Elongated-body Theory Research Articles

The hydrodynamics of eel swimming

Eels undulate a larger portion of their bodies while swimming than many other fishes, but the hydrodynamic consequences of this swimming mode are poorly understood. In this study, we examine in detail the hydrodynamics of American eels (Anguilla rostrata) swimming steadily at 1.4 L s(-1) and compare them with previous results from other fishes. We performed high-resolution particle image velocimetry (PIV) to quantify the wake structure, measure the swimming efficiency, and force and power output. The wake consists of jets of fluid that point almost directly laterally, separated by an unstable shear layer that rolls up into two or more vortices over time. Previously, the wake of swimming eels was hypothesized to consist of unlinked vortex rings, resulting from a phase offset between vorticity distributed along the body and vorticity shed at the tail. Our high-resolution flow data suggest that the body anterior to the tail tip produces relatively low vorticity, and instead the wake structure results from the instability of the shear layers separating the lateral jets, reflecting pulses of high vorticity shed at the tail tip. We compare the wake structure to large-amplitude elongated body theory and to a previous computational fluid dynamic model and note several discrepancies between the models and the measured values. The wake of steadily swimming eels differs substantially in structure from the wake of previously studied carangiform fishes in that it lacks any significant downstream flow, previously interpreted as signifying thrust. We infer that the lack of downstream flow results from a spatial and temporal balance of momentum removal (drag) and thrust generated along the body, due to the relatively uniform shape of eels. Carangiform swimmers typically have a narrow caudal peduncle, which probably allows them to separate thrust from drag both spatially and temporally. Eels seem to lack this separation, which may explain why they produce a wake with little downstream momentum while carangiform swimmers produce a wake with a clear thrust signature.

Large-amplitude undulatory fish swimming: fluid mechanics coupled to internal mechanics.

The load against which the swimming muscles contract, during the undulatory swimming of a fish, is composed principally of hydrodynamic pressure forces and body inertia. In the past this has been analysed, through an equation for bending moments, for small-amplitude swimming, using Lighthill's elongated-body theory and a 'vortex-ring panel method', respectively, to compute the hydrodynamic forces. Those models are outlined in this review, and a summary is given of recent work on large-amplitude swimming that has (a) extended the bending moment equation to large amplitude, which involves the introduction of a new (though probably usually small) term, and (b) developed a large-amplitude vortex-ring panel method. The latter requires computation of the wake, which rolls up into concentrated vortex rings and filaments, and has a significant effect on the pressure on the body. Application is principally made to the saithe (Pollachius virens). The calculations confirm that the wave of muscle activation travels down the fish much more rapidly than the wave of bending.

Kinematics and efficiency of steady swimming in adult axolotls (Ambystoma mexicanum)

The kinematics of steady swimming at a wide range of velocities was analysed using high-speed video recordings (500 frames s-1) of eight individuals of Ambystoma mexicanum swimming through a tunnel containing stationary water. Animals in the observed size range (0.135­0.238 m total body length) prefer to swim at similar absolute speeds, irrespective of their body size. The swimming mechanism is of the anguilliform type. The measured kinematic variables ­ the speed, length, frequency and amplitude (along the entire body) of the propulsive wave ­ are more similar to those of anguilliform swimming fish than to those of tadpoles, in spite of common morphological features with the latter, such as limbs, external gills and a tapering tail. The swimming speed for a given animal size correlates linearly with the tailbeat frequency (r2=0.71), whereas the wavelength and tail-tip amplitude do not correlate with this variable. The shape of the amplitude profile along the body, however, is very variable between the different swimming bouts, even at similar speeds. It is suggested that, for a given frequency, the amplitude profile along the body is adjusted in a variable way to yield the resulting swimming speed rather than maintaining a fixed-amplitude profile. The swimming efficiency was estimated by calculating two kinematic variables (the stride length and the propeller efficiency) and by applying two hydrodynamic theories, the elongated-body theory and an extension of this theory accounting for the slope at the tail tip. The latter theory was found to be the most appropriate for the axolotl's swimming mode and yields a hydromechanical efficiency of 0.75±0.04 (mean ± s.d.), indicating that Ambystoma mexicanum swims less efficiently than do anuran tadpoles and most fishes. This can be understood given its natural habitat in vegetation at the bottom of lakes, which would favour manoeuvrability and fast escape.

NOTE ON THE CALCULATION OF PROPELLER EFFICIENCY USING ELONGATED BODY THEORY

The elongated body theory has been widely used for calculations of the hydrodynamic propulsive performance of swimming fish. In the biological literature, terms containing the slope of the amplitude function at the tail end have been neglected in the calculations of thrust and efficiency, and a slope of zero has been assumed. However, some fishes, such as saithe and trout, have non-zero values of the slope near the tail end and, when this term is taken into account, the efficiency may be reduced by as much as 20 % and approaches the result given by the three-dimensional waving plate theory. The inclusion of the slope in the efficiency considerations results in an optimum ratio of the swimming speed to the wave speed that is clearly less than 1. It is suggested that the slope terms should be included in the estimation of propulsive performance for fish swimming with variable amplitude.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 2. The pressure distribution arising in two-dimensional irrotational flow from a general symmetrical motion of a flexible flat plate normal to itself

