Numerical Solutions 



cards, and a computer program has been written to use these tables, in connec- 

 tion with the ship offsets to determine the projections on the ship's centerplane 

 of: 



(a) Three lines of constant pressure on the ship's hull. The uppermost of 

 these lines is the line of zero pressure or the free surface. The bottom two 

 lines correspond to the locus of points such that the local piezometric head 

 equals 0.5H and H, where H is the draft of the ship at rest. 



(b) Three streamlines on the ship's hull. The uppermost streamline also 

 corresponds to the free surface and thus is identical to the uppermost constant- 

 pressure line. The bottom two streamlines correspond to streamlines at a 

 depth of 0.5H and H at upstream infinity. 



Since Guilloton's method yields only three streamlines and three constant 

 piezometric -head lines, several additional streamlines were interpolated. With 

 these results it was possible to obtain the variation of the velocity vector along 

 the streamlines as required to permit integration of Eqs. (l)-(3). 



NUMERICAL RESULTS AND DISCUSSION 



The numerical computation was performed on the Hydronautics, Inc. IBM 

 1130 computer. Two typical ships, series 60/.60 and series 60/.80 were used 

 in the present computation. Five speed-length ratios of 0.75, 0.80, 0.85, 0.90, 

 and 0.95 corresponding to Froude numbers 0.224, 0.237, 0.252, 0.268, and 

 0.283, respectively, were used for each ship. Each Froude number covers five 

 ship lengths- 800, 500, 200, 20, and 5 feet. Typical results of the cross-flow 

 angle /3 , shape parameter h, and momentum thickness e , along streamlines 

 are shown in Figs. 1 and 2. The cross-flow angles are shown only along the 

 stern section of these ships since the results are more reliable there. 



Cooke's criterion is that separation occurs when the cross flow is 90°. It 

 is important to note that within the range of present computation no separation 

 is found before station 19 for the series 60/.60 ship model as well as its proto- 

 type (Fig. 1, for example). However, flow separation occurs at the shoulder of 

 series 60/.80 model ships at low Froude numbers in the present calculation 

 (Fig. 2, for example). The tendency toward separation at shoulder of a series 

 60/. 80 ship is stronger for the model ship than that of the prototype; for the 

 model ship, separation occurs only near the free surface. It is to be noted that 

 the exact potential field near the bow is not known and the initial conditions 

 used at station 1/2 are only the first approximations. Thus, the present results 

 on separation at the shoulder may at best be considered as indicating the trend. 

 The exact prediction is understood to be beyond the scope of the present study. 



The cross-flow angle is larger near the stern of the ship model than that of 

 the prototype for both ships of all Froude numbers calculated. Thus, separation 

 is more likely for the ship model if it would occur after station 19. The present 

 results indicate that the values of d/h and H at a given station of the ship is much 

 larger for the model ship than that of the prototype. 



1632 



