# A Scalable Moving Boundary Treatment in the Lattice Boltzmann Method
## 5 Numerical Experiment
### Flow in a Curved Moving Pipe  
The parallel program environment of Message Passing Interface (MPI) was used to parallelize the LBM calculation code. In the calculation of static boundary problems, for example, the calculation resulted in Figure 4. In order to verify the applicability of this method in the pipeline flow with moving boundaries, the curved pipe wall model in Figure 7a was calculated. In Figure 7a, the scattered red points are the curved tube wall, and the solid black line is the calculation domain. The curved tube was approximated by a polyline segment connected by a series of points, and the movement of the points simulated the boundary motion of the curved tube. Before the simulation starts, we calculated the range of motion of these points. We set a rectangular area slightly larger than this range as the calculation domain. The advantage of this method is that the complexity of the model does not limit the computational domain. For a model of any shape, a larger rectangular area can contain the model, thereby converting the physical domain of a complex shape into a rectangular domain. The boundary influence is reflected in the flow field through boundary conditions.  
The computational domain was [800, 3360], the interval was 1. Compared with the initial fit, the displacement of the pipe wall satisfied the displacement curve of Figure 7b. The displacement of a single pipe wall was equal, and the upper and lower pipe walls remained symmetrical. In the calculation, the incoming flow velocity was maintained at 0.08, the initial \(Re\) was 1000, and the inlet position of the lower pipe wall was 50.5 at the initial time. As shown in Figure 7b, when the number of calculation steps exceeded 100,000, the displacement started with a sine function. The displacement period was 5000, and a total of eight periods were calculated. In the last displacement period, five equidistant monitoring points were selected. The flow field is shown in Figure 8.  
In the inset in Figure 7b, the red dots are the monitoring points at five moments. Figure 8a-f shows the flow fields corresponding to these five detection points, and (f) is the final field with a static boundary. We kept the legend in Figure 8 consistent. It can be seen from Figure 7b that the pipe diameters of the monitoring points (a)-(d) gradually increased, and there were two complicated flow phenomena: inside the pipe, due to the increase in pipe diameter, the velocity decreased; the pipe entrance, due to the constant incoming flow velocity, the flow rate near the inlet increased and the velocity increased. Therefore, the yellow area at the entrance decreased, the light orange area increased, and the dark orange area decreased. In the scale, the speed values represented by the three colors increased sequentially. The flow field results conformed to the flow state. At times (d) and (f), the  
Figure 7: (**a**) Schematic diagram of calculation model; (**b**) Pipe wall displacement; Inset: a partial enlarged view of the last displacement cycle. The calculation steps corresponding to point \(a\) to point \(e\) are 136,000, 137,000, 138,000, 139,000, and 140,000, respectively.  
pipe wall was at the same position, and the flow field was different, indicating that the influence of pipe wall movement on the flow field was calculated. The difference was small, indicating that the impact of pipe wall movement was relatively small compared to the impact of the incoming flow. The flow field results conformed to the flow state, with no abnormal state, and the calculation results were good, showing this method's applicability to the problem of moving pipeline flow.