7.2.2 K-ε Two-Equation Model K-ε 双方程模型[cfd-7-2-2]

K- $\varepsilon$ 湍流模型是应用最广泛的双方程涡黏性模型。它基于对湍动能 $$K$$ 和湍流耗散率 $\varepsilon$ 的方程求解。K- $\varepsilon$ 模型的历史渊源可以追溯到周培源的工作[42]。20世纪70年代期间，该模型的多种形式被提出，其中最重要的贡献来自Jones和Launder[43]、[44]，Launder和Sharma[45]，以及Launder和Spalding[46]。en

The K- $\varepsilon$ turbulence model is the most widely employed two-equation eddy-viscosity model. It is based on the solution of equations for the turbulent kinetic energy $$K$$ and the turbulent dissipation rate $\varepsilon$ . The historic roots of the K- $\varepsilon$ model reach to the work of Chou [42]. During the 1970's, various formulations of the model were proposed. The most important contributions were due to Jones and Launder [43], [44], Launder and Sharma [45] as well as due to Launder and Spalding [46].

K- $\varepsilon$ 湍流模型需要加入所谓的阻尼函数（damping functions），才能在黏性底层内一直有效直至壁面。阻尼函数的目的是保证 $$K$$ 和 $\varepsilon$ 在壁面处具有正确的极限行为，即en

The K- $\varepsilon$ turbulence model requires addition of the so-called damping functions in order to stay valid through the viscous sublayer to the wall. The aim of the damping functions is to assure proper limiting behaviour of $$K$$ and $\varepsilon$ at the wall, i.e.,

K \sim y^2 \quad \text{and} \quad \frac{\varepsilon}{K} \sim \frac{2\nu}{y^2} \quad \text{for} \quad y \rightarrow 0, \tag{7.48}

其中 $$y$$ 表示垂直于壁面的坐标。此外还可以证明，雷诺剪切应力在近壁处的行为如（参见例如[17]，第138-139页）en

where $$y$$ represents the coordinate normal to the wall. Further, it can be shown that the Reynolds shear stress behaves like (see, e.g., [17] pp. 138-139)

\tau_{ij}^R \sim y^3 \quad \text{for} \quad y \rightarrow 0,\ i \neq j. \tag{7.49}

带阻尼函数的K- $\varepsilon$ 模型也被称为低雷诺数（low Reynolds number）模型。使用最广泛的阻尼函数形式由Jones和Launder[43]、Launder和Sharma[45]、Lam和Bremhorst[47]以及Chien[48]提出。七种不同的低雷诺数K- $\varepsilon$ 模型的比较可在文献[49]中找到。en

The K- $\varepsilon$ models with damping functions are also denoted as low Reynolds number models. The most widely used formulations of the damping functions were proposed by Jones and Launder [43], Launder and Sharma [45], Lam and Bremhorst [47], and by Chien [48]. The reader may find a comparison of seven different low Reynolds number K- $\varepsilon$ models in Ref. [49].

K- $\varepsilon$ 湍流模型在数值求解上比前述Spalart-Allmaras模型（小节7.2.1）更困难。特别是，阻尼函数使湍流方程带有刚性源项。这一点，加上壁面附近为解析黏性底层所必需的高网格分辨率，要求至少采用点隐式、最好采用全隐式的时间推进格式。文献[50]给出了关于K- $\varepsilon$ 方程显式时间离散的有用提示。K- $\varepsilon$ 模型在结构网格和非结构网格上实现的例子可参见例如[51]-[59]。最后，需要特别指出，对于具有逆压梯度的流动，K- $\varepsilon$ 模型的精度会下降[49]、[17]。en

The K- $\varepsilon$ turbulence model is more difficult to solve numerically than the previously discussed Spalart-Allmaras model (Subsection 7.2.1). Particularly, the damping functions lead to turbulence equations with stiff source terms. This, and the necessary high grid resolution nearby walls (in order to resolve the viscous sublayer), requires the utilisation of at least point-implicit or better full-implicit time-stepping schemes. Reference [50] contains useful hints on the explicit time discretisation of the K- $\varepsilon$ equations. Examples of implementations of the K- $\varepsilon$ model on structured as well as on unstructured grids can be found, e.g., in [51]-[59]. Finally, it is important to note that the accuracy of the K- $\varepsilon$ model degrades for flows with adverse pressure gradient [49], [17].

