Inhomogeneity and anisotropy effects on the redistribution term in Reynolds-averaged Navier–Stokes modelling

Abstract

A channel flow DNS database at Reτ = 590 is used to assess the validity of modelling the redistribution term in the Reynolds stress transport equations by elliptic relaxation. The model assumptions are found to be globally consistent with the data. However, the correlation function between the fluctuating velocity and the Laplacian of the pressure gradient, which enters the integral equation of the redistribution term, is shown to be anisotropic. It is elongated in the streamwise direction and strongly asymmetric in the direction normal to the wall, in contrast to the isotropic, exponential model representation used in the original elliptic relaxation model. This discrepancy is the main cause of the slight amplification of the energy redistribution in the log layer as predicted by the elliptic relaxation equation. New formulations of the model are proposed in order to correct this spurious behaviour, by accounting for the rapid variations of the length scale and the asymmetrical shape of the correlation function. These formulations do not rely on the use of so-called ‘wall echo’ correction terms to damp the redistribution. The belief that the damping is due to the wall echo effect is called into question through the present DNS analysis.

Wall effects on pressure fluctuations in turbulent channel flow

“…By comparing with the exact solution obtained in the present work, it is possible to evaluate the approximation error of the method of images of Manceau et al (2001).…”

“…The approximate method of images (Manceau et al 2001) simply adds two mirror images of the channel, one above the upper wall and another below the lower wall (Fig. 4), to account for the presence of the walls, postulating that an approximation to the solution is obtained by using the freespace Green's function…”

“…The main purpose of the present algorithm is to distinguish in (1.2) the contributions of the volume integral (volume terms denoted by (·) V ) from the contributions of the surface integral (wall-echo terms denoted by (·) w ), ie to extend (2.2) as , and boundary conditions for the exact problem (2.6) whose solution (Kim 1989) is given by GKim(y, Y ; κ) (B 10b), and for the two virtual halfspace problems with only one wall present defining G±(y, Y ; κ) (B 18), which define the local wall-corrections appearing in the high-wavenumber approximation of the method of images (Manceau et al 2001), corresponding to the approximate Green's function Gmwl(y, Y ; κ) (B 42c).…”

“…Furthermore, Manceau et al (2001) neglect Stokes pressure. When analyzed under a formal Green's function framework, the approximation in the method of images used in Manceau et al (2001) is equivalent to using an approximate Green's function, instead of the exact one. By comparison with the exact solution obtained in the present work it is possible to quantify the approximation error of the method of images.…”

“…By comparison with these exact results we also quantify the error of the approximate method of images of Manceau et al (2001), as a function of the wavenumber ( §2.2.4), showing that this method is a high-wavenumber approximation. Analytical details on the Green's functions, with particular emphasis on the singular problem of xz-constant pressure fluctuations (κ = 0), are given in Appendix B.…”