Designing a positive second-order implicit time integration procedure for unsteady turbulent flows

A positivity preserving implicit procedure for first order time integration of two-equation turbulence models is extended to second-order time integration utilizing dual-time stepping. Because the time derivative of the turbulence quantities may behave as a negative source term it is treated implicitly. The procedure is applied to a non-linear k–ϵk–ϵ turbulence model using an unstructured flow solver. Numerical simulations of the unsteady flow about a square cylinder are conducted to demonstrate the positivity characteristics of the turbulence quantities and the robustness of the procedure and of the flow solver. Results from the numerical simulations favorably compare with results from experiments.