Differential Form 微分形式

一个低雷诺数K- $\varepsilon$ 模型可以写成en

A low Reynolds number K- $\varepsilon$ model can be written as

\begin{aligned} \frac{\partial\rho K}{\partial t} + \frac{\partial}{\partial x_j}\left(\rho v_j K\right) &= \frac{\partial}{\partial x_j}\left[\left(\mu_L + \frac{\mu_T}{\sigma_K}\right)\frac{\partial K}{\partial x_j}\right] + \tau_{ij}^F S_{ij} - \rho\varepsilon \\ \frac{\partial\rho\varepsilon^{*}}{\partial t} + \frac{\partial}{\partial x_j}\left(\rho v_j\varepsilon^{*}\right) &= \frac{\partial}{\partial x_j}\left[\left(\mu_L + \frac{\mu_T}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon^{*}}{\partial x_j}\right] + C_{\varepsilon1} f_{\varepsilon1}\frac{\varepsilon^{*}}{K}\tau_{ij}^F S_{ij} \\ &- C_{\varepsilon2} f_{\varepsilon2}\,\rho\,\frac{(\varepsilon^{*})^2}{K} + \phi_{\varepsilon}. \end{aligned} \tag{7.50}

右端各项分别代表守恒性扩散、涡黏性生成和耗散。此外， $\phi_{\varepsilon}$ 表示所谓的显式壁面项。Favre平均湍流应力 $\tau_{ij}^F$ 由式(7.25)给出，应变率张量 $S_{ij}$ 由式(7.3)得到。式(7.28)和(7.29)中的湍流涡黏性由下式求得en

The terms on the right-hand side represent conservative diffusion, eddy-viscosity production and dissipation, respectively. Furthermore, $\phi_{\varepsilon}$ denotes the so-called explicit wall term. The Favre-averaged turbulent stresses $\tau_{ij}^F$ are given by Eq. (7.25) and the strain-rate tensor $S_{ij}$ follows from Eq. (7.3). The turbulent eddy viscosity in Eq. (7.28) and (7.29) results from

\mu_T = C_{\mu} f_{\mu}\,\rho\,\frac{K^2}{\varepsilon^{*}}. \tag{7.51}

按式(7.24)或式(7.25)计算涡黏性时也要用到湍动能。量 $\varepsilon^{*}$ 与湍流耗散率 $\varepsilon$ 通过下式相联系en

The turbulent kinetic energy is also employed for the evaluation of the eddy viscosity according to Eq. (7.24) or Eq. (7.25). The quantity $\varepsilon^{*}$ is related to the turbulent dissipation rate $\varepsilon$ by

\varepsilon = \varepsilon_w + \varepsilon^{*}. \tag{7.52}

其中 $\varepsilon_w$ 是耗散率在壁面上的取值。方程(7.52)的定义大大简化了壁面边界条件的施加（见下文）。en

The term $\varepsilon_w$ is the value of the dissipation rate at the wall. The definition in Eq. (7.52) greatly simplifies the application of wall-boundary conditions (see further below).

在不同的K- $\varepsilon$ 模型中，常数、近壁阻尼函数以及壁面项各不相同。这里我们选择Launder-Sharma模型，因为它在很宽的应用范围内都给出良好结果[49]。对Launder-Sharma模型，常数和湍流Prandtl数为[45]en

The constants, the near-wall damping functions as well as the wall term differ between the various K- $\varepsilon$ models. Here, we choose the Launder-Sharma model because it gives good results for a wide range of applications [49]. For the Launder-Sharma model, the constants and the turbulent Prandtl number are given by [45]

\begin{aligned} &C_{\mu} = 0.09, \qquad C_{\varepsilon1} = 1.44, \qquad C_{\varepsilon2} = 1.92, \\ &\sigma_K = 1.0, \qquad \sigma_{\varepsilon} = 1.3, \qquad Pr_T = 0.9. \end{aligned} \tag{7.53}

