physical properties of a binary mixture

This appendix is using material originating from S.Laouar-Meftah's PhD thesis [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF].

The thermal physical properties of the air -CO 2 or air -H 2 O mixtures are calculated at T re f and C re f (the total pressure P is constant and equals the atmosphere pressure) by using the perfect gas law which are described below. In which, the index i designates a pure absorbing component of the mixture (CO 2 or H 2 O) whose thermal physical properties at reference state (T re f , C re f ) are given as below:

• Molar density of mixture

x i and M i stand for respectively the molar fraction and molar density of absorbing component i of the mixture.

• Mass fraction of the absorbing component i of the mixture

• Density of the mixture

• Specific heat of the mixture

• Coefficients of thermal and mass expansion Thermal expansion coefficient:

Mass expansion coefficient:

Appendix A. Calculation of the physical properties of a binary mixture

• Thermal conductivity and dynamic viscosity of the mixture The thermal conductivity (λ) and dynamic viscosity (µ) of the mixture are estimated by the formula of Wilke-Wassiljewa ( [START_REF] Poling | The properties of gases and liquids[END_REF]):

where φ air,air = φ i,i = 1 (A.8)

(A.9)

and with f is λ or µ In addition, the thermal diffusivity (α) and kinematic viscosity (ν) are obtained from the expressions:

• Mass diffusivity of a species into the environment (in this case CO 2 /H 2 O into air) is calculated by the expression of Fuller and al ( [START_REF] Poling | The properties of gases and liquids[END_REF]):

Implantation of our model in Code Saturne

We here refer to Code Saturne version 5.0.4.

B.1 Change in the directions set

Subroutine: cs_rad_transfer_dir(void) in file cs_rad_transfer_dir.c

In this file, we have changed the values associated to discrete directions and their weights for the S 8 quadrature by the ones provided in Balsara [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[END_REF]. They are presented below:

Parameter Old value New value vec[0] 0.1422555 0.1691276797 vec [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF] 0.5773503 0.5773502692 vec [START_REF] Tan | Combined radiation and natural convection in a twodimensional participating square medium[END_REF] 0.8040087 0.7987881413 vec [START_REF] Colomer | Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method[END_REF] 0.9795543 0.9709745908 weight[0] 0.0992284 0.1598388991 weight [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF] 0.1712359 0.146138939 weight [START_REF] Tan | Combined radiation and natural convection in a twodimensional participating square medium[END_REF] 0.4617179 0.173346115 #include "cs_defs.h" #include "stdio.h" #include "math.h" #include "user.h" #include "bft_error.h" #include "bft_mem.h" #include "bft_printf.h" // This function defines the necessary parameters for SLW model void setup_uservalue(void){ speca = 2; // 1: H2O if (P1 == 0){ f1 = fopen("/ALBDF_DATA/h2o_p0_1.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p0_25.txt","r"); } if (P1 == 1){ f1 = fopen("/ALBDF_DATA/h2o_p0_25.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p0_5.txt","r"); } if (P1 == 2){ f1 = fopen("/ALBDF_DATA/h2o_p0_5.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p1.txt","r"); } if (P1 == 3){ f1 = fopen("/ALBDF_DATA/h2o_p1.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p2.txt","r"); } if (P1 == 4){ f1 = fopen("/ALBDF_DATA/h2o_p2.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p4.txt","r"); } if (P1 == 5){ f1 = fopen("/ALBDF_DATA/h2o_p4.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p8.txt","r"); } if (P1 == 6){ f1 = fopen("/ALBDF_DATA/h2o_p8.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p15.txt","r"); } if (P1 == 7){ f1 = fopen("/ALBDF_DATA/h2o_p15.txt","r"); f2 = fopen("L/ALBDF_DATA/h2o_p30.txt","r"); } if (P1 > 7){ f1 = fopen("/ALBDF_DATA/h2o_p30.txt","r"); f2 = fopen("/ALBDF_DATA/h2o_p50.txt","r"); } for (i = 0; i < 500976; i++){ fscanf(f1,"%lf",Fdata1+i); fscanf(f2,"%lf",Fdata2+i); } fclose(f1); fclose(f2); } if (molecule == 2){ if (P1 == 0){ f1 = fopen("/ALBDF_DATA/co2_p0_1.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p0_25.txt","r"); } if (P1 == 1){ f1 = fopen("/ALBDF_DATA/co2_p0_25.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p0_5.txt","r"); } if (P1 == 2){ f1 = fopen("/ALBDF_DATA/co2_p0_5.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p1.txt","r"); } if (P1 == 3){ f1 = fopen("/ALBDF_DATA/co2_p1.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p2.txt","r"); } if (P1 == 4){ f1 = fopen("/ALBDF_DATA/co2_p2.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p4.txt","r"); } if (P1 == 5){ f1 = fopen("/ALBDF_DATA/co2_p4.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p8.txt","r"); } if (P1 == 6){ f1 = fopen("/ALBDF_DATA/co2_p8.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p15.txt","r"); } if (P1 == 7){ f1 = fopen("/ALBDF_DATA/co2_p15.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p30.txt","r"); } if (P1 > 7){ f1 = fopen("/ALBDF_DATA/co2_p30.txt","r"); f2 = fopen("/ALBDF_DATA/co2_p50.txt","r"); } for (i = 0; i < 55664; i++){ fscanf(f1,"%lf",Fdata1+i); fscanf(f2,"%lf",Fdata2+i [START_REF] Jeans | The Equations of Radiative Transfer of Energy[END_REF]; if (Tg < 300) Tg = 300; if (Tg > 3000) Tg = 3000; if (Tb < 300) Tb = 300; if (Tb > 3000) Tb = 3000; m = locate(T,27,Tg); l = locate(T,27,Tb); k = locate(C,70,Cabs);

Calculation of the Q-criterion

The tensor of velocity gradient ∇v can be written as: