along with the fish. The hydrodynamic drag can be 

 written as (Hoerner 1965) 



D --P^V^^U^C^, 



(3) 



where Cot is the total drag coefficient based on the 

 two-thirds power of the volume, as reference area. 

 Here drag includes gill drag, induced drag due to 

 pectoral fin extension and lift, etc. (see Magnuson 

 1978 for an excellent description of the various 

 components of drag). 



To find the relative importance of the centripetal 

 force, we divide Equation (2) by Equation (3), ob- 

 taining 



F 

 D 



V 



1/3 



RC 



+ xi 



(4) 



Dt 



showing that the fish's velocity does not influence 

 the ratio of centrifugal to drag forces in this case. 

 Equation (4) can now be used to obtain the correc- 

 tion to the force produced by the fish, since the 

 total force Tt that the fish requires for swimming 

 in a circular path in the horizontal plane is, by 

 vectorial addition. 



Tt = 



D' + F' 



(5) 



Dividing by D^ and applying Equation (1), we ob- 

 tain the required thrust increase for neutrally 

 buoyant fish, as a function of what will now be 

 called the correction factor FID, 



1 + 



(6) 



Most studies of respiration and energetics in fish 

 measure the aerobic oxygen consumption of the 

 test animals as an indication of energy consump- 

 tion. To obtain the correction required when the 

 test data are obtained in a round tank, we recall 

 that, for constant speed swimming 



E = TU 



(7) 



where E is the rate of working. Thus the ratio of 

 work (per unit time) for a continually turning fish, 

 compared with that offish swimming in a straight 

 line at the same speed, is 



E 



TU 



(8) 



The basic assumption of such respiration 

 studies is that the oxygen uptake rate Vo^ is di- 

 rectly proportional to £^, so that 



V. 



Ojf 



(9) 



V, 



OjS 



and with Equation (4) we can write this ratio in 

 terms of experimentally measurable quantities: 



V, 



Ojt 



V, 



Ojs 



V 



1/3 



1 + 2- 



RC 



^ + X 



(10) 



Dt 



w 



i.e., the equivalent oxygen uptake for a free- 

 swimming fish moving in a straight line is always 

 lower than that measured in a round tank. Also, 

 for a given turning radius, the larger the fish, the 

 greater the increase in oxygen uptake, or con- 

 versely, for a given fish, the oxygen uptake in- 

 creases with decreasing turning radius. The in- 

 crease is almost independent of swimming speed 

 except for a possible weak Reynolds number de- 

 pendence of Cd^, neglected here. 



The method shown above for obtaining the cen- 

 tripetal force by increasing the thrust is one possi- 

 bility. A different way of getting the required 

 forces is by means of hydrodynamic lift. 



Negatively buoyant fish, such as most of the 

 scombrids, produce lift by swimming continuously 

 in order to maintain a horizontal course. This de- 

 fines a hydrodynamical minimum swimming 

 speed for horizontal motion (Magnuson 1970) and 

 causes an additional effect due to swimming in a 

 curved path. 



The lift required is usually produced by the pec- 

 toral fins being placed at a small angle of incidence 

 to the direction of motion. This force is at right 

 angles to the longitudinal axis of the fish. It can 

 therefore be applied to the production of the cen- 

 trifugal force, by banking towards the center of 

 curvature (see Figure IB). Such behavior is an 

 alternative to the added asymmetric thrust calcu- 

 lated above [Equations (4)-(6)], as no direct extra 

 thrust is needed. However, the total lift force on the 

 fins is now larger (Figure IB), resulting in higher 

 induced drag (Webb 1975; Magnuson 1978), which 

 increases the thrust required. 



172 



