## ABSTRACT

Thrust and drag were calculated for ten bluefish using Newton’s law,

*F = ma*, the measured acceleration of the fish, and one essential assumption, without which only the sum of thrust and drag could be determined. Any time the thrust is zero, the drag equals the mass of the fish times its negative acceleration. For reasons given in the full text, the thrust is believed to drop to zero twice per swimming cycle, and never become negative. Zero thrust thus occurs at the negative acceleration peaks. For acceleration waveforms that are symmetric, zero acceleration lies at a level half way between the levels of the positive and negative peaks, so the peak negative acceleration is half the peak-to-peak height of the waveform. It is further assumed that the instantaneous drag thus measured at the negative acceleration peaks equals the mean drag averaged over the whole swimming cycle, which in turn equals the mean thrust.Turbulence, induced in the water tunnel by two flat metal plates placed in the stream in front of the fish, or by rotation of a mixing rod, did not change the drag, or the fish’s ability to produce thrust, as indicated by the maximal swimming speed. The attachment of two pressure transducer enclosures to the tail did not alter the maximal swimming speed.

Changes of buoyancy were produced by inflation or deflation of balloons sewn into the swim bladders of three fish. The thrust calculated from fluctuations in the forward acceleration of the fish measured at zero speed during neutral buoyancy was 0·0 to 0· 14 N. If the non-zero values are genuine, they must be balanced by rearward forces from the pectoral fins. The thrust increased linearly with speed at the rates of 0· 40, 0· 64, and 0· 71 N per m.s

^{•1}, respectively, at speeds between 0 and 0· 8 m.s^{−1}.The mean tail thrust of a negatively buoyant fish at zero water speed equalled or exceeded the body’s weight in water. Its forward component must be balanced by a rearward component of the lift force from the flapping pectoral fins. Upward components of the two forces support the weight of the fish. With 45 ml air removed from the swim bladder of a bluefish, the tail thrust diminished from 0·64 N at zero water speed to a value of 0·46 N as the water speed was increased to between 0· 2 and 0· 4 m.s

^{−1}, and then increased to 0· 71 N at a water speed of 1 m.s^{−1}.

## INTRODUCTION

The forward thrust generated by a fish tail during swimming is opposed by the force of drag on the body. This drag force depends upon the size and shape of the body, intensity of turbulence, and on the weight of the body in water. Denil (1936) studied the drag of wooden fish models. Richardson (1936) reported drag measurements obtained from dead or anaesthetized fish. Hoerner (1965) presented drag coefficients for rigid, streamlined bodies of revolution during laminar, transitional and turbulent flow. Bainbridge (1961) utilized these equations to compute the theoretical maximum speed of different fish in laminar or turbulent flow. Lighthill (1971) calculated the drag of a swimming fish from data presented in the literature. Lang (1975) measured the drag of a porpoise at one swimming speed. Webb (1975) has summarized literature available on body drag. DuBois, Cavagna & Fox (1976) measured the thrust and drag of swimming bluefish by use of an accelerometer planted in the body. Later, DuBois & Ogilvy (1978) compared the thrust generated by the tail with the force calculated from acceleration of the body. Three questions arose as a result of this last study. The first was whether the pressure transducers sewn on the tail interfered with the tail’s mechanical efficiency, limiting the maximal speed of the fish. The second was whether a wire grid placed in front of the fish induced turbulence in the water, changing the drag of the body or the efficiency of the tail. The third question was how much of the drag was attributable to the weight of the fish in water (negative buoyancy).

In the present investigation we induced turbulence, measured its intensity, and determined its effect on drag and on the maximal swimming speed of bluefish. Drag and maximal speed also were measured after pressure transducer enclosures had been attached to the tail. We compared the drag of a wooden model which had been carved in the shape of a bluefish with the drag of a swimming, flexing bluefish, in flow with differing intensities of turbulence. The effect of changing the buoyancy of bluefish upon their thrust was measured, and from this the effect of buoyancy on drag was calculated. The fish’s weight in water was resolved into a component aimed aft along the fish’s track and one at right angles to this aimed downward through the fish’s belly. The first was subtracted from the tail thrust, leaving the remainder of the thrust to balance the drag. The second was balanced by lift from the pectoral fins. From this the lift coefficient of the pectoral fins could be calculated, and from it their induced drag.

## METHODS

Ten bluefish which ranged from 0· 47 to 0· 55 m in fork length and 1· 13 to 1· 47 kg in weight were used in this study (Table 1). The fish had been caught on a hook and line and kept in a tank which was provided with running sea water. On the day of the experiment, each fish was anaesthetized by immersion for 10 –15 min in a solution consisting of 2 g tricaine methanesulphonate (Aldrich Chemical Co.) dissolved in 40 l sea water. Then the fish was placed on a V board, and the anaesthetic agent, diluted to half its previous concentration, was circulated through the mouth and gills A small midline incision was made in front of the anterior dorsal fin, and a pair miniature accelerometer (Entran Devices, model EGBL-125-5D), glued together at right angles, with a guide pin facing forward, was inserted to a depth of 1 cm, and sewn in place with the guide pin at the skin surface to provide orientation. Leading from the accelerometers was a flexible four-conductor cable (New England Wire Co.), 1 mm in diameter. This emerged from the rear of the incision and was connected to a 6-channel direct writing recorder (model 7 Polygraph, Grass Instrument Co.).

