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

Annexe Chapitre 2Reynolds stress tensor is determined. Thetransport equation of the Reynolds stresstensor written as:D ' 'u u = P + Πi j ijDtwherestresses,ij− εij(7)P is the production term of the ReynoldsijΠ represents the redistribution by theijpressure fluctuations and ε ijis the dissipationterm of the Reynolds stresses transport equations.The production term doesn’t need modelling, butclosures are required for the redistribution and thedissipation terms. For this purpose, the standardsecond order turbulence model, in homogeneousturbulence, proposed by Launder et al. [10] isused. Thus, the terms of right hand side oftransport equation (7) are written as follows:⎛⎞⎜∂U' ' j ' ' ∂UiP = − u u + u u ⎟(8)iji kj k⎜⎟⎝∂x∂xkk ⎠⎛ ⎞ ε ⎛⎞Πij= −γ⎜ P −2 ' 2ijPδij ⎟ − C ⎜uiu'j− kδij ⎟ (9)1⎝ 3 ⎠ k ⎝ 3 ⎠2ε ij= εδ ij(10)3where P = trace( P ii); C1= 1. 8 and γ = 0. 6 .According to equation (10), thedissipation rate of the turbulent energy εis assumed to be isotropic in accordancewith the local isotropy hypothesis. Thetransport equation of the dissipation rateε is modelled using a classical transportequation. In homogeneous uniformlysheared turbulence, this transportequation writes:2Dεε ' ' ∂Uε= −Cu v − C(11)1ε2εDt k ∂ykWhere C1 ε= 1. 44 and C2 ε= 1. 9 . And thetransport equations of the principal componentsthe Reynolds stress tensor are finally modeled as:DuDt'2DvDt'2Du vDt' ' ∂U= −2u v −∂y− Cε ⎛⎜uk ⎝2ε32 ⎞ ⎛− k ⎟ − ⎜ P −113 ⎠ ⎝'2γ123⎞P⎟⎠(12)⎛ ⎞= −2 ε ' 2ε − C ⎜v2− k1 ⎟3 k ⎝ 3 ⎠⎛ 2 ⎞− γ ⎜ P − P⎟(13)22⎝ 3 ⎠∂U∂yεk' ''2' '= −v− C u v − γP(14)112A two-dimensional numerical code hasbeen developed for parabolic flowresolution. The numerical method isbased on a finite-difference scheme andthe equations are solved with an explicitmethod.Because the turbulent passive scalar fluxdoes not affect the velocity field, we firststarted by validating the second orderturbulence model. The full equations ofthe second order model wereapproximated by finite difference schemeand solved. For this purpose, the inletconditions were suitably adjusted in orderto reproduce the evolution of theReynolds stress tensor. The numericalresults are confronted to the DirectNumerical Simulation (DNS) data ofRogers et al. [1] obtained inhomogeneous turbulent pure shear flowwith a constant mean shear and aconstant mean temperature gradient inthe x 2 direction. Figure 1 show goodagreement between the numerical resultsand the DNS data of Rogers et al. [1].168