A Fast, Matrix-free Implicit Method for Computing Low Mach Number Flows on Unstructured Grids

Abstract

A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and NavierStokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES+ LU-SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both inviscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU-SGS method. The numerical results obtained indicate that the use of the GM RES + LU-SGS method leads to a significant increase in performance over the LU-SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.

Low‐cost implicit schemes for all‐speed flows on unstructured meshes

“…with blocks D n,m i and C n,m i,k defined by (15)- (16). Under the assumption of a systematic evaluation of the preconditioning matrix P −1 c at the cell center i, the above relationship can also be recast under the form:…”

“…is unchanged with respect to (15)- (16). The linear system (17) can then be solved exactly, using for instance a GMRES approach, in which case the implicit scheme (17) will be referred to as a direct solver.…”

“…It was next remarked [13] the implicit stage could be further simplified by 495 introducing time increments of the flux vectors instead of products of flux Jacobian matrices and time increments of the conserved variables and relaxing these new unknowns when solving the resulting matrix-free implicit stage. Using simultaneously spectral radius simplification and flux-vector increment relaxation, Löhner et al have successfully developed a matrix-free implicit method for solving the 3D Navier-Stokes equations on unstructured grids [14], which has been further extended to unsteady flows [15] and all-speed flows using a low-Mach preconditioning technique [16,17].…”

Rotor wake and flow analysis using a coupled Eulerian–Lagrangian method

“…For the time integration, a dual time stepping method was applied with the lower-upper symmetric Gauss-Seidel scheme (Luo & Baum, 1999) to simulate the unsteady flow phenomenon at every pseudo time step. The Spalart-Allmaras turbulence model (Spalart & Allmaras, 1992) was employed for the RANS closure.…”

Helicopter Blade-Vortex Interaction Airload and Noise Prediction Using Coupling CFD/VWM Method

“…In this paper, a dual-time stepping method is applied for the time discretization to solve the governing equation. LU-SGS (Lower-upper Symmertric Gauss-Seidel) [23] is adopted for sub-iteration at every pseudo-time step. The inviscid terms are calculated by the Roe scheme [24], and the Baldwin-Lomax model is employed as the turbulence model for the Reynolds Averaged Navier-Stokes (RANS) closure.…”