## Governing Equation and Simulation Method
### Wall-Stress Model  
The wall-stress model by [PERSON] and [PERSON] [5; 7; 8] for the turbulent boundary layer was used in the present study. Stream-wise momentum and the total energy conservation equation of the wall-stress model are written in the following ordinary differential equations in the local wall-normal coordinate \(\eta\):  
\[\frac{d}{d\eta}\left[(\mu+\mu_{t,\text{wm}})\frac{dU_{\parallel}}{d\eta} \right]=0, \tag{13}\]  
\[\frac{d}{d\eta}\left[(\mu+\mu_{t,\text{wm}})U_{\parallel}\frac{dU_{\parallel} }{d\eta}+c_{p}\Big{(}\frac{\mu}{Pr}+\frac{\mu_{t,\text{wm}}}{Pr_{t,\text{wm}} }\Big{)}\frac{dT}{d\eta}\right]=0, \tag{14}\]  
where \(U_{\parallel}\) is the wall-parallel velocity component. Constants \(Pr=0.72\) and \(Pr_{t,\text{wm}}=0.90\) are the Prandtl number and turbulent Prandtl number, respectively. The solution of the wall-stress model equations shows the averaged velocity and the averaged temperature distributions in the inner-layer part of the turbulent boundary layer. The turbulent eddy viscosity in the wall-stress model \(\mu_{t,\text{wm}}\) is obtained using the following formula:  
\[\mu_{t,\text{wm}}=\kappa\rho\eta\sqrt{\frac{\tau_{w}}{\rho}}D, \tag{15}\]  
\[D=\Big{[}1-\exp(-y^{+}/A^{+})\Big{]}^{2}. \tag{16}\]  
The von Karman constant is \(\kappa=0.41\), and \(A^{+}\) is a constant that equals 17. The velocity scale \(u_{\tau}=\sqrt{\tau_{w}/\rho}\) is estimated by the instantaneous wall shear stress and density. The dimensionless wall distance \(y^{+}\) in Equation (16) is defined as:  
\[y^{+}=\frac{\rho\eta u_{\tau}}{\mu}. \tag{17}\]  
The boundary value problem of the wall-stress model Equations (13) and (14) are discretized using the one-dimensional finite volume method and solved by the Thomas algorithm. Boundary values at \(\eta=\eta_{\text{max}}=d_{\text{wm}}\) are determined from the instantaneous value of the CFD analysis (Figure 1a). A schematic of the application of the wall-stress model to the Cartesian cut-cell method is shown in Figure 1. An overview of the implementation is provided as follows:  
1. The upper bound of the wall-stress model mesh is set on the wall-normal line passing through a CFD cell center \((x_{\text{c}},\ y_{\text{c}},\ z_{\text{c}})\) that includes the cut-plane by the object. The upper bound is placed at distance \(d_{\text{wm}}\) from the wall. The distance \(d_{\text{wm}}\) corresponds to \(y^{+}\sim 300\). The location of the upper bound \((x_{\text{ub}},\ y_{\text{ub}},\ z_{\text{ub}})\) of the wall-stress model in the CFD domain is decided as follows: \[x_{\text{ub}}=x_{\text{c}}+(d_{\text{wm}}-\varphi_{\text{c}})n_{x},\quad y_ {\text{ub}}=y_{\text{c}}+(d_{\text{wm}}-\varphi_{\text{c}})n_{y},\quad z_{ \text{ub}}=z_{\text{c}}+(d_{\text{wm}}-\varphi_{\text{c}})n_{z},\] (18) where \(\varphi_{\text{c}}\) denotes the distance from the wall to the CFD cell center \((x_{\text{c}},\ y_{\text{c}},\ z_{\text{c}})\). 2. The one-dimensional non-uniform mesh for the wall-stress model is generated from the wall to the upper bound. The wall-stress model mesh is generated using the following formula: \[\eta=\exp\left[\left(r_{\text{wm}}-1\right)\left(j_{\text{wm}}+c_{1}\right) \right]+c_{2},\] (19) where \(j_{\text{wm}}\) denotes the cell number from the wall; \(r_{\text{wm}}=1.1\) denotes the rate of the increment of the grid distancing between neighboring grid points. The grid distancing increases by 10% between neighboring grid points. Constants \(c_{1}\) and \(c_{2}\) is decided to satisfy the following conditions: \[\eta=0:\text{ at the wall }(j_{\text{wm}}=0),\quad\eta=d_{\text{wm}}:\text{ at the upper bound }(j_{\text{wm}}=80).\] (20)
3.