7.1.5 Eddy-Viscosity Hypothesis 涡黏性假设[cfd-7-1-5]

湍流建模领域最重要的贡献之一由Boussinesq于1877年作出[11]、[12]。他的想法基于如下观察：湍流中的动量输运由大而高能的湍流涡所引起的混合主导。Boussinesq假设湍流剪应力像层流中那样线性地依赖于平均应变率，其比例系数就是涡黏性（eddy viscosity）。对于雷诺平均不可压缩流动（式(7.14)），Boussinesq假设可以写为en

One of the most significant contributions to turbulence modelling was presented in 1877 by Boussinesq [11], [12]. His idea is based on the observation that the momentum transfer in a turbulent flow is dominated by the mixing caused by large energetic turbulent eddies. The Boussinesq hypothesis assumes that the turbulent shear stress depends linearly on the mean rate of strain, as in a laminar flow. The proportionality factor is the eddy viscosity. The Boussinesq hypothesis for Reynolds averaged incompressible flow (Eq. (7.14)) can be written as

\tau^R_{ij}=-\rho\overline{v'_iv'_j}=2\mu_T\overline{S}_{ij}-\frac{2}{3}\rho K\delta_{ij}, \tag{7.24}

其中 $\overline{S}_{ij}$ 表示雷诺平均应变率张量（式(7.3)，另见式(7.16)）， $$K$$ 为湍动能（ $K=(1/2)\overline{v'_iv'_i}$ ）， $\mu_T$ 代表涡黏性。与分子黏性 $\mu$ 不同，涡黏性 $\mu_T$ 不是流体的物理特性，而是局部流动条件的函数。此外， $\mu_T$ 还受到流动历史效应的强烈影响。en

where $\overline{S}_{ij}$ denotes the Reynolds-averaged strain-rate tensor (Eq. (7.3), cf. also Eq. (7.16)), $$K$$ is the turbulent kinetic energy ( $K=(1/2)\overline{v'_iv'_i}$ ), and $\mu_T$ stands for the eddy viscosity. Unlike the molecular viscosity $\mu$ , the eddy viscosity $\mu_T$ represents no physical characteristic of the fluid, but it is a function of the local flow conditions. Additionally, $\mu_T$ is also strongly affected by flow history effects.

对于可压缩Favre平均与雷诺平均Navier-Stokes方程(7.19)，Boussinesq涡黏性假设写为en

In the case of the compressible Favre- and Reynolds-averaged Navier-Stokes equations (7.19), the Boussinesq eddy-viscosity hypothesis reads

\tau^F_{ij}=-\bar{\rho}\,\widetilde{v''_iv''_j}=2\mu_T\tilde{S}_{ij}-\left(\frac{2\mu_T}{3}\right)\frac{\partial\tilde{v}_k}{\partial x_k}\delta_{ij}-\frac{2}{3}\bar{\rho}\tilde{K}\delta_{ij}, \tag{7.25}

其中 $\tilde{S}_{ij}$ 和 $\tilde{K}$ 分别为Favre平均应变率和Favre平均湍动能。注意它与式(7.2)的相似性。式(7.24)和(7.25)中的 $(2/3)\rho K\delta_{ij}$ 项是为了得到 $\tau^R_{ij}$ 或 $\tau^F_{ij}$ 的正确迹（trace）所必需的。这意味着，我们必须有en

where $\tilde{S}_{ij}$ and $\tilde{K}$ are the Favre-averaged strain rate and turbulent kinetic energy, respectively. Note the similarity to Eq. (7.2). The term $(2/3)\rho K\delta_{ij}$ in Eqs. (7.24) and (7.25) is required in order to obtain the proper trace of $\tau^R_{ij}$ or $\tau^F_{ij}$ . This means that we must have

\tau^R_{ii}=-2\rho K \qquad\text{or}\qquad \tau^F_{ii}=-2\bar{\rho}\tilde{K} \tag{3}

即在 $\overline{S}_{ii}=0$ （连续方程）或 $\tilde{S}_{ii}=0$ 的情形下，以满足湍动能的关系式(7.18)或(7.21)。然而， $(2/3)\rho K\delta_{ij}$ 这一项常常被忽略，特别是在与较简单的湍流模型（如代数模型）联用时。en

in the case of $\overline{S}_{ii}=0$ (continuity equation) or $\tilde{S}_{ii}=0$ , in order to fulfil the relations Eq. (7.18) or (7.21) for the turbulent kinetic energy. However, the term $(2/3)\rho K\delta_{ij}$ is often neglected, particularly in connection with simpler turbulence models (like algebraic ones).

