Mixed analytical/numerical method applied to the high Reynolds number k-ε turbulence model

In [Comput. Fluids 32 (2003) 659], a mixed analytical/numerical method for partial differential equation with an oscillating source term was proposed. The inhomogeneous partial differential equation is split into a homogeneous one plus an ODE for the source term using the time splitting method. The homogeneous part is then integrated numerically while the source term ODE is integrated analytically. This method was demonstrated to be efficient when the source term has a time scale much smaller than the mean flow time scale. In this paper, this approach is extended to the k-ε turbulence model for high Reynolds number flows. It is found that the mixed method based on operator splitting does not converge to a stable steady state solution. We thus propose an unsplit method. Numerical experiments for homogeneous turbulent flow and for a plane jet show that the unsplit mixed method substantially improves the accuracy of a time dependent problem while it has better convergence properties for a steady flow problem.