Speed 0 m/sTail beat 0 HzStrouhal 0Power 0 % of muscleRe 0FPS 0

Drag to orbit, wheel or pinch to zoom. The dolphin circles a 22 m track; each tail stroke sheds a vortex ring (blue rings), the jet that pushes it forward. The body bends as a travelling wave, its beat rate set by speed, stroke size and Strouhal number.

Dolphin Swimming Power: Gray's Paradox, Boundary-Layer Drag and the Strouhal Tail Beat

How fast can a dolphin go on the muscle it has? Set body size, muscle power and how the water flows over the skin; see drag, the power needed, and how the tail beat settles at a Strouhal number near 0.3.

Body and muscle

Water and swimming style

Live readouts

Muscle power needed against speed

Speed through time

Propulsive efficiency against Strouhal number

Experiment: is Gray's paradox real?

For four boundary-layer states, this finds the top speed on sustained (40 W per kg of muscle) and burst (250 W per kg) power, and the muscle power per kg needed to hold 10 m/s. Gray (1936) found that a fully turbulent dolphin needs far more power than muscle was thought to give.

Independent numerical checks

Controls

Effort sets how much of the available muscle power is used. The dolphin accelerates until thrust equals drag; the speed settles at the terminal speed shown in the readouts.
Strouhal number and stroke size set the tail beat: f = St U / A. A stroke that is too fast or too slow for the speed wastes power in the wake.
Laminar fraction is how much of the body length keeps smooth, attached flow before it turns turbulent; flexible skin and body motion are thought to delay that in real dolphins.
Presets change all sliders at once. Every slider can be moved while the dolphin swims.

The science

Drag. D = ½ ρ U² S C_f(Re) FF k_U, with Re = U L / ν, ρ = 1025 kg m^-3, ν = 1.19×10^-6 m² s^-1 (seawater near 15 °C). S is the wetted area of a prolate spheroid of the same length and girth; FF = 1 + 1.5 (d/L)^1.5 + 7 (d/L)³ (Hoerner form factor); k_U raises the drag of a swimming body above a gliding one (default 1.5, typical literature range 1 to 3, still debated).

Friction coefficient. Laminar (Blasius) C_f = 1.328 / √Re; turbulent (Prandtl–Schlichting) C_f = 0.455 / (log₁₀ Re)^2.58; with laminar flow over the first fraction x/L the page uses Schlichting's correction C_f = C_f,t(Re) − Re_tr (C_f,t − C_f,l)(Re_tr) / Re. These are flat-plate results applied to a body of revolution.

Power and speed. The mechanical power to hold speed U is P = D U / η. Available power is muscle mass fraction × body mass × power per kg of muscle (illustrative defaults 25 % and 60 W kg^-1; sustained values near 40 and burst values above 200 W kg^-1 are order-of-magnitude figures, not measurements of one animal). Speed obeys m(1+0.1) dU/dt = η P / U − D(U), the 0.1 being an added-mass allowance for a slender body.

Strouhal number. St = f A / U, A the peak-to-peak tail-tip stroke. Swimming and flying animals that cruise hold St between about 0.2 and 0.4 (Taylor, Nudds and Thomas, 2003), where the wake jets are most efficient. The efficiency curve here is an illustrative bell peaking at 0.85 near St = 0.3, not a derived result.

Gray's paradox. Gray (1936) estimated that a turbulent boundary layer demands several times the power a dolphin's muscle could supply at its observed speeds. Later work points to higher muscle power than assumed, some laminar flow, and the way body motion changes the boundary layer. The table here lets you vary those and see which one removes the paradox; it does not settle the debate.

What it leaves out. Wave drag and depth, turning, leaping, buoyancy, the three-dimensional flow around flukes and flippers (the efficiency curve stands in for it), and heat. For drag the body is a rigid spheroid; on screen it is drawn bending. Typical constants only: spot-check them before quoting. Assets built with Blender; scene by Three.js.