对湍流热通量向量建模时常用的近似基于经典的Reynolds比拟[18]。于是，我们可以写en

The approximation, which is commonly used for the modelling of the turbulent heat-flux vector, is based on the classical Reynolds analogy [18]. Hence, we may write

\bar{\rho}\,\widetilde{v''_jh''}=-k_T\frac{\partial\tilde{T}}{\partial x_j} \tag{7.26}

其中湍流热导率系数（turbulent thermal conductivity coefficient） $$k_T$$ 定义为en

with the turbulent thermal conductivity coefficient $$k_T$$ being defined as

k_T=c_p\frac{\mu_T}{Pr_T}. \tag{7.27}

在方程(7.27)中， $$c_p$$ 表示定压比热系数， $$Pr_T$$ 为湍流Prandtl数。一般假设湍流Prandtl数在整个流场中为常数（对空气 $$Pr_T=0.9$$ ）。en

In Equation (7.27), $$c_p$$ denotes the specific heat coefficient at constant pressure and $$Pr_T$$ is the turbulent Prandtl number. The turbulent Prandtl number is in general assumed to be constant over the flow field ( $$Pr_T=0.9$$ for air).

把涡黏性方法应用于控制方程(2.19)或方程(7.1)的雷诺（及Favre）平均形式时，黏性应力张量式(2.15)或式(7.2)中的动力黏性系数 $\mu$ 被简单地替换为层流分量与湍流分量之和，即en

By applying the eddy-viscosity approach to the Reynolds- (and Favre-) averaged form of the governing equations (2.19) or Eq. (7.1), the dynamic viscosity coefficient $\mu$ in the viscous stress tensor Eq. (2.15) or Eq. (7.2) is simply replaced by the sum of a laminar and a turbulent component, i.e.,

\mu=\mu_L+\mu_T. \tag{7.28}

层流黏性 $\mu_L$ 例如可用Sutherland公式(2.30)计算。此外，根据式(7.26)给出的Reynolds比拟，式(2.24)或式(7.2)中的热导率系数 $$k$$ 按下式计算en

The laminar viscosity $\mu_L$ is computed, for example, with the aid of the Sutherland formula (2.30). Furthermore, according to the Reynolds analogy given by Eq. (7.26), the thermal conductivity coefficient $$k$$ in Eq. (2.24) or Eq. (7.2) is evaluated as

k=k_L+k_T=c_p\left(\frac{\mu_L}{Pr_L}+\frac{\mu_T}{Pr_T}\right). \tag{7.29}

至少从工程的角度看，Boussinesq的涡黏性概念非常有吸引力，因为它“只”需要确定 $\mu_T$ （式(7.24)或(7.25)中 $(2/3)\rho K\delta_{ij}$ 项所需的湍动能 $$K$$ ，或者作为湍流模型的副产物得到，或者干脆略去）。一旦知道了涡黏性 $\mu_T$ ，通过引入平均流动变量并把 $\mu_T$ 加到层流黏性上，我们就能容易地把Navier-Stokes方程(2.19)或(7.1)扩展用于湍流模拟。因此，Boussinesq方法成为了大量一阶湍流封闭的基础。然而，对某些应用而言，Boussinesq假设不再成立（例如见[17]第214页或[19]第111页）：

平均应变率发生突然变化的流动；
具有显著流线曲率的流动；
具有旋转和分层的流动；
管道和透平机械中的二次流；
边界层分离与再附的流动。

The eddy-viscosity concept of Boussinesq is, at least from the engineering point of view, very attractive since it requires ``only'' the determination of $\mu_T$ (the turbulent kinetic energy $$K$$ needed for the term $(2/3)\rho K\delta_{ij}$ in Eq. (7.24) or (7.25) is either obtained as a by-product of the turbulence model or is simply omitted). Once we know the eddy viscosity $\mu_T$ , we can easily extend the Navier-Stokes equations (2.19) or (7.1) to the simulation of turbulent flows by introducing averaged flow variables and by adding $\mu_T$ to the laminar viscosity. Therefore, Boussinesq's approach became the basis for a large variety of first-order turbulence closures. However, there are applications for which the Boussinesq hypothesis is no longer valid (see, e.g., [17] p. 214 or [19] p. 111):

flows with sudden change of mean strain rate,
flows with significant streamline curvature,
flows with rotation and stratification,
secondary flows in ducts and in turbomachinery,
flows with boundary layer separation and reattachment.

涡黏性方法的局限源于把湍流与平均应变场视为处于平衡状态的假设，以及其结果与系统旋转无关的假定。通过在湍流模型中加入适当的修正项，结果可以明显改善[20]、[21]。采用下面要介绍的非线性涡黏性模型，可以进一步提高预测精度。en

The limitations of the eddy-viscosity approach are caused by the assumption of equilibrium between the turbulence and the mean strain field, as well as by the independence on system rotation. The results can be notably improved by using appropriate correction terms in the turbulence models [20], [21]. Further increased accuracy of predictions can be achieved through the application of non-linear eddy-viscosity models which are described next.