NOTES 



EFFECTS OF SWIMMING PATH CURVATURE 

 ON THE ENERGETICS OF FISH MOTION 



Many respiration and other behavioral and 

 physiological studies of larger pelagic fish species 

 are carried out in round tanks (Fry 1957; Bain- 

 bridge 1958; Magnuson 1970; Neill et al. 1976). 

 These tanks have structural and space advan- 

 tages, since round tanks have the largest volume 

 to surface area of any flat-sided enclosure. 



However, circular tanks introduce an additional 

 stress factor not usually encountered by wild fish. 

 This is the centripetal force required for continued 

 motion in a curved path. This force is proportional 

 to the fish mass, and inversely proportional to the 

 path radius. The tank radii are limited by the fact 

 that respiration and heat transfer data are ob- 

 tained from the medium, resulting in a require- 

 ment of relatively small volumes so that changes 

 in the measured parameters (oxygen concentra- 

 tion, temperature, etc.) are enhanced. 



As will be shown here, this constraint on tank 

 size causes the centripetal contribution to the 

 force balance to become dominant in certain ex- 

 perimental situations. I therefore developed a cor- 

 rection factor to be applied to data collected in 

 round or annular tanks so as to make the data 

 representative of fish swimming freely in a 

 straight line, in open waters. 



Analysis 



For a neutrally buoyant fish swimming hori- 

 zontally in a straight line at a constant speed the 

 force balance in that plane (excluding forces due to 

 buoyancy and its compensation) is, in absolute 

 values, 



<CENTER OF CIRCLE 



(A) FORCES IN THE HORIZONTAL PLANE 



(B) FORCES IN THE VERTICAL PLANE 



Figure l. — Schematic description of forces acting on a fish when 

 swimming in a curved path. A. Forces in the horizontal plane. B. 

 Forces in the vertical plane. 



F, which is equal in magnitude and opposite in 

 sign to Fc , is of magnitude 



T = D 



(1) 



1 Tl2 



where T is the thrust and D is the total hy- 

 drodynamic drag. Equation (1) also describes the 

 force balance in the tangential direction for cur- 

 vilinear motion. However, an additional balance 

 must be made between the natural centrifugal 

 force Fc driving the fish to stay in a straight line 

 and the countering centripetal force F causing it to 

 remain on a cvirved path (see Figure lA). This force 



F = 



m'U 



R 



VW 

 R 



iPf-^Pu.^) 



(2) 



where m^ is the virtual mass of the fish, pf is the 

 density (averaged) of the fish, V is its volume, U is 

 the forward speed of the fish, A. is the longitudinal 

 added mass coefficient, andi? is the instantaneous 

 radius of the curved path. The added mass coefii- 

 cient k is multiplied by the density of water pw as 

 the added mass is the volume of water dragged 



FISHERY BULLETIN: VOL. 79, NO. 1, 1981. 



171 



