A Hybrid Adaptive Low-Mach-Number/Compressible Method: Euler Equations

TL;DR

This paper presents a hybrid computational method combining low-Mach-number and fully compressible Euler equations to efficiently simulate flows with significant acoustic effects without prohibitive computational costs.

Contribution

A novel multi-level hybrid algorithm that solves compressible equations globally and low-Mach-number equations locally, enabling efficient simulation of acoustic phenomena in fluid flows.

Findings

Hybrid method achieves two orders of magnitude larger time steps.

Overall computational time reduced by a factor of 8.

Successfully simulates aeroacoustic propagation from Kelvin-Helmholtz instability.

Abstract

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin-Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8.