Investigation of Relaxation Parameters and residual based truncation criteria for Reynolds averaged Navier-Stokes equations

Abstract

To solve the RANS equation and the transport equation based on one equation Spalart Allmaras model, an agglomerated non-linear multigrid scheme with implicit smoothing, based on first order approximation of the derivative ot the residual function, exists. The implicit smoothing leads to a linear system of equation that needs to be solved for every stage of the smoothing algorithm. To improve this solution method grid anisotropy is treated with the Gauss-Seidel iteration in such a way that lines with strong coupling in linear system are resolved by tridiagonal systems constructed along the direction of strong coupling. The existing methodology of solving this linear system with a line symmetric Gauss-Seidel method restricts finer meshes to reach higher CFL numbers. Therefore relaxation parameter is incorporated into the existing line symmetric Gauss-Seidel methods to investigate the change in the solution methodology. The current practice is to set the number of Gauss-Seidel sweeps to a certain number and this may lead to over-solving. Thus the extent to which the inner linear equations that needs to be solved can be truncated after the linear residual is reduced by a certain order of magnitude. In this regard too, numerical experiments demonstrate that choosing a low GS sweeps may lead to delay in converging whereas a high GS sweep value may lead to over solving thereby making it computationally expensive. Thus a truncation criteria based on a linear residual is implemented and the change is investigated. Eigenvalue analysis is done after implementation of the linear based truncation criteria to compare the asymptotic convergence of the solution method.