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

Transcription

1 Theor. Comput. Fluid Dyn. (2006) 20(2): DOI /s ORIGINAL ARTICLE Houshuo Jiang Mark A. Grosenbaugh Numerical simulation of vortex ring formation in the presence of background flow with implications for squid propulsion Received: 4 May 2004 / Accepted: 8 December 2005 / Published online: 10 March 2006 C Springer-Verlag 2006 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 finitethickness 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. Keywords Vortex ring formation Jet Background flow Squid propulsion Numerical simulation PACS Cb, cf, cb, Ft, M- 1 Introduction Vortex rings are a remarkable phenomenon of fluid motion and have attracted the attention of researchers for generations. Previous experimental, analytical, and numerical studies of the generation, formation, and evolution of vortex rings and their interactions are described in the comprehensive review by Shariff and Leonard [16]. An important finding concerning the formation process of vortex rings was made by Gharib et al. [7]. Communicated by R. D. Moser H. Jiang (B) M. A. Grosenbaugh Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA

2 104 H. Jiang, M. A. Grosenbaugh They used a piston-cylinder setup the most common way of generating vortex rings in the laboratory to investigate the formation process of vortex rings for a wide range of piston stroke to cylinder diameter (L m /D) ratios. They showed that, for large L m /D, the flow field in front of the vortex ring generator consists of a leading vortex ring followed by a trailing jet. On the other hand, for small L m /D, the flow field consists of only a single vortex ring. The transition between these two distinct states occurs at a critical L m /D ratio in the range of depending on the flow conditions. (Note that for a piston-cylinder setup, a non-dimensional time called the formation time is defined as to = Ū p t/d, whereū p is the mean piston velocity and t is the discharge time. The formation time corresponding to when the piston stops moving, tm, is equivalent to the L m /D. The critical value of L m /D is referred to as the vortex ring formation number.) In all cases of large L m /D, the vorticity field of the leading vortex ring disconnects from the trailing jet via a pinch-off process and attains a maximum circulation at this formation number. The formation number is therefore interpreted as a (universal) non-dimensional threshold formation time for vortex ring formation. Gharib et al. [7]alsoexplained the existence of the formation number based on the Kelvin-Benjamin variational principle. According to this principle, a steadily translating vortex ring has maximum energy with respect to impulse-preserving iso-vortical perturbations. Thus, a hypothesis follows that the leading vortex ring pinches off from the trailing jet when the piston device is no longer able to deliver energy at a rate compatible to the existence of a steadily translating vortex ring. Gharib et al. [7] were able to apply the Kelvin-Benjamin variational principle by considering a non-dimensional energy, which is formed from the measured impulse, circulation and energy of the observed vortex rings and successfully predicted the range of observed formation numbers. Using the same hypothesis and making two simple assumptions on the formation process, Mohseni and Gharib [11] predicted the formation number analytically by considering the non-dimensional energy of a vortex ring. Linden and Turner [10] proposed another analytical model, in which the circulation, impulse, volume, and kinetic energy of the ejected fluid were matched to the corresponding properties of a family of finite-core vortices. Several numerical studies also investigated the formation process of vortex rings. Numerical predictions of the total circulation and leading vortex ring circulation by Rosenfeld et al. [15] agree well with the experimental results of Gharib et al. [7]. The presence of a non-dimensional threshold formation time (approximately four) is confirmed for vortex rings generated by impulsive velocity programs of the piston. The numerical study extended the experimental study to additional cases, including non-impulsive velocity programs of the piston, different velocity profiles (uniform velocity profile versus the Poiseuille velocity profile), different Reynolds numbers, and different vortex generator configurations (i.e. specified discharge velocity, orifice and nozzle geometry). It was shown that the formation number is strongly dependent on the discharge velocity profile and on the piston velocity program. Thus, the universal value observed by Gharib et al. [7] is only observed for certain conditions. For the case of an impulsively formed parabolic discharge velocity profile, the formation number is Zhao et al. [19] specified a family of hyperbolic tangent axial velocity profiles (nearly uniform) with different shear layer thicknesses and a single parabolic velocity profile to numerically generate vortex rings. A non-impulsive velocity program with an initial cosine ramp up was used throughout their work. For the hyperbolic tangent inlet velocity profiles, their results are comparable with those of Gharib et al. [7] with formation numbers between 3.6 and 4.5. However, for the parabolic velocity profile, they obtained a formation number of In addition, they tried to explain a 20% variation in vortex ring circulation (with respect to different Reynolds numbers and different shear layer thicknesses) in terms of an interaction between an instability that develops in the trailing jet for large discharge times and the dynamics of the leading vortex ring. Mohseni et al. [12] numerically generated vortex rings by applying a non-conservative force of long duration. Their results again confirm the existence of a non-dimensional threshold formation time. They showed that scaling the circulation with the translational velocity and the impulse of the leading vortex ring could reduce variations in vortex ring circulation, rather than scaling with parameters associated with the forcing. They also demonstrated that increasing the spatial extent of the forcing or the magnitude of the forcing amplitude during the formation process could generate thicker rings with larger scaled circulation. Vortex rings have been used previously to understand the hydrodynamics of squid jet propulsion (Siekmann [17]; Weihs [18]). It was assumed that the pulsating efflux from squid jet orifice produces a row of vortex rings, and the thrust can be calculated based on this assumption. Weihs [18] found that a pulsed jet could provide a greater average thrust than a continuous jet of equivalent mass flux rate, if the pulses are frequent enough so that the distance between ring centerlines is less than three ring radii. However, such a mechanism of enhancing the thrust would be of no importance in squid with small orifices and low frequencies because of the absence of ring-ring interactions (O Dor [13]; Bartol et al. [4]). Recently, with the discovery of vortex ring formation number, an intriguing new question has been asked: can jet propelled organisms such as squid utilize the vortex ring formation number to enhance propulsive

