Hydrodynamic Problems Solved by Rheoelectric Analogies 

 On the lower surface of the foil, taking into account the gravity effect, 



and 



3^1 





+ F-2(^j-^ ), 



ClECx) + F-2(^2-^2J 



The first function corresponds to a nonlifting and free- of-wave- resistance 

 effect, as has already been shown (13). The second function represents the lift- 

 ing effect connected with the expression of the local pressure distribution. The 

 calculation is made by starting with the solution for F = co^ which is of an easy 

 analog determination because at the free surface B0/3n = o, and the above con- 

 ditions are of the Neumann type, with flow continuity (Fig. 3B). From this first 

 solution it is easy to define distributions of sources and sinks and of vortices in- 

 duced by the cavity, so that we can now calculate numerically the free surfaces 

 for finite Froude numbers. The iterations are then carried out as described 

 previously in the subsection Solution of the Direct Problem. 



Figure 9 shows the form of foils for the same immersion depth, at the same 

 cavity length, and for Froude numbers infinity and 3.99, as a function of the pa- 

 rameter C^/cT. The results for the infinite Froude number are given as a means 

 of comparison; it is evident that in this case the hypothesis of a finite cavity is 

 no longer valid, since the cavitation number tends to zero in both instances. 



Infinite Froude Number— On the free surface, the upper and lower surfaces 

 of the cavity, we have Bi/z/Bn = O; on the lower surface of the foil a Neumann 

 condition is imposed, -BiA'/Bn = Cl g(x). This makes rheoelectric simulation 

 easy. Figure 10 is a comparison of foils computed for different linear pressure 

 distributions with a foil fulfilling the two-term law of Tulin-Burkart. The com- 

 parison of the lift- drag ratio is favorable in the former and shows the advantage 

 of the rheoelectric method in the exploitation of pressure distribution which is 

 hardly accessible to analytical treatment. 



If a convenient pressure distribution over the upper surface of the foil is 

 imposed, it is possible to design base- vented hydrofoils with zero spray- jet drag. 

 The depressions thus imposed should be such that the cavitation formation is ex- 

 cluded upstream of the trailing edge. For this purpose a number cr^ must be de- 

 fined, which is a function of the physical characteristics of the fluid, the vapor 

 pressure, the degree of air dissolved, etc. Three foils, obtained for different 

 pressure distributions and presenting the same value of Cl/^-, are shown in 

 Fig. 11. 



Hydrodynamic Characteristics of Supercavitating Hydrofoils 

 in Unbounded Flow 



These studies were intended to test the feasibility of analog representation of the 

 singularities which arise in the solution of the direct problem of supercavitating 



381 



