# 如何在 CFD 仿真中设置入口和出口边界条件

2018年 8月 20日

### 设置内部流动的入口和出口边界

CFD 仿真通常需要大量计算，我们很自然地会想尽量减少仿真中的自由度。如果将自由度减少到极致，可能会得到一个入口边界和出口边界相交的几何结构。假设一个横截面为半圆形的管道中的 90° 管道弯头。

(1)

c_{p}=\frac{\Delta p}{\frac{1}{2}\rho U^{2}},

(2)

L_{E}=0.05D_{h}
\left(\frac{UD_{h}}
{\nu}\right)

(3)

L_{E}=4.4D_{h}\left(\frac{UD_{h}}{\nu}
\right)^{1/6}

(4)

u-iv=U/t,\hspace
{5mm}t=\sqrt{\frac{z-1}{z+1}},\hspace{5mm}
\zeta=i\log\left({\frac
{1+it}
{1-it}}\right)+\log\left({\frac
{1+t}
{1-t}}\right)

### 放置外部流动的入口和出口边界

NACA 机翼周围的湍流。剖面上侧的边界层在后缘之前分离。

(5)

\begin{align}(u,\, v)&=(U+\frac{Qx}{2\pi R^{2}},\, \frac{Qy}{2\pi R^{2}}),\hspace{12mm}\text{in 2D} \end{align}

\begin
{align}(u,\, v,\, w)&=(U+\frac{Qx}{4\pi r^{3}}, \,\frac{Qy}{4\pi r^{3}}, \,\frac{Qz}{4\pi r^{3}}), \hspace{5mm}\text{in 3D} \end{align}

(6)

p+\frac{1}{2}
\rho\left|\bf
{u}\right|^2=p_\infty+\frac{1}{2}\rho U^2\Longrightarrow c_p=\frac{p-p_\infty}{\frac{1}{2}\rho U^2}=1-\left(\frac{\left|\bf{u}
\right|}
{U}
\right)^2

(7)

\begin{align} c_p&=-\frac{2}{\pi}\left(\frac{x}{R}\right)\frac{d}{R}+O\left(\left(\frac{d}{R}\right)^2\right) \hspace{5mm}\text{in 2D} \end{align}
\begin{align}
c_p&=-\frac{1}{4}\left(\frac{x}{r}\right)\frac{d^2}{r^2}+O\left(\left(\frac{d}{r}\right)^4\right) \hspace{6.2mm}\text{in 3D} \end{align}

(8)

\begin{align} (u,\,v)& = (U+\frac{\Gamma y}{2\pi R^2},\,-\frac{\Gamma x}{2\pi R^2}),\hspace{134.4mm}\text{in 2D} \end{align}
\begin{align} (u,\,v,\,w)& = (U+\frac{\Gamma y}{4\pi R^2}\left(\frac{z+s/2}{\sqrt{R^2+(z+s/2)^2}}-\frac{z-s/2}{\sqrt{R^2+(z-s/2)^2}}\right),\,-\frac{\Gamma x}{4\pi R^2}\left(\frac{z+s/2}{\sqrt{R^2+(z+s/2)^2}}-\frac{z-s/2}{\sqrt{R^2+(z-s/2)^2}}\right) \end{align}
\begin{align}
& -\frac{\Gamma (z+s/2)}{4\pi (y^2+(z+s/2)^2)}\left(1+\frac{x}{\sqrt{R^2+(z+s/2)^{2}}}\right)+\frac{\Gamma (z-s/2)}{4\pi (y^2+(z-s/2)^2)}\left(1+\frac{x}{\sqrt{R^2+(z-s/2)^{2}}}\right), \end{align}
\begin{align}
& \frac{\Gamma y}{4\pi (y^2+(z+s/2)^2)}\left(1+\frac{x}{\sqrt{R^2+(z+s/2)^{2}}}\right)-\frac{\Gamma y}{4\pi (y^2+(z-s/2)^2)}\left(1+\frac{x}{\sqrt{R^2+(z-s/2)^{2}}}\right)),\hspace{7mm}\text{in 3D}
\end{align}

(9)

\Gamma= \pi Uc(\alpha+\beta)

(10)

\begin{align} c_p&=-\left(\frac{y}{R}\right)\frac{c}{R}(\alpha+\beta)+O\left(\left(\frac{c}{R}(\alpha\beta)\right)^2\right) \hspace{13.5mm}\text{in 2D} \end{align}
\begin{align}
c_p&=-\frac{1}{2}\left(\frac{y}{R}\right)\frac{s}{R}\frac{c}{R}(\alpha+\beta)+O\left(\left(\frac{s}{R}\frac{c}{R}(\alpha+\beta)\right)^2\right) \hspace{5mm}\text{in 3D} \end{align}

14° 迎角下 NACA 0012 机翼的二维仿真。