7.1.6 Non-Linear Eddy Viscosity 非线性涡黏性[cfd-7-1-6]

为了消除湍流与平均应变率之间平衡假设所强加的限制，Lumley[22]、[23]提议用应变张量与旋转张量的高阶乘积来扩展线性的Boussinesq方法。这可以视为一种Taylor级数展开。沿着Lumley的思路，人们提出了众多非线性涡黏性模型，例如文献[24]–[29]。en

In order to remove the restrictions imposed by the assumption of equilibrium between the turbulence and the mean strain rate, Lumley [22], [23] proposed to extend the linear Boussinesq approach by higher-order products of strain and rotation tensors. This can viewed as a Taylor series expansion. Following the idea of Lumley, numerous non-linear eddy-viscosity models were proposed, see, for example, Refs. [24]--[29].

下面我们介绍由Shih等人[25]提出的一种较新的方法。它在一般涡黏性表述中包含直至三阶的项，特别适合于旋转流（swirling flows）。正如文献[19]第194页所指出的，三次项对高精度至关重要。雷诺应力 $\tau^R_{ij}$ 可以表示为[25]、[30]（对照式(7.24)）en

In the following, we shall present one recent approach proposed by Shih et al. [25]. It includes up to third-order terms in the general eddy-viscosity formulation and is particularly suited to swirling flows. As already pointed out in [19], p. 194, cubic terms are essential for high accuracy. The Reynolds stresses $\tau^R_{ij}$ can be expressed as [25], [30] (cf. Eq. (7.24))

\begin{aligned} \rho\overline{v'_iv'_j}={}&\frac{2}{3}\rho K\delta_{ij}-C_1\frac{\rho K^2}{\varepsilon}\,2S^*_{ij}-C_3\frac{\rho K^3}{\varepsilon^2}\left[\overline{S}_{ik}\overline{\Omega}_{kj}-\overline{\Omega}_{ik}\overline{S}_{kj}\right]\\ &-C_4\frac{\rho K^4}{\varepsilon^3}\left[(\overline{S}_{ik})^2\overline{\Omega}_{kj}-\overline{\Omega}_{ik}(\overline{S}_{kj})^2\right]\\ &+C_5\frac{\rho K^4}{\varepsilon^3}\left[\overline{\Omega}_{ik}\overline{S}_{km}\overline{\Omega}_{mj}-\frac{1}{3}\overline{\Omega}_{kl}\overline{S}_{lm}\overline{\Omega}_{mk}\delta_{ij}+I_sS^*_{ij}\right] \end{aligned} \tag{7.30}

其中 $\overline{S}_{ij}$ 、 $\overline{\Omega}_{ij}$ 按式(7.3)和(7.4)。此外en

with $\overline{S}_{ij}$ , $\overline{\Omega}_{ij}$ according to Eqs. (7.3) and (7.4). Furthermore,

\begin{aligned} I_s&=\frac{1}{2}\left[\overline{S}_{kk}\overline{S}_{ll}-(\overline{S}_{kk})^2\right]S^*_{ij}\\ S^*_{ij}&=\overline{S}_{ij}-\frac{1}{3}\overline{S}_{kk}\delta_{ij}. \end{aligned} \tag{7.31}

湍动能 $$K$$ 和耗散率（dissipation rate） $\varepsilon$ 的值由低雷诺数 $$K$$ - $\varepsilon$ 湍流模型给出（参见7.2.2小节）。式(7.30)中的系数 $$C_1$$ 至 $$C_5$$ 见文献[25]、[30]。en

The values of the turbulent kinetic energy $$K$$ and the dissipation rate $\varepsilon$ are obtained from low-Reynolds $$K$$ - $\varepsilon$ turbulence model (cf. Subsection 7.2.2). The factors $$C_1$$ to $$C_5$$ in Eq. (7.30) are provided in Refs. [25], [30].

与线性涡黏性方法相比，非线性模型的计算代价只略微增大，但对复杂湍流的预测能力却有显著提高。en

In comparison to the linear eddy-viscosity approach, the non-linear models are computationally only slightly more expensive, but they offer a substantially improved prediction capabilities for complex turbulent flows.