Engineering prediction of turbulent flows heavily rely on two-equation turbulence models that provide a good compromise between versatility and computational efficiency. Specifically, the k-epsilon models are very popular and have been applied to a wide spectrum of problems of engineering application. The low Reynolds number versions of the models require a very fine grid near the wall to resolve the strong gradients of k and epsilon in this region. Therefore, the computation becomes expensive. In addition, these strong gradients make the non-linear low Reynolds number terms numerically very stiff. As a result, the time-step is limited to small values and the solution requires a large number of iterations to converge. This increases the computational cost even further. As a result, the simulation of turbulent flows using k-epsilon models becomes very computationally intensive.

I undertook a detailed study of a low Reynolds number k-epsilon model to understand the numerical issues involved in the computations. It was found that the low Reynolds number terms are formulated in a way that the methods are unstable. This limited the allowable time steps in the computations to small values, resulting in slow convergence to the steady-state solution. I modified the formulation of these terms to stabilize the methods. I also eliminated the numerical instabilities caused by low values of k by using a reduced value of free-stream dissipation. With these modifications, much larger time-steps could be taken and the steady-state solution is reached in about 10% of the time required by the original method. The modifications have very little effect on the solution.

Residual history of two-equaiton turbulence model calculations: 
conventional method (red) and new robust method (blue)