Note on Compressibility Effects for Taylor Green Vortex Analysis

The classical Taylor Green vortex problem is defined for an incompressible flow field. If the flow is computed with a compressible code at an essentially “incompressible” low Mach number (Ma=0.1 in this case), compressibility effects are deemed to be negligible for the resulting flow field. However, diagnostic quantities that depend on the spatial derivatives of the velocity field and/or the \nabla\cdot u =0 condition can show a significantly different behavior for their incompressible or compressible formulation, depending on the resolution of the problem.

The plot below compares the compressible and incompressible dissipation rate for the Taylor Green vortex problem. The difference between both formulations (the pressure term and the divergence condition) for this LES-type resolution can be very significant. It should be noted that this difference decreases as one approaches a DNS resolution.
So if an essentially incompressible problem is solved by a compressible method, care should be taken to include compressibility effects in the analysis, even for low Mach numbers.