Near-body flow dynamics in swimming fish

Several theories have been developed to explain fish swimming performance. Wu (1961, 1971a,b,c) and Lighthill (1960, 1975) have investigated the propulsion characteristics of flexible two-dimensional and three-dimensional plates using linearized potential flow theory. While these seminal works provide insight into the basic swimming propulsive mechanisms, the details of the three-dimensional flow and the dynamics of the shed vorticity are still not well understood. Several comprehensive works on fish swimming thoroughly analyze the kinematics and the dynamics of straight-line and unsteady swimming motions for a variety of aquatic species (Videler, 1993; Blake, 1983) utilizing these linear theories to estimate the propulsive dynamics. However, such methods overpredict the thrust on more realistic fish-like forms (Videler, 1981) because basic assumptions about the linear slender body theory are violated.

The study of unsteady lifting surface motion for both swimming and flying has provided insight into the details of the dynamics of fish propulsion methods. The aerodynamics of hovering insect flight (Osborne, 1951; Ellington, 1984; Ellington et al., 1996) illustrates the importance of unsteady flow mechanisms in achieving a large loading on the wing, including a delay in stall. The unsteady motions of hydrofoils have been studied analytically and numerically by several investigators (von Kármán and Burgess, 1935; Wu, 1961, 1971a,b,c; Lighthill, 1960, 1975; Chopra, 1976; Chopra and Kambe, 1977; Lan, 1979), but these methods employ linearized body boundary conditions as well as assumed wake shapes.

Most fish move with amplitudes and frequencies that exceed the limit of linear lifting surface theories. Videler and Hess (1984) showed that two species of fish exhibited high propulsive efficiency, close to the actuator disk limit, with large lateral motion amplitude. Triantafyllou et al. (1993) showed that a variety of fish and cetaceans swim with a frequency and amplitude of tail motion that are within a narrow range of Strouhal numbers, minimizing energy lost in the wake for a given momentum and increasing efficiency. This Strouhal number range corresponds to the regime of maximum stability of the vortex wake thrust jet.

Several investigators have studied the mechanisms of vorticity control achieved by unsteady motion of a body in a fluid (Taneda and Tomonari, 1974; ffowcs-Williams and Zhao, 1989; Tokomaru and Dimotakis, 1991). Gopalkrishnan et al. (1994), Streitlien et al. (1996) and Anderson et al. (1998) utilized experimental and computational methods to demonstrate active control of a shear flow with an oscillating foil, revealing energy recovery flow control mechanisms that can increase or decrease the efficiency. Direct experimental dynamic measurements of the energetic make-up and efficiency of live swimming fish are difficult to obtain, however, and may not be characteristic of natural behavior. As a result, Barrett (1996) constructed a robotic underwater flexible hull vehicle, in the shape of a tuna with a lunate tail, on which direct dynamic measurements and energetic analyses could be performed. Over small ranges of various swimming variables, the robot could achieve extremely high propulsive efficiencies and lower drag than that of the rigid body hull through mechanisms of boundary layer relaminarization and vorticity control by the tail fin (Barrett et al., 1999).

Several researchers have studied, through visualization, the wake of a naturally swimming fish. Techniques that have been used include dye visualization (Rosen, 1959; Aleyev, 1977) and visualization using a stratified layer (Rosen, 1959; McCutchen, 1976). Neither method has elucidated all the details of the flow near the body and in the wake. Stamhuis and Videler (1995) utilized experimental particle image velocimetry to capture the flow dynamics around several live swimming organisms and to analyze the energetic make-up of the wake. Anderson (1996) used experimental digital particle image velocimetry (DPIV) to visualize the wake behind a swimming giant danio (Danio malabaricus) and identified the active manipulation of shed wake vorticity to create a reverse Kármán vortex street. Müller et al. (1997) analyzed the wake of a swimming mullet (Chelon labrosus Risso) using similar DPIV techniques and concluded that the manipulation of the wake structure resulted in high propulsive efficiencies.

In the present study, we present quantitative, multipoint measurements of the flow field around a small naturally swimming fish using DPIV. The results include flow measurements for straight-line swimming and rapid maneuvering of a giant danio (Danio malabaricus). We also develop a numerical method to simulate the hydrodynamics of an actively swimming fish-like body. The characteristics of real fish swimming at large Reynolds numbers are approximated by assuming that viscous effects are confined to a thin boundary layer and wake region. The formulation allows for the satisfaction of the exact body boundary conditions and for the nonlinear evolution of the shed wakes. This inviscid approach is computationally an order of magnitude more rapid than the fully viscous numerical methods that are essential at lower Reynolds numbers, such as those developed by Liu et al. (1997) to study tadpole locomotion. Our numerical method is then applied to analyze the wake dynamics and the near-body hydrodynamics of the motions of the giant danio, which were captured using DPIV, including both continuous straight-line swimming and a 60 ° turning maneuver. We present a sample of the experimental and numerical results, which demonstrate that the controlled actuation of the caudal fin and release of bound vorticity contributes to the efficient production of a reverse von Kármán street thrust jet.

We employ digital particle image velocimetry (DPIV), first introduced by Willert and Gharib (1991), for flow measurement and visualization. The DPIV algorithm is applied to two images of the flow separated by small time dt, and small subsets of image data (interrogation windows) are compared at the same location in each image by computing the spatial cross correlation. When the particle images match well, the cross-correlation function is sharply peaked, and the location of the peak marks the displacement of the particle image pattern. The algorithm is then applied throughout the images to obtain the entire flow field.

For the DPIV algorithm to work well, there must be adequate particle seeding, randomly distributed over the entire image plane. Willert and Gharib (1991) systematically studied the seeding requirement and found that 10–20 particles per window are sufficient to ensure good correlations. For a complete discussion of general DPIV capabilities and limitations, see Willert and Gharib (1991). Unique problems arise in using DPIV to visualize flows around moving, deformable bodies. As the fish swims and boundaries move, interrogation windows cross boundaries and contain spurious image data. This biases the cross-correlation peak towards the displacement of the boundary, resulting in spurious displacement data for the fluid near the boundary. A complete discussion of the near-body DPIV techniques we employ, including their capabilities, limitations and validation, can be found in Anderson (1996).

The windowing process in DPIV filters out wavenumbers (spatial frequencies) larger than the inverse interrogation window size. This is a problem when the flow field of interest has fine-scale motions and flow reversals, such as in a vortex core. To resolve such fine-scale features correctly, the image view should be enlarged in the region of interest by moving the camera or by reducing the window size. Willert and Gharib (1991) showed that there is approximately 70 % attenuation of 32 pixel features when the data are processed with 16 pixel ×?16 pixel interrogation windows, and one should always loosely interpret displacement data for scales less than twice the window size.

We calibrated our hardware and processing techniques in the same manner as Willert and Gharib (1991) using an artificially generated test image displaced with a rotary table instead of a linear table. The algorithm is very robust up to the Nyquist limit, despite curved particle trajectories. Our relative displacement error is generally less than 5 %, and usually less than 2 % for moderate (>1 pixel) displacement.

