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

Annexe Chapitre 2equation of the scalar flux, can beconcisely written as:D ' 'u iθ = Pθi+ Πθi− εθi(3)Dtwhere P θ iis the production term, Πθiis thepressure-scalar gradient correlation and εθiis thedissipation of the scalar flux.Launder et al. [10] assumed that thedissipation term is negligible at highReynolds number but the two terms Πet ε are usually modelled together andθisimilar methods than those used in theclosure of the Reynolds Stress transportequation are used [5, 6]. Wikström et al.[6] proposed a closure of the differencebetween the pressure-scalar gradientcorrelation and the dissipation term in thefollowing general form:⎛ k ⎞' '⎜∂ΘC C u ⎟ε ' 'Π − ε = −ui i+ θmiq xθθ θθ1θ 5⎝ ε ∂ km ⎠∂U∂U' ' i' ' j+ Cθ2ujθ+ Cθ3ujθ∂x∂xji' ' ∂Θ+ C u uθ 4 i j(4)∂xjwhere Cθ1, Cθ 2, Cθ 3, Cθ 4and Cθ 5are constants,'2' 'θuq = is the scalar variance,iuk = iis the22turbulent kinetic energy and ε its dissipation rate.In homogeneous turbulence, the transport equationof q writes:Dq = Pθ − ε θDtwhereθi(5)= u'jθ 'θis a production term of the∂xjP∂Θ' '∂θ∂θscalar variance q , ε θ= σ is its∂x∂x j jdissipation rate. The transport equation of ε θismodelled according to Abe et al. [4] and Farshchiet al. [11].2D ε P Pθ θ * ε εθθ * εεθε = C + C − C − C (6)θ γ 1γ 1γ 2γ 2Dt 2qk 2qk**where C ,γ 1C , C and C are constants and Pγ 2 γ 1γ 2is the production term of the Reynolds stress.The model of the pressure-scalar gradientcorrelation proposed by Wikström et al.[6] which comprises five constants,generalizes a certain number of existingmodels. When only the first constant isretained with the value Cθ1= 3. 2 , themodel (4) reduces to the linear model oflaunder [9]. When the constant C θ 2isadded, we obtain the non linear model ofLaunder [9] with Cθ 2= 0. 5 or the nonlinear model Daly and Harlow [12] withCθ 2= 1. Wikström et al. [6] tried differentmodels using different combinations andformulations of the constants in themodel (4) as illustrated in table 1.Table 1: Different formulations of the pressure-scalargradient model (equation 4) Wikström et al. [6]Model Cθ1Cθ 2Cθ 3Cθ 4Cθ 5(a) 3.2 0.5 0.5 0 0(b) 2.5 0 0 0.35 0(c)r + 11.6r0 0 0 0.5In the model (c), Wikström et al. [6]qετθ1introduced the time ratio r = = =kεθτ Rin the formulation of the parameter Cθ1.kWhere τ = is the dynamic time scale ofεqturbulence, τ = is characteristic scalartime scale.ε θThe transport equation of the turbulentscalar flux together with the transportequations of the scalar variance q and itsdissipation rate ε θallow computing theturbulent scalar flux provided that the167