199 



FIGURE 10. Comparison of computed skin friction 

 coefficient with Vermeulen's data. 



tan6„ = 



w /u [ (u ,/u )g" - f "] 

 e e ref e ^w w 



(w /u )2 (u Vw )g" + f" 

 e e ref e w w 



(72) 



Figures 12 and 13 show a comparison of calculated 

 and experimental total velocity profiles and cross- 

 flow angle profiles along the lines C and E. Here 

 the crossflow angle is computed from 



sinB,., = 



V"e['"ref/"e'g' " ^'^ 



(73) 



As in Figures 8 through 11, again the agreement 

 between calculated results and experiment is very 

 good. The computed results follow the trend in 

 the experimental data well and indicate that the 

 present turbulence model, as in two-dimensional 

 flows, is quite satisfactory for three-dimensional 

 flows. 



Results for a Double Elliptic Ship Model 



To test our method for ship hulls, we have con- 

 sidered two separate hulls. The first one, which 

 is discussed in this section, is a double elliptic 

 ship whose hull is given analytically. The second 



one, which is discussed in the next section, is 

 ship model 5350 which has a rather complex shape. 

 Its hull is represented section-by-section in tabu- 

 lar form and contains all the features of most 

 merchant and naval vessels. It proves an excellent 

 test case to study the computational difficulties 

 associated with real ship hulls. 



The double elliptic ship model can be analytically 

 represented by 



f (x,z) 



1 -( - 



(74) 



FIGURE 11. Comparison of computed limiting crossflow 

 angle with Vermeulen's data. 



It has round edges except for the sharp corners at 

 X = ±L and z = ±H. The body of L:H:B = 1.0:0.125:0.1 

 together with the nonorthogonal coordinate nets on 

 the hull is shown in Figure 14. 



The potential-flow solutions were obtained from 

 the Douglas-Neumann computer program for three- 

 dimensional flows. To get the solutions, 120 control 

 elements on the surface were used, 12 along the x- 

 direction and 10 along the z-direction. 



Before we describe our boundary-layer calculations, 

 it is useful to discuss the pressure distribution for 

 this body shown in Figure 15. As can be seen from 

 the figure, the streamwise pressure gradient is 

 initially favorable in the bow region and then ad- 

 verse up to the midpoint of the body. This is fol- 

 lowed by a region of favorable pressure gradient and 

 then by a shape adverse pressure gradient very close 

 to the stern. The crosswise pressure gradient varies 

 in a more complex manner. Near the bow the pressure 

 decreases down from the water surface to a minimum 

 and then increases as the keel is reached. As the 

 flow moves downstream, the location of the minimum 

 pressure moves up and reaches the water surface at 

 about x/L = -0.80. The minimum pressure remains at 

 the water surface to about x/L = 0.80 and then moves 

 toward the keel. As a result, near the bow and the 

 stern, one may expect flow reversal of the crossflow 

 across the boundary layer does not reverse direction 

 from the keel to the water surface. This conclusion 

 is drawn from considering the pressure gradients only. 

 The real situation may be somewhat modified because, 

 in addition, there are the upstream effects and the 

 curvature effects on the flow characteristics. 



The boundary-layer computation starts with turbu- 

 lent flow from x/L = -0.90. We have tried to start 

 the computation from x/L = -0.97 and x/L = -0.95. 

 However, flow separation was observed at x/L = -0.90 

 near the keel due to the sharp curvature and adverse 

 pressure gradient in the bow region and can be seen 

 from Figure 15. In the previous calculations of 

 Chang and Patel (1975) and Cebeci and Chang (1977) , 

 the flow separation near the bow was not found due 

 to the orthogonal coordinate system they adopted in 

 which the second net point from the keel is so far 

 from the keel that the region of adverse pressure 

 gradient is omitted. 



In our boundary- layer calculations, we have used 

 40 points along the x-direction and 16 points along 

 the z-direction. In the normal direction, we have 

 taken approximately 40 points. The nonuniform grid 

 structure described in Cebeci and Bradshaw (1977) 

 is employed in the normal direction so that the grid 

 points are concentrated near the wall where the 

 velocity gradients are large. 



Some of the computed results for Rt = 10 are 

 shown in Figures 16 and 18. Figure 16 shows the 

 spanwise distributions of the pressure coefficients, 

 Cp, local skin-friction coefficient, Cf, the shape 



