7.3.2 Filtered Governing Equations 滤波控制方程[cfd-7-3-2]

为了去除小的湍流尺度，必须把由方程(7.76)和方程(7.77)定义的空间滤波作用于Navier-Stokes方程。滤波宽度 $\Delta$ 和滤波函数都被视为自由参数。实际上，控制方程通常并不显式滤波，而是假定网格以及离散误差定义了滤波函数 $$G$$ 。关于显式滤波的讨论见文献[116]、[117]。en

The spatial filtering, defined by Eq. (7.76) and Eq. (7.77), has to be applied to the Navier-Stokes equations in order to remove the small turbulent scales. The filter width $\Delta$ as well as the filter function are considered as free parameters. In fact, the governing equations are usually not explicitly filtered. Instead, the grid as well as the discretisation errors are assumed to define the filter $$G$$ . For the discussion of explicit filtering see Refs. [116], [117].

由于处理方式不同，下文中我们将区分Navier-Stokes方程的可压缩形式(7.1)与不可压缩形式(7.6)。en

Because of the differing treatment, we shall distinguish in the following between compressible (7.1) and incompressible (7.6) formulation of the Navier-Stokes equations.

Incompressible Navier-Stokes Equations 不可压 Navier-Stokes 方程

对牛顿流体的不可压缩流动，滤波后的控制方程(7.6)取如下形式en

For an incompressible flow of a Newtonian fluid, the filtered governing equations (7.6) take the form

\begin{aligned} \frac{\partial\overline{v}_i}{\partial x_i} &= 0 \\ \frac{\partial\overline{v}_i}{\partial t} + \frac{\partial}{\partial x_j}\left(\overline{v}_i\overline{v}_j\right) &= -\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_i} + \nu\nabla^2\overline{v}_i - \frac{\partial\tau_{ij}^S}{\partial x_j}, \end{aligned} \tag{7.81}

其中 $\nu$ 表示运动黏度系数。方程(7.81)描述了承载能量的大尺度运动的时空演化。对流项的非线性导致出现所谓的亚网格尺度应力（subgrid-scale stress，SGS）张量en

where $\nu$ denotes the kinematic viscosity coefficient. The equations (7.81) describe the temporal and spatial evolution of the large, energy-carrying scales of motion. The non-linearity of the convective term leads to the appearance of the so-called subgrid-scale stress (SGS) tensor

\tau_{ij}^S = \overline{v_i v_j} - \overline{v}_i\,\overline{v}_j, \tag{7.82}

它描述不可解析尺度的影响。为了使方程封闭，必须对SGS张量建模（见小节7.3.3）。en

which describes the effects of the unresolved scales. The SGS tensor has to be modelled (see Subsection 7.3.3) in order to close the equations.

SGS张量可以分解为三个部分[118]，即en

The SGS tensor can be decomposed into three parts [118], namely

\tau_{ij}^S = L_{ij} + C_{ij} + \tau_{ij}^{SR}. \tag{7.83}

各部分的物理含义如下：en

The individual parts have the following physical meaning:

L_{ij} = \overline{\overline{v}_i\,\overline{v}_j} - \overline{v}_i\,\overline{v}_j \tag{7.84}

为所谓的Leonard应力（Leonard stress）项，它代表产生小尺度湍流的大涡之间的相互作用。只有这一项可以由滤波速度场 $$v_i$$ 显式求出。其次，交叉应力（cross-stress）项en

is the so-called Leonard stress term and represents the interactions between large-scale eddies which produce small-scale turbulence. This term only can be evaluated explicitly from the filtered velocity field $$v_i$$ . Further, the cross-stress term

C_{ij} = \overline{\overline{v}_i\,v'_j} + \overline{v'_i\,\overline{v}_j} \tag{7.85}

描述大涡与小涡之间的相互作用。最后，en

describes interactions between large- and small-scale eddies. Finally,

\tau_{ij}^{SR} = \overline{v'_i v'_j} \tag{7.86}

为所谓的SGS雷诺应力（SGS Reynolds-stress）张量，它反映小尺度结构之间的相互作用。上述分解(7.83)如今已不再使用，主要原因是 $L_{ij}$ 和 $C_{ij}$ 在伽利略变换¹下不具有不变性。en

is the so-called SGS Reynolds-stress tensor. It reflects interactions between the small-scale structures. The above decomposition (7.83) is no longer used mainly because $L_{ij}$ and $C_{ij}$ are not invariant with respect to Galilean transformation¹.

