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Observable Divergence Theorem: Evolution Equations for Inviscid Regularization of Shocks and Turbulence

The divergence theorem of Gauss plays a central role in the derivation of the governing differential equations in fluid dynamics, electrodynamics, gravitational fields, and optics. One is often interested in an evolution equation for the large scale quantities without resolving the details of the small scales. As a result, there has been a significant effort in developing time-averaged and spatially-filtered equations for large scale dynamics from the fully resolved governing differential equations. One should realize that by starting from these fully-resolved equations (e.g. the Euler or Navier-Stokes equations) to derive an averaged evolution equation one has already taken the limit of the wave-numbers approaching infinity with no regards to our observational abilities at such a limit. As a result, obtaining the evolution equations for large scale quantities (low wave-numbers) by an averaging or filtering process is done after the fact. This could explain many of the theoretical and computational difficulties with the Euler or Navier-Stokes equations. Here, a rather different approach is proposed. The averaging process in implemented before the derivation of the differential form of the transport equations. A new observable divergence concept is defined based on fluxes calculated from observable quantities at a desired averaging scale, $α$. An observable divergence theorem is then proved and applied in the derivation of the observable and regularized transport equations. We further show that the application of the observable divergence theorem to incompressible flows results in a formal derivation of the inviscid Leray turbulence model first proposed in 1934. It is argued that such a methodology in deriving fluid evolution equations removes many of the theoretical and computational difficulties in multi-scale problems such as turbulence and shocks.