A blade-element approach is used to analyse the mechanics of the dragbased pectoral fin power stroke in an Angelfish in steady forward, rectilinear progression.
Flow reversal occurs at the base of the fin at the beginning and at the end of the power stroke. Values for the rate of increase and decrease in the relative velocity of the blade-elements increase distally, as do such values for hydrodynamical angle of attack. At the beginning and end of the power stroke, negative angles occur at the base of the fin.
The outermost 40% of the fin produces over 80% of the total thrust produced during the power stroke, and does over 80 % of the total work. Small amounts of reversed thrust are produced at the base of the fin during the early and late parts of the stroke.
The total amount of energy required during a cycle to drag the body and inactive fins through the water is calculated to be approximately 2·8 ×10−6 J and the total energy produced by the fins over the cycle (ignoring the recovery stroke) which is associated with producing the hydrodynamic thrust force, is about 1·0×10−5 J ; which gives a propulsive efficiency of about 0·26.
The energy required to move the mass of a pectoral fin during the power stroke is calculated to be approximately 2·6 ×10−7 J. Taking this into account reduces the value of the propulsive efficiency by about 4% to about 0·25. The total energy needed to accelerate and decelerate the added mass associated with the fin is calculated and added to the energy required to produce the hydrodynamic thrust force and the energy required to move the mass of the fins; giving a final propulsive efficiency of 0·18.
The kinematics, hydrodynamics and energetics of the swimming of teleosts in which undulations of the body and caudal fin are the main means of locomotion are relatively well understood (see Lighthill, 1969; Webb, 1975 a for reviews). However, little is known about the swimming of teleosts employing other mechanisms of propulsion and so we are not yet in a position to fully appreciate the diversity of strategies (mechanical, ecological and evolutionary) employed in the swimming of teleosts.
This study analyses the use by an Angelfish of the pectoral fin in locomotion. It is concluded that many simplifying assumptions can be made, and such an approach will be used in future papers to describe the influence of the gross morphological and kinematic parameters (size, fin-beat rate, etc.) on the speed and efficiency of labriform locomotion.
THE PECTORAL FIN-BEAT CYCLE IN THE ANGELFISH
During steady, slow forward rectilinear progression the Angelfish is propelled by the alternate ‘rowing’ action of its pectoral fins; the caudal fin is not active. The median long axis of the caudal fin coincides with that of the body, which is held ‘rigid’.
The wedge-shaped pectoral fins are composed of nine fin rays, separated by a highly flexible membrane. During the power stroke the long axis of the fin base makes a high angle (45–50°) with the horizontal. The dorsal fin rays are inclined caudad relative to the long axis of the fin so that the distal two-thirds of the fin makes an angle of about 90°with the horizontal.
The membrane between the fin rays is taut during the power stroke, probably due to the contraction of the fin ray inclinator muscules. The hydrodynamic force on the membrane would tend to bow the membrane between fin rays; this effect was not seen. Branching and jointing of the fin rays is restricted to the distal half of the fins where slight bending occurs. The phase difference between the most dorsal and ventral fin rays is small and the fin rotates as a unit about the median long axis of its base.
At the end of the power stroke the longitudinal axis of the fin base is inclined at about 20° to the horizontal, the anterior fin rays (morphologically dorsal) are declined ventrally about the long axis of the fin so that the distal two-thirds of the fin makes a very low angle with the horizontal and the fin (now feathered) moves forword (see Fig. 1).
An Angelfish (length from the tip of the snout to the distal end of the caudal fin = 0·08 m) was filmed from above whilst swimming steadily at a forward velocity (V) of 0·04 ms−1(V is treated as a constant once it fluctuates over a complete cycle by less than from its mean value) in a tank (1·5 m ×0 ·5 m ×0 ·5 m; maintained at 26 °C) which had a grid (2·5 cm squares) marked on the bottom. The tank was illuminated by five 1000 W quartz-iodide lamps. A John Hadland ‘Hyspeed Camera’ (Model H20/16) was used, mounted on a large pillar stand. Pan F, 16 mm film was shot at 500 frames s−1 at f 2·8. After processing the film was viewed frame by frame on an analytical projector (Vanguard Instrument Corporation Motion Analyser) and sequential tracing made of the fin motion from the image on the viewing screen.
