Numerical simulation of vortex ring formation in the presence of background flow with implications for squid propulsion

Abstract

Numerical simulations are used to study laminar vortex ring formation under the influence of background flow. The numerical setup includes a round-headed axisymmetric body with an opening at the posterior end from which a column of fluid is pushed out by a piston. The piston motion is explicitly included into the simulations by using a deforming mesh. A well-developed wake flow behind the body together with a finite-thickness boundary layer outside the opening is taken as the initial flow condition. As the jet is initiated, different vortex evolution behavior is observed depending on the combination of background flow velocity to mean piston velocity ($U/U_{p}$) ratio and piston stroke to opening diameter ($L_{m}/D$) ratio. For low background flow ($U/U_{p} =0.2$) with a short jet ($L_{m}/D =6$), a leading vortex ring pinches off from the generating jet, with an increased formation number. For intermediate background flow ($U/U_{p} =0.5$) with a short jet ($L_{m}/D =6$), a leading vortex ring also pinches off but with a reduced formation number. For intermediate background flow ($U/U_{p} =0.5$) with a long jet ($L_{m}/D =15$), no vortex ring pinch-off is observed. For high background flow ($U/U_{p} =0.75$) with both a short ($L_{m}/D =6$) and a long ($L_{m}/D =15$) jet, the leading vortex structure is highly deformed with no single central axis of fluid rotation (when viewed in cross-section) as would be expected for a roll-up vortex ring. For $L_{m}/D =6$, the vortex structure becomes isolated as the trailing jet is destroyed by the opposite-signed vorticity of the background flow. For $L_{m}/D =15$, the vortex structure never pinches off from the trailing jet. The underlying mechanism is the interaction between the vorticity layer of the jet and the opposite-signed vorticity layer from the initial wake. This interaction depends on both $U/U_{p}$ and $L_{m}/D$. A comparison is also made between the thrust generated by long, continuous jets and jet events constructed from a periodic series of short pulses having the same total mass flux. Force calculations suggest that long, continuous jets maximize thrust generation for a given amount of energy expended in creating the jet flow. The implications of the numerical results are discussed as they pertain to adult squid propulsion, which have been observed to generate long jets without a prominent leading vortex ring.