# Drag Reduction in Compressible Channel Turbulence with Periodic Interval Blowing and Suction
## 2 Computational Details
### Basic Flow and Verification  
Table 1 presents the computational settings of the basic flow (uncontrolled case) at \(Ma=0.8\) and \(Ma=1.5\). The half-width of the channel is \(\delta=1\). The bulk density \(\rho_{\rm b}\) and bulk velocity \(U_{\rm b}\) in the channel are defined, respectively, as follows:  
\[\rho_{\rm b}=\frac{1}{2\delta}\int_{0}^{2\delta}\left\langle\rho\right\rangle dy \tag{4}\]  
\[U_{\rm b}=\frac{1}{2\delta\rho_{\rm b}}\int_{0}^{2\delta}\left\langle\rho u \right\rangle dy \tag{5}\]  
For a certain variable \(\varphi\) in the flow field, \(\left\langle\varphi\right\rangle\) denotes the Reynolds average, and \(\left\{\varphi\right\}\) denotes the Favre average. The Favre average is defined as \(\left\{\varphi\right\}=\left\langle\rho\varphi\right\rangle/\left\langle\rho\right\rangle\). The bulk Reynolds number \(Re_{\rm b}\) and friction Reynolds number \(Re_{\rm\tau}\) are defined by \(u_{\rm T}\) and \(U_{\rm b}\), respectively:  
\[Re_{\rm b}=\frac{U_{\rm b}\delta}{\
u} \tag{6}\]  
\[Re_{\rm\tau}=\frac{u_{\rm\tau}\delta}{\
u} \tag{7}\]  
In this paper, the bulk Reynolds number is kept constant at \(Re_{\rm b}=3000\), and the resultant friction Reynolds number is \(Re_{\rm\tau}=198\) at \(Ma=0.8\) and \(Re_{\rm\tau}=220\) at \(Ma=1.5\) with the same \(s_{x}\). In order to validate the computational codes and the settings, the computational results are compared with the relevant references [10; 28; 29; 30]. The settings of the references are also listed in Table 1. Figure 3 provides the comparison of the velocity profile distribution for normalized average streamwise velocity \(\left\langle u^{+}\right\rangle\) and the Van Driest transformed velocity \(\left\langle u^{+}_{\rm VD}\right\rangle\). The Van Driest transformation helps mitigate the effect of compressibility when the Mach number is not very large, which can be expressed as follows:  
\[<u^{+}_{\rm VD}>=\int_{0}^{<u^{+}>}\sqrt{\frac{\left\langle\rho\right\rangle }{\rho_{\rm w}}}d<u^{+}> \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(Ma\) & \(Re_{\rm b}\) & \(Re_{\rm\tau}\) & \(L_{x}\times L_{y}\times L_{z}\) & \(N_{x}\times N_{y}\times N_{z}\) & \(\Delta x_{1}^{+}\times\Delta y_{1}^{+}\times\Delta z_{1}^{+}\) \\ \hline Ma0.8 & 0.8 & 3000 & 197 & \(2\pi\delta\times 2\delta\times 2\pi\delta\) & \(128\times 150\times 128\) & \(9.8\times 0.40\times 9.8\) \\ Ma1.5 & 1.5 & 3000 & 220 & \(2\pi\delta\times 2\delta\times 2\pi\delta\) & \(128\times 150\times 128\) & \(10.9\times 0.44\times 10.9\) \\ [PERSON] & 0.8 & 3300 & 185 & \(2\pi\delta\times 2\delta\times 2\pi\delta\) & \(101\times 140\times 101\) & \(11.6\times 0.40\times 11.6\) \\ et al. [28] & 1.5 & 3300 & 185 & \(2\pi\delta\times 2\delta\times 2\pi\delta\) & \(101\times 140\times 101\) & \(11.6\times 0.40\times 11.6\) \\ [PERSON] & 0.8 & 3000 & 190 & \(4\pi\delta\times 2\delta\times 4\pi\delta/3\) & \(256\times 120\times 128\) & \(9.32\times 0.22\times 6.22\) \\ \cline{2-7} et al. [10] & 1.5 & 3000 & 220 & \(4\pi\delta\times 2\delta\times 4\pi\delta/3\) & \(256\times 120\times 128\) & \(10.8\times 0.26\times 7.80\) \\ [PERSON] et al. [29] & 1.5 & 3000 & 222 & \(4\pi\delta\times 2\delta\times 4\pi\delta/3\) & \(144\times 119\times 80\) & \(19\times 0.1\times 12\) \\ [PERSON] et al. [30] & 1.5 & 3000 & 218 & \(4\pi\delta\times 2\delta\times 4\pi\delta/3\) & \(120\times 180\times 120\) & \(23\times 0.36\times 7.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational settings for basic flow and relevant references. In Figure 3a, though there is little difference from the literature [28] in the buffer layer due to a different \(Re_{\rm b}\) and \(Re_{\tau}\), the computational result shows good agreement with the literature [29; 30] in the viscous sublayer and the logarithmic layer. In Figure 3b, the velocity profiles of \(Ma=0.8\) and \(Ma=1.5\) are very close to the empirical expression of the incompressible flow (black dashed line).