Fig. 1 summarizes the experimental apparatus for DPIV implementation in these experiments. The measurement volume, a 51 cm × 25 cm × 28 cm glass tank, was seeded with small (20–40 μm diameter), neutrally buoyant fluorescent polymer spheres. A planar slice of the flow approximately 3 mm thick was illuminated with a 6 W argon-ion laser connected to a fiberoptic cable and light sheet optics (Dantec).

The particle flow was imaged using a high-resolution (768 pixels × 480 pixels) black-and-white CCD video camera (Texas Instruments, Multicam MC-1134P) and recorded to video disc using a Sony CRVdisc videodisc recorder (LVR-5000A). Images were acquired in real time at the camera frame rate of 30 Hz in dual-field mode, allowing full vertical resolution with no interlace. Asynchronous shuttering of the laser light with a timing box (General Pixels) synchronized to the video signal allows for higher effective frame rates. Although image pairs are collected at 15 Hz, asynchronous shuttering allows the difference between images in a pair to be very small (approximately 5 ms).

As illustrated in Fig. 1, the fish was confined within a layer of fluid approximately 3 cm thick, between the free surface and a highly permeable screen consisting of a steel wire grid (2.5 cm openings) and a fine nylon mesh (approximately 3 mm openings). The clearance between the fish and the screen and free surface was estimated to be 3 mm. The screen was located approximately 25 cm from the tank bottom. The screen confined the fish yet allowed fluid to pass through it, and the plane in which the fish was allowed to swim was sufficiently wide to eliminate any wall effects. The flow in the horizontal symmetry plane of the fish was revealed by orienting the laser light sheet at the mid-depth of the confinement layer. In general, the light sheet illuminated the fish at the caudal fork; however, in some instances, slight bending of the caudal fin caused the tips of the tail to be illuminated. The free surface distortions caused by typical swimming and maneuvering motions were observed visually to be negligible. This suggests that there was minimal interaction between the wake vorticity and the free surface.

We chose to experiment with a fast-swimming tropical species, Danio malabaricus (Jerdon), the giant danio, whose morphology is very similar to that of the pearl danio studied by Rosen (1959). The dominant fins are the caudal fin, the flexible dorsal fin and the stiffer anal fin, which acts like a keel. Anterior to the anal fin, the ventral fins flare out during maneuvering but participate little in straight-line swimming. Similarly, the pectoral fins are used in maneuvering and are generally kept flush to the body during straight-line motion.

Several fish were used with body lengths between 5 and 10 cm. The maximum span of the caudal fin was approximately 23 % of the body length. Whenever possible, data will be presented normalized with respect to fish body length L. We found the giant danio well suited to our experiments since they are steady, fast swimmers and hardy with respect to handling. The laser light sheet was oriented horizontally in the midplane so that it hit the fish near the vertical plane of symmetry. The fish were allowed to swim freely in the confinement layer, during which time the video was recorded to laser disc. The fish showed no adverse reaction to either the particles or the laser light, and the kinematic behavior of the fish was identical whether the laser was on or off.

The DPIV images obtained of the fish swimming in a straight line or undergoing maneuvering motions are processed to produce velocity field images. Each arrow in a velocity field plot represents one cross-correlation calculation, and the length of each arrow indicates the relative velocity magnitude. The images obtained were processed using 32 pixel × 32 pixel interrogation windows, shifted in eight-pixel steps, which corresponds to oversampling the available image data four times. This interrogation window dimension is sufficiently small compared with the length of the fish (0.076L) such that flow scales of size 0.15L and above can be measured accurately. The velocity data are then used to compute the component of the vorticity perpendicular to the DPIV plane, so that large-scale vorticity components along the body and in the wake may be observed and quantified.

We develop a computational tool for investigating the swimming characteristics of a three-dimensional flexible body with arbitrary motions and geometry. We consider as a canonical problem the abrupt starting from rest to a constant velocity U of a streamlined flexible body geometry undergoing prescribed undulations about its mean line, with arbitrary distribution of sharp-trailing-edged fins. Thin shear layer wakes are shed continuously from the sharp trailing edges of the body as time proceeds. With the exception of the wake, the fluid is assumed to be inviscid and irrotational, as well as incompressible, allowing for the existence of a velocity potential We define two coordinate systems, an inertial global coordinate system O,X,Y,Z, fixed in space in the fluid, and a local coordinate system o,x,y,z, instantaneously fixed in the flexible body and orthonormal to the stretched-straight mean line and body section plane. The mean periodic undulations of the body are described with reference to o,x,y,z, and the global translational and rotational motions of the body are described with respect to O,X,Y,Z. Fig. 2 details the coordinate systems used in the problem. All time and length scales are chosen to be non-dimensional with respect to the body length L=1.

We consider the computational domain to be enclosed by three surfaces, the body surface S_b, an infinitesimally thin wake sheet S_w, and a far-field boundary at infinity S_∞. Laplace’s equation for the velocity potential governs the conservation of mass of the fluid, and the unsteady Bernoulli equation describes the pressure everywhere in the field. The velocity potential can be further described by the linear superposition of a body perturbation velocity potential and a wake perturbation velocity potential each satisfying the Laplace field equation. Although the strength of the wake being shed at the body trailing edges at any time is unknown, the strengths of all previously shed wake surfaces are known, with the exception of the initial condition of the impulsive start when no previously shed wakes are present. Thus, the wake perturbation velocity potential is known, and we formulate the boundary value problem for the body perturbation velocity potential

The unsteady wake strength is a function of space and time, The pressure across the wake shear layer is continuous, and the wake surface is a material surface and can thus be represented by a smoothly varying strength dipole sheet. From Kelvin’s theorem, as the foil’s circulation changes, so does the strength of the shed wake dipole. At any time, the strength of the entire previously shed wake is known, and thus the strength of is known, except for the recently shed shear layer which smoothly separates from predetermined trailing edges. The unknown strength of the newest portion of the wake sheet is addressed through the Kutta condition as described by equation 2. Thus, the body perturbation velocity potential and the change in circulation of the various wake-shedding lifting surfaces of the body can be written in terms of surface integrals over the body surface S_b and the wake sheet surface S_w.

Representation of the time-dependent continuous potential distributions over the body and wake surfaces is developed in discrete form. The body and the wake are each divided into quadrilateral panels. The source-dipole distribution representing the body is approximated as piecewise constant over each body panel; similarly, the dipole distribution representing the thin shear layer wake is approximated as piecewise constant over each wake panel. The collocation point for each panel is the geometric center of the panel. The body grid is generated with higher panel densities in regions of presumed rapid potential variation and in regions with complex geometric considerations, such as areas of large curvature or lifting surface tips, to reduce panel skew and to increase accuracy with respect to the unsteady, continuous velocity potential distribution.

In general, we consider fish geometries which are composed of a main body, with smoothly varying elliptical cross sections, and a tail fin, with rounded leading and sharp trailing edges. On occasion, we model additional fins to investigate the effects of a more accurate geometrical representation. Modeling these fins as finite-thickness lifting surfaces with rounded leading and sharp trailing edges becomes impractical as the fins become smaller and thinner with respect to the main body. As the angle between adjacent panels at the leading edge approaches 2π, the resulting square-root singularity destabilizes the influence coefficient matrix (Katz and Plotkin, 1991).