The fish was lowered through a handport into the observation chamber situated at the centre of a wooden tunnel 5· 48 m long, octagonal in cross-section, 0·30 m between its walls, and sloping at an angle of 33·5 ° below horizontal. The tunnel was filled with sea water from a pool at the top of the tunnel. Waterflow from the pool through the tunnel was regulated by a wooden door hinged on the bottom end of the tunnel. This door was closed by tightening a steel cable with a handwinch. A monofilament line was attached to the fish jaw to prevent the fish from turning around, but this line remained slack during the swimming experiments.

Four of the fish were removed from the wooden tunnel after an initial series of accelerometer records had been made. They were re-anaesthetized, and pressure transducer enclosures, consisting of small, flat balloons glued to thin aluminium plates, were sewn to both sides of the tail. These fish were studied in the water tunnel again to determine whether the tail attachments changed their body drag or maximal swimming speed. Water speed during a run was calculated from the ratio of area of the pool surface to the cross-sectional area of a 30 cm diameter circular section of the tunnel above the fish, and measurements of rate of change of pool depth measured from the rate of decrease of pressure in the bottom of the pool and of lateral pressure at the section in the tunnel where the fish swam. The slope of this pressure drop was measured from the Grass records, and the volume of water passing the fish per second was calculated. This was then converted to water velocity in units of m.s^{−1}. The Pitot pressure calculated from the mean flow equalled the Pitot pressure measured on fee stagnation point of the pressure sphere used to measure turbulence (see below), apparently, the slower boundary layer flow was just accommodated by the difference between the cross-sectional area calculated for a circle of 30 cm diameter and the actual octagonal shape of our tunnel, 30 cm between walls.

For determination of the amount of extra thrust needed to overcome negative buoyancy and the induced drag that it causes, three fish were equipped with a latex rubber balloon placed in the opened swim bladder through a midline incision in the abdominal wall. A 1· 9 mm outer diameter (PE 200) polyethylene catheter led from the balloon to a stopcock and syringe outside the water tunnel. Thus, buoyancy of the fish could be changed between swimming runs. Neutral buoyancy was determined while the fish was in the anaesthesia tank by inflating the rubber balloon, starting from empty, until the fish just floated, or reducing the volume until the fish just sank. Similar observations could be made in the water tunnel while the fish was resting between runs. Since the fish was 2· 4 –3· 0 m below the pool surface, 24 – 30 % more air, measured at 1 atm, was needed to reach neutral buoyancy in the tunnel, though the geometric volume was the same under both conditions.

A finless wooden model 61 cm long was carved in the shape of a bluefish which had a fork length of 63 cm. Lead weights were placed inside the model until it was neutrally buoyant in sea water. The drag of the model was measured using a force transducer (Grass) at different water speeds in the tunnel. The wooden model was mounted at the centre of the tunnel using stainless steel needles embedded longitudinally in the front and rear. These needles were free to slide fore and aft in screw eyes supported in the centre axis of the tunnel by thin metal rods attached through the wall of the tunnel. A thin monofilament line connected the front of the model to the force transducer located above the pool surface. To overcome sliding friction, which was minimal, we touched the rods to vibrate the screw eyes momentarily, then let the model come to rest at a position in which force of drag was balanced by the tension on the force transducer.

### Induction of turbulence

In previous studies (DuBois *et al*. 1974, 1976; DuBois & Ogilvy, 1978), grids cut from galvanized wire fencing with 1· 27 cm spacing of 1 mm wires had been located in front of and behind the fish. Calculations showed that the amount of turbulence produced by these grids was small (see DuBois & Ogilvy, 1978). In the current study, the amount of turbulence downstream from one of these grids was determined by measuring the angle of dispersion of a stream of ink, in order to check the calculated values. After these tests, the water tunnel was modified to allow induction of additional turbulence.

The fish compartment was 1 m long. Its back retaining grid was composed of the mesh described above. In front of the fish, a new screen was made of fine fishing wire o-6i mm in diameter with 5·1 cm between wires. This screen produced even less turbulence than the previous one. A handport was installed 12· 5 cm upstream from this screen.

