# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder
## 2 Numerical Calculation Model
### Numerical Model and Solution Setting  
The computational domain is a rectangular region of \(40L\times 20L\), where the side length of the square cylinder \(L=0.054\) m; the center of the square cylinder is located at the origin of the coordinates, as shown in Figure 1. The distances from the upstream inlet, the downstream outlet, and the two sides to the cylinder center are \(10L\), \(30L\), and \(10L\), respectively. The selection of the above size parameters of the computational domain refers to the research results of [PERSON] et al. [39] and [PERSON] et al. [40]. It meets the requirement that the blocking ratio is less than or equal to \(5\%\), and the distance of \(30L\) from model to outlet can ensure the full development of the downstream flow of the square cylinder. Two rotating circular cylinders with a diameter of \(D\) are set on the rear corners of the square cylinder, and the distance between the rotating cylinders and the square cylinder is represented by \(h\). The upper and lower sides of the square cylinder are tangential to the two circular cylinder surfaces. The computational domain is discretized using unstructured grids and is divided into two regions, namely Zone 1 and Zone 2, as shown in Figure 2. Zone 1 is the region around the two circular cylinders where the circular grids can rotate with the circular cylinders. The remaining region is Zone 2, and the fixed grids near the square cylinder surface are locally refined. The interface between the rotation grid (Zone 1) and fixed grid (Zone 2) is set as a slip boundary. Figure 1: Schematic diagram of physical model and computational domain. The wind angle, rotation directions of the two circular cylinders, and arrangement of the pressure monitoring points for \(h/D=1/4\) and \(D/L=0.3\) are shown in Figure 3. The rotation directions of the two circular cylinders were designed as in the following three schemes: rotating in the downstream direction, rotating in the upstream direction, and co-rotating in the clockwise direction; each of the three schemes is represented in Figure 3a-c, respectively. The wind angle \(\theta\) is defined as in the following: when the direction of the incoming airflow and the \(x\)-axis are the same, the wind angle is set to \(0^{{}^{\circ}}\), and the flow along the clockwise direction is considered as the positive direction. When the wind angle is \(0^{{}^{\circ}}\), the two rotating circular cylinders are placed at the two rear corners of the square cylinder. A total of 32 monitoring points were arranged on the model surface, out of which 18 points were evenly arranged on the straight edges of the square cylinder, and 7 points were evenly located on the quarter arc of each rotating circular cylinder. The main purpose is to obtain the pressure distribution characteristics on each straight edge of the square cylinder and on the two quarter arcs tangent to the surface of the square cylinder. In the calculation process, the pressure coefficient of each point was monitored to reveal the underlying mechanism of the MSBC method for manipulating the wake behind the square cylinder. Figure 3: _Cont._  
Figure 2: Computational mesh arrangement and domain partition, (**a**) Computational mesh, (**b**) Computational domain partition and local refined mesh. The velocity ratio \(k\) (\(k=U_{c}/U_{\infty}\)) is the ratio of the tangential velocity of the rotating circular cylinder surface \(U_{c}\) to the uniform incoming velocity \(U_{\infty}\). The velocity ratio was adjusted by changing the rotation speed of the small cylinders, while the incoming airflow remained fixed. When \(k=0\), the two small circular cylinders were stationary; as the velocity ratio became higher, the momentum that the small cylinders delivered to the flow field became greater. The flow direction was from the left to the right, and the following boundary conditions were set.