纳维-斯托克斯方程的三种正则化模型
Three regularization models of the Navier-Stokes equations
摘要：
我们确定如何在子过滤器尺度的物理学的治疗的差异影响的性质的三个密切相关的正则化的Navier-Stokes流。SGS模型的正规化的适用性的后果也显示通过检查它们对超滤尺度属性的影响。克拉克-α模型的数值解进行比较，两个以前采用的正则化，LANS-α和勒雷-α（在Re ~ 3300，泰勒Re ~ 790）和DNS。我们推导出了Clark-alpha和Leray-alpha模型的Karman-Howarth方程。我们确认了克拉克方程的两种可能标度之一及其相关的k^（-1）能谱。在子过滤尺度下，Clark-alpha具有与Navier-Stokes相似的总耗散和特征时间，但具有更大的不稳定性。作为SGS模型，Clark再现了DNS的能谱和双折射性质。对于Leray模型，增加滤波器宽度减小了非线性，并且有效Re实质上减小。即使对于所研究的最小α值，Leray-alpha也不足以作为SGS模型。LANS能谱k^1与其所谓的“刚体”相一致，排除了在高Re下再现DNS的大尺度能谱，同时实现了分辨率的大幅降低。然而，与Clark-alpha（共享类似的Karman-Howarth方程）相比，这个相同的特征降低了它的不稳定性。克拉克被认为是最好的近似再现总耗散率和能量谱的尺度大于α，而高阶的α值较大的不稳定性最好由LANS-α再现。
英文摘要：
We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as SGS models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-alpha model are compared to two previously employed regularizations, LANS-alpha and Leray-alpha (at Re ~ 3300, Taylor Re ~ 790) and to a DNS. We derive the Karman-Howarth equation for both the Clark-alpha and Leray-alpha models. We confirm one of two possible scalings resulting from this equation for Clark as well as its associated k^(-1) energy spectrum. At sub-filter scales, Clark-alpha possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark reproduces the energy spectrum and intermittency properties of the DNS. For the Leray model, increasing the filter width decreases the nonlinearity and the effective Re is substantially decreased. Even for the smallest value of alpha studied, Leray-alpha was inadequate as a SGS model. The LANS energy spectrum k^1, consistent with its so-called "rigid bodies," precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in resolution. However, that this same feature reduces its intermittency compared to Clark-alpha (which shares a similar Karman-Howarth equation). Clark is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than alpha, whereas high-order intermittency properties for larger values of alpha are best reproduced by LANS-alpha.
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