# Numerical Simulation and Analysis of Fish-Like Robots Swarm
## 3 Results and Discussion
### The Tandem Swarm  
The thrust and drag at different spacing in the tandem swarm are shown in Figure 9. It is verified by simulations that the net force of fish is about zero. Therefore, the fishes are able to maintain the tandem formation, and the Froude efficiency \(\eta\) is useful. We calculated the non-dimensional thrust, lateral power loss coefficients and the Froude efficiency of each fish. The results are shown in Figure 10. In the tandem swarm, the performance of no.2 and no.3 fish was basically similar. In general, no.2 fish has larger thrust coefficient, less lateral power loss and higher efficiency. However, as the spacing changes, we find that the performance of the fish is not always good. We illustrate this by analyzing the flow field in Figure 11 and Table 4.  
Near the head of the fish robot, if the velocity direction of fluid in vertical direction was the same as the direction of fish swinging, the fish had better efficiency [25]. In this work, the wake indicates the vertical velocity of fluid. The color of the wake indicates the value and direction of the vertical velocity. The velocity boundary is from \(-0.25L/s\) to \(0.25L/s\), where the blue wake indicates that the vertical direction fluids velocity is negative, and the red wake indicates that the vertical direction fluids velocity is positive. The deeper the color of the wake, the greater the relative velocity. A wake flow had a diameter of approximately \(0.24L\) when it had just fallen off from the fish tail. The farther from the tail, the wake was weaker and the spacing between two wakes were larger, as shown in Figure 11a. The no.2 fish achieved the best performance in the Froude efficiency. The Froude efficiency was the highest when the spacing between fishes was \(1.0L\), and it was the lowest when the spacing was \(1.25L\). This is determined by the wake of the no.1 fish, the phase of the no.2 fish and the spacing between fishes. We measured the size of the blue wake and the red wake near the no.2 fish head when the wake had just fallen off from the tail of the no.1 fish. When the spacing is \(1.0L\) and \(1.25L\), near the head of no.2 fish, the sizes of the wake were basically equal. However, when the fish head swings downwards, the wake at the spacing of \(1.0L\) shown in Figure 11b is the same direction as the fish swings. The wake at the spacing of \(1.25L\) shown in Figure 11d is opposite to the direction of the fish swings. Similarly, when the fish head swings upwards, the wake is the same direction as the fish swings at the spacing of \(1.0L\) shown in Figure 11c. The wake is opposite to the direction of the swinging of fish at the spacing of \(1.25L\) shown in Figure 11e. The wake directions and sizes of other spacings are shown in Table 4. It is only necessary to discuss the wake in which the fish head swings downwards, because that of the upward swing is similar. Figure 9: The thrust and drag at different spacing of the tandem swarm. Figure 10: Results of the tandem swarm: (**a**) non-dimensional thrust coefficients \(C_{t}\), (**b**) non-dimensional lateral power loss coefficients \(C_{p}\), (**c**) the Froude efficiency \(\eta\). When the spacing was \(0.5L\), the wake in the opposite direction did not result in very low efficiency because the size of the wake is small. The wake became weaker as the spacing increased. Therefore, at large spacing such as \(1.75L\) and \(2.0L\), whether there was a positive or negative wake, it had little effect on efficiency. The efficiency of the single fish was about \(0.47\) which was lower than the minimum efficiency of no.2 fish. It shows that in a stable formation of fishes at a constant spacing and speed, even the effect of the worst wake was better than that of no wake. Therefore, we conclude that the wake is consistent with the direction of the fish head, which can improve swimming efficiency. In addition, the large and strong wake influences more on the Froude efficiency.