¹原书脚注:Galilean invariance means that all frames of reference which are translating uniformly with respect to each other are equivalent.(伽利略不变性指的是,彼此做匀速平移的所有参考系都是等价的。)

Compressible Navier-Stokes Equations 可压缩 Navier-Stokes 方程

若要将LES应用于可压缩流动，就必须在对方程(7.1)做空间滤波的同时施加Favre平均（小节7.1.2）。否则，滤波后的Navier-Stokes方程将包含密度与速度、温度等其他变量的乘积。于是，式(7.1)中的速度分量、能量和温度按如下方式分解en

If LES is to be applied to compressible flows, we have to apply Favre averaging (Subsection 7.1.2) together with the spatial filtering to the Equations (7.1). Otherwise, the filtered Navier-Stokes equations would contain products between density and other variables like velocity or temperature. Thus, the velocity components, the energy and the temperature in Eq. (7.1) is decomposed as

U = \tilde{U} + U''. \tag{7.87}

空间中 $\vec{r}_0$ 处的Favre滤波变量由下式给出en

The filtered variable at the location $\vec{r}_0$ in space is given by

\tilde{U}(\vec{r}_0, t) = \frac{\overline{\rho U}}{\overline{\rho}} = \frac{1}{\overline{\rho}}\int_{D}\rho(\vec{r}, t)\, U(\vec{r}, t)\, G(\vec{r}_0, \vec{r}, \Delta)\, d\vec{r}, \tag{7.88}

其中上横线表示方程(7.77)中的滤波。Favre滤波后的Navier-Stokes方程(7.1)为[111]、[113]en

where the overbar denotes the filtering in Eq. (7.77). The Favre-filtered Navier-Stokes equations (7.1) read [111], [113]

\begin{aligned} \frac{\partial\overline{\rho}}{\partial t} + \frac{\partial}{\partial x_j}\left(\overline{\rho}\,\tilde{v}_j\right) &= 0 \\ \frac{\partial\overline{\rho}\tilde{v}_i}{\partial t} + \frac{\partial\left(\overline{\rho}\tilde{v}_j\tilde{v}_i\right)}{\partial x_j} + \frac{\partial\overline{p}}{\partial x_i} - \frac{\partial\hat{\sigma}_{ij}}{\partial x_j} &= -\frac{\partial\tau_{ij}^{SF}}{\partial x_j} + \frac{\partial}{\partial x_j}\left(\overline{\sigma}_{ij} - \hat{\sigma}_{ij}\right) \\ \frac{\partial\overline{\rho}\tilde{e}}{\partial t} + \frac{\partial\left(\overline{\rho}\tilde{v}_j\tilde{e}\right)}{\partial x_j} + \frac{\partial\tilde{q}}{\partial x_j} + \overline{p}\tilde{S}_{kk} - \hat{\sigma}_{ij}\tilde{S}_{ij} &= -\mathcal{A} - \mathcal{B} - \mathcal{C} + \mathcal{D} \end{aligned} \tag{7.89}

