Inverse cascades in two-dimensional compressible turbulence. I. Incompressible forcing at low Mach number

Abstract

A fundamental property of forced, dissipative, two-dimensional incompressible Navier–Stokes (NS) systems is the dominance at long times of the longest wavelengths available to the flow. This dominance, attributed to an inverse cascade of energy with respect to enstrophy [Kraichnan, Phys. Fluids 10, 1417 (1967)], has been observed in spectrally accurate numerical simulations (see Refs. 5–7) of the incompressible NS equations. The numerical investigation of this behavior is extended to the weakly compressible regime by means of the fully compressible Fourier collocation code combox with a solenoidal forcing function that imparts no net momentum and stirs the fluid in a wave-number band in the neighborhood of kf=11. A comparison of spectral results from combox simulations with an average Mach number of 0.22 with those from identically forced incompressible simulations, at Reynolds numbers ≤700, indicates (1) the compressible and incompressible wave-number dependences in both the energy cascading and enstrophy cascading regions are nearly identical; (2) in the compressible calculation, a dual power law is also observed in density and pressure fluctuations; and (3) continued (kf=11) forcing leads to overall continued growth in the longest accessible velocity field wavelength, in both the incompressible and compressible cases.