此外，近壁阻尼函数为en

Furthermore, the near-wall damping functions read

\begin{aligned} f_{\mu} &= \exp\left(\frac{-3.4}{\left(1 + 0.02\,Re_T\right)^2}\right) \\ f_{\varepsilon1} &= 1 \\ f_{\varepsilon2} &= 1 - 0.3\exp\left(Re_T^2\right) \end{aligned} \tag{7.54}

其中 $Re_T = \rho K^2/(\varepsilon^{*}\mu_L)$ 为湍流雷诺数。（译注： $f_{\varepsilon2}$ 中的指数项原书印作 $\exp(Re_T^2)$ ,无负号；按Launder–Sharma模型的通行形式应为 $\exp(-Re_T^2)$ ,疑为原书排印疏漏,此处照录原书。）en

with $Re_T = \rho K^2/(\varepsilon^{*}\mu_L)$ being the turbulent Reynolds number.

最后，显式壁面项 $\phi_{\varepsilon}$ 和 $\varepsilon_w$ 的值定义为en

Finally, the explicit wall term $\phi_{\varepsilon}$ and the value $\varepsilon_w$ are defined as

\phi_{\varepsilon} = 2\mu_T\frac{\mu_L}{\rho}\left(\frac{\partial^2 v_s}{\partial y_n^2}\right)^2 \quad \text{and} \quad \varepsilon_w = \frac{2\mu_L}{\rho}\left(\frac{\partial\sqrt{K}}{\partial y_n}\right)^2, \tag{7.55}

其中 $$v_s$$ 表示平行于壁面的速度， $$y_n$$ 表示垂直于壁面的坐标。为了避免显式地知道壁面距离和壁面取向，通常用下面的笛卡尔张量形式[60]、[57]来计算壁面项和 $\varepsilon_w$ en

where $$v_s$$ stands for the velocity parallel to the wall, and $$y_n$$ represents the coordinate normal to the wall. In order to avoid an explicit knowledge of the wall distance and orientation, it is common to compute the wall term and $\varepsilon_w$ from the following Cartesian tensor form [60], [57]

\phi_{\varepsilon} = 2\mu_T\frac{\mu_L}{\rho}\left(\frac{\partial^2 v_i}{\partial x_j \partial x_k}\right)^2 \quad \text{and} \quad \varepsilon_w = \frac{2\mu_L}{\rho}\left(\frac{\partial\sqrt{K}}{\partial x_j}\right)^2 \tag{7.56}

以代替方程(7.55)。en

instead of Eq. (7.55).

Integral Form 积分形式

对控制体 $\Omega$ （面元为 $$dS$$ ）写成随时间变化的积分形式，低雷诺数K- $\varepsilon$ 湍流模型为en

Written in time-dependent integral form for a control volume $\Omega$ with a surface element $$dS$$ , the low Reynolds number K- $\varepsilon$ turbulence model reads

\frac{\partial}{\partial t}\int_{\Omega}\vec{W}_T\ d\Omega + \oint_{\partial\Omega}\left(\vec{F}_{c,T} - \vec{F}_{v,T}\right)dS = \int_{\Omega}\vec{Q}_T\ d\Omega. \tag{7.57}

守恒变量向量取如下形式en

The vector of the conservative variables takes the form

\vec{W}_T = \begin{bmatrix} \rho K \\ \rho\varepsilon^{*} \end{bmatrix}. \tag{7.58}

对流通量向量定义为en

The vector of the convective fluxes is defined

\vec{F}_{c,T} = \begin{bmatrix} \rho K V \\ \rho\varepsilon^{*} V \end{bmatrix}, \tag{7.59}

其中 $$V$$ 表示逆变速度（见式(2.22)）。黏性通量向量为en

where $$V$$ denotes the contravariant velocity (see Eq. (2.22)). The vector of the viscous fluxes is given by

