Great hammerhead sharks swim on their side to reduce transport costs

Animals exhibit various physiological and behavioural strategies for minimizing travel costs. Fins of aquatic animals play key roles in efficient travel and, for sharks, the functions of dorsal and pectoral fins are considered well divided: the former assists propulsion and generates lateral hydrodynamic forces during turns and the latter generates vertical forces that offset sharks' negative buoyancy. Here we show that great hammerhead sharks drastically reconfigure the function of these structures, using an exaggerated dorsal fin to generate lift by swimming rolled on their side. Tagged wild sharks spend up to 90% of time swimming at roll angles between 50° and 75°, and hydrodynamic modelling shows that doing so reduces drag—and in turn, the cost of transport—by around 10% compared with traditional upright swimming. Employment of such a strongly selected feature for such a unique purpose raises interesting questions about evolutionary pathways to hydrodynamic adaptations, and our perception of form and function.

Data loggers were equipped with a three-axis accelerometer that measures the specific force acting on the shark. The precise orientation of the accelerometer relative to the shark is unknown, and we seek the orientation of the shark relative to Earth. For the following analysis, it will be assumed that accelerometer data have been low-pass-filtered to remove the tail-beat frequency (we have used 0.25 Hz), and include only those segments for which the acceleration of the shark was practically zero (we have used a threshold of 0.02g).

Reference frames
We define three orthogonal right-handed reference frames: A, B and E. The components of the specific force acting on the shark in the three frames will be denoted (S1) The tilde marks here a transpose. Transformation matrices between the reference frames are unknown; in fact, they are the objectives of the data reduction.

Euler rotations
Consider a pair of arbitrary orthogonal reference frames, say, C and D, sharing a common origin and rotated one relative to the other through angle α about axis between D and C, is   depending on the axis about which the rotation took place. T has the properties that where tilde marks a transpose and I is the unit matrix.

Accelerations
We will assume that transformation between frames E and B can be obtained by a series of two rotations. First, about the y-axis through angle  (pitch), and then about the x-axis through angle ϕ (roll): Equivalently, by (S5). Assuming (S1), We shall assume that pitch and roll are practically independent; namely for any pair of functions f and g. We shall also assume that θ is sufficiently small to make the approximation sin cos sin cos that is, the shark spends the same time rolled right as it spends rolled left. The last assumption will be removed later on. In the interim, we assume that frame B is oriented relative to the shark in such a way that (S12) holds.

Transformation between A and B
Now, the accelerometer is connected to the body at some orientation, represented by a certain (S17) Consequently, and T a a T (S19) by (S17).
We have already established that  a a a a has a diagonal formsee (S16) and (S15). The transformation into diagonal form is known as the eigenvalue decomposition. In fact, given , and V , the matrix of the respective eigenvectors, by definition, and hence 0  TV and   1 2 3 ,,    are the terms on the diagonal in (S16).
In order to obtain the orientation of the accelerometer from 0 T we assume that it is obtained 19 through a series of three Euler rotations, roll ( 0  ), pitch ( 0  ) and yaw ( 0  ), in that order. In this case,  Combining (S7), (S17) and (S21), one will find In other words, the roll angle between B and A systems actually reflects the average roll angle and may not truly represent the rotation of the accelerometer relative to the body. If the shark has no preference angle, 0  can be interpreted as the angle between the accelerometer and the body.

Transformation between B and E
Given 0 T , the components of the specific force in the body reference frame,

Supplementary Note 2: Accelerometer data handling
The Batt Reef data logger included a three-axis accelerometer, speed sensor, and depth and temperature sensors. The former was sampled at 16 Hz; the latter two were sampled at 1 Hz. The Belize data logger (shown in Supplementary Fig. 1) included a three-axis accelerometer and depth and temperature sensors, with all three variables sampled at 8 Hz. The first three hours were discarded to minimize capture artefacts, leaving approximately 15 and 63 hours of data for the Batt Reef and Belize sharks, respectively.
Accelerometer data were low-pass filtered at 0.25 Hz to remove acceleration signals caused by the tail-beat frequency at 0.4 Hz. The data were sorted, removing segments for which total acceleration of the shark exceeded 0.02g; 98% and 46% of the data remained for the Batt Reef and Belize sharks, respectively. Orientation of the accelerometer relative to the shark (13 and 13 degrees left roll, 9 and 22 degrees pitch down and 18 and 4 degrees left yaw for the Batt Reef and Belize sharks, respectively), and orientation of the shark relative to Earth (Fig. 1 main document;   Supplementary Fig. 3) were found using the paradigm described in Supplementary Note 1. Whereas the orientation angles of the Batt Reef shark could be estimated with confidence for almost the entire deployment period (98% of records), total accelerations produced by the Belize logger were often high after low-pass filtering, so roll angles could be estimated with a high degree of confidence for just under half of the accelerometer records. We therefore presented estimates of roll and pitch angles calculated from the reduced dataset (i.e. where total shark acceleration was less than 0.02g) and also from the entire dataset (i.e. no acceleration threshold) for the Belize shark.
Note that pitch angle of the sharkmeasured between its caudo-cranial axis and the horizon -is a sum of its angle of attack (the angle between is caudo-cranial axis and the direction of swimming) and its trajectory angle (the angle between the direction of swimming and the horizon). The first constituent is akin to the pitch angle measured in the wind tunnel, and is essentially determined by the swimming speed; the second constituent reflects the yo-yo motion of the shark. Because the depth of the shark after many hours of swimming remained within a few tens of meters from its 21 initial depth, the average of the second constituent is zero, and therefore the most probable pitch angle reflects the average of the angle of attack, which is comparable with the wind tunnel measurements.
Data from the speed sensor for the Batt Reef shark are shown in Supplementary Fig. 2

The model
A fifths-scale model of the shark was constructed using CAD software based on available statistical data 5 and numerous photographs. It was printed in FullCure720. The general drawing can be found in Supplementary Figs 5 and 6; printer-ready files are available on request. The model had replaceable fins, head and neck. All fins had NACA0015 profile.
The total length of the model was 640 mm. The part of the model that went into the tunnel was 431 mm long, ending at the caudal end of the anal and second dorsal fins. Its maximal cross section area (that was used to obtain the drag and lift coefficients) was 3870 mm 2 .