With the aid of this new handport, two methods were available for induction of turbulence. Two flat plates 8· 5 cm apart, on centre, were installed with their upper ends hinged to the handport 12· 5 cm in front of the fine wire grid, so that the free ends could be lowered by wires from a position flat against the upper surface of the swimg ming tunnel to a position transverse to the water flow. The plates were 29· 2 cm long Bid 1· 9 cm across. They could be controlled from the outside of the swimming tunnel. A second method for induction of turbulence involved the use of a variable-speed portable mixer or ‘egg beater’ motor (General Electric 120 W 5-speed mixer). This rotated a 4· 8 mm diameter mixing rod, bent in a ‘zig-zag’ shape. A 2· 54 cm hole was drilled in the handport and fitted with a rubber stopper which served as a seal around the top of the mixing rod. A footing was mounted on the floor of the tunnel as a bearing for the bottom of the rod. The energy drawn by the mixer, measured by a wattmeter, was 105 W. When rotated, the bent rod swept a convoluted volume 30 cm tall with a maximum diameter of 10· 2 cm.

### Measurement of turbulence

Turbulence in a tunnel may be characterized by its intensity, scale and persistence downstream from a grid (Dryden *et al*. 1936). The fractional intensity is expressed as √ *(v*^{2}*)/U* where the numerator is the root mean square of the cross-stream component of the fluctuation velocity, and *U* is the average speed of the stream. A length, *L*, characterizes the scale of the turbulent motions. *L* near a screen is about the same as the wire size of the screen. Dryden measured the rate of decay of intensity of turbulence downstream a distance *x* from a screen whose wires were spaced by a distance *M*, centre to centre.

Dryden gave divergence angle, in degrees, downstream from a wire as angle = 134·9√(*v*^{2})/*U*. In other words, percentage turbulent velocity is the divergence angle in degrees divided by 1· 35. We measured this divergence angle using a solution consisting of ink in 50% glycerine and 50% water, injected slowly through a catheter placed in line with the stream.

Dryden also used a pressure sphere (his Fig. 12) to find the critical Reynolds number of a given tunnel. The pressure was measured at holes drilled in the front and at 22 ·5 ° off centre from the back. The critical Reynolds number occurred when the pressure coefficient (front pressure minus back pressure, divided by Pitot pressure) was 1· 22, corresponding to a drag coefficient of 0 ·3. We made a similar pressure sphere from a bowling ball 11· 4 cm in diameter, mounted it in the water tunnel with the front of the sphere 32 cm downstream from the grid (44 ·5 cm from the hinged plates), and measured the pressure coefficient at different water speeds. The critical Reynolds number was measured for different intensities of induced turbulence. This test measures directly the effect of tunnel turbulence in causing the boundary layer on the sphere to become turbulent at a lower Reynolds number than it otherwise would. The greater the turbulence the lower the critical Reynolds number is likely to be. Note that the length used in calculation of Reynolds number for a fish is the fork length, in contrast to length used for the bowling ball, namely its diameter.

### Calculation of body drag

It is given from experimental results that when a fish swims at a constant mean velocity a rhythmically recurrent pattern of acceleration is recorded using body accelerometers. The following procedure is used to calculate body drag. (1) When mean velocity is constant, mean acceleration is zero. This gives equal areas under the turves above and below zero for the recurrent patterns of acceleration and decelerate on. Therefore, zero on the accelerometer record is taken to be the height above and below which the areas are equal. (2) It is assumed that thrust is zero when the acceleration is minimal (most negative). At this point in the body cycle, drag is the only force acting on the fish. Therefore drag = mass × deceleration. Evidence that thrust is zero when acceleration is minimal is that the tail angle relative to water flow is zero, and pressure difference between the two sides of the tail is zero at this moment (DuBois & Ogilvy, 1978). (3) The mass accelerated is taken as the mass of the fish, neglecting any additional virtual mass of entrained water. (4) It is assumed that the centre of mass is not altered by bending of the body. (5) The accelerometer is assumed to remain well oriented throughout the body cycle. (6) For accelerometer records which are symmetrical with respect to time above and below zero, the magnitude of peak to valley variation in acceleration is measured and divided by 2. The magnitude of the peak instantaneous deceleration (deceleration is maximum when thrust is zero) is thereby obtained. (7) Mean drag is assumed to be equal to the instantaneous drag at this moment for the purposes of this paper. Note that the present analysis, as described above, no longer converts peak height of a sine wave to root mean square height, as we once thought necessary (Du Bois *et al*. 1976). In other words, we no longer divide by √2. Reasons for this will be given in the Discussion.

Previously, if the forward accelerometer record had an asymmetrical pattern, the accelerometer output was integrated electrically to obtain a velocity record, whose peak-to-peak amplitude and frequency were used to calculate peak-to-peak acceleration (see pp. 233–4, DuBois *et al*. 1976). In very asymmetrical accelerometer records, the frequency of the velocity record was equal to body frequency.

