Direct numerical simulation is used to study the turbulent flow over a smooth wavy wall undergoing transverse motion in the form of a streamwise travelling wave. The Reynolds number based on the mean velocity $U$ of the external flow and wall motion wavelength $\lambda$ is 10 170; the wave steepness is $2\pi a/\lambda=0.25$ where $a$ is the travelling wave amplitude. A key parameter for this problem is the ratio of the wall motion phase speed $c$ to $U$, and results are obtained for $c/U$ in the range of $-1.0$ to $2.0$ at $0.2$ intervals. For negative $c/U$, we find that flow separation is enhanced and a large drag force is produced. For positive $c/U$, the results show that as $c/U$ increases from zero, the separation bubble moves further upstream and away from the wall, and is reduced in strength. Above a threshold value of $c/U\approx 1$, separation is eliminated; and, relative to small- $c/U$ cases, turbulence intensity and turbulent shear stress are reduced significantly. The drag force decreases monotonically as $c/U$ increases while the power required for the transverse motion generally increases for large $c/U$; the net power input is found to reach a minimum at $c/U\approx 1.2$ (for fixed $U$). The results obtained in this study provide physical insight into the study of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive efficiency.
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