Title: Observable Divergence Theorem: Basic Principles and Its Application in Regularization of Shocks and Turbulence

Department of Mechanical and Aerospace Engineering/Department of Electrical and Computer Engineering

ABSTRACT: Both turbulence and shock formation in inviscid flows are prone to high wave number mode generations. This continuous generation of high wavemodes results in an energy cascade to an ever smaller scales in turbulence and/or creation of shocks in compressible flows. This high wavenumber problem is often remedied by the addition of a viscous term (parabolic term) in both compressible and incompressible flows. A regularization technique for the Burgers equation (Norgard and Mohseni, J. Phys. A 2008) was recently reported. This inviscid regularization was extended to one-dimensional compressible Euler equations in Norgard and Mohseni, SIAM Multiscale Model. Simul.   2009, 2010. Furthermore, some proof of regularization was provided in Villavert and Mohseni J. of Applied Mathematics and Computing 2012. This talk presents a formal derivation of these equations from basic principles. Our previous results are extended to multidimensional compressible and incompressible Euler equations. We define a new observable divergence based on fluxes calculated from observable quantities at a desired scale. An observable divergence theorem is then proved and applied in the derivation of the regularized equations. It is shown that the derived equations reduce to inviscid Leray flow model in the limit of incompressibility. It is expected that this technique simultaneously regularize shocks and turbulence for compressible and incompressible flows. Finally, numerical simulations are presented for the incompressible and compressible compressible l observable Euler equations including shock tube problem, 3D compressible turbulence, and shock-turbulence interaction problems.