3 Numerical simulation of vortex ring formation in the presence of background flow 105 efficiency (Gharib et al. [7]; Linden and Turner [10])? Most significantly, the theoretical analysis by Linden and Turner [10] predicts that a vortex ring produced using the highest possible L m /D ratio for the formation of a single vortex ring (i.e., under the condition of vortex ring formation number) has the maximum impulse for a given kinetic energy. If the ring is ejected behind a swimming body, it produces the maximum thrust, equivalent to highest propulsive efficiency. However, experimental studies using indirect flux measuring techniques suggest that swimming adult squid use long continuous jets with L m /D 4 (Anderson et al. [2]; Bartol et al. [4]). Also, digital particle imaging velocimetry (DPIV) has recently been used to determine the flow structure and velocities of the jets of steadily swimming adult squid (Loligo pealei) (Anderson and Grosenbaugh [3]). Direct measurements of L m /D (the ratio of jet plug length to average jet orifice diameter) were made. L m /D was measured to be for lower-bound values and for upper-bound values, based on 89 observed squid jet events. Jet frequency (i.e., the number of jet events per second) ranged from Hz in the majority of swimming sequences. The visualization of the jets of steadily swimming adult squid revealed two consistent characteristics of jet flow structure: (1) an elongated shape, and (2) the reduced and/or late development of the leading vortex ring of the jet compared to the vortex ring formation by emitting a column of fluid into still water (e.g., Gharib et al. [7]). Propulsion by individual vortex rings that formed from the edge of the jet orifice, i.e., pulsed vortex ring propulsion, was never observed in steadily swimming adult squid. All these observations evidence that steadily swimming adult squid do not use vortex ring formation number to enhance propulsion. The interested reader is referred to Anderson and Grosenbaugh [3] for the details, including squid morphology and visualizations of squid jet flow. In steady locomotion, squid swim at a relatively high velocity, not much less than their jet velocity. For example, Anderson and DeMont [1] reported an average jet velocity of 54 cm s 1 for a squid swimming steadily at 25 cm s 1. Anderson and Grosenbaugh [3] also reported similar ranges. The high swimming velocity will be transformed to a strong background flow around the squid body/jet-orifice if a frame of reference fixed on the body is considered. The apparent reduced and/or late development of the leading vortex ring in a squid jet may be due to the presence of this background flow. To better understand the effect of background flow on vortex ring formation and evolution of jet flow structure, we will study numerically the idealized problem of fluid being pushed out by a piston from a roundheaded axisymmetric tube immersed in a fully developed background flow. A fully developed background flow around the tube may have two effects. (1) The opposite-signed vorticity layer in the well-developed tube wake will interact with the vorticity layer of the jet, particularly the roll-up of the leading vortex ring of the jet. Therefore, vortex ring formation may be disrupted by the pre-existing wake. (2) The presence of background flow will alter the axial velocity profile along the tube opening. This may affect the roll-up of the vorticity layer of the jet into discrete vortices. A focus of this paper will be the evolution of jet flow structure, especially the pinch-off process of the leading vortex, under the influence of the opposite-signed vorticity layer in the wake. Implications of our results for squid jet propulsion will also be discussed, focusing on the question why would a steadily swimming adult squid use a long jet instead of a series of short pulses of equivalent total mass flux. Krueger et al. [9] studied experimentally the vortex ring formation process in the presence of a simultaneously initiated background flow and found that the formation number was reduced by background flow. The differences between the present study and Krueger et al. [9] are: (1) no wake was present at jet initiation, and (2) no boundary layer of finite thickness was present outside the tube at jet initiation, in their study. In Krueger et al. [9], the background flow and the jet flow were started simultaneously and impulsively, which is, as they pointed out, most relevant for jet propulsion devices accelerated from rest or nearly from rest. 2 Numerical methodology The numerical methodology is described in dimensional form to facilitate comparison with previous experiments of Gharib et al. [7]. However, the results will be presented in non-dimensional form. The round-headed axisymmetric body (Fig. 1) has an inner diameter, D, of 2.54 cm and an outer diameter of cm. The total length of the body is cm. In order to study the effect of nozzle geometry, we consider two different nozzle shapes. One is a sharp-wedged opening with a tip angle of 20, similar to the experimental setup of Gharib et al. [7]. The other is a squared-off opening, i.e., with a tip angle of 90. An unsteady, laminar, incompressible, and Newtonian flow is assumed. Also, the flow is assumed to be axisymmetric with respect to the geometry, and flow conditions exclude swirl or rotation (i.e., the circumferential or swirl velocity is zero). Thus, the governing equations are the axisymmetric unsteady incompressible