In the present paper mean thrust was calculated from very asymmetric accelerometer records in the following way. Some records show a small upward peak and a large upward peak. These two are averaged. Some also show a small downward trough and a large downward trough. These troughs are averaged. A third horizontal line could be drawn midway between the two that show the means of peaks and troughs, to divide the areas above and below it equally. This midline would represent zero acceleration. Then (see Fig. 1), the difference between the mean of the peaks and the mean of the troughs is measured. This value is divided by 2 to yield the deceleration used in calculation of instantaneous drag when thrust is zero. This method avoids using an electrical integrator.

## RESULTS

#### Accelerometer records

The justification for the analysis of asymmetrical accelerometer records is shown in Figs. 1A and B. Symmetric and asymmetric swimming patterns are compared during the same run. The average peak-to-valley height of the asymmetrical pattern was equal to that of the symmetrical pattern in the same run, which provides support for the method used to analyse the asymmetric records.

#### Induced turbulence

The critical Reynolds numbers resulting from the different methods of inducing turbulence are shown in Table 2.

Percentage turbulence in the swimming compartment was calculated by dividing the divergence angle of a stream of ink by 1·35. When the plates were out of the flow there was 2· 8 % turbulence at 0· 18 m.s^{−1}, 4 · 2 % at 0· 38 m.s^{−1}, and 5·6% at 1· 6m.s^{−1}. The two metal plates in the stream caused 8· 4 % turbulence at 0· 18 m.s^{−1}, 7% at 0· 38 m.s^{−1}, and 12· 4% at 1· 06 m.s^{−1}. The zig-zag rod, rotating at its fastest speed, produced 12 · 4 % turbulence at 0· 25 m.s^{−1}, and 9 · 7 % at 0· 78 m.s^{−1}.

#### Maximal swimming speeds

The water flow past the fish was controlled by a hand crank which closed or opened the outlet door. The speed was increased until the fish could not keep up with the water flow. Different fish had different maximal swimming speeds, but their maximal speed was not changed by introduction of the metal plates into the stream, or by turning on the motor which rotated the rod at low, medium or high speed (Fig. 2). Sewing the pressure transducer enclosures to the tail did not reduce the maximal swimming speed, as can be seen in Table 3.

#### Total body drag in different intensities of turbulent flow

The drag was calculated from the accelerometer record and plotted against swimming speed for six fish (Fig. 3). At any given speed the thrust, and thus the drag, were unaffected by the induction of more turbulence in the water. Two regression lines are drawn for each fish in Fig. 3. The dashed line represents the regression of drag with respect to speed for the points obtained with induced turbulence, while the solid line represents the regression for points obtained without induced turbulence. For each fish, the dashed and solid regression lines were compared using a two-tailed *F* test to determine whether the residual variances, slopes or elevations differed significantly (Snedecor & Cochran, 1967). In each of the six fish, each of the three *P* values (residual variances, slopes, or elevations) was greater than 0·05, showing that the dashed and solid lines for any one fish were not significantly different. The dissimilarity of the slopes of the solid lines for different fish in Fig. 3 led to the search for an explanation, which in turn resulted in the insertion of balloons in the fish (fish 8–10) to control their buoyancy. The outcome suggested that the differences between slopes for different fish in Fig. 3 may have been due to different buoyancies in individual fish, the more horizontal the line, the greater the assumed negative buoyancy. The results from fish 4 are not plotted in Fig. 3 because this fish did not swim fast enough to allow a complete analysis of the thrust *vs*. speed relationship.

#### Drag of a model of a bluefish

The drag of the wooden model increased with velocity, the curve being concave upwards (Fig. 4). It happened that the drag in newtons was about equal to the speed (in m. s^{−1}) squared, up to a speed of 0·8 m. s^{−1}. Drag data obtained after the two metal plates had been lowered into the stream to induce turbulence were indistinguishable from those with the plates lifted up out of the stream during the run. For example, at 0· 15 m.s^{−1} the drag was 0· 03 N, plates up or down. At 0· 70 m.s^{−1} drag was 0· 48 N both ways. At 1· 21 m.s^{−1}, the drag was 1· 12 N whether the plates were down the stream or raised out of the stream.

#### Coefficient of drag vs. Reynolds number

A common method for reporting body drag is to plot the drag coefficient against the Reynolds number. The dimensionless drag coefficient (*C*_{D}) is defined as *C*_{D}*= Fd/0·5* (*ρ**U*^{2}) *(S)*, where *F*_{D} is the drag in newtons, *ρ* is the density of sea water *U* is the velocity of the water, and S is the total wetted surface area of the fish. Reynol number, *(N*_{R}*)*, is a function of speed and is dimensionless. *N*_{R}*= Ul/v*, where *U* is the velocity in m.s^{−1}, l is the length of the fish in m, and *v* is the kinematic viscosity (viscosity of the water (μ) divided by the density of the water (ρ)) in m^{2}.s^{−1}. A loglog plot of these values for fish 8 at neutral buoyancy is shown in Fig. 5. Also plotted on this graph is a dashed line which represents an average of the data obtained from the wooden model. It was calculated using values obtained from Fig. 4.

