Elastic wave propagation governs impulse enhancement in pulsed jets through flexible nozzles

Inspired by cephalopod jet propulsion through compliant funnels, this study investigates elastic wave propagation and energy exchange in passively deforming cylindrical nozzles through three-dimensional, two-way fluid-structure interaction simulations. Flexible nozzles with varying stiffness ($Eh = 75 - 500~\mathrm{N\,m^{-1}}$, where $E$ and $h$ are Young's modulus and nozzle thickness, respectively) are subjected to a pulsatile jet inflow at $Re \sim 4000$. Increasing nozzle flexibility reduces the deformation-wave speed in accordance with Moens-Korteweg scaling, thereby prolonging the nozzle expansion phase. This delayed expansion enhances jet entrainment and elastic energy storage while suppressing early shear-layer roll-up and vortex formation. During contraction, the stored elastic energy is released, thereby enhancing jet acceleration and vortex formation. For the most flexible nozzle, the primary vortex-ring circulation increases by 52.13%, the vortex convection distance by 9.00%, and the peak outlet kinetic energy flux by a factor of 4.62 compared with a rigid nozzle. These effects collectively yield a 61.92% increase in total hydrodynamic impulse. These findings identify passive wave-speed tuning via nozzle compliance as a mechanism to enhance pulsed-jet thrust for bio-inspired underwater propulsion.