8.3.2 Modifications for Lifting Bodies 升力体情形的修正[cfd-8-3-2]

上述特征远场边界条件假设环量为零，这对亚声速或跨声速流动中的升力体并不正确。因此，远场边界必须放在离物体非常远的地方，否则流动解会不准确。如果自由来流中包含一个单独涡的影响（三维情形为马蹄涡），到远场的距离可以显著缩短（约一个数量级）。该涡假定位于升力体的中心，其强度与物体产生的升力成正比。下面我们给出涡修正的二维和三维实现。en

The above characteristic farfield boundary conditions assume zero circulation, which is not correct for a lifting body in sub- or transonic flow. For this reason, the farfield boundary has to be located very far away from the body. Otherwise, the flow solution will be inaccurate. The distance to the farfield can be significantly shortened (by one order of magnitude), if the freestream flow includes the effect of a single vortex (horse-shoe vortex in 3D). The vortex is assumed to be centred at the lifting body. The strength of the vortex is proportional to the lift produced by the body. In the following, we shall present implementations of the vortex correction in 2D and in 3D.

Vortex Correction in 2D 二维涡修正

这里要叙述的方法由Usab和Murman提出[18]。修正后自由来流速度的分量由下列表达式给出（假设可压缩流动）en

The approach, which we want to describe here, was suggested by Usab and Murman [18]. The components of the corrected freestream velocity are given by the expressions (compressible flow assumed)

\begin{aligned} u_\infty^{*}&=u_\infty+\left(\frac{\Gamma\sqrt{1-M_\infty^2}}{2\pi\,d}\right)\frac{1}{1-M_\infty^2\sin^2(\theta-\alpha)}\,\sin\theta\\ v_\infty^{*}&=v_\infty-\left(\frac{\Gamma\sqrt{1-M_\infty^2}}{2\pi\,d}\right)\frac{1}{1-M_\infty^2\sin^2(\theta-\alpha)}\,\cos\theta \end{aligned} \tag{8.24}

其中 $\Gamma$ 为环量， $(d,\theta)$ 为远场点的极坐标， $\alpha$ 为迎角， $M_\infty$ 为自由来流马赫数。环量由下式求得en

with $\Gamma$ being the circulation, $(d,\theta)$ the polar coordinates of the farfield point, $\alpha$ the angle of attack, and $M_\infty$ denoting the freestream Mach number, respectively. The circulation is obtained from

\Gamma=\frac{1}{2}\|\vec{v}_\infty\|_2\,a\,C_L \tag{8.25}

其中用到了Kutta-Joukowsky定理。式(8.25)中， $$a$$ 表示翼型弦长， $$C_L$$ 为通过表面压力积分求得的升力系数。式(8.24)中的极坐标按下式计算en

by using the theorem of Kutta-Joukowsky. In Equation (8.25), $$a$$ represents the chord length of the airfoil and $$C_L$$ is the lift coefficient evaluated by the integration of the surface pressure. The polar coordinates in Eq. (8.24) are computed as

\begin{aligned} d&=\sqrt{(x-x_{ref})^2+(y-y_{ref})^2}\\ \theta&=\tan\left(\frac{y-y_{ref}}{x-x_{ref}}\right)\,, \end{aligned} \tag{8.26}

其中 $x_{ref}$ 和 $y_{ref}$ 为参考点（涡所在位置，例如1/4弦长处）的坐标。en

where $x_{ref}$ and $y_{ref}$ are the coordinates of the reference point (location of the vortex - e.g., at 1/4 chord).

修正后的自由来流压力 $p_\infty^{*}$ 由下式给出en

The modified freestream pressure $p_\infty^{*}$ is given by

p_\infty^{*}=\left[p_\infty^{(\gamma-1)/\gamma}+\left(\frac{\gamma-1}{\gamma}\right)\frac{\rho_\infty\left(\|\vec{v}_\infty\|_2^2-\|\vec{v}_\infty^{*}\|_2^2\right)}{2\,p_\infty^{1/\gamma}}\right]^{\gamma/(\gamma-1)} \tag{8.27}

其中 $\|\vec{v}_\infty^{*}\|_2^2=(u_\infty^{*})^2+(v_\infty^{*})^2$ 。修正后的自由来流密度由状态方程得到en

with $\|\vec{v}_\infty^{*}\|_2^2=(u_\infty^{*})^2+(v_\infty^{*})^2$ . The corrected freestream density is obtained from the equation of the state

