Scaling Macroscopic Aquatic Locomotion

September 17, 2014
Aquatic swimming

Aquatic swimming: a,The organisms considered in the article span eight orders of magnitude in Reynolds number and encompass larvae (from mayfly to zebrafish), fish (from goldfish, to stingrays and sharks), amphibians (tadpoles), reptiles (alligators), marine birds (penguins) and large mammals (from manatees and dolphins to belugas and blue whales). Blue fish sketch by Margherita Gazzola. b, Swimmer of length L is propelled forward with velocity U by pushing a bolus of water14, 20, 24 through body undulations characterized by tail beat amplitude A and frequency ω. c, Thrust and drag forces on a swimmer. Thrust is the reaction force associated with accelerating (Aω2) the mass of liquid per unit depth ρL2 weighted by the local angle A/L (therefore ρLA may be understood as the mass of liquid channelled downstream). For laminar boundary layers, the drag is dominated by viscous shear (skin drag), whereas for turbulent boundary layers, the drag is dominated by pressure (pressure drag). [Figure reprinted by permission from Macmillan Publishers Ltd: see Mattia Gazzola, Médéric Argentina, and L. Mahadevan, "Scaling macroscopic aquatic locomotion," Nature Physics (2014) | doi:10.1038/nphys3078.]


Inertial aquatic swimmers that use undulatory gaits range in length L from a few millimetres to 30 metres, across a wide array of biological taxa. Prof. L. Mahadevan, working with a postdoctoral fellow Mattia Gazzola and a colleague Mederic Argentina from the University of Nice, were able to show that a handful of principles govern how virtually every animal - from the tiniest fish to birds to gigantic whales - propel themselves though the water.

Using elementary hydrodynamic arguments, the researchers uncovered a unifying mechanistic principle characterizing the locomotion of aquatic swimmers by deriving a scaling relation that links swimming speed U to body kinematics (tail beat amplitude A and frequency ω) and fluid properties (kinematic viscosity ν). This principle can be simply couched as the power law Re ~ Swα, where Re = UL/ν double greater than 1 and Sw = ωAL/ν, with α = 4/3 for laminar flows, and α = 1 for turbulent flows. Existing data from over 1,000 measurements on fish, amphibians, larvae, reptiles, mammals and birds, as well as direct numerical simulation,s are consistent with this scaling. The researchers interpret these results as the consequence of the convergence of aquatic gaits to the performance limits imposed by hydrodynamics. The study is described in a September 14 paper in Nature Physics. See also the Gazette article "All Goes Swimmingly" by Peter Reuell.