Cascade of Kinetic Energy in Three-Dimensional Compressible Turbulence

Abstract

The conservative cascade of kinetic energy is established using both Fourier analysis and a new exact physical-space flux relation in a simulated compressible turbulence. The subgrid scale (SGS) kinetic energy flux of the compressive mode is found to be significantly larger than that of the solenoidal mode in the inertial range, which is the main physical origin for the occurrence of Kolmogorov's -5/3 scaling of the energy spectrum in compressible turbulence. The perfect antiparallel alignment between the large-scale strain and the SGS stress leads to highly efficient kinetic energy transfer in shock regions, which is a distinctive feature of shock structures in comparison with vortex structures. The rescaled probability distribution functions of SGS kinetic energy flux collapse in the inertial range, indicating a statistical self-similarity of kinetic energy cascades.

Scaling laws of compressible turbulence

“…The compressible kinetic energy E c cascade follows k −2 power laws of the wave number, and the solenoidal E s follows the k −5/3 power laws. This means that the solenoidal E s dominates the energy spectrum for the wavenumber within 4 ≤ k ≤ 20 as stated in Wang et al(2013), that is E = E c + E s = k −5/3 (c c k −8/9 + c s ) ≈ c s k −5/3 , and will overall lead the total kinetic energy spectrum E to exhibit of a − 5 3 scaling as stated by Wang [7].…”

“…Hence, the energy spectrum equation (5) can be rewritten as the function of the Mach number equation (7).…”

“…The data of Figure 1 was taken from Wang [19] and reformatted, where E ρ = E(k, ρ) is total kinetic energy, and…”

“…From numerical simulation at Mach number close to 1 carried by Wang [19], the coefficient λ can be estimated as λ ≃ 1.…”

Stratified Kelvin-Helmholtz turbulence of compressible shear flows

“…Kida and Orszag (1990) studied the mechanics of energy transfer and distribution as well as an examination of 20 small-scale spectra in compressible turbulence with root-mean-square Mach numbers upto 0.9. Theoretical laws have also been advanced for the statistical behavior of turbulence quantities under the influence of compressibility effects (Shivamoggi, 1992;Lele, 1994;Shivamoggi, 2011;Wang et al, 2013). Kritsuk et al (2007) utilized an adaptive mesh refinement (AMR) algorithm along with a piecewise parabolic approach for numerical dissipation to obtain scaling tendencies at high Mach number values for both kinetic energy, density weighted kinetic energy and density power spectra.…”