Flow over traveling wavy foils in a side-by-side arrangement has been numerically investigated using the space-time finite element method to solve the two-dimensional incompressible Navier-Stokes equations. The midline of each foil undergoes lateral motion in the form of a streamwise traveling wave, which is similar to the backbone undulation of swimming fish. Based on the phase difference between the adjacent undulating foils, two typical cases, i.e., in-phase and anti-phase traveling wavy movements, are considered in the present study. The effects of lateral interference among the foils on the forces, power consumption, propeller efficiency, and flow structures are analyzed. It is revealed that the lateral interference is of benefit to saving the swimming power in the in-phase case and enhancing the forces in the anti-phase case. Some typical vortex structures, e.g., vortex-pair row, single vortex row, and in-phase and anti-phase synchronized vortex-street, are observed in the wake of the traveling wavy foils. The results obtained in this study provide physical insight into the understanding of hydrodynamics and flow structures for flow over the traveling wavy foils and swimming mechanisms relevant to fish schooling.

SUMMARYNumerical analysis is carried out to investigate viscous ow over a travelling wavy plate undergoing lateral motion in the form of a streamwise travelling wave, which is similar to the backbone undulation of swimming ÿsh. The two-dimensional incompressible Navier-Stokes equations are solved using the ÿnite element technique with the deforming-spatial-domain=stabilized space-time formulation. The objective of this study is to elucidate hydrodynamic features of ow structure and vortex shedding near the travelling wavy plate and to get into physical insights to the understanding of ÿsh-like swimming mechanisms in terms of drag reduction and optimal propulsive performance. The e ects of some typical parameters, including the phase speed, amplitude, and relative wavelength of travelling wavy plate, on the ow structures, the forces, and the power consumption required for the propulsive motion of the plate are analysed. These results predicted by the present numerical analysis are well consistent with the available data obtained for the wave-like swimming motion of live ÿsh in nature.

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