\rho_\infty^{*}=\rho_\infty\left(\frac{p_\infty^{*}}{p_\infty}\right)^{1/\gamma}\,. \tag{8.28}

将修正后的量 $u_\infty^{*}$ 、 $v_\infty^{*}$ 、 $p_\infty^{*}$ 和 $\rho_\infty^{*}$ 代入式(8.22)或式(8.23)，代替其中的 $$u_a$$ 、 $$v_a$$ 、 $$p_a$$ 和 $\rho_a$ 。en

The corrected quantities $u_\infty^{*}$ , $v_\infty^{*}$ , $p_\infty^{*}$ , and $\rho_\infty^{*}$ are inserted into Eq. (8.22) or Eq. (8.23) instead of $$u_a$$ , $$v_a$$ , $$p_a$$ , and $\rho_a$ .

上述涡修正式(8.24)-(8.28)严格来说只对亚声速流动成立。不过，实践证明对自由来流条件的这一修正在跨声速流动中同样有帮助。图8.7展示了这方面的结果，其中研究了升力系数对远场边界距离的依赖关系。远场半径取为5、20、50和99倍弦长。可以看到，不加涡修正的模拟对远场距离有强烈的依赖性；相反，带涡的模拟在约20倍弦长的距离内仍保持足够精度。这使网格单元/点数显著减少。文献[19]证明，若在涡修正中使用更高阶的项，远场边界可以只放在约5倍弦长处而不损失精度。en

The above vortex correction Eqs. (8.24)-(8.28) is strictly valid for subsonic flow only. However, the modification of the freestream conditions proved to be helpful in transonic flow as well. This is demonstrated in Fig. 8.7, where the dependence of the lift coefficient on the distance to the farfield boundary was investigated. The farfield radius was set to 5, 20, 50, and 99 chords. As we can see, simulations without the vortex correction experiences a strong dependence on the farfield distance. On the contrary, simulations with the vortex remain sufficiently accurate up to a distance of about 20 chords. This leads to a significant reduction of the number of grid cells/points. It was demonstrated in Ref. [19] that by using higher-order terms in the vortex correction, the farfield boundary can be placed only about 5 chords away without loss of accuracy.

图8.7:到远场边界的距离和单个涡对升力系数的影响

图8.7:到远场边界的距离和单个涡对升力系数的影响。NACA 0012翼型， $M_\infty=0.8$ ， $\alpha=1.25^{\circ}$ 。图例:横轴distance to farfield——到远场的距离;纵轴 $$C_L$$ ——升力系数;with vortex correction——带涡修正;without vortex correction——不带涡修正。

Vortex Correction in 3D 三维涡修正

机翼对远场边界的影响可以用马蹄涡来近似。对于可压缩流动，修正后的自由来流速度分量可由下式得到[20]、[21]en

The effect of a wing on the farfield boundary can be approximated by a horse-shoe vortex. In the case of compressible flow, the modified freestream velocity components can be obtained from [20], [21]

\begin{aligned} u_\infty^{*}&=u_\infty+\frac{\Gamma\beta^2}{2\pi}\,\mathcal{A}\\ v_\infty^{*}&=v_\infty-\frac{\Gamma}{2\pi}\left[\frac{z+l}{(z+l)^2+y^2}\,\mathcal{B}-\frac{z-l}{(z-l)^2+y^2}\,\mathcal{C}+\frac{x\beta^2}{x^2+y^2\beta^2}\,\mathcal{A}\right]\\ w_\infty^{*}&=w_\infty+\frac{\Gamma}{2\pi}\left[\frac{y}{(z+l)^2+y^2}\,\mathcal{B}-\frac{y}{(z-l)^2+y^2}\,\mathcal{C}\right], \end{aligned} \tag{8.29}