When, for an otherwise unbounded fluid, the unique irrotational flow compatible with the instantaneous motion of an immersed body has been calculated, it is straightforward to deduce the pressure field from the unsteady form of Bernoulli's equation if the body is rigid. On the other hand, if the body is flexible, a somewhat subtle analysis is required to determine the time derivative of velocity potential, ∂ϕ/∂t, which occurs in that equation. This is because no simple relationship exists between the instantaneous form of ϕ and its form at a nearby instant.In the case of two-dimensional flow, however, the two forms of ϕ for a flexible body may be related, not in general by a simple translational and/or rotational mapping as for a rigid-body motion, but by a conformal mapping. The example of a flexible flat plate is used here to illustrate this approach to calculating the pressure field.In the analysis of balistiform motion by elongated-body theory (Lighthill & Blake 1990), one part of the propulsive force on the fish has magnitude equal to P, the area integral of the pressure field just described. This area integral is shown in §3 below to take a simple form or for E, so that P itself is also not enhanced. For the relevance of these findings to the efficiency of balistiform motion, see Lighthill & Blake (1990).

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 4. Short-wavelength limitations on momentum enhancement

Elongated-body theory, used by Lighthill & Blake (1990) to investigate fish locomotion by undulatory movements of median fins, and to demonstrate momentum enhancement in the case when motile fins are attached to a rigid fish body of far greater depth, approximates local fluid motions by solutions of the two-dimensional Laplace equation. Here, a better local approximation (equation (2) below) to the three-dimensional Laplace equation for fluid motions of undulatory type is used to investigate the possibility of short-wavelength limitations on momentum enhancement. In an extreme case (fish bodies of very small width and very large depth) when on elongated-body theory the momentum enhancement factor β is predicted to become very large, short-wavelength considerations are shown to place a specific upper limit on β (see figure 2). In more general cases, this upper limit should perhaps be regarded as coexisting with other upper limits associated with either nonzero width or finite depth of fish body.Short-wavelength limitations on momentum enhancement are of some biological interest as implying the existence not only of advantages (including a reduction in body drag) but also of some competing disadvantages (limitations in propulsive force) arising from progressive reductions in the wavelength of fin undulations.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 3. Momentum enhancement in the presence of a body of elliptic cross-section

Cross-sectional shapes of many fish bodies are well approximated by ellipses. The simple elongated-body theory of balistiform locomotion was developed by Lighthill & Blake (1990) only in the limiting case when the axis ratio of the cross-section tends to zero. In that case they established that the movements of dorsal and anal fins, if attached to a rigid fish body of far greater depth, create fluid motions with substantially enhanced momentum. In this paper, standard conformal mappings are used to establish that enhancement is substantial also with elliptic cross-sections of arbitrary axis ratio, not only in balistiform locomotion with synchronous movement of two median fins but also in gymnotiform locomotion with movement of just a single (ventral) fin.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 1. Biological background, and analysis by elongated-body theory

Elongated-body theory has been fruitfully applied over twenty years to the biofluiddynamic analysis of modes of locomotion of elongated fishes by means of body flexure, with special emphasis on the anguilliform mode using undulatory body movements, and on the carangiform mode where oscillatory movements of only a fish's posterior end(including the caudal fin) exhibit phase lag of posterior movements behind anterior movements just as in an undulation yet not nearly as much as a whole wavelength is apparent at any one time. The extension of elongated-body theory to analyse the locomotion of elongated fishes with elongated median fins (dorsal and/or anal) in modes where the body (together with any caudal fin) remains rigid, being propelled forwards by undulations or oscillations of those median fins, has long been recognized as desirable but is here presented for the first time.In many large groups of fishes, evolutionary adaptation to limited environments (such as coral reefs) favoured a development of defensive ‘armour’ at the expense of speed, to such an extent that bodies became essentially inflexible, with locomotion achieved by fin movements alone. In one principal group of such fishes, however (the sub-order Balistoidei including the trigger-fishes), a later evolutionary development restored a capacity for relatively high-speed movement even though the body remained essentially rigid. The balistiform mode of locomotion, with propulsion achieved by synchronized movements of the dorsal and anal fins, exists in two alternative forms, with either undulatory or oscillatory movements of these median fins, that are analogous to the anguilliform and carangiform modes of body flexure, respectively.Analysis in this paper throws light on the puzzling question of why trigger-fishes are able to move so fast notwithstanding the modest extent of their fin movements. A form of the large-amplitude elongated-body theory, specially adapted to balistiform locomotion, allows a direct comparison of thrust and efficiency for different modes of propulsion. The conclusions in brief are that thrust is dominated by the mean rate of shedding of backward momentum at the posterior end of the fish's propulsive apparatus and that, for movements of median fins attached to a deep, essentially rigid body, this momentum is increased (above the momentum expected for the same movements of the fins ‘on their own’) by a momentum enhancement factor β of around 3 or a little more. Yet there is no such enhancement of the rate of shedding of ‘unproductive’ energy into the wake; accordingly, overall efficiency is improved. Also, especially for the undulatory mode of balistiform locomotion, sideforces are minimized so that the fish body avoids sideslip and yaw; accordingly, the body drag which fm thrust must overcome is reduced by another large factor.Alongside discussion of the Balistoidei, this paper reviews and analyses balistiform locomotion as observed in several other groups, including groups of flexible-bodied fishes that regularly use this mode as a low-energy-cost alternative to locomotion by means of body flexure. Finally, we similarly analyse gymnotiform locomotion, in which the body is again held rigid, being propelled by undulations in just a single (ventral) fin, and compare and contrast different interpretations of its advantages.