其中的各项为en

with the terms

\begin{aligned} \mathcal{A} &= \frac{\partial}{\partial x_j}\left[\overline{\rho}\left(\widetilde{v_j e} - \tilde{v}_j\tilde{e}\right)\right] \quad \text{-- divergence of subgrid-scale heat flux} \\ \mathcal{B} &= \frac{\partial}{\partial x_j}\left[\overline{q}_j - \tilde{q}_j\right] \quad \text{-- divergence of SGS heat diffusion} \\ \mathcal{C} &= \left[\overline{p S_{kk}} - \overline{p}\,\tilde{S}_{kk}\right] \quad \text{-- SGS pressure-dilatation} \\ \mathcal{D} &= \left[\overline{\sigma_{ij} S_{ij}} - \hat{\sigma}_{ij}\tilde{S}_{ij}\right] \quad \text{-- SGS viscous dissipation} \end{aligned} \tag{10}

以及en

and

\begin{aligned} \overline{\sigma}_{ij} &= \overline{2\mu S_{ij}} + \overline{\left(\mu_B - \frac{2\mu}{3}\right)\delta_{ij}S_{kk}} \\ \hat{\sigma}_{ij} &= 2\tilde{\mu}\tilde{S}_{ij} + \left(\tilde{\mu}_B - \frac{2\tilde{\mu}}{3}\right)\delta_{ij}\tilde{S}_{kk} \\ \tilde{S}_{ij} &= \frac{1}{2}\left(\frac{\partial\tilde{v}_i}{\partial x_j} + \frac{\partial\tilde{v}_j}{\partial x_i}\right) \\ \overline{q}_j &= -k\,\frac{\partial\overline{T}}{\partial x_j}, \qquad \tilde{q}_j = -\tilde{k}\,\frac{\partial\tilde{T}}{\partial x_j}. \end{aligned} \tag{7.90}

在上述方程(7.89)-(7.90)中， $$e$$ 表示单位质量的内能， $\tilde{S}_{ij}$ 是Favre滤波应变率张量， $\tau_{ij}^{SF} = \overline{\rho}\left(\widetilde{v_i v_j} - \tilde{v}_i\tilde{v}_j\right)$ 表示Favre平均的亚网格尺度应力。此外， $\mu$ 、 $\mu_B$ 和 $$k$$ 分别表示分子黏度、体积黏度和热导率；而 $\tilde{\mu}$ 、 $\tilde{\mu}_B$ 和 $\tilde{k}$ 是它们在滤波温度 $\tilde{T}$ 下的对应取值。en

In the above equations (7.89)-(7.90), $$e$$ denotes internal energy per unit mass, $\tilde{S}_{ij}$ is the Favre-filtered strain-rate tensor, and $\tau_{ij}^{SF} = \overline{\rho}\left(\widetilde{v_i v_j} - \tilde{v}_i\tilde{v}_j\right)$ represents the Favre-averaged subgrid-scale stress. Furthermore, $\mu$ , $\mu_B$ , and $$k$$ stand for the molecular viscosity, the bulk viscosity, and for the thermal conductivity, respectively. Finally, $\tilde{\mu}$ , $\tilde{\mu}_B$ , and $\tilde{k}$ are the corresponding values at the filtered temperature $\tilde{T}$ .

方程(7.89)的右端含有必须建模的项。在动量方程中，SGS应力 $\tau_{ij}^{SF}$ 被近似，而第二项即 $\left(\overline{\sigma}_{ij} - \hat{\sigma}_{ij}\right)$ 通常被忽略。在能量方程中，项 $\mathcal{A}$ 可以通过SGS应力表达[119]，项 $\mathcal{B}$ 可以忽略，项 $\mathcal{C}$ 、 $\mathcal{D}$ 可按文献[120]所建议的方式建模。en

The right-hand side of Eq. (7.89) contains terms which have to be modelled. In the momentum equation, the SGS stresses $\tau_{ij}^{SF}$ are approximated, but the second term, i.e., $\left(\overline{\sigma}_{ij} - \hat{\sigma}_{ij}\right)$ is usually neglected. In the energy equation, term $\mathcal{A}$ can be expressed through the SGS stresses [119], term $\mathcal{B}$ can be neglected, and terms $\mathcal{C}$ , $\mathcal{D}$ can be modelled as proposed in [120].