Investigation of Spectral Vanishing Viscosity for Stabilizing and Accelerating Solution of Time Spectral Equation System

Abstract

Time spectral method (TSM), as a very effective frequency domain method for analyzing temporal periodic flow field, is more popular than other frequency domain methods such as the nonlinear harmonic method (NLHM), the linear harmonic method (LHM) and the classic harmonic balance method. This is due to the fact that it is much easier to be implemented in an existing steady flow solver. But the aliasing errors caused by the nonlinear terms in the Euler/Navier-Stokes equations can have an adverse effect on the convergence rate. On the worse side, it can even lead to solution instability. This paper will present investigation of the application of the time spectral vanishing viscosity (SVV) method to stabilizing the solution of time spectral method. The spectral vanishing viscosity method has long been investigated for stabilizing spectral solution of conservation laws in space. Its application to time spectral solution of unsteady flow equations is very limited. In this investigation, the accuracy and time cost of the time spectral method with the SVV will be compared with that without the SVV and the traditional time domain dual time stepping method (DT). The stiffness of the SVV and the time spectral source terms is relieved by implicitly integrating the source terms in time using the block Jacobi method (BJ). To accelerate the convergence rate, the Lower Upper Symmetric Gauss Seidel (LU-SGS) method is used as the residual smoother for a multi-stage Runge-Kutta time integration. The following points can be drawn from this investigation: 1. Time spectral vanishing viscosity method can accelerate convergence rate and reduce time cost. 2. Time spectral vanishing viscosity method can improve the solution accuracy and fewer harmonics can be used to analyze the flow field with the SVV than without the SVV.