The dynamics of carangiform swimming motions

An investigation based on the elongated-body theory of Lighthill (1960, 1970) and Wu (1971) is made of the carangiform swimming motions of fish, a mode of propulsion in which the amplitude of body undulation becomes significant only in the posterior half, or even third, of the length of the fish and the anterior part is relatively inflexible. For typical slender fish performing undulatory swimming motions, three hydro-dynamic features are taken into account: (a) the resistance to the lateral undulatory motions as well as longitudinal (tangential) frictional resistance, (b) the forces of interaction with the water associated with its inertial (virtual-mass) response to the lateral motions of the fish and (c) the reaction forces due to the vortex sheets shed from sharp trailing edges to the rear of the section of maximum span (depth), where a great variety of fins are generally found.The active tail oscillations give rise to oscillatory side forces, to which the remainder of the body responds passively. These passive yawing motions are studied to find their amplitude, the yawing axis and any associated energy dissipation. The contribution of each of the above three forces is examined and the effects of the oscillation frequency, a slenderness parameter δ of the body, and the shape of the transverse cross-sections are considered.The present theory is further applied to predict the turning movement when the fish changes direction, the way in which the above forces act during this process being investigated. A small perturbation analysis relative to a uniform rectilinear motion not only reveals whether or not the motion is dynamically stable, but also leads to a full description of the motion that follows a given initial perturbation. For finite perturbations a numerical method is adopted to find the time development of both the direction of motion relative to the direction of the initial uniform motion and the ratio of the kinetic energy of the body to the initial kinetic energy for different magnitudes I0 of the initial impulsive perturbation in the transverse direction. The final angle of turn is obtained in terms of I0 for different values of δ. The loss of kinetic energy during the turn divided by the initial kinetic energy is found to be very small for a small angle of turn.Detailed analyses are made for rigid fish models in which the distribution of depth (or span) along the body length is quadratic with a maximum at the centre. Agreement of the present analyses with observations is fairly good at least qualitatively and some quantitative estimates are made of the side forces, both oscillatory and impulsive, exerted on the body by the tail. It is found that the vortex sheets shed from trailing edges increase the energy loss due to the yawing oscillations, in addition to the resistive dissipation, but contribute to the directional stability of rectilinear motion.

New Kinematic Data From High Speed Cine Film Recordings of Swimming Cod (Gadus Morhua)

1. We described a method to determine the values of the parameters of the body wave: speed v, wavelength λ b , wave period T b and of the sinusoidal track in space of different points of the body of a swimming fish: speed u, wavelength λ s , wave period T s and amplitude A. 2. These characteristic wave parameters were measured from high speed cine recordings of straight forward swimming behaviour of one 0.42 m Cod. 3. The instantaneous propulsive force of the tail was determined using LIGHTHILL'S (1971) large amplitude elongated body theory. 4. The same theory predicts and data in this paper affirm, that, to increase propulsive force per unit virtual mass, a fish should increase the lateral speed of the tail tip and should keep the angle between tail blade and direction of movement not far from values between -10° and + 10°. 5. Results show that swimming at higher speeds as well as acceleration is always attended with an increase of v as a result of a decrease of the period of the waves of the body. The body wavelength λ b does not change with speed. The wavelength of the tracks in space, λ s , slightly decreases with an increase of speed. 6. Relationships between some of the characteristic features of the cycles of swimming movements are discussed. With the aid of a simple model, limits of swimming abilities of the fish studied could be determined. 7. We came to the conclusion that propeller efficiency during steady swimming at cruising speeds decreases with increasing speed.

Large-amplitude elongated-body theory of fish locomotion

The elongated-body theory of the reactive forces on a fish moving in water (that is, forces resulting from the inertia of associated water movements) is extended so that a prediction of instantaneous reactive force between fish and water is obtained for fish motions of arbitrary amplitude, regular or irregular (§2). A preliminary application of the theory to the balance of reactive thrust and resistive drag in regular carangiform swimming of fishes with slender caudal fins is made (§3). Comparison with data (Bainbridge 1963) on the dace Leuciscus suggests that an important feature of this balance may be a substantial enhancement of drag for such fishes when swimming movements commence, an enhancement here interpreted in terms of a boundary-layer-thinning mechanism first suggested by Dr Quentin Bone.

Aquatic animal propulsion of high hydromechanical efficiency

Aquatic animal propulsion of high hydromechanical efficiency