When the solution of the boundary value problem has been obtained at each time step, the wake panel endpoints are convected by a velocity field described by a desingularized version of the Biot–Savart law. The desingularization technique employed (Krasny, 1986) eliminates the infinitesimal vortex sheet singularity and the associated ill-posedness. The inclusion of the desingularization is necessary to prevent nonlinear energy transfer to the highest wavenumber modes and simulation breakdown, in addition to non-physical solution growth caused by the numerical instabilities.

Similarly, the body and thin fin doublet panels are desingularized by the same technique, with arbitrary body desingularization radius δ_b. This desingularization assists in stabilizing the solution algorithm for coupled mixed boundary value problems with bodies of fine mesh discretizations. Additionally, when upstream-shed vorticity impinges on downstream body elements, non-physical free wake acceleration and deformation are avoided. These impinging wake panels pass around the outside of the body surface, often tangentially, convected by the body surface normal and tangential velocities. Without this body desingularization, free wake panels interacting with flexing body panels might convect inside the body.

The power transmitted to the fluid by the motions of the local imposed body velocity. In general, for a geometry with a number of thick bodies and thin fin surfaces, the time records of the total force F_b and the total power delivered to the fluid P_b may be quite complex, even for sinusoidal transverse motions. The total force is decomposed into a side force L_b perpendicular to the direction of straight-line swimming, which should have a zero temporal mean, and a longitudinal force T_b, which should have a non-zero mean value of thrust or drag for a mean swimming speed U in the absence of skin frictional drag modeling. The mean value of thrust generated over one period for harmonic body undulations is denoted ⁠. Similarly, the mean power transmitted to the fluid over a period of harmonic body motion is denoted ⁠. This allows us to measure the classical hydrodynamic propulsive efficiency η for swimming speed U as

The unsteady code was validated by simulating the impulsively started motion to a constant speed U of a finite-aspect ratio foil at a small angle of attack, in a fluid otherwise at rest. Convergence of the numerical method was confirmed by systematically varying the time step size dt and the number of panels k around the foil, in addition to the aspect ratio and the desingularization parameter δ_w for the wake, and comparing the steady lift coefficient with the experimental values illustrated in Fig. 17-5 of Hoerner (1985).

In the DPIV experiments, the fish exhibited varied behavior with tail double amplitudes between 0.11L and 0.21L and frequencies up to 6.2 Hz. Consequently, the Strouhal number also showed variability. Typical relative errors in the analysis of the fish and wake kinematics were 25 % for the tail double amplitude measurements (0.02–0.05L), 15 % for the frequency measurements (up to 0.9 Hz), 5 % for the speed measurement (up to 0.055 L s⁻¹) and 30 % for the Strouhal number calculation. Much of the error is simply because the fish did not always swim with a constant tail-beat amplitude or frequency and does not actually represent measurement error. The values reported here for experiments are means. The wake Strouhal numbers (St) clearly support the conclusions of Triantafyllou et al. (1993) that fish tend to swim within a narrow range of Strouhal numbers in order to exploit the natural instabilities of the thrust jet created. Although our kinematic measurements are approximate, most of the Strouhal numbers observed in our experiments are within the predicted optimal range of 0.25<St<0.4.

Several swimming cycles were captured and studied using DPIV; here, we present one particularly clear case where the fish swam directly away from the light source, thus revealing the flow on both sides of the body and in the wake without shadows. For this case, the fluid ahead of the fish was nearly stationary, and all motion can be attributed to the motion of the fish. Velocities very near the body boundary are not shown in the following results because the image of the fish itself contaminates the flow image data. Spurious data are directly removed, and adjacent data are treated specially to avoid the inclusion of contaminated data in the smoothing filter and calculations of vorticity and circulation.

Fig. 3 shows the velocity field and vorticity contours adjacent to and in the wake of the fish during steady straight swimming. The DPIV images reveal that the fish was swimming at an approximately constant mean velocity of 8.9 cm s⁻¹ (1.1 L s⁻¹), a normal observed swimming speed for the giant danio, for the 0.23 s duration of the cycle captured. The fish traveled steadily from one end of the tank to the other, passing through the imaging window in the center of the tank.

Any acceleration or deceleration of the body occurred at either end of the tank, well before or after the images shown in Fig. 3 in the center of the tank, so the flow features are likely to be typical of steady-state motion. Nearly an entire swimming cycle is shown with the successive data sets separated by approximately 0.22T, where T is the period of the tail oscillation. The figures show the fish body exactly as it appears in the first image of each image pair except that it is colored uniformly for clarity. Slight variations in the recorded body image exist due to the fact that in some images the caudal fork is illuminated by the laser sheet while in others the caudal fin tips are illuminated.

At the start of the swimming cycle (Fig. 3A), the fish tail is flexed maximally to the left. A clockwise vortex is forming with its center near the tip of the tail as a result of shed body-generated vorticity associated with the body undulation. The vortex is already well formed, although at this point the tail has not begun its rightward motion. Subsequent rightward tail motion will contribute same-signed vorticity to this vortex. Immediately to the left of the fish and slightly anterior to the tail, the flow follows the rightward velocity of the body. Slightly upstream of this position at approximately the mid-body of the fish, longitudinal flow towards the tail marks the formation of the next bound counterclockwise vortex.

In Fig. 3B, the tail has moved to the right, and the new clockwise vortex is now clearly visible in the wake. The region of longitudinal flow has moved rearward and to the right with the tail motion. In Fig. 3C, the tail is maximally displaced to the right ready to begin the return stroke. The flow to the right of the fish follows the body motion to the left which defines the circular flow that becomes the next counterclockwise vortex. Fig. 3D shows this new vortex in the wake as well as the next clockwise vortex forming along the body. The swimming cycle then concludes with the tail in the maximum downward position and the next clockwise vortex positioned at the tail tip, ready to be shed on the upstroke.

The closed circular contours shown in Fig. 3 indicate concentrated vortices arranged in a jet pattern. Body-bound vorticity cannot manifest itself, as it would require processing of the velocity field through the body of the fish. The wake organizes into a stationary reverse Kármán street: vortices align in a staggered pattern such that the net wake produces a jet flow. The maximum velocity along the centerline of the jet is approximately 90 % of the mean velocity of the fish. The flow adjacent to the fish is strongly influenced by the body undulation, as demonstrated by the clockwise flow at approximately three-quarters of the body length from the nose (Fig. 3D). This clockwise flow is the beginning of the next vortex destined to be shed by the tail into the wake. The new ‘vortex’ is formed well before encountering the tail. The increasing lateral motion of the posterior body region augments the circulation and strengthens the vortex before it is shed into the wake, just before and at the peduncle. The tail manipulates this vorticity through repositioning and shedding of additional vorticity.

