diameter. These allow a maximum path radius of 

 about 300 cm. Recalling that the longitudinal 

 added mass for a streamlined fish shape is A ~ 0.2 

 (Webb 1975) and substituting the above values 

 first into Equation (4), we obtain 



D 



= 2 



(1,530)^'^ (1.06 + 0.2) 

 300 • 0.067 



= 1.44. (28) 



This means that the centripetal force is actually 

 larger than the total drag force under these cir- 

 cumstances. The total force exerted by the fish 

 swimming in a circular path of 300 cm radius is 

 thus, from Equation (6), Tt/T = 1.76, i.e., 76% 

 greater than that required for straight swimming 

 at the same speed. This is also, from Equation (9), 

 the ratio of oxygen consimiption. The ratio F/D 

 grows essentially proportionally to fish length, so 

 that larger fish expend an even greater proportion 

 of energy in swimming in a curvilinear path. 



We now look at the ratio of oxygen consumption 

 obtained for fish swimming under the same cir- 

 cumstances while banking their fins. This is calcu- 

 lated from Equation (18), assuming that at this 

 low (close to minimum) speed the pectoral fins are 

 fully stretched so that bs/bt = 1. The ratio Dis/Ds 

 is found from Magnuson's (1978) table 6, to be 

 approximately 0.3; 



El 



V, 



©2^ 



v; 



= 1 + 0.3 



OjS 



1.09 + 0.2 • 1.025 

 1.09 - 1.025 



i.e., the increase in energy requirements is only 

 2.6%, much less than for the asymmetric thrust, 

 nonbanked type of swimming. We can conclude 

 therefore that skipjack tuna moving at low speeds 

 in circular paths will use the banked swimming 

 technique once they are adjusted. The banking 

 angle can be obtained by banking the body and 

 keeping symmetrical fin angles (as in Figure IB), 

 or by asymmetric deployment of the pectorals. An 

 experimental program to study these angles by 

 photography is underway at present, as this is the 

 most obvious and easily measured prediction of 

 the present theory. The predicted value of the 

 banking angle (from the vertical) is obtained from 

 Equation (20) as 



m I 1 -i I C7^ 



tan a = — a = tan 

 W 



-1 



RW 



, (30) 



which for the skipjack tuna example above is 12.0°. 



We see above that the banking technique is 

 much more efficient at a relatively low swimming 

 speed. However, the ratio of energy consumption 

 [Equation (18)] is proportional to the speed to the 

 fourth power, so it is important to check on its 

 values at higher swimming speeds. There is no 

 complete set of data available for such speeds, so 

 we have to make some assumptions about the be- 

 havior of the various parameters. The ratio Dis/Ds 

 is most probably not speed-dependent, as the total 

 and induced drag are both proportional to the 

 speed squared. Thus, we can use the value 0.3 

 found above. Next, we assume that fin extension is 

 the same (at each speed) for straight-line swim- 

 ming and turning (the extension can change as a 

 function of speed, but the ratio remains constant). 

 This assumption will also be checked in the ex- 

 perimental program mentioned before, as it is eas- 

 ily corrected in Equation (18). 



Skipjack tuna can move at up to 10 body 

 lengths/s, i.e., over 400 cm/s in the present case. 

 Applying Equation (18) at different speeds (still 

 for a turning radius of 300 cm), we obtain 



Vcmis 66 100 120 150 175 200 300 400 



Q-,t 



11 1.026 1.137 1.285 1.696 2.289 3.199 12.13 36.18 



66^ 



981 • 300 



+ 1 



(1) 



= 1.026, (29) 



i.e., at speeds of about 160 cm/s ( ~ 4 body lengths/ 

 s) the asymmetric thrust method, which is inde- 

 pendent of speed, becomes the better way of com- 

 pensating for circular swimming. One can there- 

 fore establish a general upper limit for the 

 centrifugal increase in oxygen consumption fi"om 

 Equation (10), with the proviso that at lower 

 speeds the banking method is more efficient. 



Next, we calculate the increase in the minimum 

 swimming speed of the tuna mentioned above, for 

 which m = 1,670 g, CLmax = 1.0, Af = 36 cm^ 

 (Magnuson 1973), pw = 1.025, k = 0.2, and it! = 300 

 cm. From Equation (25) we obtain Ut/Um — 1.093. 

 The measured minimum speed in the round tank 

 is thus 10% higher than for the same fish moving 

 in a straight line. 



Using the allometric data of Magnuson (1973), 

 we can obtain Ut/Um as a function offish size for 

 various species. This appears in Figure 2 for the 



175 



