7.3.3 Subgrid-Scale Modelling 亚网格尺度建模[cfd-7-3-3]

亚网格尺度（subgrid-scale）模型的主要任务是模拟大尺度与亚网格尺度之间的能量传递。平均而言，能量从大尺度输运到小尺度（湍流串级过程）。因此，亚网格尺度模型必须提供充分的能量耗散手段。但在某些情形下，能量也会从小尺度流向大尺度——这一过程称为反向散射（backscatter）。因此模型也应当考虑这一效应。反向散射模型可参见例如[121]。en

The main task of a subgrid-scale model is to simulate energy transfer between the large and the subgrid scales. On the average, the energy is transported from the large scales to the small ones (turbulent cascade process). Therefore, a subgrid-scale model has to provide means of adequate energy dissipation. However, in some instances the energy also flows from small to large scales - a process called backscatter. Thus, the model should account for this effect as well. Backscatter models are discussed, e.g., in [121].

过去已提出多种亚网格尺度模型，研究至今仍在继续。这些模型可分为两大类。第一类是显式地建模SGS张量 $\tau_{ij}^S$ 的方法，此时必要条件是空间离散格式引起的数值耗散必须远低于亚网格尺度耗散。大多数显式SGS模型基于涡黏性概念，下面将予以说明。此外，我们还将介绍构成所有亚网格尺度模型基础的Smagorinsky模型，并简要讨论所谓动力亚网格尺度（dynamic subgrid-scale）模型的基础。六种不同显式亚网格尺度模型的比较最近见于[122]。en

Various subgrid-scale models were proposed in the past and the research still continues. The models can be divided into two basic classes. The first one consists of approaches which model the SGS tensor $\tau_{ij}^S$ explicitly. A necessary condition is then that the numerical dissipation caused by the spatial discretisation scheme must be much lower than the subgrid-scale dissipation. The majority of explicit SGS models is based on the eddy-viscosity concept, which is explained next. Furthermore, we shall present the Smagorinsky model, which forms the basis of all subgrid-scale models. We shall also briefly discuss the basics of the so-called dynamic subgrid-scale models. A comparison of six different explicit subgrid-scale models was presented recently in [122].

第二类亚网格尺度模型是通过对流通量的适当离散来隐式地模拟SGS应力（即略去 $\tau_{ij}^S$ ）的方法。这类模型称为单调积分大涡模拟（Monotonically Integrated LES，MILES）。该方法最早由Boris等[123]提出，近来由Grinstein和Fureby大力提倡[124]、[125]。此处的条件是数值耗散要正确地模拟亚网格尺度耗散。这并不容易做到，文献[126]和[127]的研究证明了这一点。尽管如此，MILES已在多种流动问题上取得了一定成功[128]-[134]。en

The second class of subgrid-scale models consists of approaches where the SGS stresses are modeled implicitly by an appropriate discretisation of the convective fluxes (thus $\tau_{ij}^S$ is omitted). These models are referred to as Monotonically Integrated LES (MILES). The methodology was first presented by Boris et al. [123] and recently advocated by Grinstein and Fureby [124], [125]. The condition in this case is that the numerical dissipation correctly models the subgrid-scale dissipation. This is not quite easy to achieve, as the investigations in Ref. [126] and [127] prove. Nevertheless, MILES was applied with some success to a variety of flow problems [128]-[134].

Eddy-Viscosity Models 涡黏性模型

这些显式模型能够表现小尺度的全局耗散效应，但无法再现能量交换的局部细节。对不可压缩流动，涡黏性模型把SGS应力与大尺度应变率 $\overline{S}_{ij}$ 联系起来如下en

These explicit models are able to represent the global dissipative effects of the small scales, but they cannot reproduce the local details of the energy exchange. In the case of incompressible flows, the eddy-viscosity models relate the SGS stresses to the large-scale strain-rate $\overline{S}_{ij}$ as follows

\tau_{ij}^S - \frac{\delta_{ij}}{3}\tau_{kk}^S = -2\nu_T\overline{S}_{ij}. \tag{7.91}

应变率 $\overline{S}_{ij}$ 由式(7.3)用滤波后的速度分量得到。为节省计算量，涡黏性 $\nu_T$ 一般用代数关系式求值。SGS应力的各向同性部分（ $\tau_{kk}^S$ ）可以并入滤波压力[135]、单独建模[136]或忽略。en

The strain-rate $\overline{S}_{ij}$ is obtained from Eq. (7.3) by using filtered velocity components. The eddy viscosity $\nu_T$ is in general evaluated from algebraic relations in order to save numerical costs. The isotropic part of the SGS stresses ( $\tau_{kk}^S$ ) can either be added to the filtered pressure [135], modelled [136] or neglected.

