On the application of the eddy viscosity concept in the Inertial sub-range of turbulence
On the application of the eddy viscosity concept in the Inertial sub-range of turbulence
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DOI：
10.5065/d67h1ggq
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发表时间：
1966
期刊：
影响因子：
--
通讯作者：
K. Lilly
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文献类型：
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作者：
K. Lilly
结果表明，Smagorinsky提出的湍流问题的数值解中使用的涡扩散假设是一致的，在最小的可分辨尺度的数值模型的惯性子范围的存在。任意常数，Smagorinsky假设是为了统一，是一个独特的函数常数的Kolmogoroff能谱函数。另一种假设，涉及一个明确的湍流强度，作为一个可能的改进与大的空间和时间变化的湍流应力的流动。涡粘性概念在湍流惯性子区中的应用尽管流体力学家和数学家作出了不懈的努力，但获得湍流方程有用的解析解的问题仍然是艰巨的。与分析理论的缓慢进展相比，数字计算机的发展继续以快速的速度进行。因此，“蛮力”方法的解决方案的流体动力学问题变得更加有吸引力，尽管难以获得一般的结果和许多恼人的，纯粹的数值困难，数值方法已成为几乎不可缺少的解决方案的非线性二维或准二维问题，如发生在大规模的大气和海洋动力学。完全紊流的三维初边值流动问题现在才开始用计算机求解。在高雷诺数范围内，计算机能够同时解决含能标度和耗散标度仍然是完全不可想象的，但是，分辨率的极限从最大的含能标度扩展到惯性子范围，这一点并非不可能。本文的目的是描述和合理化的方法，通过一个假设的理想化的惯性子范围，模拟显式计算的运动尺度和耗散尺度之间的湍流能量交换。这些方法并不完全是新的，以前曾被Smagorinsky（1963，1965）和2 Lilly（1962）使用过，尽管在二维计算中有些不合适。它并不打算提出一个完整的理论，而只是部分合理化一些方法或配方，这些方法或配方已经取得了一定的成功，但被认为是最有效的计算模型，只有现在才能接近。为了简化起见，我们考虑等密度流体的不可压缩流动.流体运动的连续欧拉方程和连续性方程，以标准张量下标表示法，写为：-; ·。;· @（d ' -O（1））
It is shown that an eddy diffusion hypothesis suggested by Smagorinsky for use in numerical solutions of turbulent flow problems is consistent with the existence of an inertial subrange at the smallest resolvable scale of the numerical model. The arbitrary constant, assumed by Smagorinsky to be of order unity, is shown to be a unique function of the constant of the Kolmogoroff energy spectrum function. An alternative hypothesis, involving an explicit turbulent intensity, is introduced as a possible improvement for flows with large space and time variations of turbulent stress. On the Application of the Eddy Viscosity Concept in the Inertial Sub-range of Turbulence Despite persistent efforts by fluid dynamicists and mathematicians the problem of obtaining useful analytic solutions of turbulent flow equations remains formidable. By comparison to the slow progress of analytic theory, the development of digital computers continues to proceed at a rapid pace. Thus the "brute force" methods of solution of fluid dynamics problems become more attractive, in spite of the difficulty of obtaining generalized results and the many annoying, purely numerical difficulties, Numerical methods have become virtually indispensible for solutions of non-linear two-dimensional or quasitwo-dimensional problems, such as occur in large-scale atmospheric and oceanic dynamics. Fully turbulent three-dimensional initialboundary value flow problems are now just beginning to be approachable for computer solutions. It is still totally inconceivable that a computer could resolve both the energy containing and dissipative scales in a high Reynolds number regime, but it is not at all unlikely that the limits of resolution could extend from the largest energy containing scale into the inertial sub-range. It is the purpose of this paper to describe and rationalize methods of simulating the turbulent energy exchange between the scales of motion explicitly computed and the dissipation scale, through an assumed idealized inertial sub-range. The methods are not entirely new, having been used previously, although somewhat inappropriately in two-dimensional computations, by Smagorinsky (1963, 1965) and 2 Lilly (1962). It is e not intended to present a complete theory, but only to partly rationalize some methods or recipes which have already been found moderately successful, but are believed to be most valid in computational models only now becomging accessible. For purposes of simplification we consider incompressible flow of a fluid of constant density. The continuous Eulerian equations of fluid motion and the continuity equation, in standard tensor subscript notation , are written as follows: -; •. ;• @( d ' -O (1)