#### Induced vs. parasitic drag

The amount of parasitic drag depends on pressure drag and skin friction, and it would be about the same whether the fish were neutrally buoyant or negatively buoyant, and whether the fish were swimming horizontally or upward. When the fish is negatively buoyant, the fins incur an induced drag that is the backward component of the force perpendicular to the direction of flow over the lifting surface. Both parasitic and induced drag must be overcome by the component of thrust directed forward along the track. If the body is aimed at an angle upward, above the line of the track, the lift and part of the thrust balance the downward force of the weight. The remainder of the thrust balances induced and parasitic drag.

#### Measurements of parasitic drag at neutral buoyancy

Accelerometer records were obtained on fish 8, 9 and 10 at different buoyancies, produced by inflating or deflating the balloon in the swim bladder. The thrust of fish 8 at neutral buoyancy was o-o at zero speed, and it increased linearly to 0·69 N at 1 m.s^{−1} (Fig. 6). At neutral buoyancy, fish 9 and 10 both had a force of 0· 14 N at zero speed, with slopes of 0· 40 and 0· 64 N/(m.s^{−1}) respectively, over a range of speeds of zero to 0· 8 m.s^{−1}.

#### Measurements of thrust at a 33·5 ° angle during negative buoyancy

When air was removed from the balloon in the swim bladder, the fish tended to sink, and had to make swimming efforts with the tail and pectoral fins to avoid hitting the bottom. The thrust measured at zero speed in fish 8 was about 0·64 N or 65 g when 45 ml had been removed from the neutrally buoyant fish (Fig. 6). As swimming speed increased, this force diminished to a minimum at around 0· 3–0· 4m.s^{−1}, then increased again, approaching the slope of the curve describing force *vs*. speed during neutral buoyancy, a slope of 0·68 N/(m.s^{−1}). This same phenomenon occurred in reduced degree when 30 and 15 ml of air had been removed from the swim bladder of the neutrally buoyant fish (see Fig. 6). Although Fig. 6 presents data only on fish 8, data from fish 9 and 10 while negatively buoyant are superimposable on those obtained on fish 8.

The lower left-hand panel of Fig. 6 represents data of bluefish 8 while neutrally buoyant and with 15, 30 and 45 ml air removed from the swim bladder. The points have been corrected by subtraction of the values for *F*_{w} sin θ. The dashed line representing the mean curve of thrust *vs*. speed when bluefish had 45 ml air removed from the swim bladder is used in a sample analysis (F_{T},-45) in the lower right-hand panel of Fig. 6 (see Discussion). These conditions are directly analogous to a climbing airplane (see p. 238, Carter, 1932). At zero flow, the thrust of the bluefish does nothing but support some of the weight. The flapping pectoral fins support the rest of the weight, and cancel the horizontal component of the tail thrust. As the speed increases, lift is obtained from the body’s angle of attack and from the fins, *F*_{Di} decreases, *F*_{w} sin θ remains the same, and parasitic drag increases. In these three fish the balloon was not always centred exactly, causing the head or tail to tip up when the balloon was inflated, or causing the fish to float with one side up. To compensate for these positions the fish sometimes would make swimming efforts to stay upright when neutrally buoyant at zero speed. These efforts may account for the non-zero thrust values obtained for fish 9 and 10 at zero water speed.

## DISCUSSION

Several assumptions which were made in the calculation of mean body drag from the accelerometer records require further analysis.

Previously, DuBois *et al*. (1976) briefly calculated mean thrust from an accelerometer record to compare it with the drag calculated as the sum of profile and tangential drag. In the absence of information about the waveform of tail thrust, they thought it might occupy half the body cycle, and mean thrust might equal half the peak-to-peak amplitude of acceleration × mass, divided by √2, this last factor having been introduced as a method of calculating the r.m.s. value from the amplitude of a sine wave. Subsequent measurements of tail force in Fig. 2 of DuBois & Ogilvy (1978) showed a mean of 1· 0N accompanied by a peak-to-trough force of acceleration of 2·8 N, in one bluefish. In this particular fish, division of half the peak-to-trough force of acceleration by *√2* would indeed yield the measured value for mean tail thrust (1·0 N). However, this may be fortuitous, because analytical reasoning does not support taking the r.m.s. value of the accelerometer sine wave. Review of the eight other records of mean tail thrust *vs*. peak-to-trough acceleration similar to Fig. 2 of DuBois & Ogilvy (1978), shows that the mean tail thrust equals half peak-to-trough force of body acceleration divided by a factor which averages 1·70 (s.d. 0·48, s.e. 0·16). The waveform of thrust deviated from half sine waves, and there were variable pauses between the waves. In general, deceleration was greatest during these pauses, and at this time thrust was zero. Explanations for the factor 1·70 may be that the thrust is not sinusoidal, that drag varies throughout the body cycle, that thrust may arise from points of the body other than the tail, or that the effective tail pressure may be different from the one measured by placement of pressure gauges on the fork of the tail. Until more information becomes available, we prefer to omit use of the factor √ 2, and do not wish to introduce a constant such as 1· 70 to relate mean thrust to instantaneous drag.

