The barely implicit correction algorithm for low-Mach-Number flows

Abstract

A new Barely Implicit Correction (BIC) algorithm is presented for the simulation of low-Mach-number flows. This new algorithm is based on the original, introduced by G. Patnaik et al. [G. Patnaik, R. H. Guirguis, J. P. Boris and E. S. Oran, A barely implicit correction for flux-corrected transport. In: Journal of Computational Physics 71.1 (1987), pp. 120], which was a solution procedure including an explicit predictor step to solve the convective portion of the Navier–Stokes equations and an implicit corrector step to remove the acoustic limit on the integration time-step. The explicit predictor uses a high-order monotone algorithm while the implicit corrector solves an elliptic equation for a pressure correction to equilibrate acoustic waves. In this paper, we develop and extend BIC for multidimensional viscous flows. We introduce a new filter to further stabilize the algorithm and clarify the solution procedure for the inclusion of the viscous fluxes. The new algorithm is examined in three test problems with successively increased difficulty. First, a two-dimensional lid-driven cavity flow is simulated to demonstrate the ability of BIC on solving steady-state swirling flows. Using time steps at least 100 times larger than the explicit limit, good agreements are obtained for solutions when compared with an incompressible calculation by a prior work. A two-dimensional (2D) doubly periodic shear layer flow is simulated to examine the algorithm on solving a transient flow with strong vorticity gradients. Finally, vortex breakdown in three-dimensional (3D) swirling flows are used to further test the stability and performance of the new BIC algorithm. Comparisons of explicit and implicit BIC calculations of both the 2D doubly periodic shear layer and 3D vortex breakdown are presented side by side. They demonstrate that the new BIC algorithm is able to predict accurate and robust solutions using time steps varying from near the explicit stability limit to tens and hundreds of times larger. Excellent agreement is also obtained when compared with results from other algorithms. We discuss our observations of these computations and features which were found to be critical for robustly simulating low-speed, highly dynamic flows.