\vec{F}_{v,T} = \begin{bmatrix} n_x\tau_{xx}^K + n_y\tau_{yy}^K + n_z\tau_{zz}^K \\ n_x\tau_{xx}^{\varepsilon} + n_y\tau_{yy}^{\varepsilon} + n_z\tau_{zz}^{\varepsilon} \end{bmatrix} \tag{7.60}

其中的法向湍流黏性应力为en

with the normal turbulent viscous stresses

\begin{aligned} \tau_{xx}^K &= \left(\mu_L + \frac{\mu_T}{\sigma_K}\right)\frac{\partial K}{\partial x},\ \cdots \\ \tau_{xx}^{\varepsilon} &= \left(\mu_L + \frac{\mu_T}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon^{*}}{\partial x},\ \cdots \end{aligned} \tag{7.61}

在方程(7.60)中， $$n_x$$ 、 $$n_y$$ 、 $$n_z$$ 表示表面 $\partial\Omega$ 外指单位法向量的分量。源项由下式求值en

In Eq. (7.60), $$n_x$$ , $$n_y$$ , $$n_z$$ represent the components of the outward-facing unit normal vector of the surface $\partial\Omega$ . The source term is evaluated from

\vec{Q}_T = \begin{bmatrix} P - \rho\varepsilon \\ \left(C_{\varepsilon1} f_{\varepsilon1} P - C_{\varepsilon2} f_{\varepsilon2}\,\rho\varepsilon^{*}\right)\dfrac{\varepsilon^{*}}{K} + \phi_{\varepsilon} \end{bmatrix}, \tag{7.62}

其中 $$P$$ 表示湍动能的生成项，其定义为en

where $$P$$ denotes the production term of the turbulent kinetic energy. It is defined as

\begin{aligned} P &= \tau_{xx}^F\frac{\partial u}{\partial x} + \tau_{yy}^F\frac{\partial v}{\partial y} + \tau_{zz}^F\frac{\partial w}{\partial z} \\ &+ \tau_{xy}^F\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right) + \tau_{xz}^F\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right) + \tau_{yz}^F\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right) \end{aligned} \tag{7.63}

其中Favre平均湍流应力 $\tau_{ij}^F$ 由式(7.25)给出。对Launder-Sharma模型，常数、近壁阻尼函数以及壁面项均按式(7.52)-(7.56)中的定义取值。湍流涡黏性 $\mu_T$ 由方程(7.51)求得。en

with the Favre-averaged turbulent stresses $\tau_{ij}^F$ given by Eq. (7.25). The constants, the near-wall damping functions as well as the wall term follow for the Launder-Sharma model from the definitions in Eqs. (7.52)-(7.56). The turbulent eddy viscosity $\mu_T$ is obtained from Eq. (7.51).

Initial and Boundary Conditions 初始条件与边界条件

最简单的做法是用自由来流值初始化 $$K$$ 和 $\varepsilon^{*}$ 。更好的替代方案是在固体壁面附近为 $$K$$ 和 $\varepsilon^{*}$ 指定分布。该分布可以通过与湍流平板边界层的类比得到[51]。但这需要知道壁面距离，而在非结构网格上壁面距离未必容易获得。en

The simplest approach is to initialise $$K$$ and $\varepsilon^{*}$ with their freestream values. A better alternative consists of prescribing profiles for $$K$$ and $\varepsilon^{*}$ near solid walls. The profiles can be obtained from analogy to turbulent flat-plate boundary layer [51]. However, this requires the knowledge of wall distances which may not be readily available like it is the case on unstructured grids.

若采用方程(7.52)的变换，固体壁面上的正确边界条件为 $$K = 0$$ 和 $\varepsilon^{*} = 0$ ，这也意味着壁面上 $\mu_T = 0$ 。在入口边界上， $$K$$ 和 $\varepsilon^{*}$ 可以由湍流强度和长度尺度的关系式计算，即en

The proper boundary conditions at solid walls are $$K = 0$$ and $\varepsilon^{*} = 0$ , provided the transformation in Eq. (7.52) is utilised. This also implies $\mu_T = 0$ at walls. At inflow boundaries, $$K$$ and $\varepsilon^{*}$ can be computed from relations for the turbulent intensity and length scale, i.e.,

