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

Annexe Chapitre 2We may use a similar formulation thanthat used in the approximation of thecomponents of the Reynolds stress inorder to develop algebraic relations forthe turbulent passive scalar flux. Gibsonand Launder [14] proposed anapproximation of the turbulent passivescalar flux based on the assumption thatthe transport of the passive scalar flux isproportional to the transport of theparameter kq and modelled as follows:DDt' 'u θ − Diffi' '( u θ )i' 'u θ ⎛= ⎜D kqi− Diffkq ⎜⎝Dt' '' 'u θ u θi= P2k2q⎞( kq ) ⎟⎟i( P − ε ) + ( − )θε θ⎠(23)Using the second order closure(equations 3 and 4), the components ofthe turbulent scalar flux can be writtenas:' 'k ' ' ∂Θ− uθ= Λ ( 1−C ) u vθ θ 4ε ∂y' '− vθ= Λk ' ' ∂U+ Λ ( 1−C ) u θ(24)θ θ 2ε ∂yθ( 1−C )θ 4kvε'2∂Θ∂yk ' ' ∂U+ Λ ( 1−C ) uθ(25)θ θ 3ε ∂yWhere−1⎡ 1 ⎛ P ⎞ kε⎛ P ⎞⎤θ θΛ = ⎢C + 111 ⎜ − ⎟ +⎥⎢⎣2 2⎜ −⎟ (26)θ θ⎝ ε ⎠ qε⎝ εθ ⎠⎥⎦On the basis of this formulation we maycompute the turbulent passive scalarfluxes using a first order modelling. Thismodel is based on a two-equationturbulence model ( k − ε model withalgebraic relations) and on the modellingof the transport equations of the scalarvariance ( q ) and of its dissipation rateε .θFigures (3) and (4) show the numericalresults of the evolution of the Reynoldsstress components obtained from thealgebraic relations (equations 18 and 19)and compared with the numerical resultsof the second order closure. Thecomparison shows the same goodconcordance of the numerical resultsobtained with the two models with theDNS data of Rogers et al. [1].171