## Governing Equation and Simulation Method  
### Governing Equation  
The governing equation is the compressible Navier-Stokes equation. Conservation laws over a cuboid control volume (\(x_{1}\leq x\leq x_{2}\), \(y_{1}\leq y\leq y_{2}\), \(z_{1}\leq z\leq z_{2}\)) in the fluid can be written in the following integral form:  
\[ \begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\int_{z_{1}}^{z_{2}} \int_{y_{1}}^{y_{2}}\int_{x_{1}}^{x_{2}}\mathbf{Q}\mathrm{d}x\mathrm{d}y \mathrm{d}z&+\left[\int_{z_{1}}^{z_{2}}\int_{y_{1}}^{y_{2}}( \boldsymbol{E}_{\mathrm{a}}-\boldsymbol{E}_{\mathrm{d}})\mathrm{d}y\mathrm{d }z\right]_{x_{1}}^{x_{2}}\\ &+\left[\int_{z_{1}}^{z_{2}}\int_{x_{1}}^{x_{2}}(\boldsymbol{F}_ {\mathrm{a}}-\boldsymbol{F}_{\mathrm{d}})\mathrm{d}x\mathrm{d}z\right]_{y_{1}} ^{y_{2}}\\ &+\left[\int_{y_{1}}^{y_{2}}\int_{x_{1}}^{x_{2}}(\boldsymbol{G}_ {\mathrm{a}}-\boldsymbol{G}_{\mathrm{d}})\mathrm{d}x\mathrm{d}y\right]_{z_{1}} ^{z_{2}}=0,\end{split}  \tag{1}\]  
where \([f(x)]_{x_{1}}^{x_{2}}\) denotes \(f(x_{2})-f(x_{1})\). The conserved quantities \(\boldsymbol{Q}\); the advection fluxes \(\boldsymbol{E}_{\mathrm{a}}\), \(\boldsymbol{F}_{\mathrm{a}}\), and \(\boldsymbol{G}_{\mathrm{a}}\); and the viscous and conductive fluxes \(\boldsymbol{E}_{\mathrm{d}}\), \(\boldsymbol{F}_{\mathrm{d}}\), and \(\boldsymbol{G}_{\mathrm{d}}\) in Equation (1) are defined as follows:  
\[\boldsymbol{Q}= \begin{pmatrix}\rho\\ \rho u_{x}\\ \rho u_{y}\\ \rho u_{z}\\ e\end{pmatrix} ,\quad\boldsymbol{E}_{\mathrm{a}}= \begin{pmatrix}\rho u_{x}\\ \rho u_{x}^{2}+p\\ \rho u_{y}u_{x}\\ \rho u_{z}u_{x}\\ eu_{x}+pu_{x}\end{pmatrix} ,\quad\boldsymbol{F}_{\mathrm{a}}= \begin{pmatrix} \rho u_{y}\\ \rho u_{x}u_{y}\\ \rho u_{y}^{2}+p\\ \rho u_{z}u_{y}\\ eu_{y}+pu_{y}\end{pmatrix} ,\quad\boldsymbol{G}_{\mathrm{a}}= \begin{pmatrix} \rho u_{z}\\ \rho u_{x}u_{z}\\ \rho u_{y}u_{z}\\ \rho u_{z}^{2}+p\\ eu_{z}+pu_{z}\end{pmatrix} , \tag{2}\] \[\boldsymbol{E}_{\mathrm{d}}= \begin{pmatrix}0\\ \tau_{xx}\\ \tau_{yx}\\ \tau_{zx}\\ u_{x}\tau_{xx}+u_{y}\tau_{yx}+u_{z}\tau_{xx}-q_{x}\end{pmatrix} ,\quad \boldsymbol{F}_{\mathrm{d}}= \begin{pmatrix}0\\ \tau_{xy}\\ \tau_{yy}\\ \tau_{zy}\\ \tau_{zy}\\ u_{x}\tau_{xy}+u_{y}\tau_{yy}+u_{z}\tau_{zy}-q_{y}\end{pmatrix} ,\] \[\boldsymbol{G}_{\mathrm{d}}= \begin{pmatrix}0\\ \tau_{xz}\\ \tau_{yz}\\ \tau_{zz}\\ \tau_{zz}\\ u_{x}\tau_{zx}+u_{y}\tau_{yz}+u_{z}\tau_{zz}-q_{z}\end{pmatrix} ,\]where \(\rho\) is the density; \(u_{x}\), \(u_{y}\), and \(u_{z}\) are the components of the velocity; \(p\) is the pressure; and \(e\) is the total energy per unit volume. The variables \(\tau\) and \(q\) are the stress tensor and heat flux, respectively:  
\[ \begin{split}\tau_{xx}&=\frac{2}{3}\mu\left(2\frac{ \partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial y}-\frac{\partial u _{z}}{\partial z}\right),\\ \tau_{yy}&=\frac{2}{3}\mu\left(2\frac{\partial u_{y} }{\partial y}-\frac{\partial u_{z}}{\partial z}-\frac{\partial u_{x}}{\partial x }\right),\\ \tau_{zz}&=\frac{2}{3}\mu\left(2\frac{\partial u_{z} }{\partial z}-\frac{\partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial y }\right),\\ \tau_{xy}&=\tau_{yx}=\mu\left(\frac{\partial u_{x} }{\partial y}+\frac{\partial u_{y}}{\partial x}\right),\\ \tau_{yz}&=\tau_{zy}=\mu\left(\frac{\partial u_{y} }{\partial z}+\frac{\partial u_{z}}{\partial y}\right),\\ \tau_{zx}&=\tau_{xz}=\mu\left(\frac{\partial u_{z} }{\partial x}+\frac{\partial u_{x}}{\partial z}\right).\end{split}  \tag{3}\]  
\[ \begin{split} q_{x}&=-\frac{1}{\gamma-1}\frac{\mu}{Pr} \frac{\partial a^{2}}{\partial x},\\ q_{y}&=-\frac{1}{\gamma-1}\frac{\mu}{Pr}\frac{ \partial a^{2}}{\partial y},\\ q_{z}&=-\frac{1}{\gamma-1}\frac{\mu}{Pr}\frac{ \partial a^{2}}{\partial z},\end{split}  \tag{4}\]  
where \(a\) is the speed of sound; \(\mu\) is the viscosity obtained by [PERSON]'s law; and the Prandtl number \(Pr\) is 0.72. The conservation equation in integral form over a cuboid control volume that extends over the two regions of the air and the object is written in the following form using the Heaviside function \(H(x,y,z)\):  
\[