Module 1: Hydrodynamics & Swimming

The relevant dimensionless number for a dolphin-sized swimmer is the Reynolds number\(\,Re = \rho U L / \mu\,\). For a 2.5 m bottlenose dolphin cruising at\(\,U = 5\,\text{m/s}\,\) in seawater (\(\rho = 1025\,\text{kg/m}^3\),\(\mu = 1.08\times10^{-3}\,\text{Pa s}\)):

This is firmly in the turbulent regime for a rigid body; Blasius flat-plate transition occurs at \(Re \sim 5\times10^5\). Gray assumed a rigid body in fully turbulent flow and used the Prandtl 1/7-power drag law:

The required sustained power is \(P = D\cdot U \approx 325\,\text{W}\). Gray's analysis gave a higher effective drag coefficient (~0.008) because he included form drag from a rigid bluff body; he concluded a dolphin needed ~2 kW of muscle power, far above the ~280 W that dolphin muscle mass should deliver.

1.1 Resolution: Laminar Flow via Compliant Skin

Max Kramer (1957–60) and subsequent workers (Carpenter, Bushnell, Fish) have shown that dolphin skin is not a rigid surface. The epidermis contains a fine network of dermal papillae and an underlying layer of compliant, fluid-filled ridges. These features damp incipient Tollmien-Schlichting waves — the precursors to boundary-layer transition — by extracting energy from them elastically. The result is that a substantial portion of the dolphin's boundary layer remains laminar even at Re ≈ 10⁷. For laminar flow, the Blasius drag coefficient is much lower:

Real dolphin drag coefficients (measured in tow-tank experiments and inferred from glide decelerations) lie around \(C_D \approx 0.0036\), substantially below fully turbulent and above fully laminar predictions — consistent with a mixed laminar/transitional boundary layer.

Cetaceans generate thrust by oscillating a lunate tail fluke(high-aspect-ratio, crescent-shaped) in the vertical plane. Fluke amplitude, frequency, and forward speed are related via the Strouhal number:

The remarkable empirical observation, first emphasised by Triantafyllou et al. (1991, 1993) and Taylor et al. (2003), is that virtually all efficient swimming and flying animals cruise in the narrow band \(St^* \approx 0.25\text{--}0.35\) — a universal biomechanical optimum that transcends phylogeny, body plan, and medium. Bottlenose dolphins, bluefin tuna, salmon, great white sharks, pigeons, hummingbirds, and fruit flies all converge on this narrow range. Theoretical work on two-dimensional oscillating foils confirms that thrust efficiency peaks at\(St \approx 0.3\) because this is where the shed vortex street (a “reverse Kármán street”) is most coherent.

2.1 Lighthill's Elongated-Body Theory

Sir James Lighthill (1960, 1970) derived the mean thrust of an oscillating caudal fin using slender-body potential-flow theory. For an airfoil of chord \(c\)oscillating in heave with amplitude \(A\) at angular frequency\(\omega\), the mean thrust per unit span is:

The Theodorsen reduction factor accounts for wake vorticity that reduces thrust below the quasi-steady limit. At \(k \approx 0.3\) (typical cetacean cruising) the reduction factor is ~0.6. Thus thrust scales as \(\bar T \propto \rho c^2 \omega^2 A^2\). Matching thrust to drag yields the swimmer's equilibrium cruising speed.

2.2 Why Lunate?

The crescent (lunate) fluke shape minimizes induced drag for a given thrust by maximizing span-to-chord (aspect) ratio. Just as albatross wings and the horizontal tails of fast pelagic fish (tuna, marlin, swordfish) are high-aspect-ratio and often crescent-shaped, the cetacean fluke has evolved under the same hydrodynamic pressure. The key is to keep the tip vortices small. For an oscillating thin wing, the induced (vortex) drag scales as\(D_{ind} \propto L^2/(\rho U^2 b^2)\) where \(b\) is the span. High aspect ratio (large \(b\) for a given area) minimizes this.

Cetaceans frequently leap clear of the water — breaching in whales, porpoising in dolphins. One long-standing hypothesis (Au & Weihs 1980) is that porpoising is energetically favorable because ballistic flight through air (where drag is 800× lower than in water) is cheaper than swimming near the surface where wave drag adds to form and friction drag. Surface wave drag scales with the Froude number:

Above this crossover speed, the drag penalty of staying just beneath the surface becomes large enough that leaping is cheaper. Bow-riding — where dolphins align with the pressure field at the bow of a moving vessel and catch a “free ride” from the ship's displacement wave — is another exploitation of pressure gradients for drag reduction (Williams 1989).

