Pressure and work analysis of unsteady, deformable, axisymmetric, jet producing cavity bodies

Abstract

This work lays out a methodology for calculating the pressure distribution internal to a generic, deformable, axisymmetric body with an internal cavity region whose deformation expels/ingests finite jets of water. This work is partially motivated by a desire to model instantaneous jetting forces and total work required for jellyfish and cephalopod locomotion, both of which can be calculated from the internal pressure distribution. But the derivation is non-specific and can be applied to any axisymmetric, deformable body (organic or synthetic) driving fluid in or out of an internal cavity. The pressure distribution over the inner surface is derived by integrating the momentum equation along a strategic path, equating local surface pressure to known quantities such as stagnation pressure, and correlating unknown terms to the total circulation of characteristic regions. The integration path is laid out to take advantage of symmetry conditions, inherent irrotationality, and prescribed boundary conditions. The usefulness/novelty of this approach lies in the fact that circulation is an invariant of motion for inviscid flows, allowing it to be modelled by a series of vorticity flux and source terms. In this study we also categorize the various sources of circulation in the general cavity–jet system, providing modelling for each of these terms with respect to known cavity deformation parameters. Through this approach we are able to isolate the effect of different deformation behaviours on each of these circulation components, and hence on the internal pressure distribution. A highly adaptable, transparent, prototype jet actuator was designed and tested to measure the circulation in the cavity and the surrounding fluid as well as the dynamic forces acting on the device during operation. The circulation in both the jet and cavity regions shows good agreement with the inviscid modelling, except at the end of the refill phase where circulation is lost to viscous dissipation. The total instantaneous forces produced during actuation are accurately modelled by the pressure analysis during both expulsion and refilling phases of the jetting cycle for multiple deformation programs. Independent of the end goal, such as propulsion, mixing, feeding etc., the efficiency of the process will always be inversely proportional to the total energy required to drive the system. Therefore, given a consistent output, efficiency is maximized by the minimum required energy. Here it is observed (somewhat counter-intuitively) that, for both jetting and refilling, total work required to drive the fluid is lower for impulsive velocity programs with fast accelerations at the start and end of motion than sinusoidal velocity programs with smoother gradual accelerations. The underlying cause is that sinusoidal programs result in a peak in pressure (force) simultaneously with maximum deflection velocity of the deformable boundary driving fluid motion; for the impulsive programs these peaks are out of phase and overall energy consumption is reduced.