Film of one representative power stroke was chosen for analysis. The left side pectoral fin analysed moved from a postional angle (γ, the angle between the projection of the fin on to the horizontal plane and the median axis of the body; see Fig. 1) of about 110 ° to 20 ° in a time (tp, the time of the power stroke duration) of about 0·1 s. The variation in the angular velocity (ω, the angular velocity of the projection of the fin on to a horizontal plane) of the fin during the power stroke is shown in Fig. 2.
For the purpose of analysis the fin has been divided into four arbitrarily defined blade-elements (e1-e4); the lengths of which (1) were measured perpendicularly from distal border of one element to the proximal border of the next. The midpoints of measured perpendicularly from the base of the fin), values for the chord (c) at r and the mass of the elements (me) are shown in Table 1.
Relative velocities and the hydrodynamical angle of attack
During the initial phase (up to t = 3·0 ×10−2 s) of the power stroke αa has negative values at e1. From t = 3·0–7·6 ×10−2 s positive values occur; however at the end of the stroke small negative angles of αa recur.
THE FORCE BALANCE
During the power stroke a pectoral fin can be considered as being made up of a series of three-dimensional flat plates (the blade-elements) inclined at high angles (αa) to the incident flow. Experiment has shown (Fage & Johansen, 1927; Wick, 1954; Hoemer, 1958, p. 3.16) that at high (>103) Reynolds numbers (Re = υl/v, where υ is a velocity, I a characteristic length and υ the kinematic viscosity of the fluid) where inertial forces dominate, the normal force coefficient remains approximately constant at about 1 · 1 for αa±45° from the 90°position. The pectoral fin is operating at Re of the order of 103–104 (with v based on the relative water velocity) with aa between 40 and 90°, for most of the fin for most of the power stroke and therefore a value of Cn = 1 · 1 has been used in calculating dFn when αa lies between 40 and 90°.
Values of the drag force on the body and Cb (based on the total wetted surface area) were also determined experimentally in the dead fish (weight: 1·6 × 10−0 kg in water). Small lead pellets were placed in its mouth, and the pectoral fins were placed open, perpendicular to the median long axis of the body. The animal was then dropped into a tank (1·33 m high × 0·5 m wide × 0·5 m breadth) of water and filmed (at 64 frames s−1) against a grid (2·5 cm squares) as it fell. Pellets were added until a terminal velocity of descent approximately equal to V was obtained. A mean value of 0·041 ms−1 (n = 17, s.D. = 0·0082) was obtained for the terminal velocity; corresponding to a drag force of 3·24 × 10−4 N and a drag coefficient for the body of 0·086.
When the fins were amputated and the fish dropped down the tank again, a mean value 0·0406 ms−1(n = 12, s.D. = 0·012) was obtained for the terminal velocity; giving a value of 7-22 x10−s N for the drag force on the body and 0·02 for Cb (based on the total wetted surface area).
WORK, POWER AND EFFICIENCY
The effect of hydrodynamic ‘added mass’
The acceleration (a) that each blade-element imparts to its associated added mass has been calculated (from the slopes of the curves in Fig. 2 and is considered to be of positive sign for both acceleration and deceleration phases of the stroke) and multiplied by ma to obtain the force required to accelerate and decelerate the added mass (Fig. 8). Due to the complications produced by flow reversal at the base of the fin, e1 is not considered.
During the early part of the stroke (up to t = 2·0 ×10−2 s) the added mass force acts in the direction of the fin’s motion. After t = 5·0 ×10−2 s (when the hydrodynamic thrust force and added mass force are equal) the added mass force acts in the opposite direction. The impulses of the added mass forces cancel over the stroke (see Fig. 8).
In calculating Cn we have assumed :
That all of the drag force is due to pressure drag.
That the fin can be likened to a series of three-dimensional flat plates set at high angles of attack relative to the incident flow.
