and Spencer applied Green's theorem in an iterative process which alternated between 

 conditions on the free surface and those of the solid body. Fig. 2 shows the calculated 

 distributions of velocity and pressure over the surfaces of these wedges and cones. It 

 is interesting to note that the peak pressure on the body is attained close to the origin 

 of the spray jet. 



An alternative approach to the normal entry problem is to construct a model 

 flow which reproduces the main conditions of the problem. Von Karman (1929), for 

 instance, used a technique in which he approximated to the conditions of impact of 

 seaplane floats striking the water by the flow produced by the impulsive vertical motion 

 of a plate expanding in the plane of the water surface. For similar horizontal water 

 surface areas, the pressure distribution produced by this model flow approximates to 

 that over the actual submersed body. In his original paper von Karman made no 

 allowance for the rise of the free surface but methods can usually be developed to 

 introduce this refinement, if required. Wagner's (1933) paper did this for the von 

 Karman problem. 



Most of the theoretical investigations of water entry carried out during the 

 last decade have been concerned with extensions of the methods originally developed 

 by von Karman and Wagner. Thus problems have been formulated to take account 

 of the actual penetration of the body into the water. This is usually done by consider- 

 ing the flow about an expanding model consisting of the submerged portion of the 

 actual body together with its image in the plane of the undisturbed free surface. An 

 example of this type of investigation is the war-time work of Schiffman and Spencer, 

 who considered the entry of a sphere by means of an expanding lens-shaped model. 



P=0 



p s 



(b) 



Figure 3. Expanding hollow body models for water entry flow patterns. 



218 