In the numerical calculations, we approximate the body to be as close as possible to the giant danio body, including the caudal fin, and the dorsal and anal fins. The tail generates the largest wake, and a primary separation line is chosen to be the trailing edge of the tail. Although the giant danio possesses a large dorsal fin, a large anal fin and smaller ventral and pectoral fins, these edges are treated differently in the following cases. It is obvious from the experimental evidence that the tail realizes the greatest unsteady motion amplitudes, which would sustain the largest unsteady lifting forces and generate the strongest vorticity structures; however, the influence of vorticity shed upstream of the tail by secondary lifting surfaces on the flow around the tail, the total wake structure and the force on the body is unclear.

Lighthill (1960, 1975) hypothesized that if there was a large gap between the dorsal and caudal fins, any vorticity shed from dorsal or ventral lifting surfaces would behave very differently from that in the presence of a fin bridging the gap between the two fins. Specifically, for certain values of the phase between the tail and the dorsal fin motions, recovery of upstream shed energy may be achieved. Streitlien et al. (1996) showed computationally such energy recovery using an oscillating foil in a shear flow.

To clarify this point, a computational body geometry was chosen which includes the main portion of the body, the caudal fin and the smaller dorsal and anal fins (Fig. 4). The main body sections are assumed to be elliptical with a major-to-minor axis ratio of =2.2, where the major axis corresponds to the height of the body. A curve-fitting technique is used to determine the profile shape of the body with L=1.0, and is given simply by:

The numerical grid in the region of the caudal peduncle, where the body and tail intersect, is artificially smoothed. This simplification allows rapid repanelization of the structured grid and ease of relative panel motion. The differences in the peduncle contraction region between the numerical form and the actual giant danio are assumed to have a small effect on the hydrodynamic similitude, as this region sustains a small hydrodynamic load and serves to transmit force from the oscillating tail to the center of mass of the body (Lighthill, 1960, 1975). Additional geometric differences between the numerical body representation and the live giant danio include the zero-thickness representation of the anal and dorsal fins and the omission of the smaller ventral and pectoral fins. The smaller ventral and pectoral fins are assumed to have minimal effects on propulsion and are neglected in this analysis. The roles of the anal and dorsal fins in propulsion are investigated by implementation of the wake-shedding scheme.

Fig. 5 presents a case of typical straight-line swimming, revealing the nonlinear wake sheet deformations and interactions. Clean separation of the main wake from the trailing edge of the caudal fin is shown, and the wake rolls up behind the tail, forming a reverse Kármán vortex street structure and thrust jet, under the self-influence of the shed vorticity and also the body perturbation velocity influence. The thrust jet is composed of connected vortex-ring-like structures, which are evident from the concentric contours of dipole strength on the deforming wake surface. The wakes shed from the vortex-lattice lifting surfaces of the thin dorsal and anal fins are also evident, with strength which oscillates due to the unsteady fin undulations caused by the traveling backbone wave. While the thin fin trailing edges are not entirely vertical, they are close to the leading edge of the caudal fin and removed from the main body, which reduces the roll-up effects caused by the influence of oscillating body perturbation velocities and the self-influence of the shed vorticity. Thus, the interaction of the vortex sheets shed from the thin fins with the caudal fin wake occurs before the thin fin wakes have significantly tightened into vortex lines. These unrolled thin fin wakes may then affect more of the flow around the leading edge of the caudal fin than if they had an opportunity to deform significantly, increasing the likelihood of energy recapture by the caudal fin through increased leading-edge suction.

A kinematic history of giant danio straight-line swimming motions was obtained from experimental sequences of images taken over three swimming cycles using DPIV, such as that shown in Fig. 3. From this information, we derived an analytical representation of the swimming motions. Giant danio swim using carangiform motion. Carangiform swimmers possess tapering tails which terminate in a well-defined caudal fin. Carangiform motion is characterized by a smooth, amplitude-modulated traveling wave moving along the length of the fish backbone with a phase speed c_p=ω/k_w, which is usually different from the swimming speed U. In this description, k_w=2π/λ is the wavenumber, for a wavelength of the backbone perturbation λ, and ω is the circular frequency of oscillation. The transverse amplitude of the backbone motion a(x) increases gradually along its length, such that the primary propulsive motions are confined to the tail region.

Our computational geometry is employed to investigate the straight-line swimming of the giant danio as obtained during DPIV experiments. The kinematic variables that prescribe the motion are those calculated from experimental DPIV images for the sequence shown in Fig. 3, specifically: swimming speed U=1.1 L s⁻¹; tail-beat frequency f=3.3 Hz; tail-tip double amplitude A=0.16L; backbone wavelength λ=1.1L; Strouhal number St=0.45; phase angle between pitch and heave of the tail motion ϕ=95 °; and tail angle of attack α=6 °.

Focusing on the midbody plane flow dynamics, Figs 6 and 7 highlight many prominent in-plane flow features which develop during the course of a straight-line swimming period. Fig. 6 illustrates the midbody plane velocity vectors and streamlines at eight discrete intervals over the swimming cycle. Velocity vectors densely grid the plane, scaled in size by the magnitude of the local velocity, so that far from the fish they may appear as points. In-plane streamlines are superimposed on the velocity vector grid to clarify the direction and the structure of the flow perturbations. Although the streamlines in this representation are continuous, the magnitude of the fluid velocity decreases greatly in this reference frame as the distance from the fish body increases. While the velocity vectors in this plane, in general, may have three-dimensional vertical components, careful analysis reveals that the flow is strongly two-dimensional in most regions along the body at this depth for all separation schemes and body geometries. While the flow around the top and bottom of the fish body shows strong three-dimensional behavior, sectional plane profiles along the body length show that the flow in the midplane is entirely longitudinal and has no vertical components. The complexity of the flow at other depths is the subject of continuing investigation.

The color contours of the velocity vectors are correlated to the sign and magnitude of the vertical vorticity component ω_z. The red vorticity indicates a clockwise or positive circulation, and the blue vorticity indicates a counterclockwise or negative circulation. The green areas are irrotational. These vorticity contours compare favorably with the DPIV experimental results shown in Fig. 3, with similar wake strength and developing wake dynamics. The formation of a steady thrust jet is evident through the unsteady dynamics of the oscillating caudal fin, continuously shedding and manipulating wake vorticity of oscillating strength. Streamline trajectories are determined using a fourth-order Runge–Kutta spatial integration with linear interpolation between grid points. The absence of small-scale turbulent fluctuations in the numerical representation of the fluid dynamics obviates the need for higher-order integration and interpolation schemes. The rectangular grid relevant length is dL=0.01 based on the body length L=1.

Fig. 7 illustrates the thrust jet region and the near-body flow dynamics by showing the dynamic pressure contours at midbody depth, at regular intervals over the course of the swimming cycle. The dynamic pressure contours reveal low-pressure regions formed upstream, along the midsection region and contraction region of the fish body length, which are then manipulated by the oscillation of the afterbody. Specifically, the controlled actuation of the caudal fin intercepts these low-pressure regions as they progress posteriorly in the local body frame of reference and pass into the region on the inside of the maximum lateral excursion of the caudal fin. As the caudal fin is swept to the other side through the low-pressure region, the separation from the sharp trailing edge of the caudal fin contributes to the formation of a reverse Kármán vortex street. This vorticity control and production interaction mechanism contributes to the efficient production of the thrust jet comprising the wake of the straight-line swimming motions, as is evident from the contours of dynamic pressure or momentum in the fluid (Fig. 7).

