177 



E(H) 



(H-1) (H+2) 

 J(H) = E(H) - D(H) , 

 N(H) = -^ E H 



6-6 



(22) 



Then Eqs. (17), (18), and (20) are reduced to 

 simultaneous differential equations in 36ii/35, 

 3H/35, and 3e/3C; 



^ilF" ^ ^itt ^ ^ilf = <^i <-^'^-3' 

 where 



ai =1, bi = 0, cj = 0, 



di = -Jtangf^^ - J-eiitanefi - jeiisec2e|^ 



^ dr\ -^ ^ 3ri ^ ^ 3ri 



i) IT 



- (H+2)^-^ + KiBiid-Ctan^B) 



+ T / pu2 , 



wi e 



(23) 



a2 = EtanB, b2 = E'e^tanB, C2 = Eeusec^g, 



2„3in 



Bjj 3U 

 d2 = -2Etane-- — ^7^ - Ctan ^„ 



C'Biitan'^B- 



3H 



3n 



20611 tanBsec^g-^ - (l+H+Ctan2B)K2ei 1 



3n 



+ 2KiE6iitanB + x , / pU^tanB , 

 wl e 



33 = N, b3 = N'Bii, C3 = 0, 



(24) 



3B11 



3n 



'3n 



3n 



1 8"e 



Ug 35 ' 



(The ' means differentiation with respect to H.) 

 If Ludwieg-Tillmann' 3 skin friction formula, 



Eq. (12) , is used, all the coefficients of Eq. (24) 



are known at earlier 5 coordinate. 



This formulation is the same as that of Cumpsty 



and Head (1967) . 



Numerical Calculations and Discussions 



Numerical calculations were carried out for GBT-125 

 at Rg=10 . First, 18 streamlines were traced inter- 

 polating the 254 X 2 descrete values of velocity, 

 obtained by the surface source method, and xj coor- 

 dinates were determined. 



The differentials with respect to ri were numeri- 

 cally determined along the n axis which was defined 

 by bending short segments orthogonally to the xj 

 axis. This is the main difference from Cumpsty- 

 Head's original calculations. It>r such calculations 

 as 3Ue/3ri, 39ix/3ti» and so on, the differentials 

 with respect to n should be carried out as care- 

 fully as possible. Most numerical errors stem from 

 ■these terms. 



e../*«io* 



1.^ 



FIGURE 10. Comparisons of momentum thickness (GBT-125) 



0.5 X 10"**, 1.4, and 0.0 were used for the 

 initial values of Qn, H and B at S.S. 9Js(x=-0.85) . 

 These values were obtained from Burl's two- 

 dimensional formula assuming the flow is turbulent 

 just from P.P. (see Figure 1) . Fortunately they 

 do not seriously affect the calculations. 



About 200 steps were taken and Eq. (23) is 

 integrated with respect to 5 by Lunge-Kutta-Gill' s 

 method (five points for each step) . 



In Figures 10, 11, and 12, calculated results of 

 9n, H, and B along typical streamline Nos. 5, 9, 

 and 11 are shown along with experimental results. 

 The skin friction is shown in Figure 8. Streamline 

 No. 5 generates a simple, quasi-two-dimensional 

 curve on the hull surface and it may be expected 

 the flow can be truly represented by the present 

 framework. On the other hand streamline No. 11 

 passes through a region where the boundary layer is 

 rather thin and also through a bilge corner where 

 pressure increments were observed. 



The experimental values of the streamwise momen- 

 tum thickness, 9ii, of streamline No. 11 were much 

 greater than those calculated around S.S.I. This 

 discrepancy can be related to the fact that S.S.I 

 of streamline No. 11 corresponds to the position 



H 



STREAMLINE NO. 5 



NO. 9 



CALCULATED 

 MEASURED 



1.5 



1.0 



NO. 11 



_] 1 -1- 



F.P. 9 



! 7 6 5 4 3 2 1 A. P. 

 FIGURE 11. Comparisons of shape factor (GBT-125). 



