The performance of a flapping foil for a self-propelled fishlike body

Abstract

Several fish species propel by oscillating the tail, while the remaining part of the body essentially contributes to the overall drag. Since in this case thrust and drag are in a way separable, most attention was focused on the study of propulsive efficiency for flapping foils under a prescribed stream. We claim here that the swimming performance should be evaluated, as for undulating fish whose drag and thrust are severely entangled, by turning to self-propelled locomotion to find the proper speed and the cost of transport for a given fishlike body. As a major finding, the minimum value of this quantity corresponds to a locomotion speed in a range markedly different from the one associated with the optimal efficiency of the propulsor. A large value of the feathering parameter characterizes the minimum cost of transport while the optimal efficiency is related to a large effective angle of attack. We adopt here a simple two-dimensional model for both inviscid and viscous flows to proof the above statements in the case of self-propelled axial swimming. We believe that such an easy approach gives a way for a direct extension to fully free swimming and to real-life configurations.

Figure 1

A cartoon for the virtual body (gray) and the tail propulsor (red) with a sketch of the exchanged forces and the oscillatory trajectory of the tail pivot point. The details of the flapping motion are reported in the inset. See also the animation of the swimming fishlike model and the related vortex wake in the Supplementary Video online.

Figure 2

(a) Time history of the locomotion speed for $A_{\mathit{TE}}=1$ and three different values of $A_h/A_{\mathit{TE}}$. (2b) Mean steady state swimming velocity U/L and phase velocity c/L (dashed line) against $A_h/A_{\mathit{TE}}$ for different peak-to-peak trailing edge oscillation amplitudes ($A_{\mathit{TE}}=1$ and 1.5). Inviscid numerical results for zero resistance of the virtual body.

Figure 3

(a) Mean steady state swimming velocity U/L and phase velocity c/L (dashed line), (b) cost of transport of the whole body and (c) efficiency of the propulsor against $A_h/A_{\mathit{TE}}$ for different peak-to-peak trailing edge oscillation amplitudes ($A_{\mathit{TE}}=1$ and 1.5). Viscous and inviscid numerical solutions for a prescribed virtual body resistance.

Figure 4

(a) Cost of transport of the whole body (blue) and efficiency of the propulsor (red) as function of the Strouhal number St. (b) Feathering parameter $\mathrm{\Theta }$ (blue) and maximum angle of attack ${\alpha }_m$ (red) for the inviscid case as function of the Strouhal number. Comparison between $A_{\mathit{TE}}=1$ and 1.5 for the inviscid case.