Fish are known to have outstanding capabilities for fast-starting and maneuvering. Fish can turn through 180 ° on a radius considerably less than their body length, whereas man-made underwater vehicles require several body lengths to execute a similar turn. Experimental measurements of live fish maneuvering and fast-starting can be found in Weihs (1972), Blake (1983), Harper and Blake (1990) and Domenici and Blake (1997). A thorough kinematic analysis of unsteady turning and maneuvering motions can be found in Videler (1993). In the present study, we employ our numerical method to simulate transient fish-like motions, such as turning and maneuvering, and then compare the simulation results with experimentally observed fish turning motions. By examining the near-body flow and the wake produced by the turning motions of the fish, we extend the concepts of vorticity shedding and manipulation by the tail to explain fish maneuvering performance.

The flow patterns around a swimming giant danio performing a 60 ° turn were captured on video. Some of the velocity-field data were missing because of fish shadows. In Fig. 8, horizontal slices of the flow are shown with the fish viewed from above. The laser is located at the top of each figure. The fish of length L=8.32 cm begins the maneuver coasting in a straight line at a speed U=1.50 L s⁻¹, having recently completed another turn; then it rapidly contorts its body into a tight ‘C’-curve and subsequently recoils to resume straight-line swimming at U=1.58 L s⁻¹ in a direction 60 ° towards the right as viewed by the fish from the starting direction. A total of 25 images were taken during the turn, at 0.0333 s intervals, for a total of 0.8 s. The backbone mean line positions of the fish were obtained from 13 of these image graphically, at 0.0667 s intervals, by manual discretization of the backbone at each time into 20 equal arc length segments. The backbone locations were later used to analyze the trajectory of the fish for numerical simulation. The fish intersected the laser plane at mid-body depth for the duration of the turn.

The DPIV results for this maneuver are shown in Fig. 8. The region immediately to the left of the fish was in its shadow, so these data were removed. Shortly preceding the position shown in Fig. 8A, the fish executed a turn originating in the upper left-hand corner of Fig. 8A and resumed straight-line swimming. Residual vorticity from these movements remains above and to the left of the fish. In Fig. 8B,C, the fish begins to bend, moving its head towards the new swimming direction while the tail moves in the opposite sense, forming a characteristic ‘C’-shape. The fluid motion follows that of the contorting body, moving directly towards the fish on the concave side of the body and directly away from the fish on the convex side of the body, at the points of maximum curvature.

In Fig. 8D, we see the flow organize into two circular-like flows, one centered at the tail and one near the head, as the tightening of the body into a ‘C’-shape nears completion. In Fig. 8E, the tail has begun to move towards the right of the viewed image, shedding in the wake the previously formed counterclockwise vortex. At this instant, the effect of bound vorticity near the head is obscured by reflection of light from the body. The next several frames are not presented because these body reflections obscured a large portion of the flow. However, analysis of the video record shows that during the stroke of the tail, sweeping downwards in the viewed image as the fish body heads to the left, the counterclockwise vortex sheds into the wake. During the next stroke of the tail, which sweeps back upwards in the viewed image as the fish continues moving to the left, the clockwise vortex initially associated with the front region of the body moves posteriorly and sheds in the wake.

The net result is shown in Fig. 8F: a strong vortex pair forms which provides a jet directed slightly downwards and to the right. The vortices in this pair are packets of counter-rotating large-scale wake vorticity, and Fig. 9 shows the vorticity contours at the same instant. The asymmetric shapes of the contours of the two vortices that make up the turning jet are due to the unsteady, non-periodic body motions and velocity. Additional vorticity shown in this figure, but not labeled with contour strength values, was created by the fish before the initiation of the turning maneuver. The entire turning sequence, from a straight-line coasting trajectory following a previous turn to a steady swimming trajectory 60 ° to the right as viewed by the fish from the starting direction, takes slightly more than 0.5 s. The vortices comprising the jet have average non-dimensional circulation Г_e^*=Г_e/LU=0.43, which is 42 % greater than the typical wake vortex strength for steady swimming at similar speeds. The core radius of the jet vortices is more than double that of those produced in straight swimming. The jet is 0.34L wide, with maximum jet velocity of u_j≈U oriented approximately 60 ° to the left of the initial starting position as seen by the fish. Interestingly, the fish velocity after the turn is roughly the same as the jet velocity. Also, the direction of the jet is roughly in the direction required by horizontal momentum balance. Three-dimensional effects are significant, and the vigorous motion of the fish somewhat disrupts our two-dimensional planar results. After completing the turn, the fish began to swim steadily at U=1.5 L s⁻¹.

We impose on the fish model the body motions observed in live fish, including the movements of the tail as well as the trajectory of the fish. The shape and position of the fish were only known at 13 time steps during the maneuver; an accurate wake picture could not be produced if these 13 positions were the sole simulation input with such a large time differential between body positions. An interpolative scheme was therefore necessary to calculate the location of the fish at intermediate time steps, from which the fish turn could be simulated.

Fig. 10 details the coordinate systems used. The O,X,Y,Z coordinate system is global and fixed. The o,x,y,z coordinate system is a coordinate system whose origin is fixed at the nose of the fish. The x-axis always passes through the tail of the fish. Thus, the flexing motion of the fish can be described in a local coordinate system, while its trajectory can be described in the global coordinate system. The z coordinates of the fish body depth were assumed to be invariant as the plane of the laser sheet intersected the fish at midbody depth for the duration of the turning motion.

The 13 discretized fish backbone images were then analyzed as follows. First, the position of the nose in the global coordinate system was recorded as a function of time, serving as the origin of the o,x,y,z coordinate system. The angle θ that the o,x,y,z coordinate system made with the global coordinate system was also recorded as a function of time, by constructing the x-axis through the fish tail at each time interval. The experimental data of fish head and tail locations during the turning maneuver can be seen in Fig. 11A.

A periodic motion was assumed for the local flapping motion of the fish. As the fish had a fairly uncambered mean line shape at both the first time step and the last time step of the turn, it was assumed that the local shape of the fish mean line at the extrapolated time of 0.8667 s would be equal to the original mean line shape. In this way, the local x and y coordinates of the mean line, as well as the thickness distribution h, could be expanded as Fourier sine series as functions of time. Additionally, as the local y coordinate and the thickness h are always zero at the nose and tail, these could be expanded as Fourier sine series as functions of backbone arc length, while the local x coordinate could be expanded as a linear function plus a Fourier sine series as a function of backbone arc length. Thus, the local behavior of the backbone can be found at any time during the simulation.

The orientation of the local coordinate system within the global frame of reference is similarly assessed. Expansion of the global X, Y and θ coordinates in terms of Fourier series in time yields the trajectory and inclination of the local coordinate system within the global reference frame for any time during the simulation. Details of the complete analytical description of the unsteady motion can be found in Appendix B. From this analytical description of the motion, we can produce a simulation with an arbitrary number of time steps, and the difficulties in accurately reproducing the body motion and the resulting wake are alleviated by having a sufficient number of time steps in the simulation. As an example, Fig. 11B shows the backbone trajectory and flexing behavior of the giant danio over the entire simulation employing 50 intermediate time steps of information.

