This study focuses on the hydrodynamic interaction between two or three human swimmers in competitive swimming. Although the swimming performance of a single swimmer has been widely examined, studies on the interaction between multiple competitive swimmers are very rare. Experiments showed evidence that the drag of a swimmer could be modified by the existence of the other adjacent competitors (Chatard & Wilson. 2003 Med. Sci. Sports Exerc. 35, 1176–1181. (doi:10.1249/01.MSS.0000074564.06106.1F10.1249/01.MSS.0000074564.06106.1F)). The following questions arise: (1) what mechanism determines the interaction; (2) which position experiences drag reduction or drag increase; (3) how much can drag be reduced or increased in a formation? According to the authors' knowledge, such questions have not been addressed by any published literature. Therefore, the main purpose of this study is to find the mechanism of the hydrodynamic interaction between human swimmers and to quantify this interactive effect by using a steady potential flow solver. The free-surface effect was fully taken into account in our calculations. We firstly calculated the wave drag of a swimmer swimming solely in an open swimming pool. Then we calculated the wave drag of the same swimmer when he/she swam in the wake region of one or two leading swimmers. The results showed that the hydrodynamic interaction made a significant contribution to the drafter's wave drag. By following a leading swimmer, a drafter at wave-riding positions could save up to 63% of their wave drag at speed of 2.0 m s−1 and lateral separation of 2.0 m. Particularly, when a drafter is following two side-by-side leaders, the drag reduction could even be doubled. To the authors' knowledge, this study is the first to demonstrate that the hydrodynamic interaction between human swimmers can best be described and explained in terms of wave interference effect on the free water surface. When the wave cancellation effect is observed, the wave drag of a drafter could be minimized, and this wave cancellation effect can be achieved only when the drafter is in a wave-riding position.

In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated on p. 317 of [7] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined with respect to the periodic perturbation, which implies that the 3-D transonic shock is unstable in general.

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The determination of optimal values for three parameters required in the original particle swarm optimization algorithm is very difficult. It is proposed that two new parameters simulating the harmony search strategy can be adopted instead of the three parameters which are required in the original particle swarm optimization algorithm to update the positions of all the particles. The improved particle swarm optimization is used in the location of the critical slip surface of soil slope, and it is found that the improved particle swarm optimization algorithm is insensitive to the two parameters while the original particle swarm optimization algorithm can be sensitive to its three parameters.