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

可观测散度定理：一种推导多尺度冲击问题平均方程的新技术

• 批准号: 1134229
• 负责人: Kamran Mohseni
• 金额: $ 30.01万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1134229&HistoricalAwards=false
• 关键词: Observable; Divergence; Theorem; new; technique

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.

1134229 Mohseni 多尺度问题（例如复杂流动中的冲击）建模的目标是导出大尺度量的演化方程，而不解决小尺度的细节。该提案旨在开发一种推导流体方程的新技术，该技术能够在不引入粘性耗散的情况下规范冲击形式的不连续性。这是通过定义可观测通量和可观测散度来实现的。然后将可观测散度定理应用于无粘流体流的质量、动量和能量守恒。导出一组方程，称为可观测欧拉方程，它们满足可观测尺度 alpha 上的守恒定律。可观察的尺度通常取决于我们观察流体特性的能力。这是数值模拟中的分辨率尺度或实验中设备的最小可分辨尺度。如果可观测尺度接近零，则经典欧拉方程将被恢复。这项工作旨在从理论上、计算和物理上理解可观测性及其在单相流体冲击问题中的应用。虽然所提出的想法是在流体冲击正则化的背景下进行测试的，但这一举措有可能应用于各种其他多尺度问题，例如弹性、磁流体动力学、多相流等。湍流空气和流体动力学预测将帮助大多数相关技术的制造商降低机器成本并提高其性能。考虑到此类问题在我们社会中发挥的作用，预计会产生重要的社会经济影响。本科生研究助理将通过 REU 的补充支持寻求，并且预计来自这些领域。这项工作的成果将丰富 PI 现有的学科课程，扩大学生的多学科接触。将开发一个网站向公众传播信息。