The viscous effects of the body are again assumed to be confined to a thin boundary layer near the body and in a small wake described by shear layers. The Reynolds numbers Re of the giant danio before and after the turning motion are approximately Re=10 400 and Re=11 000, respectively. At these values, we would expect the boundary layer to be laminar or within a transition region, as illustrated by experimental data for the drag coefficients of axisymmetric streamlined bodies with varying L/d ratios (where d is the diameter of the streamlined body) found in Fig. 6.22 of Hoerner (1965). It is reasonable to assume that the near-body viscous effects on the body are small in comparison with the effects of unsteady vorticity generated by separation, such as that occurring at the sharp trailing edges of the body which generate wakes. Again, the tail is assumed to generate the largest wake, and a primary separation line is chosen to be the trailing edge of the tail. The large dorsal and anal fins are also included in the course of our investigation, to examine the influence of upstream-generated vorticity on the flow around the tail, the overall wake structure and the unsteady body forces generated.

The body shape can be described at each time step, and the imposed normal velocities at each panel midpoint are known; hence, the boundary integral equation can be solved to find the solution of the unsteady boundary value problem at each time step, as described above.

The turning motion of the giant danio is simulated using the method described above. It was observed in the DPIV experiments that the near-tail flow dynamics are affected by the motions of the dorsal and anal fins during the maneuver. These interactions are less apparent during straight-line swimming, in which the dorsal and anal fin lateral motions are smaller. As a result, the accurate modeling of the fish geometry during the turning maneuver was deemed crucial to resolving the flow dynamics around the tail and around the dorsal and anal fins with their large excursions.

The simulation time step size dt=0.01 s was chosen to be small enough to describe the details of the motion completely and to resolve wake-body dynamics and wake self-interactions. Fourier expansions of the mean line shapes in the local coordinate system reference frame employ 15 temporal modes and 10 spatial modes. Similar expansions of the local coordinate system trajectories within the global coordinate system employ 10 temporal modes. Numerical desingularization radii for the wake and body were chosen to be δ_w=0.02 and δ_b=0.025, respectively, based on a body length of L=1.0, selected through a convergence process ensuring optimal stability of the numerical solution.

To eliminate the influence of any starting vorticity shed by the caudal, dorsal and anal fins during the impulsive start of the numerical simulation, a ramping-up time interval was added to the initial portion of the simulation. The initial conditions of the simulation are such that the fish mean line is locally perturbed to assume the shape of the giant danio at the start of the turning maneuver. The global position of the geometry is such that it translates at a constant velocity, equal to the velocity at the start of the turning maneuver. At the conclusion of the ramping-up time interval, the position, velocity and shape of the geometry are identical to those of the giant danio at the initial frame of the experimental DPIV data. The total simulation length of 160 time steps thus modeled the 0.800 s turning maneuver, with 0.800 s of initial coasting motion before the turn was initiated to eliminate unwanted starting vorticity dynamics.

The results of the simulation are shown in Figs 12 and 13. These are taken at 0.100 s intervals, shown from t=−0.100 s before the start of the turning maneuver until the turn is nearly completed at t=0.600 s. These frames are representative of time steps t_s=70 to t_s=140 at 10-time-step intervals.

The sequence of images shown in Fig. 12 details the two-dimensional flow patterns around the turning fish in a plane at midbody depth on the fish. In the initial plots of the sequence, at the extreme top of the fluid plane there is residual vorticity resulting from the starting motions in the simulation. This is caused by the translation of the giant danio geometry from the top of the frame into its starting position. The turning maneuver produces additional vorticity which is largely unaffected by the decaying starting vorticity. The sequence of images shown in Fig. 13 illustrates the evolution of dynamic pressure contours during the turn.

From both Figs 12 and 13, it can be seen that as the fish contorts itself into a tight ‘C’-shape, the flow is organized into three circular patterns, around the head, the midbody and the tail fin, respectively. As the fish begins to sweep its tail towards its left side, the tail fin sheds positive vorticity, as negative bound vorticity moves from the contraction region towards the tail. A large low-pressure region coming from the right of the body crosses over to the left, passing over the tail and eventually pairing with the previous vortex to form a strong jet, which turns the fish. The negative vorticity in the contraction region moves posteriorly to the tail during the stroke of the tail to the left, and a new low-pressure region is formed below the fish. This negative vorticity is released into the wake during the subsequent return stroke to the right, as the second low-pressure region passes over the leading edge of the tail fin. As the tail concludes its stroke to the right and passes through the second region of high fluid momentum, it recovers energy from the jet through the enhanced separation from the trailing edge of the tail, reinforcing the negative vorticity that is being shed into the wake and resulting in additional loading on the tail, which propels the body forward. The dynamics of the maneuver are shown in Fig. 14. The forces on the fish body during the turn are depicted as a function of time, with the total force decomposed into its X and Y components of the global coordinate system (Fig. 14A). In addition, the fluid force vectors acting on the fish center of mass at each time step of the simulation are shown, with the mean line curves of the fish backbone superimposed on the plot every four time steps (Fig. 14B). The forces are moderately large at the outset of the maneuver, as the fish halts the oscillatory motions of straight-line swimming and releases residual body-bound vorticity into the wake, then increase sharply as the fish contorts into a tight curve and sweeps its caudal fin to the left. The oscillatory forces gradually subside as the turn is completed and straight-line swimming commences again. It is evident from the vector time history that the forces on the body allow it to turn in a short distance relative to its length.

Comparison between the experimental data and the numerical simulation results shows good quantitative and qualitative agreement. Wake vortex strength and placement compare well, with a thrust jet width of approximately 1.3 times the maximum body width. Maximum computed numerical vorticity strengths are within 20 % of experimental values. Simulation results reveal that the side forces are distributed evenly between the tail and the main body, while the tail sustains over 90 % of the longitudinal force. The detailed patterns shown in Fig. 15 elucidate the complex processes of bound vorticity release and thrust wake formation in the experimental results and those of the simulation. The near-body and wake streamline patterns near the trailing-edge separation line of the caudal fin show qualitatively similar features.

Fig. 16 summarizes the development and control of vorticity by the flexible body undulation during the swimming cycle. This model simplifies the complex three-dimensional nature of the near-body flow, but accurately portrays the dominant features of the mid-depth plane flow and the principal mechanisms involved in manipulation of the large-scale two-dimensional vorticity features in the wake. In Fig. 16A, clockwise (negative) vorticity is shed into the wake. Along the contraction region of the body, counterclockwise (positive) vorticity is fully developed, due to the local undulation of the body and the coordinated upward stroke of the caudal peduncle as seen by the reader. Fig. 16B reveals positive bound vorticity moving towards the tail tip during the upward stroke of the tail. New negative bound vorticity develops in the midbody region of the fish length, due to the downward midbody motion. Fig. 16C illustrates the shedding of the positive bound vorticity into the wake, while negative vorticity is fully developed in the contraction region, owing to the local body contortion and coordinated downward caudal peduncle motion. In Fig. 16D, the cycle is complete and recommences. The alternating-sign free vortices in the wake form a reverse Kármán street, resulting in the development of a thrust jet. Optimal propulsion is achieved through proper selection of the frequency and spacing of the wake vortices. The process of the formation of wake vortices through the release of body-generated vorticity and subsequent tail manipulation as revealed in Figs 6, 7, 15 and 16 is an important result of the present paper, extending the analysis and conclusions of Triantafyllou et al. (1993).

