Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in R3\mathbb{R}^3

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in $\mathbb{R}^3$. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for barotropic flows, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in $L^2$, which is independent of the viscosity coefficient $\mu>0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^\gamma$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $\mu>0$, for some fixed $q_1 >\gamma$ and $q_2 >2$, where $\gamma>1$ is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations for barotropic fluids in $\mathbb{R}^3$. Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $\mu>0$, that is in the high Reynolds number limit.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1008.154

A Comparison Study of Two Methods for Elliptic Boundary Value Problems

In this paper, we perform a comparison study of two methods (the embedded boundary method and several versions of the mixed finite element method) to solve an elliptic boundary value problem

Kubo Combinatorics for Turbulence Scaling Laws

We present an extension to Kolmogorov's refined similarity hypothesis for universal fully developed turbulence. The extension is applied within Z. She and E. Leveque's multifractal model of inertial range scaling and its generalizations. Our modification rectifies an apparent gap between the implicit continuum of length scales in Obukhov's conception of a turbulent energy cascade, and scaling law models derived from Kolmogorov's refined similarity hypothesis that lack infinite divisibility. The development has relevance to universal fully developed turbulence, a state we describe explicitly in terms of the coupling between velocity fluctuations and averaged energy dissipation at all orders. This description is unique and leads to a reparametrization of the She-Leveque model that preserves its original forecasts and is infinitely divisible.Comment: 8 pages, 3 figures, 1 tabl

Modeling and Simulation of Fluid Mixing Laser Experiments and Supernova

The three year plan for this project is to develop novel theories and advanced simulation methods leading to a systematic understanding of turbulent mixing. A primary focus is the comparison of simulation models (both Direct Numerical Simulation and subgrid averaged models) to experiments. The comprehension and reduction of experimental and simulation data are central goals of this proposal. We will model 2D and 3D perturbations of planar interfaces. We will compare these tests with models derived from averaged equations (our own and those of others). As a second focus, we will develop physics based subgrid simulation models of diffusion across an interface, with physical but no numerical mass diffusion. We will conduct analytic studies of mix, in support of these objectives. Advanced issues, including multiple layers and reshock, will be considered