Efficient implementation of the \(k-{\sqrt{k}} L\) turbulence model with the discontinuous Galerkin method

We present the approaches to implementing the \(k-{\sqrt{k}} L\) turbulence model within the framework of the high-order discontinuous Galerkin (DG) method. We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations. In order to enhance the robustness of approaches, some effective techniques are designed. The HWENO (Hermite weighted essentially non-oscillatory) limiting strategy is adopted for stabilizing the turbulence model variable k. Modifications have been made to the model equation itself by using the auxiliary variable that is always positive. The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods. Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations.