# Numerical Simulation and Analysis of Fish-Like Robots Swarm
## 3 Results and Discussion  
### Correctness Validation  
To approach the stable hydro-environment, we choose laminar flows, and the fluid velocity is \(1L/s\) and the \(Re\) number is 1000. We did some work to ensure grid- and time-independence at the \(Re\) number and the correctness of the simulation method. In order to evaluate the grid dependence of the solutions, we built the coarse and fine unstructured grids that produce \(M1(33,399 cells)\), \(M2(70,553 cells)\) and \(M3(191181 cells)\), respectively. The grid-independence analysis is performed by comparing the thrust of single fish obtained with the three meshes. As shown in Figure 3, the thrust of the single fish obtained with \(M2\) and \(M3\) are almost identical and the maximum relative difference is less than 1%. Thus, a simulation of time-independence was achieved with \(M2\). The same mesh scale as \(M2\) was adopted in the following simulations of fishes swarm. In order to evaluate the time dependence of the solutions, we built the small and big \(\Delta T\), and set \(\Delta T1(0.001)\), \(\Delta T2(0.0001)\) and \(\Delta T3(0.00001)\), respectively. The time-independence analysis was performed by comparing the thrust of single fish obtained with the three \(\Delta T\). As shown in Figure 4, the thrust of the single fish obtained with all \(\Delta T\) were almost identical. \(\Delta T2\) is adopted in the following simulations. In order to evaluate the correctness of simulation method, we simulated a single fish under conditions in Carling [39], and the results are shown in Figure 5b. In Figure 5, [PERSON]'s model and results are presented on the left and ours on the right. Because of the difference of geometric models, the asymptotic mean velocity of fish and the convergence time were different, but the overall trend was consistent. We found that [PERSON] et al. [46] used the same geometric model as we did, and their asymptotic mean velocity was very close to our result. Therefore, we have reason to believe that our method is correct and the method is adopted in the following simulations. In addition, it is important to note that only when the net force of the fish body is zero, the fishes are able to maintain formation configurations and the Froude efficiency is useful [44,45]. If the net force was not zero, the thrust force and the drag force were not equal, which means the fish was accelerating or decelerating. Thus the constant inlet velocity cannot indicate the swimming speed, and the fishes can not maintain formation configurations. Because the net force of the fish was influenced by the \(St\) number, we conducted a series of simulations at different \(St\) to find the value that can ensure the net force to be zero. Firstly, a single fish  
Figure 4: Result of time-independence analysis. Figure 5: Result of method correctness analysis: (**a**) model, (**b**) image of velocity change with time. Figure 3: Result of grid-independence analysis. was considered. The \(St\) number is tested from 0.35 to 0.6 shown in Table 1. We find that the net force of the fish body is zero when the \(St\) number is 0.45 as shown in Figure 6.  
Then, the diamond swarm and the rectangle swarm were considered. The \(St\) number was tested from 0.15 to 0.625, and the spacing between fishes is changed from 0.5\(L\) to 2\(L\) shown in Table 2. We calculated the thrust force and drag force of each fish and the average net force, and a part of the results are shown in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Re** & **Fluid Velocity** & **St** \\ \hline
1000 & 1\(L/s\) & 0.15, 0.3, 0.4, 0.45, 0.5, 0.625 & 0.5L, 0.75L, 1.0L, 1.25L, 1.5L, 1.75L, 2.0L \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The \(St\) number and spacing in the diamond or rectangle swarm test. Figure 6: Results of simulations at different \(St\): a single fish. Figure 7: Results of the average force when the spacing is 1\(L\): (**a**) the diamond swarm, (**b**) the rectangle swarm.