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Center for Nonlinear Studies |
Background: In inviscid compressible flows, only the sum of kinetic and internal energy is a global invariant. The idea of a cascade, a central notion in incompressible turbulence, is therefore without physical basis since kinetic energy is not conserved separately. Compressible flows allow for an exchange between kinetic and internal energy through two mechanisms: viscous dissipation and pressure dilatation. While the former process is localized to the smallest scales just like in incompressible turbulence, the latter is a hallmark of compressibility and can a priori allow for an exchange at any scale through compression and rarefaction. Results: We present the first direct evidence that mean kinetic energy cascades conservatively beyond a transitional ``conversion'' scale-range despite not being an invariant of the compressible flow dynamics. We use high-resolution three-dimensional simulations of compressible hydrodynamic turbulence on $10243$ grids. The key quantity we measure is pressure dilatation cospectrum, $E^{PD}(k)$, where we provide the first numerical evidence that it decays at a rate faster than $k^{-1}$ as a function of wavenumber. This is sufficient to imply that mean pressure dilatation acts primarily at large-scales and vanishes at small scales beyond the transitional "conversion'' scale-range. Discussion: Our results and the physical picture we are advancing might seem counter-intuitive at first. After all, a hallmark of compressible turbulence is the formation of shocks and the generation of sound waves. Such phenomena involve compression and rarefaction at all scales and are not restricted to large scales. However, our results concern global pressure dilatation, $-\langle P\grad\bdot\bu\rangle$, and not the pointwise quantity. We show that while pressure dilatation has large values at small scales in the vicinity of shocks, such small-scale contributions vanish when averaging over the flow domain due to cancellations between compression and rarefaction regions. This is joint work with Shengtai Li (T-5) and Hui Li (T-2).
Host: Peter Loxley, loxley@lanl.gov