Runge-Kutta/Implicit Scheme for the Solution of Time Spectral Method

This paper investigates the Runge-Kutta implicit scheme applied to the solution of the time spectral method for periodic unsteady flow simulation. Several explicit and implicit time integration schemes including the Runge-Kutta scheme, Block-Jacobi SSOR (symmetric successive over relaxation)scheme and Block-Jacobi Runge-Kutta/Implicit scheme are implemented into an in-house code and applied to the time marching solution of the time spectral method. The time integration is coupled with Full Approximation Storage (FAS) type multi-grid method for convergence acceleration. The in-house code is based on the finite volume method and solves the RANS (Reynolds Averaged Navier-Stokes) equations on multi-block structured mesh. For spatial discretization the 3rd/5th order WENO (weighted essentially nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is used in the simulations. The time integration schemes for the solution of the time spectral method are tested with two different compressor cascades with periodically oscillating inlet boundary conditions. The first case is a low speed compressor stator with inlet flow angle varying with time. The second case is a high speed compressor rotor with inlet boundary condition profile to simulation the influence of upstream wakes. The results show that for moderate frequencies and wave mode numbers, the Block-Jacobi Runge-Kutta/Implicit scheme shows favorable convergence behavior compared to the other schemes. However, for extremely high frequencies and wave mode numbers such as in the simulation of high rotating speed compressors, the advantage of the Block-Jacobi Runge-Kutta/Implicit scheme over the explicit Runge-Kutta scheme is totally lost.