For αa> 40° (true for most of the fin for most of the power stroke duration time) it is reasonable to assume that the flow has separated from the rear surface of the fin and that the drag force is almost entirely due to pressure drag, while the skin friction can be neglected (Prandlt & Tietgens, 1957, p. 90).
It could be argued that elements e1-e3 are not equivalent to three-dimensional flat plates as they are bounded (e1 by the side of the body and the proximal border of e2, e2 by the distal border of e1and the proximal border of e3 and e3 by the distal border of e2 and the proximal border of e4) and that only e4 experiences a flow regime similar to that of a three-dimensional flat plate in free-flow. However, as e4 is responsible for over 80% of the thrust and work produced by the fin the application of one value of Cnto 04 and another to e1–e3 (based on curve A in Fig. 6 for instance) makes little difference to the final results.
Laminar boundary layers are relatively prone to separation as the transfer of momentum from the relatively rapidly moving outer flow to the slower moving fluid near the body surface is slow and ineffective. The broad based pectoral fins could disturb and locally disrupt the laminar boundary layer causing it to separate, in a region downstream of the pectoral fins, thereby increasing the drag on the body. The drag of circular and square flat plates set at high angles to the incident flow does not differ appreciably from free-flow values when they are attached to streamlined bodies; however, the drag of the streamlined body can be increased by a factor of five if the plate is placed (as the pectoral fins are) at the shoulder (Hoerner, 1958). The inactive pelvic fins could also produce a degree of interference.
Although it is not the subject of this paper we may note another mode of swimming, using the caudal fin, that is employed by the Angelfish for high speed movements. In this mode the pectoral fins are tightly folded against the body, and may well produce no interference drag. The coefficient of the body may then well be as low as 0·01.
In most discussions of aquatic animal locomotion, consideration is confined to rectilinear locomotion at constant speed. In steady motion the fluid acceleration is zero and therefore no added mass forces arise. However, the force-producing surfaces which produce the exchange of momentum with the surrounding water are subject to unsteady motion, even when propelling a body forward at a constant speed. Many if the thrust-producing devices exployed by aquatic animals (e.g. the parapodia of the Polychaeta, the metapodial limbs of the Dytiscidae, Gyrinidae and Hydrophilidae, the limbs of some aquatic Heteroptera and Trichoptera, the paddle-like limbs of the Portunid crabs, the antennae used as oars in nauplii, some Conchostraca, Cladocera and Ostracoda) are hydrodynamically ‘bluff-bodies’ which have a large associated added mass in unsteady motion.
The retarding effect of the entrained added mass greatly reduces the efficiency of a paddling appendage, by increasing the amount of energy required to pull it through the water. The energy required to produce the hydrodynamic force of a pectoral fin is of the same order as that needed to accelerate and then decelerate the entrained added mass. The propulsive efficiency of the fin ignoring fin inertia and added mass is calculated to be 0 · 26. When they are taken into account a value of 0 · 18 is calculated, a reduction of about 31%.
Webb (1973) has described the kinematics of a lift-based mechanism of labriform locomotion in Cymatogaster aggregata. On the basis of respirometric data and an analogy with the hovering flight of birds, Webb (1975b) concluded that the work required to rotate the pectoral fins would be high. This study indicates that this is probably not the case: as the mean energy required to move the actual mass of the fin is only about 7 % of the value of the added mass term for the specimen of P. eimekei studied here. Although respirometric data gives valuable experimental data on the total energy cost of locomotion in animals, it provides little information as to the mechanical principles involved.
In calculating the energy required to produce the hydrodynamic drag force, the force needed to overcome the mass of the fin and the added mass we have assumed that no interactions with the lateral sides of the body occur. The value of η ″ = 0·18 should be regarded as a lower limit as it is possible that the influence of the sides of the body aids the deceleration of the fin and its associated added mass.
In the final stages of the power stroke the fin can be regarded as a decelerating body which is approaching a relatively large boundary and therefore study of the effect of a small boat decelerating as it approaches the side of a large ship and the effect of a ship decelerating as it enters shallow water is relevant. The analytical studies of Koch (1933) and Prohaska (1947) indicate that the added mass of the smaller body should increase when the kinetic energy in the unsteady flow around it is increased as it approaches the larger body. However, it has been found experimentally (Saunders, 1957) that a decrease in added mass occurs around a small tug as it decelerates when approaching a large ship.
