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In this work, we develop a Reynolds stress model along the line of the approach presented by Commun. Nonlinear Sci. Numer. Simul. 9, 543 (2004)]], aiming to assess the role and contribution of the mean spin tensor in turbulence modeling. Here, the constitutive functional for the Reynolds stress depends on the mean spin tensor as well as the mean stretching tensor and its Jaumann derivative, the turbulent kinetic energy K , and the turbulent dissipation rate epsilon , which is at the complexity level of p=1,m=1 , and n=0 of a rate-type constitutive equation for the Reynolds stress proposed in the aforementioned paper. The explicit form for the Reynolds stress is obtained with recourse to the representation theorem and the theory of invariants developed in modern rational continuum mechanics, and, as an approximation, a nonlinear cubic K-epsilon model is worked out in which the model coefficients are analytically identified based on the experimental results of Tavoularis and Corrsin [J. Fluid Mech. 104, 311 (1981)]]. In addition, numerical results based on this model, in the forms of employing the Jaumann derivative and the Oldroyd derivative, respectively, for homogeneous turbulent shear flow and fully developed turbulent flow over a backward-facing step, are presented in comparison with those obtained based on a few previously proposed linear and nonlinear K-epsilon models, showing reasonably good agreement with the experimental results and the DNS data concerned and a better performance than the previously developed quadratic models.

SUMMARYThe dynamic boundary conditions for vorticity, derived from the incompressible Navier-Stokes equations, are examined from both theoretical and computational points of view. It is found that these conditions can be either local (Neumann type) or global (Dirichlet type), both containing coupling with the boundary pressure, which is the main difficulty in applying vorticity-based methods. An integral formulation is presented to analyse the structure of vorticity and pressure solutions, especially the strength of the coupling. We find that for high-Reynolds-number flows the coupling is weak and, if necessary, can be effectively bypassed by simple iteration. In fact, even a fully decoupled approximation is well applicable for most Reynolds numbers of practical interest. The fractional step method turns out to be especially appropriate for implementing the decoupled approximation. Both integral and finite difference methods are tested for some simple cases with known exact solutions. In the integral approach smoothed heat kernels are used to increase the accuracy of numerical quadrature. For the more complicated problem of impulsively started flow over a circular cylinder at Re = 9500 the finite difference method is used. The results are compared against numerical solutions and fine experiments with good agreement. These numerical experiments confirm our thoeretical analysis and show the advantages of the dynamic condition in computing high-Reynolds-number flows.