4 106 H. Jiang, M. A. Grosenbaugh 20 cm r 0 U axisymmetry axis x -20 cm -40 cm zoom 200 cm piston velocity U p sharp-wedged opening piston velocity U p squared-off opening Fig. 1 Sketch of the simulation setup and computational domain. The round-headed axisymmetric body with a sharp-wedged or squared-off opening at the posterior end has an inner diameter, D, of 2.54 cm. The piston velocity, U p, is cm s 1.The constant background flow, U, models the swimming velocity of the axisymmetric body. For standard vortex ring formation, the background flow, U, is set to zero Navier-Stokes equations together with the continuity equation. Throughout the present numerical simulations, the fluid density, ρ,is kg m 3, and the fluid kinematic viscosity, ν, is m 2 s 1. The size of the computational domain is 240 cm in the axial, x, direction, and 20 cm in the radial, r, direction (Fig. 1). For the simulation of vortex ring formation without background flow, zero pressure boundary conditions are imposed on the outer boundaries. When simulating the formation process of vortex ring under the influence of background flow, a velocity inlet boundary condition is imposed on the left outer-boundary of the domain, which creates a constant background flow, U, around the body. Owing to the axisymmetry assumption, only a meridian plane (θ = constant) of the domain is discretized into a mesh and a symmetry condition is specified on the symmetry axis. An axisymmetric setup is used because two-dimensional flow features dominate the flow, and any three-dimensionality will have limited effects on the overall results. Assuming axisymmetry also greatly reduces computational costs. In the present work, the governing equations with the above-described computational domain and simulation setup are solved by a commercially available, finite-volume code, FLUENT TM (version 4.5). Specifically, the SIMPLEC method is used for pressure velocity coupling. The quadratic upwind interpolation (QUICK) scheme is used for spatial interpolation. Temporal discretization is a first-order implicit scheme. A deforming mesh model built into FLUENT TM is employed to simulate the flow due to piston motion inside the cylindrical body (i.e., the piston face is set as a moving wall boundary). The deforming mesh model allows for simulating flows in domains with geometry that changes with time. Mesh deformation is included by modifying the conservative equations for mass and momentum so that the convective fluxes are evaluated relative to the velocity of the control volume. FLUENT TM (version 4.5) uses a solution algorithm suggested by Demirdzic and Peric [6] for its deforming mesh model. In this model, a prescribed boundary and grid deformation is needed as part of the problem definition. We prescribe the domain discretization at each time step by reading a series of grid files (e.g., the starting, intermediate and ending grid descriptions) and allowing FLUENT TM to interpolate to find the grid positions at in-between times. For example, for the case of L m /D = with piston velocity U p = cm s 1, the input grid files consist of the grid descriptions at t = 0, t = 1.0sand t = 2.5 s representing the left-to-right motion of the piston (Fig. 2). The grid distribution around the body is not altered with time. Only the grid distribution inside the body is deformed to reflect the piston motion. No new control volumes are created as the piston moves and the basic grid topology remains the same throughout the calculation. At the initial time, the piston velocity is suddenly raised to a constant velocity U p, i.e., an impulsive velocity program is used. Then the piston moves at U p until it reaches the position of maximum stroke where the piston stops. After the piston stops, the simulation continues allowing the flow to evolve for specified time. In the present study, U p = cm s 1 for all cases. It is worth noting that if our purpose is only to study the effect of background flow on vortex ring formation process, then specifying an axial velocity profile of certain shape at an inlet would be enough. The method would be similar to what Rosenfeld et al. [15] and Zhao et al. [19] have done for different types of velocity profiles. However, an important purpose of our work in understanding squid jet propulsion is to calculate the jet thrust, the drag force acting on the body, and the mechanical power expended by the piston in pushing out the fluid. A curvilinear body-fitted coordinate system is used, so that the body shape is represented smoothly. In order to determine a suitable mesh size, a mesh sensitivity test has been performed by simulating the same case for five meshes with , , , and nodes in the axial and radial

5 Numerical simulation of vortex ring formation in the presence of background flow 107 (a) (b) (c) Fig. 2 Grid distribution around the round-headed axisymmetric body with a sharp-wedged opening at the posterior end at time a t = 0, b t = 1.0 sandc t = 2.5 s, for the case of L m /D = with piston velocity U p = cm s 1. The mesh has nodes in the axial and radial directions, respectively. Every fifth node is shown in each direction for clarity Γ 0 241x51 241x x x x201 t 0 Fig. 3 Mesh sensitivity test on the evolution of total circulation for the case of L m /D = 6.00, U p = cm s 1 with the squared-off nozzle-geometry. (See Sect. 3 for the definitions of the non-dimensional circulation, Ɣ0, and of the non-dimensional formation time, t0.) directions, respectively. The evolution of the total circulation for the five meshes is shown in Fig. 3.Itisshown that the difference among the results obtained for the finest three meshes is small. Therefore, the mesh is employed throughout this work. 3 Validation: Standard vortex ring formation In order to validate the numerical method, we simulate standard vortex ring formation without background flow using the sharp-wedged nozzle-geometry. Starting from a zero-background-flow initial condition, the flow behind the axisymmetric body is created exclusively by the motion of the piston inside the body. The flow field is computed for four piston strokes corresponding to L m /D = 2.10, 4.05, 6.00 and The Reynolds number, Re, isdefinedbyre = U p D/ν.Time,t, is scaled using t 0 = U p D t (1)

