# A hydrodynamics assessment of the hammerhead shark cephalofoil

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Sep 2, 2020

There have been various hypotheses directed toward explaining the function of the hammerhead cephalofoil. In this study, we examined the suggestions that this structure provides a hydrodynamic advantage by increasing maneuverability and by producing lift similar to a cambered wing. Our analysis revealed several important points that cast light on these ideas: (1) Although some regions of high and low pressure were visible at a zero attack angle, when we examined surface pressure contours, there was a lack of an overall net dorso-ventral pressure gradient. (2) Mean pressure differences across dorsal and ventral surfaces of the cephalofoil were typically small and, in most cases, were higher on the dorsal surface. (3) Pressure and velocity cutting planes did not indicate significant pressure or velocity differentials between dorsal and ventral surfaces except at some angle of inclination with respect to flow. This was true even in distal regions where (as naturalists and biologists have previously suggested) in parasagittal section the structure’s profile most closely bears superficial resemblance to a man-made cambered wing. (4) Lift coefficients at α = 0° were negative for all but two species. Taken together, these results do not support a cambered wing hypothesis.

Although our results suggest that significant lift is not produced during horizontal swimming in Sphyrnids, the generation of lift at positive angles of attack during routine swimming can nevertheless be important. For example, the Leopard shark, Triakis semifasciatus and the Bamboo shark, Chiloscylium punctatum, held the body at + 11 and 9°, respectively, during horizontal, steady swimming in a swim tunnel, to counteract the antero-ventral reaction force created by the heterocercal caudal22. However, this does not appear to be the case for Sphyrnids. In many observations of in-situ shark swimming by one of the authors (GRP), hammerhead sharks are often observed swimming parallel to flow and often in close proximity to a featureless, horizontal substrate when foraging for prey.

To provide additional insight regarding potential hydrodynamic function, we examined lift and drag across a wide range of attack angles for all species. Modern day, man-made, cambered foils are generally typified by C-shaped, parabolic drag polars; drag values tending to increase concurrently with lift values as attack angle becomes more extreme. Drag polars depicted in this study, indicated that drag typically increased at a faster rate than lift. This effect is imparted by boundary layer separation at the trailing edge of the foil progressing increasingly toward its center. This continues until flow separation ultimately occurs to such a great extent that the lifting efficiency (the ratio of lift to drag) is undermined, and the foil stalls23. Lift polars typically exhibit an overall positive linear slope progressing to a maximum (the stall point), inflecting suddenly, and taking on a sharp downward trajectory thereafter.

In previous work, no notable differences were observed across the drag polars of sphyrnid species, and little evidence of wing-like hydrodynamic properties of the cephalofoil among small sphyrnids was cited11. In contrast, we observed substantial differentiation across sphyrnid species with regard to both drag and lift polars as well as substantial interfamilial differences. In our examination of lift and drag coefficients, we observed a pattern whereby curves were grouped broadly by slope. These groups corresponded with sharks featuring discretely different head morphologies: (1) carcharhinids, (2) hammerheads possessing small cephalofoils, i.e., tudes, media, tiburo and corona, (3) hammerheads with intermediate cephalofoils, i.e. mokarran, zygaena and lewini, and (4) a large cephalofoil (E. blochii). Across species, slope tended to decrease with decreasing aspect ratio of the head. The above groupings precisely mirror phylogenetic groupings as determined by morphological, isozyme and mtDNA sequence data24.

Despite its common name (winghead shark) the E. blochii cephalofoil generated the greatest amount of drag and produced, at low angles of attack, the least amount of lift. It may be noteworthy that the E. blochii head morphology displayed the greatest rate of change in lift coefficient as attack angle changed. Additionally, at the highest attack angles, the lift coefficients were the greatest of any species. The results for this species likewise do not support the cambered wing lift hypothesis. However, the relatively rapid change in lift generated at positive attack angles implies that the E. blochii cephalofoil in particular may provide a hydrodynamic advantage via an increase in maneuverability. It may be significant that the E. blochii diet was found to consist of about 93% teleost fishes, apparently of the family Clupeidae25, whereas other hammerhead species feed predominantly on stingrays, crabs, and other bottom-dwelling organisms26. The predominance of highly mobile clupeids in the diet of E. blochii may reflect the greater mobility that its cephalofoil provides.

