Accuracy of high order density based compressible methods in low mach vortical flows

Abstract

A new, well posed, two-dimensional two-mode incompressible Kelvin{Helmholtz instability test case has been chosen to explore the ability of a compressible algorithm, Godunov-type scheme with the low Mach number correction, which can be used for simulations involving low Mach numbers, to capture the observed vortex pairing process due to the initial Kelvin{Helmholtz instability growth on low resolution grid. The order of accuracy, 2nd and 5th , of the compressible algorithm is also highlighted. The observed vortex pairing results and the corresponding momentum thickness of the mixing layer against time are compared with results obtained using the same compressible algorithm but without the low Mach number correction and three other methods, a Lagrange remap method where the Lagrange phase is 2nd order accurate in space and time while the remap phase is 3rd order accurate in space and 2nd order accurate in time, a 5th order accurate in space and time nite di erence type method based on the wave propagation algorithm and a 5th order spatial and 3rd order temporal accurate Godunov method utilising the SLAU numerical ux with low Mach capture property. The ability of the compressible ow solver of the commercial software, ANSYS Fluent, in solving low Mach ows is also examined for both implicit and explicit methods provided in the compressible ow solver. In the present two dimensional two mode incompressible Kelvin{Helmholtz instability test case, the ow conditions, stream velocities, length-scales and Reynolds numbers, are taken from an experiment conducted on the observation of vortex pairing process. Three di erent values of low Mach numbers, 0:2, 0:02 and 0:002 have been tested on grid resolutions of 24 24, 32 32, 48 48 and 64 64 on all the di erent numerical approaches. The results obtained show the vortex pairing process can be captured on a low grid resolution with the low Mach number correction applied down to 0:002 with 2nd and 5th order Godunovtype methods. Results also demonstrate clearly that a speci cally designed low Mach correction or ux is required for all algorithms except the Lagrange-remap approach, where dissipation is independent of Mach number. ANSYS Fluent's compressible ow solver with the implicit time stepping method also captures the vortex pairing on low resolutions but excessive dissipation prevents the instability growth when explicit time stepping method is applied.