Development of fifth-order slow return-to-isotropy and pressure transport models and their applications to computation of turbulent free shear flows

Abstract

A set of new models for the slow pressure-strain correlation and the dissipation tensor in the Reynolds stress equation and the decay term in the dissipation equation is presented. In order to reproduce the significant role of the third invariant $I\!I\!I_b(=b_{ij}b_{jk}b_{ki})$ of the Reynolds stress anisotropy $b_{ij}(=u_iu_j/(2k)-(1/3\delta_{ij})$ in the return-to-isotropy process, the former two terms are represented by fifth-order series expansions in powers of $b_{ij}$. The strong realizability condition for the non-negative component energies and an asymptotic requirement for the two-dimensional turbulence are utilized to reduce the number of model constants produced. A Reynolds number correction is then included by considering an energy-weighted average time scale of eddies over the three-dimensional energy spectrum. It is found that the present dissipation model after the Reynolds number correction yields very good prediction in comparison with available direct numerical simulation data. The performance of the present set of models is compared with those of other three models for a number of experiments of homogeneous turbulence. Based on the r.m.s. error estimation, the present set of models predicts the experimental data within $\pm7.3\%$, which is at least twice better than the presently best model of Sarkar & Speziale(1990, Physics of Fluids A, vol.2, p.84). In particular, the nonlinear behavior of the return-to-isotropy in the phase space and the different rate of return-to-isotropy depending on the sign of $I\!I\!I_b$ are nicely described by the present model. And then, a closure model of the pressure transport term in the Reynolds stress equation is developed by using a potential theory and a number of experimental evidences. The proposed model consists of a bulk convective transport term and a triple velocity correlation term. The former represents a contribution from the large coherent turbulence, and it is expressed in terms of the difference...