Kean Kiera, Layton William, Schneier Michael
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
Philos Trans A Math Phys Eng Sci. 2022 Jun 27;380(2226):20210054. doi: 10.1098/rsta.2021.0054. Epub 2022 May 9.
We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Let denote a selected time scale. Away from walls, interpreting an early suggestion of Prandtl, we set [Formula: see text]In the near-wall region analysis suggests replacing the traditional [Formula: see text] ([Formula: see text] wall normal distance) with [Formula: see text] giving [Formula: see text]This specification of [Formula: see text] results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct [Formula: see text], balancing energy input with energy dissipation. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.
我们证明了剪切流通用涡粘性模型中总（粘性加模拟湍流）能量耗散的一个估计。近壁平均粘性与有效全局粘性之比是该估计中的关键参数。然后将此结果应用于单方程URANS湍流模型，该模型中此比值取决于湍流长度尺度的设定。该模型由普朗特于1945年推导得出，是柯尔莫哥洛夫于1942年推导的两方程模型的一个组成部分，并且是许多用于预测湍流的非定常雷诺平均模型的核心。设 表示选定的时间尺度。在远离壁面处，根据普朗特早期的一个建议，我们设定[公式：见正文]在近壁区域，分析表明用[公式：见正文]替换传统的[公式：见正文]（[公式：见正文]为壁面法向距离），得到[公式：见正文]这种对[公式：见正文]的设定产生了一个具有正确近壁渐近性的更简单模型。其能量耗散率的尺度不大于物理上正确的[公式：见正文]，使能量输入与能量耗散达到平衡。本文是主题为“物理流体动力学中的数学问题（第2部分）”的一部分。