The results of this study show that the top speed and drag of bluefish are independent of turbulence in the intensity range from 6 to 12 %.

A comparison between the coefficients of drag of a neutrally buoyant swimming bluefish (fish 8) and those of a wooden model of a bluefish, over a range of Reynold numbers, shows that at an *N*_{R} of 4· 2 × 10^{5} the drag coefficients were equal, but at *N*_{r}*=* 1 × 10^{5}, the drag coefficient of the swimming bluefish was about three times that of the wooden model. Fish 8 swimming at neutral buoyancy and the model both fall between the theoretical curves of *C*_{D}*vs. N*_{R} for *l/d* ratios between 1·8 and 5, curves which are shown on our Fig. 5. The lines for *l/d* ratios are taken from Fig. 22 of Hoemer (1965), and were based on his equation no. 24 for laminar, and no. 28 for turbulent, flow. When fish were not at neutral buoyancy, their drag coefficients at a low Reynolds number were much higher owing to the induced drag.

One possible explanation for the absence of a change of drag coefficient with induction of turbulence is that for a streamlined body with an *1/d* ratio between 3 and 5, the coefficients of drag for laminar and turbulent flow at any particular Reynolds number over the range between 10^{5} and 10^{6} are almost equal. Other possible explanations are that the turbulence already present in the water, without added induction of turbulence, disturbed the boundary layer around the body, or that swimming motions may have done so. It is also possible that vibration of the wooden model may have disturbed its boundary layer.

The presence of a high drag coefficient at low Reynolds numbers may depend on increased drag of a flexing body, whose curvatures presented to the water may create a large pressure drag. But we do not know whether the body is more widely flexed at slow swimming speeds. From data presented by Bainbridge (1963), Lighthill (1971) calculated a drag coefficient of 0·04 at a Reynolds number of 1·0 × 10^{5}, as contrasted with a value of 0·01 predicted for a streamlined body with an *l/d* ratio of 4. Our data on bluefish 8 show a *C*_{D} of 0·08 at a Reynolds number of1·0 × 10^{B} (see Fig. 5).

The rear of a rigid body towed through the water presents a convex surface moving away from the fluid through which it passes, and the boundary layer tends to separate from this surface. Conversely, the rear of a flexing streamlined body such as that of a swimming fish moves sideways and approaches the water, so that a fish has two advancing convex surfaces, one advancing forward at the front, the other advancing transversly at the rear. Thus the boundary layer may not separate the rear of the body of the swimming fish.

It was surprising to find that the bluefish could swim as fast in highly turbulent water as in less turbulent flow. It may be that the main factor determining forward propulsion is muscular force, rather than flow conditions next to the tail. The fish tail may thrust as well against water which is turbulent as it does against still water. This would explain why the maximal swimming speed was not reduced when the pressure transducer enclosures were attached to the tail. The cine films, and representative tail tracings in Fig. 3 of DuBois & Ogilvy (1978), suggest to us that the convex surface behind the leading edge of the tail advances toward the water beside and in front of the tail as the tail strokes sideways. If true, this implies that the tail, unlike an efficient aerofoil, has a positive camber with a negative angle of attack. Thus, induction of turbulence may have caused little loss of the already low mechanical efficiency of the tail. However, there is insufficient information about the tail’s curvature in relation to its path through the water to draw a general conclusion that the camber and angle of attack are, indeed, opposite in sign, and therefore this question merits further investigation.

Another remarkable finding was that the fish maintained a straight track through highly turbulent water. The eddies produced by the electric mixer, operated at in full power of 105 W, were intense and of large scale. This mixer, placed in a bucket, hurls water over the edges. Yet the turbulence did not prevent the fish from maintaining a true course, even while swimming at a high speed. However, if one considers that the natural habitat of bluefish is turbulent sea water, their ability to guide themselves through this turbulence becomes more acceptable. When ink was injected into still water in a tidepool, it stayed in one place, settling gradually to the bottom. But when it was injected under the surface of water some distance off the beach, it spread out 10 cm in 5 s, and pieces of seaweed 1 cm long rotated in this water, showing that the scale of turbulence included a diameter of 1 cm. Bluefish swim through this turbulent water. Flexibility of the body and stability of their guidance systems may allow the bluefish to thread their way through vortices of various diameters and intensities. For example, vortices having a diameter less than half the body length would tend to cancel out over the length of the fish, and vortices greater than half the body length could be negotiated by the fish because a bluefish can turn around 180° in half its body length. That is, a 2-foot-long fish can turn, with difficulty, in a 1-foot diameter tunnel and therefore thread its way through large as well as small vortices.

