Hydrodynamic interactions of low-aspect-ratio oscillating panels in a tip-to-tip formation

The vertical, tip-to-tip arrangement of neighbouring caudal fins, common in densely packed fish schools, has received much less attention than staggered or side-by-side pairings. We explore this configuration using a canonical system of two trapezoidal panels (aspect ratio ${\textit{AR}}=1.2$ ) that pitch about their leading edges while heaving harmonically at a Strouhal number $St=0.45$ and a reduced frequency $k=2.09$ . Direct numerical simulations based on an immersed-boundary method are conducted over a Reynolds-number range of $600\leq {\textit{Re}}\leq 1\times 10^{4}$ , and complementary water-channel experiments extend this range to $1\times 10^{4} \leq {\textit{Re}}\leq 3\times 10^{4}$ . Results indicate that when the panels oscillate in phase at a non-dimensional vertical spacing $H/c\leq 1.0$ with $c$ denoting the panel chord length, the cycle-averaged thrust of each panel rises by up to 14.5 % relative to an isolated panel; the enhancement decreases monotonically as the spacing increases. Anti-phase motion instead lowers the power consumption by up to 6 %, with only a modest thrust penalty, providing an alternative interaction regime. Flow visualisation shows that in-phase kinematics accelerate the stream between the panels, intensifying the adjacent leading-edge vortices. Downstream, the initially separate vortex rings merge into a single, larger ring that is strongly compressed in the spanwise direction; this wake compression correlates with the measured thrust gain. The interaction mechanism and its quantitative benefits persist throughout the entire numerical and experimental Reynolds-number sweep, indicating weak ${\textit{Re}}$ -sensitivity within $600\leq {\textit{Re}}\leq 3\times 10^{4}$ , and across multi-panel systems. These results provide the first three-dimensional characterisation of tip-to-tip flapping-panel interactions, establish scaling trends with spacing and phase, and offer a reference data set for reduced-order models of vertically stacked propulsors.