6 108 H. Jiang, M. A. Grosenbaugh (a) (b) (c) (d) Fig. 4 Standard vortex ring formation with the sharp-wedged nozzle-geometry for Re = 3,871 with different maximum strokes of the piston. Axisymmetric vorticity field for a L m /D = 2.10, b L m /D = 4.05, c L m /D = 6.00, and d L m /D = at the non-dimensional formation time t0 = Non-dimensional contour levels: min = with increment = where t 0 is termed the formation time after Gharib et al. [7]. Vorticity is non-dimensionalized by U p/d, and circulation, Ɣ, is non-dimensionalized accordingly using Ɣ 0 = Ɣ U p D (2) Figure 4 presents the instantaneous vorticity field for the four different piston strokes at the same formation time (i.e., t 0 = 24.0). ForL m/d = 2.10, only a single vortex ring is created (Fig. 4a). For L m /D = 4.05, only a single vortex ring is created but with a stronger intensity than for L m /D = 2.10 (Fig. 4b). Also, no vorticity is left behind the vortex ring for L m /D = 2.10, and a nearly negligible vorticity field is left behind the vortex ring for L m /D = For both cases, the vorticity created at the opening during the ejection is (almost entirely) entrained into the forming vortex ring. When the stroke ratio is increased to L m /D = 6.00, a prominent vorticity field develops behind the downstream traveling vortex ring (Fig. 4c). This indicates that not all of the vorticity created at the cylinder opening during the ejection is entrained into the leading vortex ring and that the leading vortex ring pinches off at an earlier time from the vorticity tail behind it. For L m /D = 15.00, a strong trailing jet is present behind the leading vortex ring (Fig. 4d). One can visually see (and confirm through calculations of the circulation) that the leading vortex rings among the three cases of L m /D = 4.05, 6.00, and are approximately the same size, suggesting that only a certain fraction of the vorticity created at the opening during the ejection can be entrained to form the leading vortex ring for large stroke ratios. All of these observed features of the vortex ring formation process from the present numerical simulations are in good agreement with those measured by Gharib et al. [7] and simulated by Rosenfeld et al. [15]. The total discharged circulation and the leading vortex ring circulation can be calculated from the vorticity data. The non-dimensionalized vorticity above the contour level of (non-dimensional) is integrated over the whole area outside the opening of the body to form the total discharged circulation. To calculate the leading vortex ring circulation, the non-dimensional vorticity above the contour level of is integrated over the area that the vortex ring occupies. Such calculations have been done for L m /D = 2.10, 4.05, 6.00 where a well-separated leading vortex ring forms (Fig. 4a, b, and c). In Fig. 5, the non-dimensional total and leading vortex ring circulations are plotted as functions of the formation time for L m /D = The results are consistent with those for the nozzle case studied by Rosenfeld et al. [15](seeFig.6 of their paper). Based on the maximum value of the vortex ring circulation, the formation number is estimated to be Vortex ring formation under the influence of background flow 4.1 Initial flow condition The vorticity layer originating from the finite thickness boundary layer on the external surface of the nozzle due to the background flow is of great relevance to our problem. Thus, the external quasi-steady flow field

7 Numerical simulation of vortex ring formation in the presence of background flow 109 Γ 0 total circulation vortex ring circulation Fig. 5 Standard vortex ring formation for Re = 3,871 with different maximum strokes of the piston. Non-dimensional total and leading vortex ring circulations, Ɣ 0, as functions of the formation time, t 0,forL m/d = 6.00 =6 =24 =60 =300 =600 Fig. 6 Evolution of the streamlines around the sharp-wedged opening, which shows the build-up of the quasi-steady background flow around the body. The background flow U is equal to 0.50U p. In order to be consistent with results from the cases with piston motion, the streamfunction (m 3 s 1 ) is scaled by U p D 2 /8 instead of UD 2 /8. The contour levels are zero for the streamline in black, positive (min = with increment = ) for the streamlines in red, and negative (max = with decrement = ) for the streamlines in blue must be fully developed before the impulsive piston motion is started. The build-up of this external quasisteady state is an asymptotic process driven by viscous diffusion; hence, a long calculation time is needed. Six initial flow conditions have been calculated, namely U = 0.20U p,0.50u p and 0.75U p, respectively, with two different nozzle geometries (i.e., sharp-wedged and squared-off). The process of building the quasi-steady flow field around the body with the sharp-wedged nozzle-geometry is illustrated in Fig. 6 for U = 0.50U p. It takes a relatively long time to develop a quasi-steady flow field. The flow field corresponding to the time when the calculation is terminated (measured in terms of the non-dimensional formation time t 0 = ) is taken as the initial flow condition (i.e., the time or the formation time is reset to zero) for later simulations when the piston motion is included. Physically, the wake behind the axisymmetric cylinder (before the piston starts moving) will not be axisymmetric at the background flow velocities used in this study. There could be turbulence, vortex shedding,

8 110 H. Jiang, M. A. Grosenbaugh Fig. 7 Evolution of the axisymmetric vorticity field for the case of L m /D = 15.00, Re = 3,871, and U/U p = 0.00 with the sharp-wedged nozzle-geometry. The time t is scaled using Eq. (1), and the vorticity is scaled by U p /D. Note that positive vorticity contours are in red with non-dimensional contour levels of min = with increment = and that negative vorticity contours are in blue with non-dimensional contour levels of max = with decrement = and even swirl in the mean flow. However, we assume that these three-dimensional effects are small and that the dominant flow processes, including the laminar boundary layer flow along the outside of the cylinder and the formation of the leading vortex ring and trailing jet, are axisymmetric processes. When the piston does start moving, the wake will be blown downstream away from the jet nozzle. Because of these reasons, we feel that neglecting three-dimensionality in our initial conditions will have little effect on the overall results. 4.2 Evolution of the vorticity field The evolution of the vorticity field for four different levels of background flow, namely U = 0, 0.20U p, 0.50U p and 0.75U p, is shown in Figs. 7 10, respectively. The four cases have the same L m /D ratio (=15.00), Reynolds number (=3,871) and nozzle geometry (i.e., sharp-wedged). For a zero-background-flow, a leading vortex ring is formed, though it leaves behind a noticeable trailing jet due to a large L m /D ratio (Fig. 7). We identify a leading vortex ring in cross-section by its closed vorticity contours and rotating flow (as shown later in Fig. 17) about a single central axis. Increasing the background flow velocity to U = 0.20U p causes a leading vortex ring to form, though the shape of the ring is slightly distorted due to the interaction with the weak, opposite-signed vorticity layer in the initial cylinder wake (Fig. 8). For U = 0.50U p, the vorticity layer from the initial cylinder wake is strong enough that its interaction with the opposite-signed jet vorticity layer produces significant effects (t0 = in Fig. 9). After the piston

