Fluid Mechanics of Swimming Propulsion 



boundary layer over a rigid, smooth surface of a similar body in this Reynolds- 

 number range is definitely turbulent. K the skin friction is evaluated on this 

 basis, then the power required to maintain such high speeds would violate sev- 

 eralfold the rule of thumb in biology that a pound of strong muscle can deliver 

 only up to 0.01 horsepower. More recently, the speed of porpoises has been 

 investigated carefully, under well-controlled conditions, by Lang (1962, 1963). 

 Another interesting study is that of migratory salmon by Osborne (1960). Ac- 

 cording to this careful investigation, a detailed estimate again led to one of the 

 two conclusions: either (1) these creatures have a much smaller drag than 

 could be achieved with similar, rigid bodies, or (2) the power output per gram 

 of muscle is much larger than observed from physiological experiments on 

 warm-blooded animals — these being known as the paradox of Gray (1948, 1949). 

 These puzzling conclusions have stimulated fluid dynamicists to explore various 

 other possibilities, such as the effect of compliant skin and the effects of mu- 

 cous surface and additives on frictional drag, the studies of the former being 

 so far inconclusive. 



The purpose of this paper is to discuss some of the hydrodynamic aspects 

 of swimming propulsion. No attempt is made here to venture beyond this scope. 

 It may well be that only after some extensive efforts are made in basic research 

 can the final chapter be written of this most interesting story. 



I. SWIMMING MOTION AT LARGE REYNOLDS NUMBERS 



When the Reynolds number R is large, the swimming motion depends pri- 

 marily on the inertial effects which can be evaluated based on the potential 

 theory. Viscosity of the fluid is unimportant, except in its role of determining 

 the vorticity shed into the wake, and of producing a thin boundary layer, and 

 hence a skin friction at the body surface. A large number of swimming objects 

 employ in propelling themselves the commonly observed mode of body motion 

 which can be characterized by a wave of lateral displacement moving down the 

 body from head to tail. As the body attains a forward momentum, the propul- 

 sive force pushes the fluid backward with a net total momentum equal and op- 

 posite to that of the action, while the frictional resistance of the body gives 

 rise to a forward momentum of the fluid by entraining some of the fluid sur- 

 rounding the body. The momentum of reaction to the inertia forces is concen- 

 trated in the vortex wake due to the small thickness and amplitude of the un- 

 dulatory trailing vortex sheet; this backward jet of fluid expelled from the body 

 can however be counterbalanced by the momentum in response to the viscous 

 drag. When a self-propelled body is cruising at a constant speed, the forward 

 and backward momenta exactly balance; they can nevertheless be evaluated 

 separately. 



THRUST; ENERGY BALANCE 



In order to visualize why a waving form of the body motion is desirable 

 for swimming propulsion, we consider the energy balance for the typical case 

 of a planar flexible body of negligible thickness, performing a general unsteady 

 motion of small amplitude, achieving a forward velocity u (t), which may 



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