Route to transition in propulsive performance of oscillating foil

Transition in the propulsive performance and vortex synchronization of an oscillating foil in a combined heaving and pitching motion is numerically investigated at a range of reduced frequencies (0.16 $\ensuremath{\le}{f}^{*}\ensuremath{\le}$ 0.64), phase offsets (${0}^{\ensuremath{\circ}}$ $\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}$ ${315}^{\ensuremath{\circ}}$), and Reynolds number ($1000\ensuremath{\le}\text{Re}\ensuremath{\le}16\phantom{\rule{0.16em}{0ex}}000$). Focusing on the common case of $\text{Re}=1000$, the drag to thrust transition is identified on a $\ensuremath{\phi}\ensuremath{-}{f}^{*}$ phase map. Here, the range of ${90}^{\ensuremath{\circ}}$ $\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}$ ${225}^{\ensuremath{\circ}}$ depicted a drag-dominated regime for increasing reduced frequency. However, thrust-dominated regimes were observed for $\ensuremath{\phi}<$ ${90}^{\ensuremath{\circ}}$ and $\ensuremath{\phi}>$ ${225}^{\ensuremath{\circ}}$, where increasing the reduced frequency led to an increased thrust production. The isoline-depicting drag-thrust boundary was further observed to coincide with transitions in the characteristic near-wake modes with increasing reduced frequency, which ranged from $\mathrm{2P}+\mathrm{2S}$ to 2P and reverse von K\'arm\'an modes. However, evaluation of the wake with changing phase offsets at individual reduced frequencies only depicted effects on the spatial configuration of the vortex structures, while the number of vortices shed in one oscillation period was unchanged. The existence of similar wake modes with significantly different propulsive performance clearly suggests that transitions of the wake topology may not always be a reliable tool for understanding propulsive mechanisms of fish swimming or development of underwater propulsion systems. We further assessed a possible route to drag production via investigation into the mean velocity fields at increasing phase offset and at intermediate reduced frequencies ranging from 0.24 to 0.40. This revealed bifurcation of a velocity jet behind the foil on account of the wake topology and dynamics of shed vortex structures. The changes posed by increasing $\ensuremath{\phi}$ on wake structure interactions further hints at potential mechanisms that limit the achievement of optimum efficiency in underwater locomotion.

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