The balance
The model was placed on a six-component string balance. Measurement resolution was 1 μv, which is equivalent to approximately 0.42, 0.36 and 0.26 grams of lift, side-force and drag; the accuracy was about 2 grams in lift and side force, and about 1 gram in drag. The lift and side force measured during the experiment were of the order of 1 kg; the drag was of the order of 100 grams. The data were acquired at 5 KHz. Data were low-pass filtered at 4 Hz, and block averaged with 500 samples per block.

Experiments
In each experiment, the shark was set at a constant bank angle (0,10,…,90 deg) and its orientation relative to the flow (equivalent to the pitch angle of a free swimming shark) was changed between minus 15 and plus 15 degrees, at the rate of approximately 0.5 degree per second. This setup is shown in Supplementary Fig. 7.

Supplementary Note 4: Shark Cost Of Transport
Energy expenditure per distance swam (commonly termed the "cost of transport", COT) is defined as: where v is the swimming speed, D is the hydrodynamic drag, 0 P is the standard metabolic rate,  the hydrodynamic propulsion efficiency, and m  the chemo-mechanical efficiency of the muscles.
There are a few intricate details in using this equation to estimate the cost of transport of a 'real' shark from wind-tunnel data. These details are described below.
Lift and drag are commonly expressed in terms of the respective coefficients, in which  is the density of water, and S is an arbitrary reference area, chosen here as the maximal cross section area of the body. The lift coefficient depends mainly on the angle between the surface that generates the lift (as a fin) and the swimming direction; the drag coefficient depends mainly on the lift coefficient (see "Supplementary Note 5" below).
S was correlated with the fork length of the shark, l, The propulsion efficiency (  ) depends on the swimming gait of a shark. We have no simple means to estimate it; most studies indicate that its value ranges between 0.7 and 0.8. We have used 0.75, but changing its value within the acceptable range has no qualitative effect on Fig. 3c-d in the main document ( Supplementary Fig. 8).

26
The muscle efficiency ( m  ) depends on the loading and the contraction rate. Again, we have no simple means to estimate it, and therefore we took a constant 24 Joule per mmol ATP after Kushmerick and Davies 19 .
There is no consensus about the standard metabolic rate ( 0 P ) of large sharks 20 Fig. 9).

Supplementary Note 5: Drag Theory
Hydrodynamic forces acting on a swimming shark can be conveniently (albeit somewhat ambiguously) divided into lift L, drag D, thrust T and buoyancy B. For simplicity, we will assume that the thrust is generated mainly by the caudal, anal and the second dorsal fins, and is directed along the swimming path; whereas lift and drag are generated by all other fins and by the body of the shark, they are directed perpendicular and parallel to the swimming path, respectively. When swimming at constant speed along a straight horizontal path, all forces cancel out with gravity, G: Lift and drag are commonly expressed in terms of the respective coefficients, L C and D C with in which  is the density of water, and S is an arbitrary reference area, chosen here as the maximal cross section area of the body. The lift coefficient depends mainly on the angle between the surface that generates the lift (as a fin) and the swimming direction; the drag coefficient depends mainly on the lift coefficient. It is commonly approximated by a parabola: (S5) 0 D C is the parasite (zero lift) drag coefficient, associated with friction between the body and water; 2 L KC is the induced drag coefficientthe cost of lift generation. An example can be found in Supplementary Fig. 10.
The weight of the shark in water, GB  , is commonly expressed in terms of the sinking factor, β, in which m is the displaced mass of the shark, and g is the acceleration of gravity.
At a given swimming speed, the combination of (S29), (S26) and (S30), determines the lift coefficient needed to counteract gravity; the combination of (S2), (S27) and (S5), determines the thrust needed to maintain that speed. In turn, the lift coefficient determines the angle between the lift generating surfaces and the flow.
The induced drag depends on the horizontal span of the lift generating surfaces, b, and on the distribution of lift along these surfaces, reflected in the numerical coefficient K k :  Fig. 3b in the text.

Experiments: the effect of configuration
We believe that the purpose of the second dorsal and the anal fins is to facilitate propulsion.
Consequently, the drag measurements that appear in the text were made with the fins removed.
Restoring the fins has no qualitative effect on the results, as shown on the right plate in Supplementary Fig. 11.
We also examined the influence of the cephalofoil on the results, by replacing it with a traditionally-shaped shark head. The results remained qualitatively the same. They are shown in Supplementary Fig. 12.

Corrections
In conducting the experiments, we were careful to preserve the similarity in the Reynolds number, which affects both the parasite drag coefficient 0 D C and the lift coefficient at which the flow separates from the lifting surfaces (approximately 2.5 in Supplementary Fig. 10). Consequently, the drag coefficient could have been used practically 'as is' in estimation of the cost of transport. We did not measure the drag of the caudal fin and the drag of the gills. The former can be accounted for by the hydrodynamic propulsion efficiency. The drag of the gills was accounted for by increasing the measured value of 0 D C (approximately 0.17) by 0.02.