# Drag Reduction in Compressible Channel Turbulence with Periodic Interval Blowing and Suction
## 3 Results and Discussions  
### Controlled Cases  
The details of the cases with blowing/suction control are presented here. As illustrated in Figure 4, the controlled region and the uncontrolled region are staggered in the streamwise direction of the infinitely long channel. The blowing strips are implemented in the controlled region on the bottom wall and the suction strips are placed symmetrically on the top wall. A control unit is designated as the computational domain (blue dashed box), with the controlled region positioned centrally within this domain. The streamwise length of the controlled region is \(L_{c}\), and the area ratio of the controlled region and the computational domain \(\alpha\) is defined as follows:  
\[\alpha=\frac{L_{\rm c}}{L_{x}} \tag{9}\]  
Figure 4: Schematic diagram of the top/bottom wall of the physical model. The blue dashed box represents the computational domain. The control intensity \(I\) is defined as follows:  
\[I=\left\{ \begin{array}{l}\frac{v_{\rm B}}{u_{\rm co}}\left(\rm{blowing}\right)\\ \frac{v_{\rm B}}{u_{\rm co}}\left(\rm{suction}\right)\end{array} \right. \tag{10}\]  
In the controlled region, the wall-normal velocity \(v_{\rm w}\) is adjusted to the \(v_{\rm B}\) (blowing velocity) or \(v_{\rm S}\) (suction velocity), while it is kept at zero in the uncontrolled region. The simulation employs velocity boundary conditions rather than modeling the real blowing/suction devices because this paper primarily investigates the impact mechanism of blowing/suction control on channel turbulence. This may lead to a specific discrepancy from the real-world conditions. However, owing to the small blowing/suction intensity, it can guarantee effective wall attachment performance for blowing/suction. At each Mach number, one basic flow case and four controlled cases are calculated. C1 and C6 are the basic flow cases, C2 and C7 are the cases with global uniform blowing/suction, and the others are the interval blowing/suction cases with various \(\alpha\). For controlled cases, the product of \(I\) and \(\alpha\) is an unchanged constant, so guaranteeing the control (blowing/suction) flow rate remains fixed in C2-C5 or C7-C10. The control effect parameters of these cases are listed in Table 2, where the subscript \"B\" represents the blowing side, \"S\" represents the suction side, and \"0\" represents the uncontrolled case. The drag reduction rate \(DRR\) refers to the drag reduction/augment when it is positive/negative. The net energy-saving rate \(S\) is determined by subtracting the loss of blowing power from the \(DRR\). Furthermore, the control gain \(G\) reflects the amplification effect of the blowing control on drag reduction. Their definitions [17] are expressed as follows:  
\[C_{\rm f}=\frac{\tau_{\rm w}}{0.5\rho_{\rm w}U_{\rm b}^{2}} \tag{11}\]  
\[DRR_{\rm B/S}=\frac{C_{\rm f0}-C_{\rm f,B/S}}{C_{\rm f0}} \tag{12}\]  
\[DRR=\frac{DRR_{\rm B}+DRR_{\rm S}}{2} \tag{13}\]  
\[S=\frac{C_{\rm f0}-C_{\rm f,B/S}-P_{\rm B}}{C_{\rm f0}} \tag{14}\]  
\[G=\frac{U_{\rm b}(C_{\rm f0}-C_{\rm f,B})}{P_{\rm B}} \tag{15}\]  
where \(P_{\rm B}\) is the input power of the blowing and is defined as  
\[P_{\rm B}=\frac{1}{2}inv_{\rm w}^{2}\approx\frac{\alpha\langle\rho_{\rm w} \rangle\langle v_{\rm w}\rangle^{3}}{2} \tag{16}\]  
\(\dot{m}\) is the wall mass flow rate, and \(\dot{m}\propto\alpha\cdot I\). According to the results from [PERSON] et al. [10] and [PERSON] et al. [13], a larger blowing/suction velocity results in a higher/smaller \(DRR\). Nonetheless, in the cases with interval control, the blowing area will also influence the control efficacy, which was not considered in the aforementioned work. Because of the micro-blowing, the input power \(P_{\rm B}\) is minimal; therefore, the net energy-saving rate \(S\) is nearly equivalent to the \(DRR_{\rm B}\) in all cases. It can also be concluded that, under the same control flow rate \(\dot{m}\) (C2-C5 or C7-C10), \(Re_{\rm r,b}\), \(DRR_{\rm B}\), \(DRR_{\rm S}\), \(DRR\), and \(S\) are almost unchanged.