出流边界[cfd-8-4]

针对Navier-Stokes方程的数值入流、特别是出流（也称为开放，open）边界条件的实现，已提出了多种方法[22]-[26]。这里我们集中讨论为叶轮机械应用而发展的方法。合适的无反射入流与出流边界条件可参见例如[27]-[30]。Giles[31]以及Hirsch和Verhoff[32]为Euler方程提出了无反射边界条件，适用于物体与入流或出流平面之间距离较短的求解域。en

Various approaches were devised for the implementation of numerical inlet, and in particular, of outlet (also named open) boundary conditions for the Navier-Stokes equations [22]-[26]. Here, we will concentrate on methodologies, which were developed for turbomachinery applications. Suitable non-reflecting inlet and outlet boundary conditions were described, e.g., in [27]-[30]. Giles [31], and Hirsch and Verhoff [32] suggested non-reflecting boundary conditions for the Euler equations, which are intended for domains with a short distance between the body and the inlet or the outlet plane.

在某些情形下，入流和出流边界除速度外，其压力梯度和温度梯度也是周期性的。例如在换热器的模拟中就会遇到这类流动[33]。针对槽道LES的周期入流和出流边界条件的实现见[34]-[36]。en

In certain cases, the inlet and outlet boundary are additionally periodic with respect to the velocity as well as the pressure and temperature gradient. This type of flow is encountered, for example, in the simulation of heat exchangers [33]. The implementation of periodic inlet and outlet boundary conditions was presented in [34]-[36] for LES in channels.

Subsonic Inlet 亚声速入流

常用的做法是给定总压、总温和两个气流角。有一个特征变量需要从流动域内插值。一种可能的做法是利用外传黎曼不变量（Riemann invariant）[30]，其定义为en

A common procedure consists of the specification of the total pressure, total temperature, and of two flow angles. One characteristic variable has to be interpolated from the interior of the flow domain. One possibility is to employ the outgoing Riemann invariant [30], which is defined as

\mathcal{R}^{-}=\vec{v}_d\cdot\vec{n}-\frac{2c_d}{\gamma-1}\,, \tag{8.32}

其中下标 $$d$$ 表示域内状态（参见图8.6a）。黎曼不变量用来确定边界上的绝对速度或声速。实践中发现，选取声速会得到更稳定的格式，对低马赫数流动尤其如此。因此，我们令en

where the index $$d$$ denotes the state inside the domain (cf. Fig. 8.6a). The Riemann invariant is used to determine either the absolute velocity or the the speed of sound at the boundary. In practice, it was found that selecting the speed of sound leads to a more stable scheme, particularly for low Mach-number flows. Therefore, we set

c_b=\frac{-\mathcal{R}^{-}(\gamma-1)}{(\gamma-1)\cos^2\theta+2}\left\{1+\cos\theta\sqrt{\frac{\left[(\gamma-1)\cos^2\theta+2\right]c_0^2}{(\gamma-1)(\mathcal{R}^{-})^2}-\frac{\gamma-1}{2}}\right\} \tag{8.33}

其中 $\theta$ 为相对于边界的气流角， $$c_0$$ 表示滞止声速。于是en

with $\theta$ being the flow angle relative to the boundary, and $$c_0$$ denoting the stagnation speed of sound. Hence,

\cos\theta=-\frac{\vec{v}_d\cdot\vec{n}}{\|\vec{v}_d\|_2} \tag{8.34}

以及en

and

c_0^2=c_d^2+\frac{\gamma-1}{2}\|\vec{v}_d\|_2^2\,, \tag{8.35}

其中 $\|\vec{v}_d\|_2$ 表示内点 $$d$$ 处的总速度（图8.6a）。式(8.34)中的单位法向量 $\vec{n}$ 假设指向域外。en

where $\|\vec{v}_d\|_2$ denotes the total velocity at the interior point $$d$$ (Fig. 8.6a). The unit normal vector $\vec{n}$ in Eq. (8.34) was assumed to point out of the domain.

边界上的静温、静压、密度或绝对速度等量按如下方式求值en

Quantities like the static temperature, pressure, density, or the absolute velocity at the boundary are evaluated as follows

\begin{aligned} T_b&=T_0\left(\frac{c_b^2}{c_0^2}\right)\\ p_b&=p_0\left(\frac{T_b}{T_0}\right)^{\gamma/(\gamma-1)}\\ \rho_b&=\frac{p_b}{RT_b}\\ \|\vec{v}_b\|_2&=\sqrt{2c_p\left(T_0-T_b\right)}\,, \end{aligned} \tag{8.36}

其中 $$T_0$$ 和 $$p_0$$ 为给定的总温和总压， $$R$$ 和 $$c_p$$ 分别表示气体常数和定压比热。入口处的速度分量通过按两个（二维为一个）给定的气流角分解 $\|\vec{v}_b\|_2$ 得到。en

where $$T_0$$ and $$p_0$$ are the given values of total temperature and pressure, $$R$$ and $$c_p$$ represent the specific gas constant and the heat coefficient at constant pressure, respectively. The velocity components at the inlet are obtained by decomposing $\|\vec{v}_b\|_2$ according to the two (one in 2D) prescribed flow angles.

Subsonic Outlet 亚声速出流

在叶轮机械中，出口处通常给定静压。亚声速出流边界的处理方式与式(8.23)的出流条件非常相似，只是把环境压力 $$p_a$$ 换成给定的出口静压。en

In turbomachinery, the static pressure is usually prescribed at the outlet. The subsonic outlet boundary can be treated in a way quite similar to the outflow condition in Eq. (8.23). Only the ambient pressure $$p_a$$ is replaced here by the given static exit pressure.

虚单元（点）中的流动变量可以通过对边界处和内点 $$d$$ 处的状态作线性外推得到。en

Flow variables in the dummy cells (points) can be obtained by linearly extrapolating the states at the boundary and at the interior point $$d$$ .