与方程(7.91)类似的关系式也适用于可压缩Navier-Stokes方程。Favre平均SGS应力张量的分量近似为en

Relation similar to Eq. (7.91) applies also in the case of compressible Navier-Stokes equations. The components of the Favre-averaged SGS stress tensor are approximated as

\tau_{ij}^{SF} - \frac{\delta_{ij}}{3}\tau_{kk}^{SF} = -2\overline{\rho}\,\nu_T\tilde{S}_{ij} + \left(\frac{2\overline{\rho}\,\nu_T}{3}\right)\frac{\partial\tilde{v}_k}{\partial x_k}\delta_{ij}. \tag{7.92}

应变率张量 $\tilde{S}_{ij}$ 的分量由方程(7.90)给出。en

The components of the strain-rate tensor $\tilde{S}_{ij}$ are given in Eq. (7.90).

Smagorinsky SGS Model Smagorinsky SGS 模型

Smagorinsky模型[75]基于平衡假设，即小尺度把从大尺度接收到的能量全部、瞬时地耗散掉。该代数模型取如下形式en

The Smagorinsky model [75] is based on the equilibrium hypothesis which implies that the small scales dissipate entirely and instantaneously all the energy they receive from the large scales. The algebraic model assumes the form

\nu_T = \left(C_s\Delta\right)^2|\overline{S}|, \tag{7.93}

其中 $|\overline{S}| = \left(2\overline{S}_{ij}\overline{S}_{ij}\right)^{1/2}$ 为应变率张量的大小， $$C_s$$ 为Smagorinsky常数（Smagorinsky constant）。Lilly[137]求得的理论值为 $C_s \approx 0.18$ 。但Smagorinsky常数依赖于流动类型，例如在剪切流中 $$C_s$$ 须减小到约0.1。方程(7.93)中的滤波宽度 $\Delta$ 通常取平均网格尺寸的两倍，即 $\Delta = 2\left(\Delta x_1\,\Delta x_2\,\Delta x_3\right)^{1/3}$ 。en

where $|\overline{S}| = \left(2\overline{S}_{ij}\overline{S}_{ij}\right)^{1/2}$ is the magnitude of the strain-rate tensor and $$C_s$$ denotes the Smagorinsky constant. The theoretical value found by Lilly [137] is $C_s \approx 0.18$ . However, the Smagorinsky constant depends on the type of the flow. For example, in shear flows $$C_s$$ has to be reduced to approximately 0.1. The filter width $\Delta$ in Eq. (7.93) is usually chosen to be twice the average grid size, i.e., $\Delta = 2\left(\Delta x_1\,\Delta x_2\,\Delta x_3\right)^{1/3}$ .

为了体现近壁处小尺度增长的减弱，必须减小涡黏性 $\nu_T$ 的取值。于是，Smagorinsky模型——方程(7.93)——按Van Driest阻尼修改为en

In order to account for the reduced growth of the small scales near walls, the value of the eddy viscosity $\nu_T$ has to be reduced. Thus, the Smagorinsky model Eq. (7.93) is modified according to Van Driest damping as

\nu_T = \left[C_s\Delta\left(1 - e^{-y^{+}/25}\right)\right]^2|\overline{S}|, \tag{7.94}

其中 $y^{+}$ 表示无量纲壁面距离（壁面距离的计算参见例如文献[39]）。en

where $y^{+}$ represents the dimensionless wall distance (for the computation of wall distances see, e.g., Ref. [39]).

Smagorinsky模型计算代价低、易于实现，但它有若干严重缺点：

在有平均剪切的层流区域中它耗散过强；
在壁面附近以及层流-湍流转捩处需要特殊处理；
参数 $$C_s$$ 没有唯一定义；
没有模拟能量反向散射过程。

The Smagorinsky model is numerically cheap and easy to implement. However, it has several serious disadvantages:

it is too dissipative in laminar regions with mean shear;
it requires special provisions near walls and at laminar-turbulent transition;
the parameter $$C_s$$ is not uniquely defined;
the process of energy backscatter is not modelled.

由于这些缺陷，人们提出了各种其他方法（参见例如[111]）。下面这些动力模型非常流行。en

Because of these shortcomings, various other approaches were proposed (see, e.g., [111]). Very popular are the following dynamic models.

