Hydrodynamic scaling law in undulatory braking locomotion

Flow over a fish-like airfoil is numerically investigated to elaborate the hydrodynamics of the undulatory braking locomotion for an elongated eel-like body or long-based fin. For undulation with low frequency, we find that boundary layer separation occurs in a parameter region with wakes in which two vortex pairs are formed per undulatory period. The physical mechanism of separation is governed by the slip (the ratio of swimming-to-body-wave speed), and the critical value of the slip in an inertial flow regime is approximately 4/3 rather than 1, which is independent of steepness (or amplitude). The relationship between pressure drag and relative velocity (between phase speed and free stream velocity) changes from linear to quadratic, corresponding to two different flow structures; this happens due to boundary layer separation, and the piecewise scaling relationship between pressure drag and relative velocity is explicitly clarified. Considering the viscosity effects, the separation criterion and the scaling relationship in the case of an undulatory brake are both synthetically modified using the Reynolds number, with all the required parameters clearly expressed. The results of this study provide physical insight into understanding the flow structures and hydrodynamics of the undulatory braking locomotion, which has instructional significance to brake design.