This article addresses two acceleration techniques in the context of high‐order methods: p‐multigrid and local Mach number preconditioning. The flux reconstruction method is used as the spatial discretization scheme and the flow of interest is modeled by the two‐dimensional Euler and Navier–Stokes equations in both steady and unsteady settings. The Weiss and Smith low Mach number preconditioner is used together with dual time stepping in order to perform unsteady simulations. The p‐multigrid uses the third order explicit Runge–Kutta (RK3) scheme as the smoother and the non‐linear LU‐SGS implicit method as the coarse level solver. The algorithm performance is compared against both the explicit RK3 and the LU‐SGS approaches. The use of the Mach number preconditioning significantly increases the efficiency of the p‐multigrid method. For unsteady simulations, the preconditioner helps with the efficiency of the p‐multigrid with larger physical time steps. In most cases, the preconditioned p‐multigrid approach is comparable to or faster than the implicit LU‐SGS algorithm and requires less memory, specially for p>2$$ p>2 $$ schemes.
The present work addresses several aspects associated with an efficient use of the high-order Spectral Difference (SD) method for the simulation of compressible flows. Flows of interest are assumed to be adequately modeled by the two-dimensional (2-D) Euler or the 2-D Navier-Stokes equations. Issues associated with the use of an implicit time integration scheme, limiter formulations and curved boundary approaches are discussed. Order of accuracy studies are considered for literature test cases to measure the effective accuracy of the SD method. The results provide data for discussions regarding the coupling of high-order mesh boundaries, implicit time-marching, viscous effects and limiter techniques for the high-order SD method. In particular, the need for the use of high-order meshes at wall boundaries for adequate simulation of flows using high-order SD methods is highlighted and discussed. Furthermore, typical aerospace airfoil flows are also addressed, both with inviscid and viscous formulations, in order to demonstrate the capability implemented.
Flow between rotating concentric cylinders, or the Taylor Couette flow, has been studied extensively because of its rich physics, ranging from axisymmetric steady laminar flow, to fully developed turbulent flow. In the present study, we advocate the use of this problem as a benchmark case for scale-resolving simulation, such as large eddy simulation (LES) and direct numerical simulation (DNS). The problem is attractive because of its simple geometry, simple boundary conditions, and complex physics involving wall-shear induced and centrifugal instability. Unlike the well-known fully developed channel flow, this problem has a curved wall boundary, and it is unnecessary to add a source term to the governing equations to sustain the fully developed turbulent flow. A p-refinement study for Re = 4000 is performed first to establish DNS data, including the time history of enstrophy, which can be used as an accuracy and resolution indicator to evaluate numerical methods, and is orders of magnitude faster than using the mean flow quantities and Reynolds stresses to evaluate solution quality. Finally, an hp-refinement study is performed to establish the relative accuracy and efficiency of high-order schemes of various accuracy.
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