9 Numerical simulation of vortex ring formation in the presence of background flow 111 Fig. 8 Evolution of the axisymmetric vorticity field for the case of L m /D = 15.00, Re = 3,871, and U/U p = 0.20 with the sharp-wedged nozzle-geometry. See Fig. 7 legend for additional explanations starts moving, the two vorticity layers roll up into two vortex rings of opposite signs: (1) the roll-up vortex ring from the vorticity layer of the cylinder wake (called the wake vortex ), and (2) the roll-up vortex ring from the jet vorticity layer (called the leading jet vortex ). Initially, the formation of the wake vortex occurs downstream of the formation of the leading jet vortex. At this background flow, the leading jet vortex appears to be stronger than the wake vortex. As the simulation proceeds, the wake vortex is pulled back toward the jet nozzle due to the strength of the leading jet vortex. The vorticity of the wake vortex is carried radially inward by the leading jet vortex. The two opposite-signed vortex rings cancel each other at their peripheries. The interaction between the two opposite-signed vortex rings greatly distorts the shape of the leading jet vortex. It appears that the deformed leading jet vortex never pinches off and even connects backwards with the trailing jet at t0 = (vorticity contours not shown). As a consequence, the leading jet vortex appears to eventually collapse (t0 = 19.8and24.0inFig.9). When the background flow velocity is increased to U = 0.75U p, the interaction between the oppositesigned vorticity layer from the initial cylinder wake with the jet vorticity layer appears more intense (t0 = in Fig. 10). The two vorticity layers roll up into the wake vortex and the leading jet vortex, respectively. As with the previous case, the wake vortex initially forms downstream of the leading jet vortex. However, the wake vortex now seems to be stronger than the leading jet vortex. The wake vortex pulls the vorticity of the leading jet vortex forward away from the nozzle and radially outward. During this process, the two opposite-signed vortex rings cancel each other at their peripheries. As a result, most of the vorticity of the leading jet vortex is destroyed. What remains at large t0 is a band of deformed vorticity rather than a classic roll-up vortex ring (t0 = 19.8 and 24.0 in Fig. 10) as previously defined. As we show later in Fig. 17, the band of deformed vorticity is associated with a region of high shear, as opposed to a region of rotating flow about a

10 112 H. Jiang, M. A. Grosenbaugh Fig. 9 Evolution of the axisymmetric vorticity field for the case of L m /D = 15.00, Re = 3871, and U/U p = 0.50 with the sharp-wedged nozzle-geometry. See Fig. 7 legend for additional explanations single central axis (the small regions of rotating flow upstream of the region of high shear seen in Fig. 17 are associated with the secondary vortices of the trailing wake). Similar simulations using the sharp-wedged nozzle-geometry have been done for a smaller piston stroke ratio (i.e., L m /D = 6.00). The results corresponding to t0 = 24.0 are plotted in Fig. 11. A prominent leading vortex ring is formed under a low-level background flow (e.g., U = 0.20U p ), though the shape of the ring is slightly distorted (Fig. 11b). Clear pinch-off of the vortex ring from the trailing jet is observed for both cases of U = 0 and 0.20U p. Increasing the background velocity to U = 0.50U p causes the leading vortex ring to be greatly deformed because of its interaction with the wake vortex of opposite sign. The deformed leading vortex ring pinches off and never reconnects backwards with the trailing jet (compared to the case with L m /D = 15.00). This is because of earlier stopping of the jet (at t0 = 6.00 instead of t 0 = 15.00). At a high-level of background flow (e.g., U = 0.75U p ), the leading jet vortex initially rolls up from the jet vorticity layer. Simultaneously, the wake vortex of opposite sign rolls up from the vorticity layer of the initial cylinder wake. Most of the vorticity of the leading jet vortex is destroyed through its interaction with the stronger wake vortex. What remains at t0 = 24.0 (Fig.11d) is a band of vorticity that is similar in structure and flow characteristics to that described for the case of U = 0.75U p with L m /D = (i.e., the band of vorticity has a saddle point appears to be a shear layer as opposed to a classic roll-up vortex ring). The band of leading vorticity is isolated in this case (L m /D = 6.00) only because the vorticity of the trailing jet wake has been destroyed by the opposite-signed vorticity produced by the background flow. This process is different from the classic pinch-off process, where the leading vortex ring pulls ahead of the trailing jet once

