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Center for Nonlinear Studies |
Seventy years have passed since Kolmogorov's successful theory of incompressible turbulence, yet there is a stark absence of an analogous theory to describe compressible flows. I will briefly discuss the key notions of an inertial-range and the cascade in turbulence. I will present the relevant background and some of the difficulties that have prevented an extension of these concepts to compressible flows. I will then go over some recent developments in this regard. Some of my main conclusions which emerge from this work are: (1) any scale-decomposition aimed at probing inertial-range dynamics must satisfy an “inviscid criterion,” i.e. it must guarantee that viscous effects on the dynamics of large-scale flow are negligible. (2) The inviscid criterion naturally leads to a density-weighted coarse-graining of the velocity field. Such a coarse-graining method is already known in the literature as Favre filtering; however its use has been primarily motivated by appealing modelling properties rather than underlying physical considerations. (3) The non-linear transfer of kinetic energy to small scales is dominated by scale-local interactions. In particular, the results preclude transfer of kinetic energy from large-scales directly to dissipation scales, such as into shocks, in high Reynolds number turbulence as is commonly believed. (4) If the pressure dilatation co-spectrum decays fast enough, I will show that mean kinetic and internal energy budgets statistically decouple beyond a transitional ``conversion'' range. (5) In collaboration with Shengtai Li, Hui Li, and Hao Xu at LANL, we use high-resolution numerical simulations of forced and decaying compressible turbulence to compute, for the first time, pressure dilatation co-spectrum and verify statistical decoupling. (6) The analysis establishes the existence of an ensuing inertial range over which mean subgrid-scale (SGS) kinetic energy flux becomes constant, independent of scale. Over this inertial range, mean kinetic energy cascades locally and in a conservative fashion, despite not being an invariant. Time permitting, I will then discuss rigorous constraints on the scaling of density, velocity, and pressure spectra and structure functions.
Host: Peter Loxley, loxley@lanl.gov