205 



-0.6 r 



-0.4 



-0.2 



-too 



FIGURE 22. Pressure distribution for the bow region 

 of ship model 5350. 



bow is a saddle point with the streamlines of the 

 inviscid flow converging on A along the line BA and 

 diverging along an orthogonal direction. It is 

 known that the boundary-layer equations can always 

 be solved at B but that at A the situation is more 

 complicated and furthermore it is still not entirely 

 clear what their role is in relation to the general 

 solution. It is likely, however, that provided no 

 reversed flow occurs at A in the component of the 

 solution along the direction BA, then separation can 

 be avoided along this line by appropriate choice of 

 design. Furthermore, if separation does occur, its 

 effect may be limited. The recently developed Cebeci- 

 Stewartson procedure (1977) , however, can be applied 

 to the present problem but there are some hurdles to 

 be overcome . 



Of particular difficulty is the choice of coordi- 

 nate system on which to compute the solution and to 

 join it with the already well-established method 

 downstream of CD. We have seen that in the case of 

 the prolate spheroid (see Cebeci, Khattab, and 

 Stewartson (1978) ) it is helpful to have a mesh 

 which is effectively Cartesian near the nose and the 

 methods which were used to produce it in the earlier 

 study are applicable to any body which can be repre- 

 sented by a paraboloid of revolution in the neighbor- 

 hood of the nose. Now here we have a paraboloid near 

 B but not one of revolution, but we believe that the 

 necessary generalization is possible. The mesh now 

 has to match with that which has proved convenient 

 downstream of CD. Again we believe that a smooth 

 transition can be achieved by building into the 

 mesh sides, right from CBA, an appropriate spacing 

 such that the points of a uniform mesh on CD are 

 also points of this mesh although not, of course, at 

 a constant value of one of the coordinates. Our 



evidence for this is based on a successful scheme 

 that we have already worked out for the prolate 

 spheroid, Cebeci, Khattab, and Stewartson (1978). 



Other aspects that need further study include the 

 condition at the water-line section. It has been 

 usual to assume that the normal velocity is zero at 

 the undisturbed free surface. This is not quite 

 correct and the error may have implications for the 

 nature of the solution near A and especially the 

 question of separation along BA. Even if separation 

 does occur, it may be possible to handle the post- 

 separation solution, since it probably extends only 

 over a limited region of the ship, by means of an 

 interaction theory, i.e., modifying the inviscid 

 flow by means of a displacement surface. 



Viscous-Inviscid Flow Interaction 



The present boundary-layer calculations are done 

 for a given pressure distribution obtained from an 

 inviscid flow theroy. In regions where the boundary- 

 layer thickness is small, the inviscid pressure dis- 

 tribution does not differ much from the actual one; 

 as a result, the boundary-layer calculations are 

 satisfactory and agree well with experiment, see, 

 for example, the papers by Cebeci, Kaups , and Moser 

 (1976) and by Soejima and Yamazaki (1978) . When 

 the boundary-layer thickness is large, which is the 

 case near the stern region, the effect of viscous 

 flows on the inviscid pressure distribution must 

 be taken into account. One possible way this can 

 be done is to compute the displacement surface for 

 a given inviscid pressure distribution and iterate. 

 Such a procedure is absolutely necessary to account 

 for the thickening of the boundary layer as was 

 observed by Soejima and Yamazaki (1978) . 



Prediction of Wake Behind Ship Hulls 



The present boundary- layer calculations can be done 

 up to some distance close to the stern; after that, 

 flow separation occurs. Since one, and probably the 

 biggest, reason why one is interested in boundary- 

 layer calculations on ship hulls, is the calculation 

 of drag of the hull , additional studies should be 

 directed to perform the calculations in the separated 

 region and in the wake behind the ship. Recent calcu- 

 lation methods developed and reported by Cebeci, 

 Keller, and Williams (1978) for separated flows by 

 using inverse boundary- layer theory and recent calcu- 

 lation methods developed and reported by Cebeci, 

 Thiele and Stewartson (1978) for two-dimensional 

 wake flows are appropriate for these purposes . 



PRINCIPAL NOTATION 



Van Driest damping parameter, see 



(18b) 

 constants 

 local skin-friction coefficient in 



streamwise direction 

 constants 



transformed vector potential for ijj 

 transformed vector potential for ((> 

 metric coefficients 

 net spacing in ri-direction 

 boundary-layer shape factor along 



streamwise direction, 6*/9jj 



Ai,A2,A3,A;+ 

 Cf 



Ci ,C2 ,C3,C^ 

 f 



g 



hi,h2 



hj 

 H,Hii 