Fig. 17 summarizes the development and control of vorticity by the flexible body deformation during the turning maneuver. This model simplifies the complex three-dimensional nature of the near-body flow, but accurately portrays the dominant features of the mid-depth plane flow and illustrates the principal mechanisms involved in manipulation of the large-scale two-dimensional vorticity. In Fig. 17A, the fish ceases its straight-line swimming undulations and releases body-bound vorticity associated with straight-line swimming motions into the wake (not shown). In Fig. 17B, the fish initiates its backbone curve. In the center of the body, a pair of oppositely signed bound vortices develop, owing to the midbody translation to the left and the opposite motion of the head and tail. In Fig. 17C, the local contortion of the backbone into a ‘C’-shape is complete, and the pair of body-generated, oppositely signed bound vortices separate as one of the vortices moves closer to the tail. Straightening of the body in a wave-like motion commences in Fig. 17D, which releases counterclockwise vorticity into the wake through manipulation by the caudal fin. As straightening continues, the clockwise bound vorticity travels down the length of the body towards the tail and is released into the wake. Initiation of straight-line swimming motions completes the release of clockwise vorticity into the wake, as shown in Fig. 17E, which then pairs with the counterclockwise vorticity to form a thrust jet in the wake. The rapid formation of this thrust jet accomplishes the turning of the fish. It is remarkable that no uncontrolled vorticity appears to be shed during the turning process.

Thus, control of bound vorticity and wake formation by the body and the tail fin for thrust vectoring to perform turning maneuvers is achieved through large localized backbone and tail actuation. This mechanism of vorticity control allows for the efficient manipulation of near-body flows and empowers the generation of large, short-duration maneuvering forces.

We have presented detailed experimental measurements of the flow around a live, naturally swimming fish and performed numerical simulations on a fish model performing the same motions. Using DPIV, we have clarified the principal propulsive mechanisms used by the fish. Three-dimensional unsteady flexible-body numerical simulations are corroborated against the experimental observations and then used to elucidate further details of the formation of the thrust jet. Specifically, fish control body-generated vorticity through body flexure and active manipulation by the caudal fin. Our results indicate that the structure of the jet wake is obtained through the regulated release of body-generated vorticity and its constructive interaction with vorticity formed and shed by the oscillating caudal fin.

We also presented detailed results for the near-body flow dynamics around a live fish executing an unsteady, rapid turning motion. Through precise body actuation, the fish regulates the formation and controlled release of body-generated vorticity, resulting in the production of a pair of counter-rotating vortices and, hence, a thrust jet. Thus, the fish is able to redirect its generated thrust rapidly and achieve its desired trajectory.

It has long been known that fish swimming in a straight line produce an unsteady wake in the form of a reverse Kármán street. However, it has generally been assumed that the wake vortices originate from the action of the tail as a lifting surface. Our results indicate that the tail fin has a secondary role in the generation of wake vortices. Vorticity is generated well upstream of the tail by the undulations of the body. Upon arriving at the tail, body-generated vorticity is favorably affected by the motion of the tail: it is reinforced by the tail, while vorticity shed at the trailing edge of the tail merges with and amplifies each body-generated vortex. The undulation of the fish body and the caudal fin motion are synchronized such that vorticity formation and evolution are actively controlled for both straight-line swimming and maneuvering.

The discrete forms of the Kutta condition (equation 24) and the kinematic body boundary condition (equation 25) can then be employed to arrive at a simple system of linear equations for at any given time t, expressed as equation 4.

Given the backbone shape and body thickness distribution at a discrete number of time steps for arbitrary fish swimming motions, we can determine the details of the local backbone undulations and the global body trajectory as functions of time using a series expansion technique. This allows the generation of the smooth trajectory and body shape needed for simulation.

 
• A

  total lateral excursion of the tail at the caudal peduncle

 
• A_n

  Fourier coefficients of lateral unsteady motion spatial expansion

 
• A_nm

  Fourier coefficients of lateral unsteady motion spatial and temporal expansion

 
• 

  aspect ratio of three-dimensional lifting surface

 
• a(x)

  amplitude of the backbone motion along the length of the body x in the o,x,y,z coordinate system

 
• B_n

  Fourier coefficients of longitudinal unsteady motion spatial expansion

 
• B_nm

  Fourier coefficients of longitudinal unsteady motion spatial and temporal expansion

 
• C_n

  Fourier coefficients of unsteady body thickness distribution spatial expansion

 
• C_nm

  Fourier coefficients of unsteady body thickness distribution spatial and temporal expansion

 
• C_P

  pressure coefficient

 
• C_Xp

  Fourier coefficients of the X-coordinate temporal expansion for unsteady body motions

 
• C_Yp

  Fourier coefficients of the Y-coordinate temporal expansion for unsteady body motions

 
• C_θp

  Fourier coefficients of the θ-coordinate temporal expansion for unsteady body motions

 
• c_p

  phase speed of propagating backbone wave

 
• c₁

  linear backbone wave amplitude coefficient

 
• c₂

  quadratic backbone wave amplitude coefficient

 
• d

  diameter of streamlined body from Hoerner (1965) 

 
• dl

  elemental unit of length along a vortex segment for Biot–Savart integration

 
• dL

  velocity grid length spacing based on body length L

 
• 

  tangent vector for circulation contour integration

 
• ds

  elemental unit of surface area

 
• dt

  experimental DPIV image interval or computational time step

 
• F_b

  force on the thick body surfaces

 
• F_b,thin

  force on the vortex-lattice thin body surfaces

 
• f

  frequency of tail oscillation (Hz)

 
• H

  fish body maximum total depth

 
• h

  fish body thickness distribution at midbody depth along the length of the body in o,x,y,z coordinate system

 
• i

  index of observation panel number in discrete representation of boundary integral formulation

 
• j

  index of field panel number in discrete representation of boundary integral formulation

 
• k

  number of body panels

 
• k_w

  wavenumber of the backbone wave

 
• L

  fish body length

 
• L_m

  Fourier coefficients of the temporal expansion of the backbone arc length

 
• L_o

  original arc length of the backbone in unsteady motion Fourier expansions

 
• L_b

  component of total body force Fb in the lateral direction, perpendicular to the direction of swimming

 
• M

  number of temporal modes in the Fourier expansions for unsteady body motion local trajectories

 
• m

  index of unsteady motion temporal Fourier expansion

 
• 

  body outward normal vector

 
• ∂n_b

  derivative in the direction of the body outward normal vector

 
• 

  vortex-lattice thin body surface normal vector

 
• 

  derivative in the direction of the thin fin outward normal vector

 
• nw

  number of wake panels

 
• ∂nw

  derivative in the direction of the wake surface normal vector

 
• O,X,Y,Z

  inertial reference frame

 
• o,x,y,z P

  local body-fixed coordinate system

 
• P

  number of temporal modes in the Fourier expansions for unsteady body motion global trajectories

