FISHERY BULLETIN: VOL 77. NO 3 



some distance away, which is not disturbed by the 

 fish's passage. 



The drag coefficient C^ has been found experi- 

 mentally to be a similar function of Re for various 

 shapes. Thus, for all shapes tested (Hoerner 1965, 

 chapter 3 >, the coefficient C p is a decreasing func- 

 tion up to Re 200-300, being a constant for larger 

 Re up to the turbulent regime which starts at 

 about Re = 500,000, where a new, lower, constant 

 value is obtained. Thus when Re>200, hydro- 

 dvnamic drag becomes proportional to the veloc- 

 ity squared, as the other factors in Equation ( 1) do 

 not change for a fixed body. 



The Re>200 regime was examined by the au- 

 thor ( Weihs 1974 ) previously and will therefore be 

 mentioned here for comparisons only. When 

 Re<200, two main regimes are observed, that of 

 Re<10 and 10<Re<200. The low Re range, in 

 which velocity plays a much more important role 

 than the inertial (acceleration) force effects, has 

 C[) <x Re '. Strictly speaking, this regime ends 

 when Re = 1, but experiments have shown that 

 this relationship can be applied up to Re = 10 with 

 good accuracy. Vlymen (1974) actually used this 

 relationship at Re = 30, where C^, estimated by 

 the low Re formula, still is within lOVf of the exact 

 value, for his analysis of larval anchovy swim- 

 ming. The remaining range is a transition regime 

 where Cp gradually changes from the low Re form 

 to the high Re iC^, cc Re") form. 



As continuous swimming in anchovy larvae is 

 observed mainly in the yolk-sac stage (2.8-5 mm 

 long), the low Re regime is relevant. Thus, Re = 8 

 corresponds to a 3 mm long larva moving at 3 

 mm's, and a 5 mm long larva, moving at 5 mm/s, 

 experiences a Re<30. 



Applying the relationship for low Re (C^ « 

 Re ' ), Equation ( 1) can be written 



D = - p ACu 



(2) 



where C is a numerical constant depending on the 

 shape. 



When a fish is actively swimming by means of 

 body and fin oscillations, the drag force is affected, 

 both through the increase in frontal area A, and 

 through the change in the numerical coefficient C. 

 Thus, for example, C = 20.37 for a disc moving in a 

 direction normal to its plane, and is equal to 13.6 

 when moving parallel to its plane. Experimental 

 data collected by Webb ( 1975) show that the drag 

 coefficient for fish swimming at high Re can be up 

 to four times that of a rigidly gliding fish. For the 



598 



low Re regime considered here, the increase in 

 drag coefficient caused by swimming motions does 

 not exceed a factor of 2, for a long slender body ( Wu 

 et al. 1975). This increase is written as 



D, 



p ACau = oiKu 



(3) 



where a is the ratio of swimming to gliding drag 

 and the subscript ,s stands for active swimming. 

 The equation of motion for fish, while actively 

 producing a thrust T is 



du_ 

 dt 



+ aKu 



(4) 



where m is the fish mass (slightly augmented by 

 the added mass at higher Re), and K is defined in 

 Equation (3). 



The energy ( E ) required to traverse a distance L 

 within time t is 



£ = ( Tudt. 

 b 



(5) 



There is no available information on the depen- 

 dence of propulsive efficiency on swimming speed 

 for larval anchovy, so that we have to perform the 

 calculations on the energy required, and not the 

 energy actually expended by the fish. However, for 

 comparison of different swimming modes, these 

 are equally applicable. 



The purpose of the present study is to compare 

 the effectiveness of continuous and intermittent 

 swimming of larval fish so that we start by finding 

 the energy, E, , required for swimming at constant 

 speed (7; over the same distance L. From Equation 

 (5i, 



E,. = TAL T = T^L 



(61 



where the subscript c stands for continuous 

 swimming. Defining the energy expenditure per 

 unit distance as £ = E/L we obtain, by applying 

 also Equation (4), 



E, = T, = aKU,. 



(7) 



Intermittent .swimming, on the other hand, re- 

 quires energy only during the beating part of the 

 cycle so that the total energy for one beat-and- 

 glide cycle is described by 



E„ = / Tudt 







o ^ 



mu^ + aKu^jdl <8 