Not all teleosts which swim in the labriform mode employ the drag-based mechanism for direct thrust production; many (e.g. Serranidae, Scorpididae and Scaridae) ‘clap’ their pectoral fins against the sides of their body to create backwardly directed jets which propel the animal forward (to be discussed in a future paper).
Little data is available on the propulsive efficiency of fusiform fish, swimming at low speeds. Webb (1971) considered the speed ratio V/Vw (forward swimming speed/backward speed of the propulsive wave) to be representative of the propulsive efficiency of undulatory swimming and found values of about 0 · 05 for trout (Salmo gairdneri) swimming at about 0·05 m s−1 (Re = 1·5 × 104); a value about three and one half times less than that calculated for the Angelfish at a similar speed and Reynolds number. If we assume that the two measures of propulsive efficiency and directly comparable, we can conclude that the pectoral fin drag-based mechanism propulsion is an adaption to slow swimming, when the efficiency of the undulatory mode is very low. However, considerations other than energetics could e×plain the use of the pectoral fins in low speed swimming. Were the Angelfish to swim in the undulatory mode at low speed it would be more conspicuous to predators and less manoeuvrable.
Although drag-based mechanisms of propulsion are common among the aquatic invertebrates, little detailed work has been done. Nachtigall (1961, 1977) has described the kinematics of rowing in water beetles swimming at Reynolds Numbers of the same order as the Angelfish studied here (5·0–8·0 ×103). He estimated a propulsive efficiency of 0 · 3 for the metapodial limbs of Acilius sulcatus during the power stroke, based on the fraction of the total impulse available for propulsion and losses due to vorticity. His estimate is of the same order as η for the Angelfish (0·26), but was not calculated on the same basis.
Clark & Tritton (1970) analysed the dynamics of parapodial swimming in certain polychaetes ; however, their model was not used to estimate propulsive efficiency. Lochhead (1961, 1977) has reviewed the literature on the locomotion of the Crustacea and has pointed out that much work has to be done before we understand the drag mechanisms of propulsion many of them employ.
I would like to thank Dr K. E. Machin, Professor Sir James Lighthill, F.R.S. and Mr C. P. Ellington for their advice and encouragement. I am grateful to Mr G. G. Runnalls for his expert advice and assistance with photography and the N.E.R.C. for financial support.
Quantities relating to a pectoral fin blade-element.
ei denotes the ith pectoral fin blade-element
Ne the number of elements
I length of an element
r distance from the base of the fin to the midpoint of an element
c chord of the fin at the midpoint of an element
vn normal velocity component of an element relative to the water
vs spanwise velocity component relative to the water
v resultant relative velocity for an element
v average resultant relative velocity
a acceleration of an element
αa hydrodynamical angle of attack of an element
dA area of an element
Cn normal force coefficient of an element
dFn normal force acting on an element
me the mass of a blade-element
ma the added mass of an element
dT forward directed component of thrust acting on an element
dW rate of working of an element
W power required to produce the hydrodynamic force on an element
Wa power required to accelerate the added mass of an element
Ef the energy required to move the mass of a blade-element
ρ the water density
R the total length of the fin
ω angular velocity of the fin
y positional angle of the fin
Pt impulse of the thrust acting on the fin
mean power produced during the power stroke
tp time of duration of the power stroke
t0 total fin-beat cycle time
V velocity of the body
Sw total wetted surface area of the body and inactive fins
Cb drag coefficient of the body
k a constant concerned with the normal force coefficients
Pb impulse of the drag force acting on the body
Eo energy dissipated during a cycle in dragging the body and inactive fins through the water
E(tot) total energy required to produce the total hydrodynamic thrust force
Eα (tot) total energy required to move the added mass of the fin
Ef(tot) average total energy required to move the mass of the fin
η propulsive efficiency (excluding inertial terms)
η ′propulsive efficiency (including the effect of fin inertia)
η ″propulsive efficiency (including the effect of fin inertia and added mass)
Re Reynolds number