The increased maneuverability hypothesis can be interpreted in concert with the hydrodynamic data generated as part of this study. Relevant to this study are findings regarding greatly enhanced hypaxial musculature in hammerheads relative to carcharhinid (typical) sharks10. This musculature may indicate the importance of the head in facilitating prey capture by generating rapid shifts in trajectory. As the results presented here show, if the head were depressed to the maximum possible extent indicated in the aforementioned study (− 15°), the reaction force produced would be substantial. In larger sphyrnids, the downward moment produced would necessarily be large (owing to the large surface area of the cephalofoil). In smaller sharks, a change in trajectory might be facilitated just as easily using a smaller cephalofoil (thus lessening the tradeoff between cephalofoil utility and its associated drag) given their smaller overall mass.

Previous research has concluded that the cephalofoil likely has a negative effect on stability27. Its position at the far anterior end of the shark does increase its mechanical advantage substantially, and our results confirm that the magnitudes of the reaction forces produced increase rapidly as attack angle deviates from level. The inference here is that the cephalofoil may serve as a forward rudder under active control of hypaxial and epaxial musculature, thus providing for rapid dives and ascents. It should be noted that the cephalofoil may actually confer stability during turning, as it was observed that sphyrnid sharks did not roll during sharp turns, as did their carcharhinid counterparts28. Prey detection, via an increased number of Ampullae of Lorenzini, and prey capture do seem to be the prevailing directions in current thought regarding cephalofoil function. Larger hammerhead species are known to prey disproportionately on skates and rays, and increased maneuverability could confer an advantage in avoiding such prey defenses. We caution, however, that it remains uncertain from our results and other literature presently available whether or not the structure provides a prey-capture advantage via increased maneuverability.

The drag coefficients estimated in this study can be used to make ecophysiological comparisons between hammerhead and typical carcharhinid sharks. An avenue for making these comparisons is to examine cephalofoil drag coefficients across species at the same lift coefficients (CL = 0.2). Relative to the cephalofoil of other hammerhead species, the E. blochii cephalofoil generated 6.4 × (S. corona) to 12.6 × (S. lewini) greater drag. Additonally, the E. blochii cephalofoil generated 22.6 × , 32.3 × , and 39.7 × greater drag when compared with C. limbatus, C. leucas, and N. brevirostris, respectively. Finally, comparing drag coefficients of sharks possessing typical head morphologies with the remaining hammerhead species revealed that the cephalofoil generated 1.9 × (S. tiburoC. limbatus comparison) and 6.2 × (S. coronaN. brevirostris comparison) increased drag.

An important implication of the increased drag in hammerheads is the concurrent increase in energy expenditure necessary to maintain forward motion. This is especially relevant in obligate ram-ventilating sharks such as hammerheads. During level, unaccelerated cruising, sufficient force (i.e. thrust) must be imparted to a body to overcome the resistive force of drag and move that body through a fluid. In this instance, the required thrust is equal to the drag force23. Calculation of the thrust required can be easily accomplished by the equation:

$$T_{R} = , D , = 1/2 , rho v^{2} AC_{D}$$

where TR is required thrust, D is the drag force, ρ is the fluid density (1.023026 × 103 kg/m3 for seawater at 25 °C and 35 ppt salinity), υ is the fluid velocity (held constant at 1 m/s), CD is drag coefficient, and A is the respective planform area (outline of an area as seen from above) of each head model.

Thrust required for forward motion requires power generation and hence energy expenditure on the part of the fish. Using data obtained from CFD and the above equation, it is possible to calculate and compare the difference in drag force (and corresponding thrust required) for differing shark head morphologies. For these calculations we chose to compare adult S. lewini and C. limbatus because Reynolds’ numbers were comparable (1.25 × 105 and 1.32 × 105, respectively). Thus, for S. lewini and C. limbatus a drag force of approximately 9.34 and 1.007 Newtons, respectively was calculated. Conversion from Newtons to pounds-force (where 1 N = 0.2248 lbf) yields 2.099 and 0.226 lbf for S. lewini and C. limbatus, respectively.

