It is well known that the incorporation of two-equation turbulence models into explicit time-marching schemes, especially the low Reynolds versions, encounters various numerical difficulties. This paper sets out to review the numerical techniques presently used and to analyse the problems encountered. A numerical approach is proposed for the implementation of the low-Reynolds-number (LRN) k-∈ models into an explicit Runge-Kutta time-marching solver. In this method, a semi-implicit treatment of the source terms of the k and ∈ equations is introduced, and modified residuals of the two equations are used in order to adopt implicit residual smoothing. These techniques ensure the positivity of k and ∈ in the computation, even with uniform initial conditions and local time stepping. The asymptotic behaviour of the turbulence quantities during time marching is analysed, providing insight into the stability and convergence characteristics of the k and ∈ equations using explicit and implicit schemes with and without the damping terms. It is found that the time step size has a strong influence on the evolution of the turbulence quantities and the stability of the solution, whether implicit or explicit schemes are used. The time step size determined by the stability analysis is a function of k, ∈ and their damping terms. This analysis also provides a guide for the selection of time step size for the turbulence equations. It is found that the selection of time step size is an important factor in ensuring realistic values of k, ∈ and μt in the flow field during time marching from uniform initial conditions. Numerical solutions of several flow test cases are given and the computed results are compared with experimental measurements or theoretical solutions for validation of the method. It is shown that this method is reasonably efficient and robust.
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