11 Numerical simulation of vortex ring formation in the presence of background flow 113 Fig. 10 Evolution of the axisymmetric vorticity field for the case of L m /D = 15.00, Re = 3,871, and U/U p = 0.75 with the sharp-wedged nozzle-geometry. See Fig. 7 legend for additional explanations its circulation reaches a given steady value. Because of the interaction with the wake vortex, the circulation associated with the band of leading vorticity in Fig. 11d never reaches a steady-state. Similar simulations have been performed using the squared-off nozzle-geometry. The results repeat the observed features for the cases of the sharp-wedged nozzle-geometry (figures not shown). 4.3 Total and leading vortex ring circulations The total discharged circulation can be calculated for each case with background flow from the vorticity data at sampled times. The leading vortex ring circulation can also be calculated for those cases in which the leading vortex ring can pinches off from the trailing jet. The method for calculating the circulations is the same as that for the standard vortex ring formation described in Sect. 3. The results for U = 0.20U p with piston stroke ratios of L m /D = 6.00 are shown in Fig. 12, andthe results for U = 0.50U p with L m /D = 6.00 are shown in Fig. 13. The initial ramp of the total circulation

12 114 H. Jiang, M. A. Grosenbaugh Fig. 11 Vortex ring formation with the sharp-wedged nozzle-geometry for L m /D = 6.00, Re = 3,871 under the influence of different background flows. Axisymmetric vorticity field for a U/U p = 0, b U/U p = 0.20, c U/U p = 0.50, d U/U p = 0.75 at the non-dimensional formation time t0 = See Fig. 7 legend for additional explanations Γ 0 total circulation vortex ring circulation Fig. 12 Vortex ring formation with the sharp-wedged nozzle-geometry for Re = 3,871 under the influence of a low-level background flow U = 0.20U p. Non-dimensional total and leading vortex ring circulations Ɣ 0 as functions of the formation time t 0 for L m /D = Note that the time t is scaled using Eq. (1) and that the circulation Ɣ is scaled using Eq. (2) curve in each figure corresponds to when the piston is moving. The discontinuity is due to piston stopping. For U = 0.20U p with L m /D = 6.00, the formation number is estimated to be 4.1 based on the maximum value of the vortex ring circulation (Fig. 12). This formation number is slightly higher than that for U = 0 with L m /D = 6.00 (Fig. 5). For U = 0.50U p with L m /D = 6.00, the formation number is estimated to be 2.3 based on the maximum value of the leading vortex ring circulation (Fig. 13). We did not calculate a formation number for U = 0.75U p with L m /D = 6.00 because we could not identify a classic roll-up vortex ring with steady (or nearly steady) total circulation. Krueger et al. [9] derived a scaling for vortex ring formation in the presence of background flow, in which the time t is scaled such that t = (U p + U) t (3) D and the circulation Ɣ is scaled such that Ɣ = Ɣ (U p U)D (4)

13 Numerical simulation of vortex ring formation in the presence of background flow 115 Γ 0 total circulation vortex ring circulation Fig. 13 Vortex ring formation with the sharp-wedged nozzle-geometry for Re = 3,871 under the influence of a moderate-level background flow U = 0.50U p. Non-dimensional total and leading vortex ring circulations Ɣ0 as functions of the formation time t0 for L m/d = Note that the time t is scaled using Eq. (1) and that the circulation Ɣ is scaled using Eq. (2) In this paper, t and Ɣ are called respectively the modified formation time and the modified nondimensional circulation. Applied to our simulations, this new scaling successfully collapses the total-circulation data during the initial ramp-up stage (before the piston stops) for the lowest three values of background flow (i.e., U = 0, 0.20U p and 0.50U p ) (Fig. 14). The total-circulation data for the case of U = 0.75U p fails to collapse onto the other three curves, indicating that the solution is not self-similar (Fig. 14). This may be due to the fact that the initially formed roll-up vortex of the jet is destroyed by its interaction with the strong, opposite-signed vorticity layer of the initial cylinder wake. The residual of the jet vorticity layer is a vorticity band (instead of a roll-up vortex), which translates downstream at a speed that is faster than a roll-up vortex (as seen by comparing Fig. 11d with Fig. 11a, b, c). In Fig. 15, the vortex ring circulation and the total circulation non-dimensionalized according to (4) are plotted as functions of the modified formation time for L m /D = 6.00 and U = 0.50U p. From this figure, the modified formation number is estimated to be 3.4. In our simulations, the weak, opposite-signed vorticity layer associated with the lowest level of background flow (U = 0.20U p ) seems to promote the roll-up of the leading vortex ring of the jet and therefore increase the formation number. Using the new scaling of (3) and (4), the modified formation number is estimated to be 4.9 for the case of L m /D = 6.00 and U = 0.20U p.the reason may be that the translation of the leading vortex ring of the jet is retarded by its interaction with the weak, opposite-signed vorticity layer of the initial cylinder wake. Thus, more vorticity from the trailing jet can be entrained into the forming leading vortex ring. On the other hand, the weak, opposite-signed vorticity layer has limited ability to cancel the vorticity of the leading vortex ring of the jet at the periphery. 4.4 Explanation for the observed squid jet flow pattern The time evolution of the axial velocity profile at the jet opening is plotted in Fig. 16 for four cases with different values of the background flow (i.e., U/U p = 0, 0.20, 0.50 and 0.75) for L m /D = with the sharp-wedged nozzle-geometry. At the start of the piston motion, each case shows a small peak in the velocity profile near the inside wall of the cylinder that is greater than the piston velocity. At the same time, the velocity at the centerline is smaller than the piston velocity. After a short time, this peak disappears leading to a centerline velocity that is larger than the piston velocity. The time it takes for the peak to disappear varies for the four different values of the background flow the larger the background flow, the longer the time (Fig. 16). For all four cases, a parabolic-like velocity profile eventually develops at the jet opening as the piston is continuously pushed forward (t0 = 15.0inFig.16). (The boundary layer becomes thick because the final position of piston is very close to the jet opening.) The overall characteristics of the axial velocity profile at the jet openingare similar to thoseobserved by Didden [5] in an experimental study and by several authors in numerical studies (e.g., Heeg and Riley [8]; Rosenfeld et al. [15]) for situations without background flow. With the background flow included, the merged axial velocity profile at and close to the jet opening (Fig. 16c, d) forms a strong shear layer (a wake-like flow). This shear layer is present even if the jet velocity