\left(T_u\right)_{\infty} = \frac{\sqrt{\dfrac{2}{3}K_{\infty}}}{\|\vec{v}_{\infty}\|_2}, \qquad \left(l_T\right)_{\infty} = \frac{C_{\mu}K_{\infty}^{3/2}}{\varepsilon_{\infty}^{*}}, \tag{7.64}

这里假定了 $\varepsilon_{\infty}^{*} = \varepsilon_{\infty}$ 。在叶轮机械中， $\left(l_T\right)_{\infty}$ 取为平均径向叶片间距的 $10^{-3}$ 至 $10^{-2}$ 倍[61]。在出口边界上， $$K$$ 和 $\varepsilon^{*}$ 的值由计算域内部外推。en

where we assumed $\varepsilon_{\infty}^{*} = \varepsilon_{\infty}$ . In turbomachinery, $\left(l_T\right)_{\infty}$ is chosen between $10^{-3}$ and $10^{-2}$ times the mean radial blade spacing [61]. The values of $$K$$ and $\varepsilon^{*}$ are extrapolated from the interior at outflow boundaries.

Wall functions 壁面函数

如前所述，低雷诺数模型要求壁面处的网格非常细。标准条件是第一个节点（或单元质心）距壁面 $y^{+} \le 1$ 。为了降低湍流方程的刚性并节省网格点/单元数量，常采用 $10 \le y^{+} \le 100$ 的较粗网格。在这种情形下，K- $\varepsilon$ 模型——方程(7.50)或方程(7.57)——在不含阻尼函数（ $f_{\mu} = f_{\varepsilon1} = f_{\varepsilon2} = 1$ ； $\varepsilon_w = 0$ ）和壁面项（ $\phi_{\varepsilon} = 0$ ）的情况下使用。这就是所谓的高雷诺数（high Reynolds number）湍流模型。显然，第一个节点（单元质心）与壁面之间的距离必须由所谓的壁面函数（wall functions）来弥合。壁面函数给出紧邻壁面的节点（单元质心）处 $$K$$ 和 $\varepsilon^{*}$ 的值。湍流方程不在壁面本身以及第一层节点（单元）上求解。壁面函数有多种形式，一般基于对数壁面律。一个例子是Spalding的函数[62]，它同时模拟了黏性底层、过渡区和对数层。高雷诺数模型的实现可参见例如[63]-[67]或[58]。en

As we already noted, the low Reynolds number models require very fine grids at walls. The standard condition is that the first node (or cell centroid) should be located at the distance $y^{+} \le 1$ from the wall. In order to reduce the the stiffness of the turbulence equations and to save a number of grid points/cells, coarser grids with $10 \le y^{+} \le 100$ are often employed. In such a case, the K- $\varepsilon$ model Eq. (7.50) or Eq. (7.57) is applied without the damping functions ( $f_{\mu} = f_{\varepsilon1} = f_{\varepsilon2} = 1$ ; $\varepsilon_w = 0$ ) and the wall term ( $\phi_{\varepsilon} = 0$ ). We speak here of a high Reynolds number turbulence model. Apparently, the distance between the first node (cell centroid) and the wall has to be bridged by the so-called wall functions. The wall functions deliver the values of $$K$$ and $\varepsilon^{*}$ at the node (cell centroid) adjacent to the wall. The turbulence equations are not solved at the wall itself and at the first layer of nodes (cells). Various formulation of the wall functions are used, in general based on the logarithmic wall-law. One example is the function of Spalding [62], which models the viscous sublayer, the transition region as well as the logarithmic layer. Implementations of high Reynolds number models were described, e.g., in [63]-[67] or [58].

只要网格不太粗，使用壁面函数对附着边界层可以得到相当准确的结果。它还允许采用纯显式时间推进格式。然而，对分离流动而言，壁面函数的使用就非常成问题了。en

The application of the wall functions leads (provided the grid is not too coarse) to reasonably accurate results for attached boundary layers. It also allows the utilisation of purely explicit time-stepping schemes. However, the use of wall functions becomes highly questionable for separated flows.