 
• 

  column vector of panel normal velocities for source-dipole potential panel distribution

 
• P_b

  power transmitted to the fluid by the body

 
• 

  temporal mean of P_b over an integral number of steady cycles

 
• P_ij

  source distribution influence of panel j on panel i

 
• P₁

  partial column vector of panel normal velocities for thick body panel distribution

 
• P₂

  partial column vector of panel normal velocities for vortex-lattice thin fin panel distribution

 
• p(x)

  vertical coordinate of the mean line along length of body in o,x,y,z coordinate system

 
• p

  index of wake panels for panel normal velocity calculation

 
• 

  matrix of influence coefficients for source-dipole potential panel distribution

 
• Q_ij

  dipole distribution influence of panel j on panel i

 
• Q_iw

  dipole distribution influence of new wake panel

 
• Q₁₁

  sub-matrix of influence coefficients for source-dipole potential panel distribution; thick body panels’ influence on thick body panels

 
• Q₁₂

  sub-matrix of influence coefficients for source-dipole potential panel distribution; vortex-lattice surface panels’ influence on thick body panels

 
• Q₂₁

  sub-matrix of influence coefficients for vortex-lattice potential panel distribution; thick body panels’ influence on vortex-lattice surface panels

 
• Q₂₂

  sub-matrix of influence coefficients for vortex-lattice potential panel distribution; vortex-lattice surface panels’ influence on vortex-lattice surface panels

 
• Re

  Reynolds number

 
• R_o

  core radius of observed wake vortices

 
• r

  magnitude of distance between two points

 
• 

  vector of magnitude r from a vortex element to the field point for Biot–Savart integration

 
• r_ij

  magnitude of distance between collocation points of panel i and panel j

 
• r_iw

  magnitude of distance between collocation points of panel i and new wake panel w

 
• S

  all bounding surfaces of computational domain

 
• S_b

  all body surfaces in computational domain, subset of S

 
• S_w

  all wake surfaces in computational domain, subset of S

 
• S′_w

  new wake panels shed adjacent to trailing edge

 
• S_∞

  far-field surface at in computational domain, subset of S

 
• St

  Strouhal number

 
• s

  backbone arc length coordinate for unsteady motion parametization

 
• 

  tangent vector to vortex element for Biot–Savart integration

 
• T

  period of one cycle of straight-line swimming motions

 
• T_b

  thrust component of total body force F_b in the direction of swimming

 
• 

  temporal mean of T_b over an integral number of steady cycles

 
• TE

  trailing edge of lifting surfaces

 
• t

  time

 
• t_p

  total number of arc length segments for unsteady motion parametization

 
• t_s

  time step number

 
• t_t

  total number of time steps of experimental data for unsteady motion parametization

 
• U

  swimming speed

 
• u_j

  velocity of fluid in thrust jet

 
• 

  velocity field in the X,Y plane of the O,X,Y,Z coordinate system for circulation contour integration.

 
• 

  velocity of the body at point on Sb or a panel at time t

 
• 

  velocity induced by vortex element at a point with respect to O,X,Y,Z coordinate system

 
• X

  longitudinal coordinate of o,x,y,z origin with respect to O,X,Y,Z

 
• x

  longitudinal coordinate along length of body in o,x,y,z coordinate system

 
• x(z)LE

  longitudinal coordinate of the leading edge of the tail as a function of the height z in o,x,y,z coordinate system

 
• x(z)TE

  longitudinal coordinate of the trailing edge of the tail as a function of the height z in o,x,y,z coordinate system

 
• 

  three-dimensional coordinate with respect to O,X,Y,Z

 
• 

  three-dimensional coordinate with respect to O,X,Y,Z on a vortex-lattice thin body surface three-dimensional coordinate of the collocation point of the observation panel with respect to O,X,Y,Z

 
• 

  three-dimensional coordinate of the collocation point of the field panel with respect to O,X,Y,Z

 
• 

  three-dimensional coordinate with respect to O,X,Y,Z at the trailing edge of lifting surfaces

 
• 

  three-dimensional coordinate with respect to O,X,Y,Z of the collocation point of the wake panel adjacent to the trailing edge of unknown strength.

 
• Y

  lateral coordinate of o,x,y,z origin with respect to O,X,Y,Z

 
• y(x,t)

  transverse motion of the fish backbone as a function of time t and length along the body x in the o,x,y,z coordinate system

 
• z(x)

  vertical coordinate along length of body in the o,x,y,z coordinate system

 
• α

  angle of attack of the tail

 
• Γ

  circulation

 
• Γ_e

  circulation of observed wake vortices

 
• Γ_e^*

  non-dimensional circulation of observed wake vortices

 
• Γ_v

  wake vortex element circulation strength

 
• 

  total velocity potential field at point x at time t

 
• 

  body perturbation velocity potential field at point x at time t

 
• 

  wake perturbation velocity potential field at point x at time t

 
• 

  wake perturbation potential field at the trailing edge of lifting surfaces

 
• ϕ

  phase angle between the pitch and heave motion of the tail

 
• ϕ_b

  column vector of body perturbation velocity potentials

 
• ϕ_bb

  partial column vector of thick body perturbation velocity potentials

 
• ϕ_bf

  partial column vector of vortex-lattice thin body perturbation velocity potentials

 
• 

  body perturbation velocity potential on a panel at at time t

 
• 

  body perturbation velocity potential on a trailing edge panel at on the top of the lifting surface at time t

 
• _{top panel}

  body perturbation velocity potential on a trailing edge panel at on the bottom of the lifting surface at time t

 
• _{bottom panel}

  wake perturbation velocity potential field on a panel at at time t

 
• 

  wake perturbation velocity potential at a panel adjacent to the trailing edge of a lifting surface

 
• 

  gradient operator

 
• 

  divergence operator

 
• _TE

  jump in body perturbation potential field at the trailing edge of lifting surfaces

 
• δ_b

  body panel desingularization radius

 
• δ_w

  wake panel desingularization radius

 
• λ

  wavelength of the backbone wave

 
• η

  propulsive efficiency

 
• ρ

  density of the fluid

 
• θ

  angular rotation of the local coordinate system o,x,y,z about z axis with respect to O,X,Y,Z

 
• 

  angular rotational velocity of the local coordinate system o,x,y,z about z axis with respect to O,X,Y,Z

 
• ω

  circular frequency of oscillation of the tail

 
• ω_z

  vertical component of vorticity

 
• 

  three-dimensional coordinate with respect to O,X,Y,Z

The financial support of the Office of Naval Research under contract N00014-96-1-1141 monitored by P. Purtell, T. McMullen and J. Fein, the Office of Naval Research under grants N00014-89-J-3186 and N00014-93-1-0774, the Advanced Research Project Agency under contracts N00014-92-1726 and N00014-94-1-0735 and the Sea Grant Program under Grant Number NA46RG0434 is gratefully acknowledged.