Low Mach Preconditioning for Harmonic Balance Solvers

Abstract

Density-based solvers suffer from a decrease in solution quality and convergence rate as the Mach number decreases. Previous studies have demonstrated that low Mach preconditioning improves the performance of density-based steady-state solvers at low Mach numbers. Applying a similar procedure, this study presents a low Mach preconditioning approach for harmonic balance solvers, which enhances the solution quality and optimizes convergence rates of frequency-domain computations in the low Mach regime. The convergence rate is improved by multiplying the time derivative with a preconditioning matrix, equalizing the acoustic and convective propagation velocities. However, for unsteady computations, the preconditioner introduces a frequency dependence to the acoustic propagation velocity. Thus, frequency-dependent preconditionining parameters are required for harmonic balance computations. Excessive numerical dissipation in the Roe solver causes inaccuracies at low Mach numbers. To address this, a preconditioned Roe solver for unsteady computations is employed, ensuring correct Mach number scaling in the artificial diffusion. The preconditioned harmonic balance solver is applied to a low Mach cylinder and the results are compared with non-preconditioned computations and experimental data. This comparison highlights shortcomings of the non-preconditioned Roe scheme in predicting unsteady flows and demonstrates that preconditioned computations are in excellent agreement with the experiments across all Mach numbers. Furthermore, the use of a frequency-dependent preconditioner is shown to achieve the fastest convergence.