A.6 薄剪切层近似

A.6 薄剪切层近似

高 Reynolds 数流动中，黏性应力仅在物体周围的狭窄区域影响流动。因此，可合理假设壁面法向梯度占主导，忽略其他方向的梯度 $ [21, 22] $。这就是 Navier–Stokes 方程的薄剪切层（TSL）近似。

以一般曲线网格上的 Navier–Stokes 微分方程（A.4）说明。假设如图2.4所示，壁面位于 $ \eta = const. $，则从扩散项中去掉全部沿流向 $ (\partial/\partial\xi) $ 与横流方向 $ (\partial/\partial\zeta) $ 的导数。三维 TSL Navier–Stokes 方程在曲线坐标系 $ (\xi, \eta, \zeta) $ 中的微分形式为

$$ \frac{\partial\vec{W}^{*}}{\partial t}+\frac{\partial\vec{F}_{c,1}}{\partial\xi}+\frac{\partial\vec{F}_{c,2}}{\partial\eta}+\frac{\partial\vec{F}_{c,3}}{\partial\zeta}=\frac{\partial\vec{F}_{v,2}}{\partial\eta}+\vec{Q}^{*}.\tag{A.51} $$

守恒变量矢量 $ (\vec{W}^{*}) $、对流通量 $ (\vec{F}_{c,1/2/3}) $ 及扩散通量 $ (\vec{F}_{ v,2}) $，分别由式（A.5）—（A.7）给出。源项 $ (\vec{Q}^{*}) $ 也保持式（A.8）的形式。但黏性应力张量分量（式（A.10））与热通量变为

$$ \begin{aligned}\tau_{xx}&=2\mu\eta_{x}\frac{\partial u}{\partial\eta}+\lambda\left(\eta_{x}\frac{\partial u}{\partial\eta}+\eta_{y}\frac{\partial v}{\partial\eta}+\eta_{z}\frac{\partial w}{\partial\eta}\right),\\\tau_{ y y}&=2\mu\eta_{ y}\frac{\partial v}{\partial\eta}+\lambda\left(\eta_{x}\frac{\partial u}{\partial\eta}+\eta_{ y}\frac{\partial v}{\partial\eta}+\eta_{z}\frac{\partial w}{\partial\eta}\right),\\\tau_{zz}&=2\mu\eta_{z}\frac{\partial w}{\partial\eta}+\lambda\left(\eta_{x}\frac{\partial u}{\partial\eta}+\eta_{ y}\frac{\partial v}{\partial\eta}+\eta_{z}\frac{\partial w}{\partial\eta}\right),\\\tau_{x y}&=\tau_{yx}=\mu\left(\eta_{ y}\frac{\partial u}{\partial\eta}+\eta_{x}\frac{\partial v}{\partial\eta}\right),\\\tau_{xz}&=\tau_{zx}=\mu\left(\eta_{z}\frac{\partial u}{\partial\eta}+\eta_{x}\frac{\partial w}{\partial\eta}\right),\\\tau_{ y z}&=\tau_{zy}=\mu\left(\eta_{z}\frac{\partial v}{\partial\eta}+\eta_{ y}\frac{\partial w}{\partial\eta}\right),\\\Theta_{x}&=u\tau_{xx}+v\tau_{xy}+w\tau_{xz}+k\eta_{x}\frac{\partial T}{\partial\eta},\\\Theta_{ y}&=u\tau_{ y x}+v\tau_{ y y}+w\tau_{ y z}+k\eta_{ y}\frac{\partial T}{\partial\eta},\\\Theta_{z}&=u\tau_{zx}+v\tau_{zy}+w\tau_{zz}+k\eta_{z}\frac{\partial T}{\partial\eta}.\end{aligned}\tag{A.52} $$

式（A.52）中，对笛卡尔坐标的偏导数近似为

$$ \begin{array}{l}\displaystyle\frac{\partial u}{\partial x}\approx\eta_{x}\frac{\partial u}{\partial\eta},\\\displaystyle\frac{\partial u}{\partial y}\approx\eta_{ y}\frac{\partial u}{\partial\eta},\\\displaystyle\frac{\partial u}{\partial z}\approx\eta_{z}\frac{\partial u}{\partial\eta},\quad\text{等等。}\end{array}\tag{A.53} $$

这与薄剪切层假设一致。