Time-Accurate Solution of Unsteady Flows in an Implicit Solver Using Block LUSGS Method

Abstract

Implicit solvers based on the lower-upper symmetric Gauss–Seidel (LUSGS) method are widely used in Computational Fluid Dynamics (CFD) codes to obtain steady-state solutions using large false-transient time steps. However, the method destroys time accuracy and thus cannot be directly used for true transient problems. For such problems, the LUSGS method has to be used with a dual time-stepping algorithm. In this paper, a modified method that preserves time accuracy and is called the block LUSGS (B-LUSGS) solver is employed to compute unsteady flows using larger time steps. This method is compared with the mentioned dual time-stepping technique used for transient problems. Numerical solutions obtained for Stokes' second problem simulation underline the superior temporal accuracy of the block LUSGS method in computing unsteady flows, even at CFL numbers close to 30. Simulations of 2D laminar and 3D turbulent vortex shedding downstream of a circular cylinder further validate the modified method in accurately predicting unsteady separated flows at a CFL number of 100. Comparisons of overall simulation CPU times show that the B-LUSGS method is at least six times computationally cheaper than the dual time-stepping algorithm that uses the traditional LUSGS solver. Abbreviations: BDF, Backward differentiation formula; CFD, Computational fluid dynamics; CFL Courant–Friedrichs–Lewy; CPU, Central processing unit; DTS, Dual time-stepping; LES, Large-eddy simulation; LUSGS, Lower-upper symmetric Gauss–Seidel; NASA, National Aeronautics and Space Administration; PDE, Partial differential equation; RAM, Random access memory; RANS, Reynolds-averaged Navier–Stokes; SA, Spalart–Allmaras; SST, Shear stress transport.