4.1 Biomimetic Skin Surfaces

Two radically different drag-reduction strategies have evolved in swift aquatic predators:

• Shark denticles (riblets): tiny V-shaped scales of dentine that align with the flow, break up turbulent vortices near the wall, and reduce skin friction by 5–10% at high Re. These work by restricting lateral movement of turbulent structures.
• Dolphin compliant dermis: an elastic dermal layer that absorbs energy from incipient Tollmien-Schlichting waves and delays laminar-turbulent transition. These work by postponing turbulence altogether.

Both strategies have inspired engineering applications: aircraft manufacturers have experimented with riblet films that yield 2–8% fuel savings, and compliant coatings for ship hulls have been investigated since Kramer's original patent.

The power required to swim at speed \(U\) is\(P_{\text{hydro}} = D\,U = \tfrac{1}{2}\rho U^3 C_D A_{wet}\). However the metabolic power \(P_{\text{met}}\) the animal must expend to produce this hydrodynamic power must include two additional terms: the basal metabolic rate (BMR) and the mechanical-to-chemical efficiency \(\eta_{\text{musc}} \approx 0.20\text{--}0.25\) of vertebrate muscle:

The cost of transport \(COT = P_{\text{met}}/(M_{body} U)\) is the energy required to move 1 kg of body mass 1 m. BMR contributes a term \(\propto 1/U\); the drag term contributes \(\propto U^2\). The sum has a minimum at the optimal cruising speed:

Empirically measured minimum COT for a bottlenose dolphin is roughly 1.3 J/(kg m), about half the mass-specific COT of a human swimmer and about 3× better than a running terrestrial mammal of comparable size. Among cetaceans, blue whales achieve the lowest mass-specific COT known for any animal: ~0.4 J/(kg m) at their preferred speed of ~3 m/s. This energetic efficiency enables the enormous migrations discussed in Module 7.

5.1 Wave Drag and the Optimum Submergence Depth

Surface-wave drag, as noted, peaks near Froude number Fr ≈ 0.5. For a body swimming at depth \(h\) beneath the surface, the wave resistance decays roughly as\(e^{-4\pi h/\lambda_w}\) where \(\lambda_w = 2\pi U^2/g\) is the wavelength of the Kelvin-wave pattern. A dolphin swimming at \(U = 5\,\text{m/s}\)generates \(\lambda_w \approx 16\,\text{m}\); to avoid wave drag it should stay at depths \(h \gtrsim \lambda_w/(4\pi) \approx 1.3\,\text{m}\). Observations show cetaceans routinely cruise at exactly this depth or below.

Cetacean locomotion is powered by the epaxial (dorsal) and hypaxial (ventral) lumbar muscles, which derive directly from the spinal flexors of their terrestrial ancestors. These muscles, aggregated into massive epaxial and hypaxial blocks along the spine, contract alternately to generate the dorsoventral bending of the tail stock.

6.1 Tendon Energy Storage

The cetacean tail stock contains large elastic tendons and collagenous septa that store elastic energy during each half-stroke and release it during the reversal. An elastic spring of stiffness \(k\) and natural frequency \(\omega_0=\sqrt{k/m}\)resonates with driven oscillations at \(\omega \approx \omega_0\). Observed tail beat frequencies match the estimated natural frequency of the muscle-tendon system, suggesting resonant energy recovery analogous to the Achilles tendon in running mammals. A cetacean cruising at its preferred gait is thus moving in synchrony with the resonance of its own tail.

6.2 Burst vs Cruise Mechanics

Dall's porpoise (Phocoenoides dalli) reaches top bursts of ~55 km/h (15 m/s) and the orca ~56 km/h. Burst speeds require a shift from pure aerobic, slow, oxidative muscle fibers (type I) to anaerobic, fast, glycolytic fibers (type IIb). The cost of this mode is lactate accumulation and restricted duration; all cetaceans show a sharp separation between cruising (aerobic) and sprinting (anaerobic) regimes.

6.3 Allometric Scaling of Swimming Speed

Large cetaceans are not, in the absolute sense, the fastest swimmers. Despite a blue whale's immense mass, its cruise speed is only ~3–4 m/s. Speed scales with body length roughly as \(U \propto L^{1/2}\) across swimming animals (Bainbridge relation for fish extended to cetaceans). The physical basis is that muscle cross-section scales as \(L^2\), force as \(L^2\), and drag as\(L^2 U^2\); setting force equal to drag gives \(U \propto L^0\)— constant. Empirically the scaling is closer to \(L^{0.4}\) due to shape and Reynolds corrections.