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

1134229模拟多尺度问题(如复杂流动中的激波)的莫森目标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。这一建议旨在找到一种新的技术，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的流体方程。这是通过定义可观测通量和可观测散度来实现的。然后，一个可观测的散度定理被应用于无粘流的质量、动量和能量守恒。导出了一组方程，称为可观测欧拉方程，其中它们满足可观测尺度下的守恒定律。可观察的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。如果能观尺度趋近于零，经典的欧拉方程将被恢复。这项工作的目的是从理论上、计算上和物理上理解可观测性及其在含激波的单相流体问题中的应用。虽然提出的想法在流体中的激波正则化的背景下进行了测试，但这一倡议有可能应用于各种其他多尺度问题，如弹性、磁流体动力学、多相流等。湍流气动和流体动力学预测的不确定性的减少将有助于大多数相关技术的制造商降低其机器成本并提高其性能。考虑到这些问题在我们的社会中所起的作用，预计会产生重要的社会经济影响。本科生研究助理将通过补充的REU支持来寻找，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，扩大学生对多学科的接触。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.