其中 $\Gamma$ 表示环量， $$(x,y,z)$$ 为远场点的笛卡儿坐标， $$l$$ 为半展长。此外，式(8.29)中假设流动沿正 $$x$$ 方向，机翼沿 $$z$$ 轴布置。式(8.29)中的 $\mathcal{A}$ 、 $\mathcal{B}$ 和 $\mathcal{C}$ 为en

where $\Gamma$ denotes the circulation, $$(x,y,z)$$ the Cartesian coordinates of the farfield point, and $$l$$ stands for the half span, respectively. Furthermore, in Eq. (8.29) it was assumed that the flow is in the positive $$x$$ -direction with the wing being oriented along the $$z$$ -axis. The terms $\mathcal{A}$ , $\mathcal{B}$ and $\mathcal{C}$ in Eq. (8.29) read

\begin{aligned} \mathcal{A}&=\frac{z+l}{\sqrt{\psi_+}}-\frac{z-l}{\sqrt{\psi_-}}\\ \mathcal{B}&=1+\frac{x}{\sqrt{\psi_+}}\\ \mathcal{C}&=1+\frac{x}{\sqrt{\psi_-}}\,. \end{aligned} \tag{8.30}

其中缩写定义为en

The abbreviations are given by

\begin{aligned} \psi_+&=x^2+\beta^2(z+l)^2+y^2\beta^2\\ \psi_-&=x^2+\beta^2(z-l)^2+y^2\beta^2\\ \beta&=\sqrt{1-M_\infty^2} \end{aligned} \tag{8.31}

其中 $M_\infty$ 为自由来流马赫数。环量 $\Gamma$ 仍用式(8.25)计算，此时 $$a$$ 表示平均弦长。修正后的压力（ $p_\infty^{*}$ ）和密度（ $\rho_\infty^{*}$ ）分别由公式(8.27)和(8.28)求得。式(8.22)或式(8.23)中的 $$u_a$$ 、 $$v_a$$ 、 $$w_a$$ 、 $$p_a$$ 和 $\rho_a$ 替换为其修正值 $u_\infty^{*}$ 、 $v_\infty^{*}$ 、 $w_\infty^{*}$ 、 $p_\infty^{*}$ 和 $\rho_\infty^{*}$ 。en

with $M_\infty$ being the freestream Mach number. The circulation $\Gamma$ is calculated using Eq. (8.25), where $$a$$ represents the mean chord. The corrected values of pressure ( $p_\infty^{*}$ ) and of density ( $\rho_\infty^{*}$ ) are obtained from the formulae (8.27) and (8.28), respectively. The quantities $$u_a$$ , $$v_a$$ , $$w_a$$ , $$p_a$$ , and $\rho_a$ in Eq. (8.22) or Eq. (8.23) are replaced by their corrected values $u_\infty^{*}$ , $v_\infty^{*}$ , $w_\infty^{*}$ , $p_\infty^{*}$ , and $\rho_\infty^{*}$ .

式(8.29)中修正速度分量 $v_\infty^{*}$ 和 $w_\infty^{*}$ 的表达式，在涡线穿过出流边界的位置会变为无穷大。这些点是en

The expressions for the corrected velocity components $v_\infty^{*}$ and $w_\infty^{*}$ in Eq. (8.29) becomes infinite at locations, where the vortex lines cross the outflow boundary. These are the points

\begin{aligned} z&=+l\,,\quad y=0\,,\\ z&=-l\,,\quad y=0\,, \end{aligned} \tag{9}

以及 $x=x_{farf}$ 。为避免数值奇性，文献[21]建议把en

and $x=x_{farf}$ . In order to avoid the numerical singularity, in Ref. [21] it was suggested to constrain the values of

(z+l)^2+y^2\quad\text{and}\quad(z-l)^2+y^2 \tag{10}

在式(8.29)中的取值限制在1/4展长，即 $$l/2$$ 以内。这一措施把涡线 $$z=l$$ 和 $$z=-l$$ 周围 $$l/2$$ 距离内对速度 $v_\infty$ 和 $w_\infty$ 的修正量减小。en

in Eq. (8.29) to the 1/4 wingspan, i.e., $$l/2$$ . This measure reduces the corrections to the velocities $v_\infty$ and $w_\infty$ within the distance $$l/2$$ around the vortex lines $$z=l$$ and $$z=-l$$ .

文献[21]给出的数值结果表明，应用式(8.29)的涡修正后，升力系数和阻力系数对远场距离的敏感性降低。研究发现，到远场边界 $7\cdot l$ 的距离足以获得准确的结果。en

The numerical results presented in [21] indicate a reduced sensitivity of the lift and drag coefficient with respect to the farfield distance, if the vortex correction in Eq. (8.29) is applied. It was found that a distance of $7\cdot l$ to the farfield boundary is sufficient for accurate results.