Dynamic SGS Models 动力 SGS 模型

动力SGS模型在计算方程(7.91)或(7.92)中的涡黏性 $\nu_T$ 时，采用与Smagorinsky模型（方程(7.93)）相同的关系式。区别在于：事先调整的Smagorinsky常数被一个在空间和时间中动态演化的参数所取代，即en

The dynamic SGS models employ the same relation as the Smagorinsky model (Eq. (7.93)) for the evaluation of the eddy viscosity $\nu_T$ in Eq. (7.91) or (7.92). The difference is that the Smagorinsky constant (adjusted a priori) is replaced by a parameter, which evolves dynamically in space and in time. Hence,

\nu_T = C_d(\vec{r}, t)\,\Delta^2|\overline{S}|. \tag{7.95}

参数 $$C_d$$ 依据湍流最小尺度的能量含量来计算。为此，Germano等[138]提出采用第二个滤波——所谓的测试滤波（test filter） $\hat{\Delta}$ 。测试滤波的宽度必须大于作用于控制方程的滤波宽度 $\Delta$ （通常 $\hat{\Delta} = 2\Delta$ ）。把测试滤波作用于已滤波的方程，便得到所谓的亚测试尺度应力（subtest-scale stresses） $\tau_{ij}^{ST}$ en

The parameter $$C_d$$ is computed based on the energy content of the smallest scale of the turbulence. For this purpose, Germano et al. [138] proposed to employ a second filter - the so-called test filter $\hat{\Delta}$ . The width of the test filter has to be larger than that of the filter $\Delta$ applied to the governing equations (usually $\hat{\Delta} = 2\Delta$ ). The application of the test filter to the filtered equations leads to the so-called subtest-scale stresses $\tau_{ij}^{ST}$

\tau_{ij}^{ST} = \widehat{\overline{v}_i\,\overline{v}_j} - \hat{\overline{v}}_i\,\hat{\overline{v}}_j. \tag{7.96}

亚测试尺度应力通过Germano恒等式（Germano identity）[139]与SGS应力 $\tau_{ij}^S$ （方程(7.83)）相联系en

The subtest-scale stresses are related to the SGS stresses $\tau_{ij}^S$ (Eq. (7.83)) via the Germano identity [139]

\hat{L}_{ij} = \tau_{ij}^{ST} - \hat{\tau}_{ij}^S = \widehat{\overline{v}_i\,\overline{v}_j} - \hat{\overline{v}}_i\,\hat{\overline{v}}_j, \tag{7.97}

其中 $\hat{L}_{ij}$ 表示与测试滤波相关的Leonard应力，它代表长度介于滤波宽度 $\Delta$ 与测试滤波宽度 $\hat{\Delta}$ 之间的尺度对雷诺应力的贡献。en

where $\hat{L}_{ij}$ denotes the Leonard stresses associated with the test filter. It represents the contribution to the Reynolds stresses by the scales whose length is intermediate between the filter width $\Delta$ and the test filter width $\hat{\Delta}$ .

若用涡黏性方法——方程(7.91)连同方程(7.95)——来表示方程(7.97)中的亚测试尺度应力与SGS应力，便得到en

If we express the subtest-scale and SGS stresses in Eq. (7.97) using the eddy-viscosity approach Eq. (7.91) together with Eq. (7.95), we obtain

\hat{L}_{ij} - \frac{\delta_{ij}}{3}L_{kk} = -2C_d M_{ij} \tag{7.98}

其中en

with

M_{ij} = \hat{\Delta}^2|\widehat{\overline{S}}|\,\widehat{\overline{S}}_{ij} - \left[\Delta^2|\overline{S}|\,\overline{S}_{ij}\right]^{\wedge}. \tag{7.99}

记号 $[\,]^{\wedge}$ 表示方括号中的整个项都经过测试滤波。参数 $$C_d$$ 可以利用Lilly的最小二乘极小化[140]从方程(7.98)导出，这给出en

The notation $[\,]^{\wedge}$ means that the whole term enclosed in the square brackets is test-filtered. The parameter $$C_d$$ can be derived from Eq. (7.98) by using the least-squares minimisation of Lilly [140]. This leads to

C_d(\vec{r}, t) = -\frac{1}{2}\frac{L_{ij}M_{ij}}{M_{mn}M_{mn}}. \tag{7.100}

上述表述(7.100)在数学上是不自洽的，因为在方程(7.98)中参数 $$C_d$$ 被拿到了测试滤波之外。因此在实际中，方程(7.100)的分子和分母都在均匀方向上做系综平均，即en

The above formulation (7.100) is mathematically inconsistent since the parameter $$C_d$$ was taken outside the test filter in Eq. (7.98). In practice, the numerator and denominator in Eq. (7.100) are therefore ensemble-averaged in the homogeneous directions, i.e.,

C_d(\vec{r}, t) = -\frac{1}{2}\frac{\left\langle L_{ij}M_{ij}\right\rangle}{\left\langle M_{mn}M_{mn}\right\rangle}. \tag{7.101}

改进的动力SGS模型由Ghosal等[141]、Carati等[142]、Piomelli和Liu[143]以及Held[90]等人提出。en

Improved dynamic SGS models were proposed, e.g., by Ghosal et al. [141], Carati et al. [142], Piomelli and Liu [143], and Held [90].