Skin friction on a moving wall and its implications for swimming animals

Estimating the energetic costs of undulatory swimming is important for biologists and engineers alike. To calculate these costs it is crucial to evaluate the drag forces originating from skin friction. This topic has been controversial for decades, some claiming that animals use ingenious mechanisms to reduce the drag and others hypothesizing that undulatory swimming motions induce a drag increase because of the compression of the boundary layers. In this paper, we examine this latter hypothesis, known as the 'Bone-Lighthill boundary-layer thinning hypothesis', by analysing the skin friction in different model problems. First, we study analytically the longitudinal drag on a yawed cylinder in a uniform flow by using the approximation of the momentum equations in the laminar boundary layers. This allows us to demonstrate and generalize a result first observed semi-empirically by G. I. Taylor in the 1950s: the longitudinal drag scales as the square root of the normal velocity component. This scaling arises because the fluid particles accelerate as they move around the cylinder. Next we propose an analogue two-dimensional problem where the same scaling law is recovered by artificially accelerating the flow in a channel of finite height. This two-dimensional problem is then simulated numerically to assess the robustness of the analytical results when inhomogeneities and unsteadiness are present. It is shown that spatial or temporal changes in the normal velocity usually tend to increase the skin friction compared with the ideal steady case. Finally, these results are discussed in the context of swimming energetics. We find that the undulatory motions of swimming animals increase their skin friction drag by an amount that closely depends on the geometry and the motion. For the model problem considered in this paper the increase is of the order of 20 %.