(1) Navier-Stokes 方程组
守恒形式控制方程：
(3.2.1)\[\frac{\partial \mathbf{U}}{\partial t} +
\frac{\partial \mathbf{F}}{\partial x} +
\frac{\partial \mathbf{G}}{\partial y} +
\frac{\partial \mathbf{H}}{\partial z} = J\]
守恒变量 (conserved variables) 解向量：
(3.2.2)\[\mathbf{U} = [\rho, \rho u, \rho v, \rho w, \rho E]^T\]
相应的原始变量 (primitive variables): \(\rho, u, v, w, p\)。
其中, \(p\) 是压力, \(\rho\) 是密度, \(V\) 是速度模长。
比内能 (specific internal energy) \(e\),
比总能 (specific total energy) \(E=e+\frac{V^2}{2}\)
比焓 (specific enthalpy) \(h=e+p/ \rho\),
比总焓 \(H=h+\frac{V^2}{2}\)。
对于量热完全气体，比热比 \(\gamma\) 为常数,
\(R\) 为普适气体常数，\(T\) 为温度，有：
(3.2.3)\[e=c_vT=\frac{RT}{\gamma-1}=\frac{p}{(\gamma-1)\rho}\]
(3.2.4)\[p = \rho R T\]
(3.2.5)\[h = \gamma e\]
(3.2.6)\[H = \gamma E - \frac{1}{2}(\gamma-1)V^2 = E + p/\rho\]
从守恒变量计算压力的公式：
(3.2.7)\[p = (\gamma-1)[(\rho E) - 0.5 \rho V^2]\]
通量向量：
(3.2.8)\[\begin{split}\mathbf{F} = \left[
\begin{array}{lr}
\rho u \\
\rho u^2+p-\tau_{xx} \\
\rho uv-\tau_{xy} \\
\rho uw-\tau_{xz} \\
\rho Eu + pu - k\frac{\partial T}{\partial x}
-u \tau_{xx} -v \tau_{xy} -w \tau_{zx}
\end{array}
\right]\end{split}\]
(3.2.9)\[\begin{split}\mathbf{G} = \left[
\begin{array}{lr}
\rho v \\
\rho uv-\tau_{xy} \\
\rho v^2+p-\tau_{yy} \\
\rho vw-\tau_{yz} \\
\rho Ev + pv - k\frac{\partial T}{\partial y}
-u \tau_{xy} -v \tau_{yy} -w \tau_{yz}
\end{array}
\right]\end{split}\]
(3.2.10)\[\begin{split}\mathbf{H} = \left[
\begin{array}{lr}
\rho w \\
\rho uw-\tau_{zx} \\
\rho vw-\tau_{yz} \\
\rho w^2+p-\tau_{zz} \\
\rho Ew + pw - k\frac{\partial T}{\partial z}
-u \tau_{zx} -v \tau_{yz} -w \tau_{zz}
\end{array}
\right]\end{split}\]
源项向量：
(3.2.11)\[\begin{split}\mathbf{J} = \left[
\begin{array}{lr}
0 \\
\rho g_x \\
\rho g_y \\
\rho g_z \\
\rho(ug_x+vg_y+wg_z)+\rho \dot {q}
\end{array}
\right]\end{split}\]
其中, \(\tau\) 是粘性应力张量, 形式为 Eq.3.1.13。
\(\tau_{ij}\) 是粘性应力张量的 \(i,j\) 分量。
式 Eq.3.1.13 中 \(\mu\) 是分子粘性系数，对于量热完全气体，有
Sutherland’s Formula:
(3.2.12)\[\mu = \mu_0 \left(\frac{T}{T_0}\right)^{3/2} \frac{T_0+110}{T+110}\]
其中, \(\mu_0\) 和 \(T_0\) 是标准海平面条件下的分子粘性和温度 (单位: K)。
\(\mathbf{g}\) 是体积力向量, \(g_{i}\) 是体积力的 \(i\) 分量。
\(k\) 是热导率, 热流通量 \(\dot{\mathbf{q}}=-k \nabla T\)。
(3) 通用坐标系 NS 方程组
假定计算是在贴体正交网格坐标系 (通用坐标系, generalized coordinates) 下进行，
且网格刚性地固联于物体做非定常运动。
通用坐标系下的无量纲三维非定常可压缩 Navier-Stokes 方程组:
