Observable Divergence Theorem: A new technique for deriving averaged equations for multi-scale shock problems

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1134229 MohseniAn objective of modeling of multiscale problems, such as shocks in complex flows, is to derive an evolution equation for large scale quantities without resolving the details of the small scales. This proposal aims at a new technique for deriving fluid equations capable of regularizing discontinuities in the form of shocks without the introduction of viscous dissipation. This is achieved by defining observable fluxes and observable divergence. An observable divergence theorem is then applied to the conservation of mass, momentum, and energy of an inviscid fluid flow. A set of equations, called the observable Euler equations, are derived where they satisfy the conservation laws at the observable scale, alpha. The observable scale is often dictated by our ability to observe a fluid property. This is the resolution scale in numerical simulations or the minimum resolvable scale of an apparatus in an experiment. The classical Euler equations will be recovered if the observable scale approaches zero. This effort is aimed towards theoretical, computational, and physical understanding of the observability and its application to single phase fluid problems with shocks. While the proposed ideas are tested in the context of shock regularization in fluids, this initiative has the potential to be applied to a wide variety of other multi-scale problems such as elasticity, magnetohydrodynamics, multi-phase flows, etc. Reduction in uncertainty of turbulent aero and hydrodynamic predictions will help manufacturers of most related technologies to reduce the cost of their machines and enhance their performances. Considering the role that such problems play in our society, important socio-economical impacts are expected. Undergraduate research assistants will be sought via supplementary REU support, and can be expected to come from these fields. The PI's existing disciplinary courses will be enriched with results from this work, expanding student multidisciplinary exposure. A web site will be developed to disseminate information to the general public.