forces. Thus, the minimum swimming speed for 

 moving in a curved path is always higher than 

 that for swimming along a straight line. This 

 further increases the effort required to swim in 

 circular tanks above and beyond the thrust correc- 

 tions found before, when the fish wants or has to 

 move at minimum speed. 

 At the minimum speed Um 



W = V2p^Af^ 



u  



max m 



(19) 



where W is the submerged weight of the fish and 

 A/- is the fin lifting area. If the fish turns (when it is 

 confined to a round tank) and still remains at 

 minimum speed, the pectoral fins are banked. The 

 force produced on the fins is then (see Figure IB) 



P = 



W'^ + F^ 



or 



(20) 



(21) 



P is produced while the fins are at CLmax so that 

 P = V^p^Af^^^^U,' (22) 



where Ut is the minimum speed for a horizontal 

 turn. This is a function of the turning radius, 

 through F [see Equation (2)]. 



Substituting Equations (22), (19), and (2) in 

 Equation (21) we have 



(23) 



U,y __^^ 2ma^X,J,,) (V^^ ^^^^ 



u. 



m, 



Pw^^fLm.. \U„ 



from which = 



1/2 



(25) 



where B ^ "'^' ^ ^'"'"^ i. 



From Equation (25) we see that Ut>Um for all 

 174 



finite values of R and that the minimum swim- 

 ming speed grows as the radius decreases. The 

 increase in minimum speed also is reflected in the 

 total effort required for swimming in the round 

 tank, as the rate of working goes up as the cube of 

 the velocity. The induced drag changes [Equation 

 (11)] are included since the fins are producing 

 maximum lift coefficients both in straight-line 

 swimming and in turning 



V, 



Ojf 



V, 



minimum speed = 



U, 



OjS/ 



U„ 



(26) 



Results and Discussion 



To estimate the actual significance of the vari- 

 ous corrections developed in the previous section, 

 numerical values of the parameters are now sub- 

 stituted for a negatively buoyant species, the skip- 

 jack tuna. 



Most of the quantities appearing in the oxygen 

 consvunption ratios [Equations (10), (18), and (26)] 

 are easily measurable. The added mass and drag 

 coefficients (A. and Cot) are more difficult to ob- 

 tain, as both may also be dependent on the turning 

 radius (due to higher drag when turning and addi- 

 tional added mass effects due to the fish body cur- 

 vature). However, no such information is presently 

 available, so both these quantities have to be esti- 

 mated from data on rigid engineering structures. 



The most complete set of data for estimation of D 

 appears in Magnuson (1978, table 6) for skipjack 

 tuna. These data will now be used to obtain a 

 typical value of the correction factor Equation (4). 

 The total drag for a 44 cm skipjack tuna swimming 

 at 66 cm/s was estimated to be 19,780 dyn. The 

 mass of the fish is approximately 1.67 kg (Naka- 

 mura and Uchiyama 1966), and with an average 

 density of 1.09 (Magnuson 1978, table 3) the vol- 

 ume of the fish is 1,530 cm^. The drag coefficient 

 Cot (which is different from Magnuson's drag co- 

 efficient because of the different reference area) is 

 found, from Equation (3), to be 



2 • 19,780 



'Dt 



1.025 • (1,530) 



2/3 



66' 



0.067 . (27) 



Magnuson's (1978) data are partially based on the 

 study of ram ventilation by Brown and Muir (1970) 

 which was carried out in the holding tanks of the 

 National Marine Fisheries Service Kewalo Re- 

 search Facility in Honolulu, which are of 7.3 m in 



