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Annexe Chapitre 2First and second order modelling of turbulent scalar transportin homogeneous turbulenceH. Ayed 1,2* , J. Chahed 1 , and V. Roig 21 Ecole Nationale d’Ingénieurs de Tunis BPN°37, 1002 Le Belvédère, Tunis, Tunisia,2 Institut de Mécanique des fluides de Toulouse, UMR CNRS 5502, Av. C. Soula, 31400, Toulouse, France*Corresponding author. E-mail: ayed@imft.frAbstractThe present work aims at developing a full second order model for turbulent transportof a passive scalar. A turbulent Prandtl number formulation is obtained by reducing thesecond order modelling. A first order turbulence model is than proposed with analgebraic formulation of the turbulent scalar flux. The model is tested in homogeneousshear flow with an imposed mean scalar gradient. The Direct Numerical Simulation(DNS) data by Rogers et al. [1] were used in the validation of the first and second ordermodelling of the turbulent passive scalar transport. The comparison of the numericalresults with DNS data shows good agreements.Keywords: Passive scalar, turbulent flux, homogeneous shear flow, turbulence modelling1. IntroductionThe prediction of the transport of scalarssuch as temperature, chemical species orpollutants in turbulent flows is of greatimportance in the design and the controlof industrial fluid systems. During thelast decades, an important progress in thestatistical modelling of the turbulenttransport of passive scalar has beenachieved. Number of recent works wasdevoted to the development of first andsecond order turbulence models, (Naganoet al. [2]; Abe et al. [3,4]; Kawamura etal. [5]; Wikström et al. [6]). The modeldeveloped by Nagano et al. [2] isdeveloped for heat transfer problems inboth wall and free turbulent flows. Abe etal. [3,4] proposed a modified version ofNagano model. Furthermore, the modelcan also take into account the lowReynolds number effects in the near-wallregion and is also applicable to complexheat transfer fields with flow separationand reattachment by introducing theKolmogorov velocity scale. Kawamura etal. [5] proposed a transport model for theturbulent heat flux by introducing theinteraction between the mean temperatureand velocity gradients in rapid term ofturbulent scalar flux equation.On the other hand, many experimentaland numerical simulation works were165
Annexe Chapitre 2devoted to the study of the scalartransport in homogeneous turbulence(Rogers et al. [1], Maekawa et al. [7],Tavoularis et al. [8]). As a result,complete data bases concerning thestatistical description of the turbulentcorrelations involving the turbulent scalarfluctuation are now available. Inhomogeneous turbulence, the meanvalues of the velocity and of the passivescalar fields are known and fixed throughthe flow domain. Furthermore, inhomogeneous turbulence the diffusion isneglected. In the frame work of thesehypotheses, it becomes possible to focusthe analysis on the budget of turbulentcorrelations (production, dissipation,redistribution) and many numericalworks used these basic turbulence data inorder to validate statistical turbulencemodelling involving scalar transportclosure.Recently, Wikström et al. [6] proposed anew algebraic relation based on secondorder closure for the turbulent passivescalar transport and applied tohomogeneous turbulence. This modelmay be regarded as one of the mostcomplete closure of the pressure-scalargradient correlation.The present work aims at developing afull second order model for turbulenttransport of passive scalar flux inhomogeneous turbulence. After thevalidation of the second order model, afull first order modelling is obtained byreducing the second order model.2. Second order modelling of the transportequationsWithout transfer, the Reynolds averagedtransport equation for the mean scalar, Θ ,for incompressible flows reads :D ∂ ⎛⎞⎜∂ΘΘ = σ − u'θ ⎟∂'iDt xi ⎝ ∂xi⎠where σ is the molecular diffusivity.(1)In equation (1) appears the scalar'flux, u'iθ, which originates from theaveraging of the nonlinear term in thetransport equation of the total scalar field.This term plays a role analogous to thatof the Reynolds stress tensor in the meanflow equation. The exact transportequation for scalar flux u' θ'i, may beanalytically, it is expressed as (Launder[9]):DDt⎛' '' '⎜∂Ui ' 'u θ = − u θ + u uiji j⎝ ∂xj1 '+ pρ'∂θ−∂xi( σ + ν )∂Θ ⎞⎟∂xj ⎠' '∂ui∂θ∂x∂x'⎛''∂u⎞⎜∂i ' ' ' ' ui' ∂θ− u u θ −νθ−σu ⎟i ji∂xj ⎝∂xj∂xj ⎠1 ' '+ pθδ ij(2)ρThe first terms in right hand of equation(2) represents the production of theturbulent scalar flux, due to theinteraction of Reynolds stresses with themean scalar gradient and to the scalarflux interacting with the mean velocitygradient. The second term is the pressurescalargradient correlation, the third termis the rate of dissipation and the last termrepresents the turbulent and moleculardiffusions. In homogeneous turbulencethe diffusion is zero and the transportjj166
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N° d’ordre : 2454Thèseprésent
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RemerciementsLe travail présenté
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ResuméCe travail vise l’amélior
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Sommaire2.5.2 Equations de transpor
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Sommaire4.6.4 Profils de vitesse mo
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Nomenclatureu' ' ( S )Liu LjLk , in
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Chapitre 1: Introduction générale
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Chapitre 1: Introduction générale
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Page 168 and 169: BibliographieBibliographieAbbas M.,
Page 170 and 171: Bibliographiefor simulating hydrody
Page 172 and 173: Bibliographieflows. Journal of Turb
Page 174 and 175: BibliographieOxygen sensor manuel,
Page 176 and 177: BibliographieVoinov O. V., 1973. Fo
Page 180 and 181: Annexe Chapitre 2equation of the sc
Page 182 and 183: Annexe Chapitre 210864u'v' v' 220-2
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Page 186 and 187: Annexe Chapitre 2Figure (5) shows,
Page 188 and 189: Annexe Chapitre 2[3] K. Abe, T. Kon
Page 190 and 191: Annexe Chapitre 3xgy sh 1 h 2z1 2-b
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