#### Theoretical analysis of lift, drag and weight at different angles of attack

When neutrally buoyant, a fish can swim horizontally or at any angle upward or downward incurring only parasitic (pressure plus skin) drag. There would be no induced drag because the weight of the fish in water would be zero. However, if the fish has positive weight in water, a condition which may result from either compression or absorption of gas in the swim bladder, the weight in water must be supported by lifting forces arising from the pectoral fins and tail. These forces are analysed below.

When the fish is swimming horizontally with its pectoral fins not horizontal, there is, in addition to the parasitic drag on the pectoral fins, a force perpendicular to the flow over the fins. The angle of flow is the resultant of the water’s downward velocity caused by the fin and the forward velocity of the fish. The force can be resolved vectorially into a vertical component (lift) and a horizontal component (induced drag) directed backward along the track. When the average velocity remains constant, the mean tail thrust must equal and overcome the mean total drag, which is the sum of parasitic and induced drag, since the component of weight directed backward along the track is zero.

At zero water flow or low speed, a fish which is heavy in water could overcome the tendency to sink by moving its pectoral fins forward and backward while changing their angle of attack, like a humming bird hovering. Or the fish could obtain some lift from its tail thrust provided the head was inclined upward, and also assuming that forward translation of the body was arrested by a force from the pectoral fins.

Our water tunnel was built at an angle of 33· 5° to make the water flow rapidly from the pool. When placed in this stream, a neutrally buoyant fish would have the same total body drag as a fish neutrally buoyant swimming horizontally. This drag, shown in Fig. 6, is labelled ‘Neutral’ in the lower right-hand panel.

When swimming through water flowing downward at an angle of 33-5°, fish which are heavy in water would have to overcome parasitic drag, induced drag, and the component of weight directed backward along the track. The weight of the fish in water, *F*_{w}, is opposed by an equal and opposite vertical force which is the sum of three components. These are the vertical components of (1) force of the tail along the track of the fish due to the angle of the track above horizontal; (2) lift from the pectoral fins and body; (3) the vertical component of lift from the tail caused by any inclination of the angle of the body relative to the track of the fish through the water. This condition for the fish is analogous to that for a climbing aircraft except that fish can change the angle of the fins relative to the body. The pectoral fins act as a wing.

The amount of lift that can be obtained from the pectoral fins depends on their profile area, S, the angle of attack, and the water speed past the fins, *V*. For a fish swimming along a horizontal track the vertical lift, *F*_{L}, and induced drag, *F*_{Di}, from the pectoral fins can be calculated as follows: *F*_{L}*= C*_{L} × 5 × ρ *V*^{2}*/2*, where *C*_{Di} is the dimensionless NACA absolute lift coefficient, and *p* the density of sea water. Furthermore, *F*_{Dl}*=C*_{Di} × S × ρ*V*^{2}, where *C*_{Di}is the drag coefficient. A value for 5 of 41 cm^{2} was obtained as the area outlined by the cast of the fins of a bluefish whose fork length was 0· 63 m. When scaled down to a fish of 0· 51 m fork length, the area, *S*, would be 27 cm^{2}. Sea water has a density of 1022 kg.m^{−3}. Since the pectoral fins resemble thin foils with concave lower surfaces, values which vary with the angle of attack (α) can be obtained for *C*_{L} from Fig. 3, p. 15–08 and the relationships between *L*_{c} and *C*_{L} given on p. 15–07, in Kent (1967). The induced drag coefficient C_{Di} is given by *= C*_{Di} =C_{L}^{2}/π*A* where *A* is the aspect ratio, defined as the wing span squared divided by the wing area.

One may calculate the angle of attack of the pectoral fins necessary to support a weight in water of 45 g at different horizontal swimming speeds. Assume *V* is 1·0 m.s^{−1}, S is 27 × 10^{−4} m^{2}, ρ is 1022 kg.m^{−3} and *F*_{L} is 0· 44 N. Now *C*_{L}*= F*_{L}*/(S ρV*^{2}×0·5). Substituting, *C*_{L} = 0·44/(27 × 10^{−4} × 1022 × 1^{2}×0·5) = 0·32. Because of the fin’s positive camber this value for *C*_{L} occurs at a low angle of attack, between 0° and 1°. Assuming that the aspect ratio of bluefish pectoral fins is 3, the induced drag coefficient is about 0·0109. Substituting, *F*_{Di} = 0· 0109 × 27 × 10^{−4} × 1022 × 0· 5 × 1^{2} = 0· 015 N or 1·5 g.

At a slower swimming speed, the angle of attack (α) of the pectoral fins required to support a weight of 45 g would be greater.

