HYDRODYNAMIC FORCES 



137 



bow of a fast ship without forefoot emergence, severe 

 shocks (slams) to commercial vessels, and heavy pres- 

 sures, occur as the bottom strikes water on its downward 

 motion after emergence. This is similar to impacts 

 sustained when seaplanes land. As a consequence the 

 principles developed in the seaplane field can be used 

 with only a few modifications. 



Seaplanes generally have V-shaped bottoms in order to 

 alleviate landing shocks. The sections are often modi- 

 fied by giving them curvature. Quantitative evaluation 

 of the impact forces is based on two theories which, for 

 brevity, will be referred to as the "expanding-plate" and 

 the "spray-root" theories. 



The first originated with von Karman (1929) and was 

 given its full development by Wagner (1931). In it the 

 water flow jiattern at the wedge penetrating the water 

 surface is taken to be comparable at each instant with 

 the flow about a flat plate of the same width as the wetted 

 width of the wedge. This is shown in Fig. 31. On the 

 basis of this analogy, the velocities and accelerations, the 

 resultant prcssiu'es, and the total water reaction are ob- 

 tained. This theory has been used by Mayo (1945), 

 Benscotter (1947), and Milwitzky (1948) for the analysis 

 of the entire seciuence of seaplane impact; i.e., in de- 

 fining the wedge penetration, velocity and acceleration 

 during impact as functions of time. 



Were the water surface not disturbed, the wetted semi- 

 breadth^* Co would be connected with the instantaneous 

 draft z by 



Co = 2/tan fi, (61) 



where fi is the angle of deadrise. However, the water ou 

 the sides of a wedge rises as the wedge displacement in- 

 creases. Wagner (1931) found, on the basis of the ex- 

 panding-plate analogy, that the actual wetted semi- 

 breadth is 



C = tCo/2 (62) 



Likewise, if the vertical velocity of the falling wedge 

 s = To, the rate of propagation C at the edge of a wetted 

 area in the horizontal direction is 



C = - Vo/tan (3 



(63) 



Thus, at small angles of deadrise /3, the horizontal veloc- 

 ity of the point S at which the water meets a body sur- 

 face is very large. Fig. 31. The local pressure at the 

 point is approximately 



V. 



= PCV2 = \ p 



f'o/tan 



(64) 



This simple expression shows that the peak of the unpact 

 pressure increases rapidly in magnitude with decrease of 

 the angle of deadrise /J. 



The foregoing relationships have been given for a sim- 

 ple V- wedge with straight sides. Wagner (1931) has 

 shown that, if another body form can be represented 



Fig. 31 Representation of water flow pattern induced by falling 

 wedge by flow about lower half of a flat plate of half-breadth C 

 equal to instantaneous wetted half-breadth C of wedge (follow- 

 ing Wagner, 1931) 



by the following series 



y = B^x + Byx- -I- B.x^ + ... + B„.v"+' (65) 

 the corresponding ratio I'o/C becomes 



v„/C =- Bo + BiC + - B.r- 4- -J B^C^ + 



(66) 



^ The notation often u.scil in aeronautical literature is retained 

 in this section. 



M. A. Todd (1954) and Bledsoe (1956) used this prin- 

 ci])le in evaluating the impact pre.ssures on ship bot- 

 toms. Small initial deadrise and a sharp turn of the 

 bilge necessitated use of a few terms with high powers. 



7.2 Spray Root Theory. The expanding-plate theory 

 correctly indicates the rise of the water surface on the 

 sides of a wedge due to pressures generated in water. 

 It also shows that maximum pressure occurs near the 

 edge of the wetted area and indicates with gootl approxi- 

 mation the total force exerted liy water on a plate. It 

 does not, howe\'er, describe in detail the local water flow 

 phenomena occurring at the edge of the wetted area. 

 These phenomena have been described by the "spray 

 root" theory of Wagner (1932). Existence of high pres- 

 siu'es near the edge of a wetted area is connected with for- 

 mation of the spray jet in which water is quickly accele- 

 rated to a high velocity. 



The complete mathematical solution of the time- 

 dependejit flow about two sides of a penetrating wedge 

 with two spray roots, i.e., two regions of spray genera- 

 tion, has not been possible. Wagner (1932) treated 

 three substitute steady-state flows about flat plates. 

 These are valuable in describing local details of the water 

 flow in the spray root and in giving the resultant pres- 

 sure distribution over a flat planing surface. Further- 

 more, a numerical solution has been given in the case of 

 a penetrating wedge with two spray roots (J. D. Pierson, 

 1950, 1951). 



The work of Wagner was written in German and is 

 difficult to understand because of extreme condensation 