Hence, our calculations indicate that, possessing a cephalofoil requires an almost 10 × increase in thrust for S. lewini compared with a similarly sized C. limbatus. It is noteworthy that the drag difference in this example is conservative since S. lewini has a higher reference length and higher Reynolds number. It would be anticipated that the greater thrust and energy necessary for S. lewini to swim at the same speed as C. limbatus would result in increased food consumption to offset increased metabolism and, potentially, a cascade of physiological changes that would accompany those increases. However, various compensating mechanisms could offset the increased energy requirement of possessing a cephalofoil. For instance, hammerhead sharks may reduce cruising speeds, enhance static lift mechanisms, and/or possess a more efficient metabolism. In actuality, a general trend toward higher metabolic rates has been shown in ram-ventilating sharks15,29. Sphyrnid metabolic rates are typically high as well, with rates as high as 168 mg O2 kg−1 h−1 in S. tiburo and 189 mg O2 kg−1 h−1 in S. lewini30,31,32. Thus, it seems unlikely that sphyrnids offset drag-related energetic loss via lower metabolism. To the contrary, metabolic rates are higher in sphyrnids (which may exacerbate the problem of drag-related energy loss).

An interesting behavior has been observed in S. mokarran33 which may be relative to the results presented herein. These sharks have been reported to swim in-situ on their sides up to 90% of the time. This change in orientation during swimming may provide increased lift via the repositioning of the large dorsal fin, which is estimated to reduce the energetic cost of swimming by approximately 10%33. While it is not known if this behavior is widespread across Sphyrnids, hammerheads in general possess relatively large dorsal fins (relative to other shark species) and adopting this swimming behavior could be related to possession of a hydrodynamically costly cephalofoil.

There was limited evidence to support the suggestion that the cephalofoil produces trailing vortices similar to those generated by aircraft wings11. Wing tip vortices, also called lift-induced vortices, result from air flowing from below the wing where pressure is high, around the tip and onto the top of the wing where pressure is low, in a circular motion. The absence of significant trailing vortices observed during this study at zero angle of attack reflects the absence of pressure differences between dorsal and ventral surfaces of the cephalofoil. However, to better understand cephalofoil vorticity, time-accurate simulations should be used to view flow patterns across a temporal gradient. The possibility exists that the energetic cost of possessing a cephalofoil may be offset by increased swimming efficiency through active flow control. Interaction of the cephalofoil with appropriately placed downstream appendages, such as the hammerhead pectoral fins, may significantly transform flow ahead of the caudal fin and cause beneficial interactions with the caudal. An examination of the potential for these interactions between the cephalofoil and fins, as well as how changes in orientation of the fins relative to the cephalofoil during undulatory swimming might alter this interaction, would necessitate CFD analysis of the entire hammerhead body during active swimming which was beyond the scope of this study.

The results of this study suggest that the hammerhead cephalofoil functions as a foil insofar as it operates as a symmetric foil, or thin plate requiring alteration of its attack angle for the production of lift. It does not appear to possess sufficient camber to generate lift at α = 0. Our analysis suggests that the possession of a cephalofoil may increase maneuverability. In light of evidence presented regarding active control of cephalofoil attack angle via hypaxial and epaxial musculature10, we suggest that this structure may function as a forward rudder (and perhaps as a fluid dynamic brake) at the anterior end of the animal facilitating more rapid changes in position in the water column and increased maneuverability during the final moments of prey capture. It is important to recognize that the Spalart–Allmaras turbulence model21 can reasonably predict drag and lift at small to moderate angles of attack, but may not be well-suited for high angles of attack. However, this would not alter our observation that there was little to no significant lift generated at zero attack angle. It also should not invalidate the potential relative differences in maneuverability observed in this study between species at low to moderate angles of attack. Relative to this is the observation that electrically stimulated S. lewini have been observed to maximally depress and elevate their cephalofoil ca 15° and 30°, respectively. Anecdotal observations in situ by one of the authors (GRP) reveals that these sharks rarely flex their cephalofoil to the maximum extent during routine swimming. Interestingly, these data suggest that hammerhead sharks may pay a greater energetic price for their head morphology when compared with typical sharks. However, the hydrodynamic price of possessing a cephalofoil, may be offset by the increase in prey detection and capture that this unusual structure may provide. Finally, in the face of an almost complete absence of information regarding the hydrodynamic role of the cephalofoil, and the fact that no computational fluid dynamics study of fish swimming existed prior to this work, we suggest that this research be viewed as a first approximation of the hydrodynamic and energetic costs and benefits of possessing a cephalofoil. Ultimately, we hope that it will stimulate additional research using CFD in the biological sciences and, in particular, to study fish swimming hydrodynamics.