Stalling speed occurs at an angle of attack of about 15 ± At this angle, *C*_{L} is 1· 2. The stalling speed of the pectoral fins would be *V*_{s}*=* √{0· 44/(1· 2× 27 × 10^{−4} × 0· 5 × 1022)} = 0· 52 m.s^{−1}. Owing to increased velocity of the water around the fins, this corresponds to a body speed of 0· 46 m/s. (Dr Yves Parlange calculated that the flow over the fins was 13·3 % greater than the velocity of the body.) Below this speed we would expect the 45 g overweight fish to have to flap his pectoral fins.

In our water tunnel, inclined at 33· 5°, the conditions are analogous to an aeroplane climbing. If the fish weighs 45 g in water, and swims with its long axis in line with its track through the water, then all the force to support the weight must come from the pectoral fins and tail. Of the weight, a component *F*_{W} sin θ is aimed back along the track and is balanced by part of the tail thrust, while a component *W* cos θ perpendicular to this aims downward through the fish’s belly, to be balanced by lift from the actoral fins. It is here assumed that the pectoral fins are close enough to the centre (negative) buoyancy for any couple due to the offset between them and the centre of buoyancy to be unimportant. In Fig. 6 the part of the thrust, 0· 24 N, that balances the component of the weight along the thrust line has been subtracted from the total. The pectoral fins must supply 0· 37 N. This requires a lift coefficient of 0· 27 at 1 m.s^{−1}, which implies an induced drag coefficient of 0· 0077, and an induced drag of 0· 0107 N at 1 m.s^{−1}; 0· 0427 N at 0· 5 m.s^{−1}. The induced drag has been added to F_{W(}, sin θ on Fig. 6 and the curve obtained is labelled *‘F*_{Di}*+ F*_{w} sin θ.*’* It has not been continued below 0· 46 m.s^{−1}, which is the body speed at the stalling speed of the pectoral fins at this load. The thrust minus (F_{Di} + *F*_{w} sin θ) gives the parasitic drag labelled ‘*F*_{D} parasitic’. It is lower than the parasitic drag measured at neutral buoyancy labelled ‘neutral’. We do not know why.

To achieve full support of the weight at zero speed, the body would have to be angled upwards to take advantage of tail thrust, or else the pectoral fins moved forwards and backwards as in hovering, or some combination of these two would have to be used. Tail thrust, measured from the accelerometer records, equalled or exceeded the total weight of the fish in water at zero speed, as shown by the curves in Fig. 6.

When the buoyancy is negative in the tunnel inclined at 33· 5° the fish must overcome the component of weight directed along the track *(F*_{w} sin θ). Future experiments might be designed to measure the fish’s weight in water and to photograph the area and angles of attack of the body and fins as a function of weight, water speed, and the angle of water flow with respect to horizontal.

The accelerometer records appear to reflect the tail thrust, and this in turn generates lift and overcomes drag. At low speed the thrust required increases with the weight in water, as expected. At high speed it does not increase with the weight in water. This is unexplained.

At low speeds, the negatively buoyant fish tended to sink in our inclined tunnel. Indeed, when fish no. 8 was 45 g heavy, it would have required the same thrust to swim at 0· 8 m.s^{-1} as it did to stay up at zero speed. When they were this heavy, some fish did not even attempt to stay off the bottom until the water flow was fast enough to provide sufficient lift from the pectoral fins.

The authors wish to thank Mr Eugene Tassinari and Mr Lewis Lawday of the MBL Supply Department for catching the bluefish used in this study. Mr Arthur Edberg of the Pierce Laboratory Shop carved the wooden model of the bluefish. Dr Yves Parlange calculated velocity of flow over the pectoral fins, and their effective aspect ratio, S, based on their dimensions and shape. Dr Paul Webb pointed out to us that drag would be measurable from the accelerometer record, at the moment when thrust is zero, if thrust really is zero at that instant. We wish to thank the unknown referee(s) of this manuscript for many valuable, though anonymous, suggestions which have greatly improved the analysis and exposition of the data. Preliminary results were reported in abstract form (Ogilvy & DuBois, 1978). This research was supported in part by grant no. HL 17487 from the National Institutes of Health.

## REFERENCES

*J. exp. Biol*

*Simple Aerodynamics and the Airplane*

*Am. Trav. Publ. Belg*

*J. exp. Biol*

*J. exp. Zool*

*(Pomatomus saltatrix)*

*J. exp. Biol*

*Fluid Dynamic Drag*

*Swimming and Flying in Nature*

*Proc. R. Soc. Lond*. B

*(Pomatomus saltatrix)*during laminar and induced turbulent flow conditions (abstract)

*Biol. Bull. Mar. Biol. Lab., Woods Hole*

*J. exp. Biol*

*Kent’s Mechanical Engineer’s Handbook*, 12th ed., power volume

*Statistical Methods*

*Hydrodynamics and Energetics of Fish Propulsion*