14 116 H. Jiang, M. A. Grosenbaugh Γ t U/U p =0 U/U p =0.50 U/U p =0.20 U/U p =0.75 Fig. 14 Non-dimensional total circulation as a function of the modified formation time for L m /D = 6.00 with four different values of the background flow using the sharp-wedged nozzle-geometry. The time t is scaled using Eq. (3), i.e., the modified formation time, and the circulation Ɣ is scaled using Eq. (4) Γ t total circulation vortex ring circulation Fig. 15 Vortex ring formation with the sharp-wedged nozzle-geometry for Re = 3,871 under the influence of a moderate-level background flow U = 0.50U p. Non-dimensional total and leading vortex ring circulations Ɣ as functions of the modified formation time t for L m /D = Note that here the time t is scaled using Eq. (3) and that the circulation Ɣ is scaled using Eq. (4) and the background flow velocity are equal. Specifically, the existence of the thin wall of the jet opening causes the boundary layer to separate from the outer wall of the cylinder for large background flow. The separation creates a recirculation region near the exit of the jet opening and stronger shear than would otherwise be expected, but only near the wall (r/d = 0.5) and only close to the exit. This shear layer becomes stronger when the background flow is increased. Further downstream this shear dissipates and the flow is again dominated by the outer flow formed by the background flow, U, and the jet centerline velocity, U centerline.in contrast, the outer flow has weaker shear at increased background flow. The result is that for low background flow, the classical shear layer instability (i.e., the trailing-jet instability driven by U U centerline ) develops behind the vortex ring at large time (Fig. 7). For high background flow, the flow close to the jet exit is dominated by shear from the separation bubble and an instability close to the jet opening develops, but it begins to dissipate downstream as the classical shear layer instability reemerges (Fig. 10). Figure 17 shows the instantaneous velocity vector fields at the formation time t0 = 12.0 for the four cases, in a frame of reference moving at the convective velocity of the vortical structures in the outer flow, namely (U + U centerline )/2. For U/U p = 0, almost no vortical flow structure formed behind the leading vortex ring at this instantaneous time (Fig. 17a); significant vortical structures will develop at later times (Fig. 7) as expected for the strong shear developed with no background flow. With an increase of the background flow velocity to U/U p = 0.50 or 0.75, a series of vortical flow structures emerges starting from the jet opening and dissipates downstream (Fig. 17c, d).

15 Numerical simulation of vortex ring formation in the presence of background flow 117 (a) (b) r/d r/d =0 =0.60 =1.80 u x (r)/u p (c) u x (r)/u p (d) =6.00 =13.2 =15.0 =24.0 r/d r/d u x (r)/u p u x (r)/u p Fig. 16 Time evolutions of the axial velocity profile at the body opening under the condition of L m /D = with four different values of the background flow. The sharp-wedged nozzle-geometry is used. The Reynolds number Re = 3,871. a U/U p = 0, b U/U p = 0.20, c U/U p = 0.50, and d U/U p = Note that the formation time t0 is defined by Eq. (1) As pointed out previously, a steadily swimming adult squid is known to employ a long, continuous jet for individual jet events. For steadily swimming adult squid, the observed wavy flow structure as well as the chain formation of vorticity blobs (Anderson and Grosenbaugh [3]) could be explained by such a scenario rather than being produced by a series of short puffs with short time intervals in between to form a series of vortex rings. Here, we suggest that the flow pattern of a steadily swimming adult squid is formed due to the instability in the wake-like flow close to the jet-orifice. In contrast to the classical shear layer instability, this instability seems to be enhanced by increasing the U/U p ratio. Squid jetting involves a periodic (at a rate of 0.6 to 1.3 Hz) opening and closing of the jet orifice to produce a long continuous jet (Anderson and DeMont [1]; Anderson and Grosenbaugh [3]). The size of the jet-orifice is the smallest at the beginning of a squid jet event. This might explain why the observed squid jet flow does not have a prominent leading vortex ring at moderate background flow velocities, in contrast to our simulation. On the other hand, regularly distributed vortical flow structures, such as what is seen in Fig. 3B of Anderson and Grosenbaugh [3], will not be formed

16 118 H. Jiang, M. A. Grosenbaugh (a) U/U p =0 U p (b) U/U p =0.20 U p (c) U/U p =0.50 U p (d) U/U p =0.75 U p Fig. 17 Instantaneous velocity vector fields at the formation time t 0 = 12.0 under the condition of L m/d = with four different values of the background flow. Here the velocity vectors are seen in a frame of reference moving at the convective velocity of the vortical structures in the outer flow, namely (U + U centerline )/2. The sharp-wedged nozzle-geometry is used. The vectors shown are those interpolated onto a regular grid. The Reynolds number Re = 3,871 even if the squid were to use a series of short puffs with short time intervals in between. To demonstrate this, we have done several controlled numerical experiments involving three different piston velocity programs described in the following. (1) Continuous jet: piston velocity = { Up 0s t 2.5s (i.e. 0 formation time t ) 0 2.5s < t 4.0s (i.e < formation time t )

