Analyse expérimentale et modélisation du transfert de matière et du ...

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