# A Scalable Moving Boundary Treatment in the Lattice Boltzmann Method
## 5 Numerical Experiment
### Flow Past a Moving Cylinder  
Consider the problem of moving boundary flow. In initialization, the fluid in the two-dimensional calculation domain is static, the cylinder moves to the right at a constant speed, and the four boundaries are pressure boundary conditions. Suppose the cylinder is considered static. The problem of the moving boundary can be transformed into the corresponding problem of the stationary boundary. When these two flows are fully developed, there should be similar flow field conditions around the cylinder, and the force conditions are the same. Through the analysis of the flow field and the force around the cylinder at the stationary boundary and the moving boundary, the reliability of the simulation of the moving boundary in this paper can be effectively verified. In the moving cylinder problem, the initial flow field is static, the cylinder moves at a constant horizontal speed from right to left and the fluid moves under the influence of the cylinder. The calculation domain is [2400, 300], the time and space intervals are 1, the cylinder diameter is 20, and \(Re\) is 100. After a period of time, the fluid lateral velocity cloud diagram is shown in Figure 5a. In Figure 5a, most of the fluid area is in a static state, and the area of fluid movement extends from the proximal end to the distal end of the cylinder. Transform the transverse velocity flow field, that is, \(u=u_{0}-u\) where, \(u\) is the lateral velocity of the fluid and \(u_{0}\) is the velocity of the cylinder movement, which is also the flow velocity at the inlet in the static problem. The transformed result is shown in Figure 5b. There is a uniform rightward flow at the left end entrance in the static problem, and the lateral velocity cloud diagram of the flow field is shown in Figure 5c. Compared with Figure 5b, due to the different fluid motion states corresponding to the two flow fields,the fluid motion states are different at the far end of the cylinder. At the proximal end of the cylinder, the cylinder has a disturbing effect on the fluid, and the flow field formed at the proximal end of the cylinder is basically the same. Calculate the lift coefficient \(C_{l}\), the drag coefficient \(C_{d}\), and the vortex separation frequency \(S_{t}\) around the cylinder in the two problems respectively, as shown in Table 1. In the moving boundary problem, the velocity of the cylinder is used instead of the entry velocity. The absolute value of the difference between the result of the moving cylinder and the result of the stationary cylinder is multiplied by 100% to calculate the relative error. The lift coefficient and drag coefficient can reflect the force on the cylinder in the flow field, and the vortex separation frequency can reflect the separation of the vortex at the back of the cylinder. It can be seen from Table 1 that the lift and drag coefficients of the two flow fields are the same as the vortex separation frequency, and the relative error value is less than 5%. It shows that in the two problems, the force on the cylinder is the same, indicating the algorithm's accuracy for the moving boundary problem. In order to verify the application of the method in this paper under the medium \(Re\), the mobile airfoil No. 0012 at an angle of attack of 7 degrees was calculated. In addition, the flow field of the stationary airfoil in the uniform incoming flow was simulated. The calculation domain was [2500, 1000], the grid interval and time interval were both 1, the airfoil length was 200, and the \(Re\) of the two simulations was 1000. In the static boundary simulation, the airfoil faced horizontally to the left, and the uniform incoming flow velocity at the inlet was Mach 0.1. In the moving boundary simulation, the airfoil faced horizontally to the left, and the moving speed horizontally to the left was Mach 0.1. Figure 6a shows the flow field simulated by the moving boundary, with \(u=u_{0}-u\) to obtain the flow field in Figure 6b, and Figure 6c is the flow field simulated by the static boundary.