Observable Divergence:
A Method for Regularization of Continuum Problems including Shocks and
Turbulence

By: Kamran Mohseni
University of Florida

ABSTRACT: A challenge in many non-dissipative continuum problems,
including shocks and turbulence, is the formation of sharp gradients
in the vector field. To this end, both turbulence and shock formation
in inviscid flows are prone to high wave number mode-generation. The
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 regularized
by the addition of a viscous term in both compressible and
incompressible flows.  An inviscid regularization technique for the
multi-dimensional Burgers equation (Norgard and Mohseni, SIAM MMS
2008, 2009) was recently reported where a unique solution is proved to
exist at all times and approach the entropy solution of the inviscid
Burgers equation in some limit. This inviscid regularization was
extended to one-dimensional compressible Euler equations (Norgard and
Mohseni, SIAM MMS 2010). 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 the inviscid Leray flow model in the limit of
incompressibility. It is expected that this technique simultaneously
regularize shocks and turbulence for fluid flows. Finally, numerical
simulations are presented for the compressible 1D, 2D, and 3D problems
with and without shocks or turbulence.