arXiv 1010.26122010-10-13physics.flu-dyn

可观测发散定理：激波和Turbulence无粘正则化的发展方程

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

Kamran Mohseni

摘要：

高斯的发散定理在流体动力学、电动力学、引力场和光学中的控制微分方程的推导中起着核心作用。人们往往对大尺度量的演化方程感兴趣，而不解决小尺度的细节。因此，在从完全解析的控制微分方程发展大尺度动力学的时间平均和空间滤波方程方面，已经有了很大的努力。人们应该认识到，从这些完全解析的方程（例如欧拉方程或纳维-斯托克斯方程）出发，导出一个平均的演化方程，就已经得到了接近无穷大的波数的极限，而不考虑我们在这样一个极限的观测能力。因此，通过平均或滤波过程获得大尺度量（低波数）的演化方程是在事后完成的。这可以解释欧拉或纳维-斯托克斯方程的许多理论和计算困难。在这里，提出了一个相当不同的方法。平均过程在导出输运方程的微分形式之前实现。一个新的可观测发散的概念定义的基础上计算的通量从可观测量在一个所需的平均尺度，$α$。然后证明了一个可观测的发散定理，并应用于可观测的正则化输运方程的推导。我们进一步表明，不可压缩流的可观测发散定理的应用结果在正式推导的无粘勒雷湍流模型首次提出于1934年。有人认为，这种方法在推导流体演化方程消除了许多理论和计算的困难，在多尺度问题，如湍流和冲击。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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