A Low-Mach, Low-Reynolds Preconditioning Scheme With Particular Attention To Viscous Time-Stepping

Industrial design and optimization processes rely increasingly on powerful and well-engineered CFD tools. Compressible solution methods, which perform very well at transonic and supersonic flow speeds, display a dramatic degradation of convergence as well as of solution quality as the incompressibility limit is approached (very low flow speeds). In many technical applications, especially in turbomachinery, the flow conditions vary strongly within the computational domain and with time. When the incompressibility limit is approached, a large disparity arises between the smallest and largest eigenvalues of the systems characteristic matrix. In order to overcome these problems low-Mach preconditioning methods have been devised to rescale the eigenvalues of the characteristic matrix of the system of governing equations and, hence, reduce the large inequality in the acoustic and convective flow speeds. Frequently, low flow speeds are observed in low Reynolds environments. As noted by several authors, often instability issues in these regions, such as cavities and boundary layers, arise by preconditioning. Often, this problem is due to an overestimation of the maximum allowable timestep size. In fact, in a low Reynolds regime, the influence of viscous effects on time marching schemes predominates. An important role in the determination of viscous time steps plays the von Neumann number (VNN), whereas the Courant Friedrichs Lewy criteria (CFL) influences the inviscid time step behaviour. The objective of this work is the presentation of the theoretical background and results of a consistent Low-Mach preconditioning scheme based on a preconditioner proposed by Turkel, which has been extended to a wide range of Reynolds numbers. Furthermore, the interaction of low-Mach preconditioning, the von Neumann and Courant Friedrichs Lewy numbers on the convergence history and quality of results will be discussed. Its implementation is illustrated using DLR’s in-house CFD code TRACE. To prove the robustness and correctness of the algorithm, we discuss a set of test cases like the lid-driven cavity at different Reynolds numbers influenced by CFL and VNN