# Drag Reduction in Compressible Channel Turbulence with Periodic Interval Blowing and Suction
## 2 Computational Details  
### DNS Method and Physical Model  
The governing equations of the numerical simulation are the compressible N-S equations. Hereafter, the variables with asterisk superscripts are dimensional quantities, and the variables without asterisks are dimensionless quantities. \(\rho\), \(u\), \(v\), \(w\), \(T\), and \(p\) are used to represent the density, the streamwise velocity, the normal velocity, the spanwise velocity, the temperature, and the pressure, respectively. The variables in the equations are non-dimensionalized using reference quantities such as the length \(L^{*}\), the density \(\rho_{\infty}^{*}\), the velocity in the \(X\) direction \(u_{\infty}^{*}\), the temperature \(T_{\infty}^{*}\), and their combinations. The dimensionless NS equations can be expressed as follows:  
\[\frac{\partial\mathbf{Q}}{\partial t}+\frac{\partial\mathbf{F}_{x}(\mathbf{Q})}{\partial x }+\frac{\partial\mathbf{F}_{y}(\mathbf{Q})}{\partial y}+\frac{\partial\mathbf{F}_{z}(\bm {Q})}{\partial z}=\frac{\partial\mathbf{G}_{x}(\mathbf{Q})}{\partial x}+\frac{\partial \mathbf{G}_{y}(\mathbf{Q})}{\partial y}+\frac{\partial\mathbf{G}_{z}(\mathbf{Q})}{\partial z}+S \tag{1}\]  
where \(\mathbf{Q}\) is the conserved variable vector, \(F(\mathbf{Q})\) is the inviscid flux vector, \(\mathbf{G}(\mathbf{Q})\) is the viscous flux vector, \(\mathbf{S}\) is the source term, and the subscripts \(x\), \(y\), and \(z\) denote the components of \(\mathbf{F}\) and \(\mathbf{G}\) in the streamwise (\(X\)), normal (\(Y\)), and spanwise (\(Z\)) directions, respectively. The components of \(\mathbf{Q}\), \(\mathbf{F}\), \(\mathbf{G}\), and \(\mathbf{S}\) are defined as follows:\[\mathbf{Q}= \begin{pmatrix}\rho\\ \rho u\\ \rho v\\ \rho w\\ E\end{pmatrix} \text{, }F_{x}(\mathbf{Q})= \begin{pmatrix}\rho u\\ \rho u^{2}+p\\ \rho uv\\ \rho uw\\ u(E+p)\end{pmatrix} \text{, }F_{y}(\mathbf{Q})= \begin{pmatrix}\rho v\\ \rho uv\\ \rho v^{2}+p\\ \rho vw\\ \rho v(E+p)\end{pmatrix} \text{, }F_{z}(\mathbf{Q})= \begin{pmatrix}\rho w\\ \rho uw\\ \rho vw\\ \rho w^{2}+p\\ w(E+p)\end{pmatrix} \text{, }F_{z}(\mathbf{Q})= \begin{pmatrix}\rho w\\ \rho uv\\ \rho vw\\ \rho w^{2}+p\\ w(E+p)\end{pmatrix} \text{, }\] \[\mathbf{G}_{x}(\mathbf{Q})= \begin{pmatrix}0\\ \tau_{xx}\\ \tau_{xy}\\ \tau_{xz}\\ k\frac{\partial T}{\partial x}+u\tau_{xx}+v\tau_{xy}+w\tau_{xz}\end{pmatrix}  \text{, }\mathbf{G}_{y}(\mathbf{Q})= \begin{pmatrix}0\\ \tau_{xy}\\ \tau_{yy}\\ \tau_{yz}\\ k\frac{\partial T}{\partial y}+u\tau_{xy}+v\tau_{yy}+w\tau_{yz}\end{pmatrix}  \text{, } \tag{2}\] \[\mathbf{G}_{z}(\mathbf{Q})= \begin{pmatrix}0\\ \tau_{xz}\\ \tau_{yz}\\ \tau_{zz}\\ k\frac{\partial T}{\partial z}+u\tau_{xz}+v\tau_{yz}+w\tau_{zz}\end{pmatrix}  \text{, }\mathbf{S}= \begin{pmatrix}0\\ \rho s_{x}\\ 0\\ 0\\ \rho us_{x}\end{pmatrix} \]  
where \(E\), \(s_{x}\), and \(k\) represent the internal energy, the external body force, and the thermal conductivity. \(\tau\) is the viscous stress tensor, and its subscripts denote the viscous stress in different directions. Hoam-OpenCFD (Version 1.10.4), developed by [PERSON] et al. [27], is used in this paper, and the direct numerical simulation (DNS) method is applied in the computation. The software uses the finite difference method (FDM) to acquire detailed flow field information via high-precision numerical simulation. It is highly appropriate for the examination of mixing layers, wall-bounded turbulence, jet flows, homogeneous isotropic turbulence (HIT), and numerous intricate unsteady flows. The temporal term \(\partial\mathbf{Q}/\partial t\) is advanced using the third-order Runge-Kutta method with a TVD property, and the inviscid term \(\partial F(\mathbf{Q})/\partial\mathbf{x}\) and the viscous term \(\partial G(\mathbf{Q})/\partial\mathbf{x}\) (\(\mathbf{x}=(x,y,z)^{T}\)) are calculated using the seventh-order upwind difference and the sixth-order central difference, respectively.