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

多尺度问题建模的一个目标，如复杂流动中的冲击，是在不解决小尺度细节的情况下推导出大尺度量的演化方程。该建议旨在提出一种新的技术，用于推导能够在不引入粘性耗散的情况下以激波形式正则化不连续性的流体方程。这是通过定义可观测通量和可观测发散来实现的。一个可观察到的发散定理，然后应用到无粘流体流的质量，动量和能量守恒。一组方程，称为可观察的欧拉方程，推导出它们满足守恒定律在可观察的尺度，α。可观察尺度通常由我们观察流体性质的能力决定。这是数值模拟中的分辨率尺度或实验中仪器的最小可分辨尺度。当可观测尺度趋于零时，经典的欧拉方程将被恢复。这项工作的目的是对理论，计算和物理的可观测性及其应用单相流体问题的冲击理解。虽然所提出的想法在流体中的冲击正则化的背景下进行测试，这一举措有可能被应用到各种各样的其他多尺度的问题，如弹性，磁流体力学，多相流等减少不确定性的湍流空气动力学和流体动力学预测将有助于制造商的大多数相关技术，以降低其机器的成本，提高其性能。考虑到这些问题在我们社会中所起的作用，预计会产生重大的社会经济影响。本科研究助理将通过补充REU支持寻求，并可以预期来自这些领域。PI现有的学科课程将丰富这项工作的成果，扩大学生的多学科接触。将建立一个网站，向公众传播信息。

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.