17 Numerical simulation of vortex ring formation in the presence of background flow 119 (2) Three pulses (Program 1): U p 0s t 0.8s (i.e. 0 formation time t0 4.8) 0 0.8s < t < 0.9s (i.e. 4.8 < formation time t0 < 5.4) U p 0.9s t 1.75s (i.e. 5.4 formation time t0 piston velocity = 10.5) s < t < 1.85s (i.e < formation time t0 < 11.1) U p 1.85s t 2.7s (i.e formation time t ) 0 2.7s < t 4.0s (i.e < formation time t0 24.0) (3) Three pulses (Program 2): U p 0s t 0.8s (i.e. 0 formation time t0 4.8) 0 0.8s < t < 1.3s (i.e. 4.8 < formation time t0 < 7.8) U p 1.3s t 2.15s (i.e. 7.8 formation time t0 piston velocity = 12.9) s < t < 2.65s (i.e < formation time t0 < 15.9) U p 2.65s t 3.5s (i.e formation time t ) 0 3.5s < t 4.0s (i.e < formation time t0 24.0) These three piston velocity programs have the same total piston stroke length (i.e., L m /D = 15.00) and, thus, the same total mass flux. However, the latter two programs each consists of three short puffs with short time intervals in between. The difference between Program 1 and Program 2 is the time delay between puffs, 0.1 s versus 0.5 s. Figure 18 shows the evolution of the vorticity field for U/U p = 0.50, Re = 3,871 with the sharp-wedged nozzle-geometry using the Three pulses (Program 2) piston-velocity-program. The vortices originating consecutively from the three short puffs are not regularly distributed, which appears different from the regular wavy structure that is behind the leading vortex ring in the case of using the Continuous jet (t0 = 4.8, 6.0, 9.0, 12.0, and 15.0 in Fig. 9, see also Fig. 17c). Vortex pairing (Panton [14]) and interaction with the opposite-signed vorticity layer in the initial cylinder wake may account for the formation of the complex vortical flow field seen in Fig. 18. A large number of successive pulses may or may not lead to a steady-state pattern of regularly spaced vortices. However, for squid, which have limited fluid capacity and low pulsing frequency, the results of simulations with a limited number of puffs appear to be more relevant. Here, our simulations of three short puffs with equal short time intervals in between were not able to achieve the regular spacing of vortical structures that is observed in squid swimming (Anderson and Grosenbaugh [3]). 5 Jet thrust The present simulation study is able to calculate the instantaneous axial velocity profile, u x (r, t), andthe instantaneous pressure profile, p(r, t), at the jet opening. With these data and by applying the momentum theorem, the jet thrust can be calculated as: D/2 D/2 Thrust(t) = 2πρ u 2 x (r, t)rdr + 2π [p(r, t) p ]rdr (5) 0 where p = 0 is the ambient pressure at the centerline (i.e., x ). The first term is the momentum flux term; the second term is the nozzle pressure term. Generally, the contribution from the nozzle pressure term to the jet thrust is much less than from the momentum flux term. We are also interested in two other forces acting on the body: the drag acting on the external surface of the body and the normal force on the piston (as sketched in the upper panel of Fig. 19). The drag is calculated as the axial component of the surface integral of the shear stress and pressure over the external surface of the body. The piston force, F piston, is calculated as the surface integral of the pressure over the piston outer-surface. The calculated forces are normalized by π 4 ρu 2 p D2,and the normalized forces are denoted by. These forces are calculated for all the cases considered. Figure 19 presents an example of the calculated forces for the continuous jet case of L m /D = 15.00, Re = 3,871, and U/U p = 0.50 with the sharp-wedged nozzle-geometry. Abrupt changes are observed to occur in the force history with regard to the abrupt changes in the piston velocity program and the final position of the piston inside the cylinder. 0

18 120 H. Jiang, M. A. Grosenbaugh Fig. 18 Evolution of the axisymmetric vorticity field for the case of L m /D = [Three pulses (Program 2)], Re = 3,871, and U/U p = 0.50 with the sharp-wedged nozzle-geometry. See Fig. 7 legend for additional explanations Based on the force history calculated for each case, a net-impulse to energy ratio is calculated for a time period of interest as: net impulse [Thrust(t) Drag(t)]dt = (6) energy Fpiston (t)u piston (t)dt where U piston (t) is the piston velocity program. (Note that the body is not self-propelled but tethered so that [Thrust(t) Drag(t)]dt 0.) The calculated ratio is then normalized by 1/Up. Figure 20 shows the nondimensional net-impulse to energy ratio as a function of L m /D for four different values of the background flow (i.e., U/U p = 0, 0.20, 0.50 and 0.75) with the sharp-wedged nozzle-geometry. The time period of interest is 0 t0 L m/d. The ratio increases with increasing L m /D. For constant L m /D, the ratio is inversely proportional to the background flow. We have also calculated the ratios for cases of L m /D = with the three different piston velocity programs combined with the two different nozzle-geometries, for constant background flow of U/U p = 0.50 and constant Re = 3,871. Here, the time period of interest is 0 t The results are listed in Table 1. The ratio for the continuous jet piston-velocity-program