# Turbulent Drag Reduction Characteristics of Bionic Nonsmooth Surfaces with Jets
## 2 Physical Model Establishment and Hypothesis Verification
### Calculation Model  
The inlet velocity of the simulation was set to 50.1 m/s, which is consistent with available experiments. The fluid in this study was considered incompressible, and there was also no temperature difference in the flow, so it was not necessary to consider the energy conservation equation. Therefore, only the mass conservation equation and the momentum conservation equation were established.  
The mass Equation (continuity equation) is as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{3}\]  
The momentum Equation (viscous fluid motion equation) is as follows:  
\[\frac{\partial(\rho u)}{\partial t}+\text{div}(\rho uu)=-\frac{\partial p}{ \partial x}+\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{ \partial y}+\frac{\partial\tau_{xx}}{\partial z}+F_{x} \tag{4}\]  
\[\frac{\partial(\rho v)}{\partial t}+\text{div}(\rho vu)=-\frac{\partial p}{ \partial y}+\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{ \partial y}+\frac{\partial\tau_{zy}}{\partial z}+F_{y} \tag{5}\]  
\[\frac{\partial(\rho w)}{\partial t}+\text{div}(\rho vu)=-\frac{\partial p}{ \partial z}+\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{ \partial y}+\frac{\partial\tau_{zz}}{\partial z}+F_{z} \tag{6}\]  
The SST K-Omega turbulence model is a two-equation eddy viscosity model. The model accounts for the transfer of turbulent shear stress, enabling the model to accurately predict the beginning of flow and the separation position under negative pressure gradients [43; 44]. The eddy viscosity will also not be over-predicted due to the consideration of turbulent shear stress. Therefore, this model is well suited for simulating flow in the viscous sublayer [45; 46] and was adapted to enclose the equations in this paper. The turbulent kinetic energy equation and specific dissipation rate equation are as follows:  
\[\rho\frac{Dk}{Dt}+\rho\frac{\partial(ku_{i})}{\partial x_{i}}=\frac{\partial}{ \partial x_{j}}(T_{k}\frac{\partial k}{\partial x_{j}})+G_{k}-Y_{k} \tag{7}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Grid Quantity** & **Skin Friction Coefficient** \\ \hline
102,505 & 0.00275 \\
196,784 & 0.00301 \\
221,027 & 0.00311 \\
251,215 & 0.00325 \\
308,794 & 0.00323 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Grid independence.  
Figure 6: Inflation layer.  
Figure 7: Grid settings of models.  
\[\rho\frac{D\omega}{Dt}+\rho\frac{\partial(\omega u_{i})}{\partial x_{i}}=\frac{ \partial}{\partial x_{j}}(T_{\omega}\frac{\partial\omega}{\partial x_{j}})+G_{ \omega}-Y_{\omega}+D_{\omega}, \tag{8}\]  
where \(G_{k}\) is the turbulent kinetic energy produced by the laminar velocity gradient; \(G_{\omega}\) is the turbulent kinetic energy produced by the \(\omega\) equation; \(T_{k}\) and \(T_{\omega}\) are the diffusion ratios of \(k\) and \(\omega\), respectively; \(Y_{k}\) and \(Y_{\omega}\) are the turbulence generated by the diffusion; and \(D_{\omega}\) is the orthogonal divergence term.