(3.2.14)\[\frac{\partial \hat{\mathbf{U}}}{\partial t} +
\frac{\partial (\hat{\mathbf{F}}-\hat{\mathbf{F}}_{\nu})}{\partial \xi} +
\frac{\partial (\hat{\mathbf{G}}-\hat{\mathbf{G}}_{\nu})}{\partial \eta} +
\frac{\partial (\hat{\mathbf{H}}-\hat{\mathbf{H}}_{\nu})}{\partial \zeta} = 0\]
其中，笛卡尔坐标 \(x, y, z\) 与通用坐标 \(\xi(x,y,z,t), \eta(x,y,z,t), \zeta(x,y,z,t)\)
的转换雅可比矩阵 (Jacobian of the transformation) 为:
(3.2.15)\[J = \frac{\partial (\xi, \eta, \zeta, t)}{\partial (x,y,z,t)}\]
通常认为通用坐标系中, 两个网格中心间的空间坐标差量 \(\Delta \xi, \Delta \eta, \Delta \zeta\) 等于 1。
Note
为了简单起见，将坐标转换雅可比矩阵的模记为 \(J\), 数值上等于网格单元体积的倒数。
守恒变量向量 (vector of conserved variables):
(3.2.16)\[\hat{\mathbf{U}} = \mathbf{U}/J = [\rho, \rho u, \rho v, \rho w, \rho E]^T/J\]
无粘通量 (inviscid flux):
(3.2.17)\[\begin{split}\hat{\mathbf{F}} = \frac{1}{J}
\left[\begin{array}{c}
\rho U \\ \rho U u + \xi_x p \\ \rho U v + \xi_y p \\
\rho U w + \xi_z p \\ \rho H U - \xi_t p
\end{array}\right],
\hat{\mathbf{G}} = \frac{1}{J}
\left[\begin{array}{c}
\rho V \\ \rho V u + \eta_x p \\ \rho V v + \eta_y p \\
\rho V w + \eta_z p \\ \rho H V - \eta_t p
\end{array}\right],
\hat{\mathbf{H}} = \frac{1}{J}
\left[\begin{array}{c}
\rho W \\ \rho W u + \zeta_x p \\ \rho W v + \zeta_y p \\
\rho W w + \zeta_z p \\ \rho H W - \zeta_t p
\end{array}\right]\end{split}\]
逆变速度 (contravariant velocities):
(3.2.18)\[\begin{split}\begin{array}{l}
U &= \xi_x u + \xi_y v +\xi_z w + \xi_t \\
V &= \eta_x u + \eta_y v +\eta_z w + \eta_t \\
W &= \zeta_x u + \zeta_y v +\zeta_z w + \zeta_t
\end{array}\end{split}\]
粘性通量 (viscous flux):
(3.2.19)\[\begin{split}\hat{\mathbf{F}}_v = \frac{1}{J}
\left[\begin{array}{c}
0 \\
\xi_x \tau_{xx} + \xi_y \tau_{xy} + \xi_z \tau_{xz} \\
\xi_x \tau_{xy} + \xi_y \tau_{yy} + \xi_z \tau_{yz} \\
\xi_x \tau_{xz} + \xi_y \tau_{yz} + \xi_z \tau_{zz} \\
\xi_x b_x + \xi_y b_y + \xi_z b_z
\end{array}\right]\end{split}\]
(3.2.20)\[\begin{split}\hat{\mathbf{G}}_v = \frac{1}{J}
\left[\begin{array}{c}
0 \\
\eta_x \tau_{xx} + \eta_y \tau_{xy} + \eta_z \tau_{xz} \\
\eta_x \tau_{xy} + \eta_y \tau_{yy} + \eta_z \tau_{yz} \\
\eta_x \tau_{xz} + \eta_y \tau_{yz} + \eta_z \tau_{zz} \\
\eta_x b_x + \eta_y b_y + \eta_z b_z
\end{array}\right]\end{split}\]
(3.2.21)\[\begin{split}\hat{\mathbf{H}}_v = \frac{1}{J}
