Michael Triantafyllou, Dick Yue and colleagues at the Massachusetts Institute of Technology have been studying mechanical and computational models of swimming fish for the last 10 years. With Qiang Zhu, in the Journal of Fluid Mechanics in October, they described how their 3D model based on simplifications of the Navier-Stokes equation shows how swimming fish use their tails to re-capture energy shed by their bodies. The team has also compared their 3D model with two classic 2D theoretical fish models to see how well the 2D models describe how tuna fish and danios swim.
First, Zhu and colleagues use their simulations to show how tuna fish and danios exploit `vorticity control' to improve their thrust or their swimming efficiency. When a fish's tail moves back and forth, it causes rotation in the water, called vorticity. For most fishes, the body also moves back and forth and also produces vorticity. The simulations show that when the body vorticity hits the tail vorticity, they interact, either strengthening or weakening each other. If the body vorticity is clockwise when the tail vorticity is also clockwise, they strengthen each other, increasing the total vorticity and thrust. However, if the rotations are in opposite directions, the vortices weaken each other, reducing the amount of energy in the wake. Ideally all of the energy would produce forward motion, rather than a wake where the energy is mostly wasted. So reducing the energy in the wake when the vortices interact increases the fish's swimming efficiency. Zhu's calculations show that tunas improve their thrust while danios improve their swimming efficiency, both through vorticity control.
Zhu's model also addresses a problem in simple models of swimming fish: are fish very tall or very long? Two widely used theories from the 1960s analyse a 2D horizontal slice through a swimming fish. The difficulty with analysing a 3D fish in two dimensions is that it requires assumptions about how the fluid flows near the top and bottom of the fish. One 2D theory suggests that if fish are very tall, most of the fluid flows longitudinally down the fish's body. An alternative theory assumes that fish are long and short, and almost all fluid moves laterally and wraps over the top and bottom. Zhu's calculations show that flow around the tuna is largely longitudinal, as if the fish were tall,but that flow around the danio is both longitudinal and lateral, like a combination of the two theories. The difference lies in the different swimming motions. Tunas move their bodies side-to-side relatively little, but danios move side-to-side much more, which forces more fluid laterally.
In the end, it seems that tunas can be modelled by one of the earlier 2D models, but neither simplified theory captures the danio's mode of swimming. Since danios are much more similar to an average fish than tunas, Zhu's simulations suggest that 2D theories might not describe how ordinary fish may re-use energy to boost their efficiency.
