173 



constant assumption can be employed unless the radii 

 of curvature are not significantly small; the dis- 

 placement effects are important near the stern and 

 should be taken into account in higher order calcu- 

 lations [Hatano and Hotta (1977)]. 



Velocity Profiles 



In order to calculate the boundary layer equations 

 by an integral method it is convenient to represent 

 velocity profiles by analytical functions which in- 

 clude several parameters. 



The most commonly used formulae are based on a 

 1/n-power law and on a wall-wake law. The former 

 has a definite merit of simplicity. The latter, 

 developed by Coles (1956) , has more freedom than 

 the 1/n-power law and can be expected to represent 

 velocity profiles more exactly. 



Mager ' s expression is well known as the three- 

 dimensional velocity profile model based on a 1/n- 

 power law, Mager (1951) . He gave the streamwise 

 and crossflow velocity profiles as 



qi/U^ = 



6 

 1/n 



1/n 



q2/U = tani 

 where n is a variable parameter. 



1- 



(3) 



(4) 



If velocity profiles are represented by Eqs. (3) 

 and (4) , the boundary layer thickness-parameters 

 6*, Sii and shape factor H are 



^ / ' (Ui 

 "e 



qi)dc 



n+l 



ni 



^ ^\'"l - ^l"^^ = (n+l)"n+2) 



6, 



(5) 



&1 



n+2 



'11 



and Eqs. (3) and (4) can be written in other forms, 

 H-1 



qi/"e 



11 



H-1 

 H(H+1) 



(6) 



q /U = tang b 



H-1 



H-1 

 2 



11 



H(H+1) 



[1 



H-1 



'll «'«+l' (7) 



If 6 and 6* are integrated and B is determined 

 from measured velocity profiles, then velocity pro- 

 files represented by Mager' s model can be calculated 

 from Eqs. (6) and (7) and can be compared with the 

 measured profiles. 



Figure 5 shows the comparisons of them. It can 

 be safely pointed out that Mager 's model is employ- 



5(mm) 



40 r 



IJVue 



FIGURE 5. Velocity profiles represented by 

 Mager 's model. 