\left[\begin{array}{c}
0 \\
\zeta_x \tau_{xx} + \zeta_y \tau_{xy} + \zeta_z \tau_{xz} \\
\zeta_x \tau_{xy} + \zeta_y \tau_{yy} + \zeta_z \tau_{yz} \\
\zeta_x \tau_{xz} + \zeta_y \tau_{yz} + \zeta_z \tau_{zz} \\
\zeta_x b_x + \zeta_y b_y + \zeta_z b_z
\end{array}\right]\end{split}\]
切应力张量 (shear stress) 和热流 (heat flux) 以张量形式给出
(含爱因斯坦求和假设):
(3.2.22)\[\tau_{ij} = \frac{M_{\infty}}{Re_R}
\left[\mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)
+ \lambda \left( \frac{\partial u_k}{\partial x_k} \right) \delta_{ij} \right]\]
(3.2.23)\[b_i = u_k \tau_{ik} - \dot q_i\]
(3.2.24)\[\dot q_i = -\left[ \frac{M_{\infty} \mu}{Re_R \; Pr (\gamma-1)} \right]
\frac{\partial a^2}{\partial x_i}\]
无量纲化参考变量: 来流密度 \(\tilde{\rho}_{\infty}\), 来流声速 \(\tilde{a}_{\infty}\),
来流分子粘性 \(\tilde{\mu}_{\infty}\) (~ 代表有量纲参数)。
(3.2.25)\[\begin{split}\begin{array}{l}
\rho = \tilde{\rho}/\tilde{\rho}_{\infty}, u=\tilde{u}/\tilde{a}_{\infty},
v=\tilde{v}/\tilde{a}_{\infty}, w=\tilde{w}/\tilde{a}_{\infty},
p=\tilde{p}/(\tilde{\rho}_{\infty} \tilde{a}_{\infty}^2) \\
\rho_{\infty}=1, p_{\infty}=1/\gamma, \\
u_{\infty}=M_{\infty}\cos\alpha\cos\beta,
v_{\infty}= - M_{\infty}\sin \beta,
w_{\infty}=M_{\infty}\sin\alpha\cos\beta \\
e = \tilde{e}/(\tilde{\rho}_{\infty} \tilde{a}_{\infty}^2),
a = \tilde{a}/\tilde{a}_{\infty},
T = \tilde{T}/\tilde{T}_{\infty} = \gamma p/\rho = a^2 \\
e_{\infty} = \frac{1}{\gamma(\gamma-1)} + \frac{M_{\infty}^2}{2},
a_{\infty} = 1, T_{\infty}=1\\
x = \tilde{x}/\tilde{L}_R, y = \tilde{y}/\tilde{L}_R,
z = \tilde{z}/\tilde{L}_R, t = \tilde{t}\tilde{a}_{\infty}/\tilde{L}_R
\end{array}\end{split}\]
几何特征长度 \(\tilde{L}\) (建议以 m 为单位),
几何特征长度在网格中的数值 \(L_\text{ref}\) (无量纲, 建议数值上乘1000, 即表现为以 mm 为单位),
程序中使用的参考长度 \(\tilde{L}_R=\tilde{L}/L_\text{ref}\)。
那么, 雷诺数 \(Re=\tilde{\rho}\tilde{V}_{\infty}\tilde{L}/\tilde{\mu}_{\infty}\),
\(Re_R = Re / L_\text{ref}\)。
\(M_\infty=\tilde{V}_{\infty}/\tilde{a}\),
\(\tilde{V}_{\infty} = \sqrt{ \tilde{u}_{\infty}^2 + \tilde{v}_{\infty}^2 + \tilde{w}_{\infty}^2 }\)。
无量纲分子粘性系数:
(3.2.26)\[\mu = \tilde{\mu}/\tilde{\mu}_{\infty} =
T^{\frac{3}{2}}
\left[
\frac{1+\frac{\tilde c}{\tilde{T}_{\infty}}}
{T+\frac{\tilde c}{\tilde{T}_{\infty}}}
\right]\]
其中, Sutherland’s constant \(\tilde{c}=198.6 ^{\circ}R = 110.4 ^{\circ}K\)。