Vortex Rings in Bio-inspired and Biological Jet Propulsion

Pulsed-jets are commonly used for aquatic propulsion, such as squid and jellyfish locomotion. The sudden ejection of a jet with each pulse engenders the formation of a vortex ring through the roll-up of the jet shear layer. If the pulse is too long, the vortex ring will stop forming and the remainder of the pulse is ejected as a trailing jet. Recent results from mechanical pulsedjets have demonstrated that vortex rings lead to thrust augmentation through the acceleration of additional ambient fluid. This benefit is most pronounced for short pulses without trailing jets. Simulating vehicle motion by introducing background co-flow surrounding the jet has shown that vortex ring formation can be interrupted, but only if the co-flow is sufficiently fast. Recent in situ measurements on squid have captured vortical flows similar to those observed in the laboratory, suggesting thrust augmentation may play a role in their swimming performance. Likewise, recent measurements with a mechanical self-propelled pulsed-jet vehicle (“robosquid”) have shown a cruise-speed advantage obtained by pulsing. Introduction Vortex rings are surprisingly common flow phenomena. They may be generated by surface tension effects, buoyant plumes, flow separation from the unsteady motion of a solid boundary, or momentum driven jets [1]. In the case of the latter, vortex rings are generated by the transient ejection of a jet from a tube or orifice, which leads to the roll-up of the jet shear layer into a toroidal ring that propagates downstream under its own self-induced velocity in accordance with Helmholtz laws of vortex motion. In effect, any unsteady jet flow can lead to vortex rings or flow structures that resemble vortex rings. This is apparent in a wide range of natural and man-made flows ranging from synthetic jet actuators [2] to volcanic eruptions. Unsteady jets in the form of a series of jet pulses are a common method of aquatic propulsion utilized by squid and jellyfish among others. Recent investigations of squid and jellyfish locomotion have demonstrated vortex ring formation during pulsed jet propulsion for several squids and jellyfishes. Parallel studies of mechanically generated pulsed jets have revealed key advantages of pulsed jet propulsion and their relationship to vortex ring formation. This paper discusses key features of vortex ring formation by pulsed jets and their relationship to propulsion, recent results of studies of biological and biomimetic pulsed-jet propulsion, and ongoing work by the authors investigating pulsed-jet performance in different scenarios. Background on Mechanical and Biological Pulsed Jets Vortex Ring Generating Mechanisms. A simple method for generating a transient jet (jet pulse) is to eject a finite amount of fluid from a nozzle in a time by translating a piston through a distance L as shown in Fig. 1(a). This device is known as a piston-cylinder mechanism and L/D is the stroke-ratio of the pulse. The functional form of the jet velocity variation, , is known as the velocity program. When the external fluid is initially quiescent, the jet is called a starting jet. p t ! " t U J !"#$%&'()*%)+&*'%&')$%"),'&-%./.01)2./3)45)678859)::)7;<=7>? .%/*%')$@)-@@:ABBCCC3(&*'%@*D*&3%'@ E)678859),F$%(),'&-)GHI/*&$@*.%(J)+C*@K'F/$%" L%/*%')$#$*/$I/')(*%&')7885B+':B87 !""#$%&'()#$*)*$+*,-#./#01$(#/2#3/4(*4()#/2#('%)#010*$#516#7*#$*0$/,83*,#/$#($14)5%((*,#%4#146#2/$5#/$#76#146#5*14)#9%('/8(#('*#9$%((*4#0*$5%))%/4#/2#('* 087"%)'*$:#;$14)#;*3'#<87"%31(%/4)#=(,>#?9%(@*$"14,>#999-((0-4*(-#ABC:#DEF-DDF-DGE-DHIHGJHFJHK>DF:HL:EMN Ejecting pulses periodically while ensuring a period of no flow between pulses provides a vortex ring with each pulse. This situation is commonly termed a fully-pulsed jet or fully-modulated jet [3, 4]. Practically speaking, fully-pulsed jets require a separate inlet to generate continuous pulsing, but for simplicity, this is not shown in Fig. 1. Figure 1. Mechanical Vortex Ring Generation: (a) Pulsed-jet generated by a piston-cylinder mechanism; (b) Synthetic jet generated by an oscillating piston (or membrane) in a cavity. A related method for generating jet pulses is a synthetic jet, which is illustrated in Fig. 1(b). In this case the jet driver oscillates in place so that the average mass flux of the jet is zero. Outside the Stokes limit, such devices exhibit a net momentum flux and positive thrust is generated [5, 6]. Synthetic jets can generate continuous thrust indefinitely using a vanishingly small plenum, but the boundary motion generating the jet is time-reversible, making them ineffective in the Stokes limit. There are numerous biological analogs to the mechanical methods for generating pulsed jets. Two canonical cases are squid and jellyfish. A schematic of typical squid locomotion is illustrated in Fig. 2(a). The squid propulsion cycle begins by expanding the mantle, a large muscular organ enclosing an internal cavity. Expanding the mantle draws fluid into the mantle cavity via ducts surrounding the head. Circular muscles in the mantle then contract, which increases the fluid pressure in the mantle, closes the inlet valves, and forces the fluid out of the funnel (nozzle) [7]. The funnel may be directed in a hemisphere below the body of the squid so that the squid may be propelled forward or backward, hover in place, or turn in place. Squid also exhibit active control over the funnel diameter during jet ejection [7, 8, 9]. Jellyfish, in contrast, have only one orifice used for both fluid ingestion and ejection, as shown in Fig. 2(b). For propulsion, jellyfish expand the bell, drawing fluid into the sub-umbrellar cavity through the bell margin. Contracting the umbrella forces fluid out the same orifice and propels the jellyfish in the opposite direction [10]. (It should be noted that some species of jellyfish exhibit non-uniform bell contraction, leading to a combination of jet-type and rowing-type propulsion [11].) Recent observations using both dye visualization and digital particle image velocimetry (DPIV) have demonstrated vortex ring formation by jet pulses from both squid and jellyfish propulsion [12, 13, 14]. Although both squid and jellyfish have features not present in the mechanical analogs (most notably, variable exit diameters), it is apparent that squid propulsion is functionally most similar to fully-pulsed jets while jellyfish are most similar to synthetic jets. In particular, both jellyfish and synthetic jets use the same orifice for fluid ingestion and ejection, and both use timereversible boundary motion. In the interest of brevity, the remainder of this paper will focus on starting jets and fully-pulsed jets. !"# $%&%&'()*+,-&.//(0,1*(2+-3,.