Reynolds-Stress Modeling of 3-D Secondary Flows with Emphasis on Turbulent Diffusion Closure

The purpose of this paper is to evaluate the impact of the explicit dependence of turbulent- diffusion on the gradients of mean-velocity on 3-D detached and secondary ows prediction. Following recent theoretical works of Younis et al. (Proc. Royal Society Lon. A 456 (2002) 909-920) and Gatski (Theor. Comp. Fluid Dyn. 18 (2004) 345-369), we propose a triple-velocity correlation model including the effects of the spatial gradients of mean velocity. A model for both the slow and the rapid part of the pressure-diffusion term was also developed and added to wall-normal-free Reynolds-stress model. The present model is validated against 3-D detached and secondary ows. Further developments especially on the echo-terms which should appear in the formulation of pressure-velocity correlation are discussed. P ij is ne- glected. The model proposed by Lumley 3 (1978) was obtained by considering weakly anisotropic and in- homogeneous ow and can be considered as an extension of the Hanjalic-Launder (hereafter HL) closure. 3 Numerous previous assessments for the triple-velocity correlation 4n8 based on a priori comparisons with experimental or DNS data, have shown that the HL and the Lumley 3 models give the best overall results, in 1-D plane channel ow 9,10 and in 3-D boundary-layers. 11 Nevertheless, in general, these 3 closures (DH, HL and Lumley) give quite similar results. Taking into account that all of these models are bilinear in the Reynolds stresses and their gradients, Gatski 8 did not consider this conclusion surprising. Futhermore, recent theoretical work 8,12 demonstrated the dependance on spatial gradients of mean velocity of the triple- velocity correlation. Concerning the pressure-diffusion term, there exist few studies and most of them take into account only the slow part of the pressure-velocity correlation. Indeed, the pressure-velocity correlation is often neglected not only because of the lack of experimental or DNS data, but also because this term does not seem important in plane channel ow . 9 As early as 1969, Hirt 13 proposed a function of the Reynolds stresses gradients and therefore modeled only the slow part of the pressure-velocity correlation. The only theoretical closure for the slow part was established in 1978 by Lumley 3 for weakly inhomoheneous ows . This homogeneous ex- pression was however used in inhomogeneous ows by several authors. 14n17 In 1996, Demuren et al. 16 used an elliptic relaxation approach with the Lumleyy 3 closure for the slow-part of the pressure-velocity corre- lation and proposed a rapid part closure for the pressure-diffusion term. More recently, Suga 18 developed a rapid-part pressure-velocity correlation model and used the DH proposal to model both the triple-velocity correlation and the slow-part pressure-diffusion terms. In the present work, we develop a turbulent-diffusion closure including mean-velocity gradients both for the triple-velocity correlation and the pressure-diffusion. This complete model was added to the low- Reynolds-number second-moment closure developed by Gerolymos and Vallet 19 (hereafter GVnRSM) where a unit-vector pointing in the turbulence inhomogeneity direction 20 is used to obtain a Reynolds-Stress model (RSM) completely independent of wall topology which can be used to simulate arbitrary 3-D complex geome- tries.