Hydrodynamic Problems Solved by Rheoelectric Analogies 



Design of a Supercavitating Wing — The boundary conditions on a section 

 y = cte are the following: 



(a) = 0, on z = :■ ■•, " • ■' '■"■ '^ •' . -^^ 



(b) 0"^ = <^£(y)' on z"^ = -d, xj < X < 00 ■■ "7 , ' ' '' 



(c) cjf - cp^(y), on z~ = -d, Xj. < X < 00 



(12) 



CsCy) 



4^~ = 4>i (y) + — 5 — g(x- Xj ,y), on z = - d, xj < X < x^ 



x->o) J \ dn dn / .-•..•- 



Xf 



When cr is zero, the growth of the cavity thickness is simulated by a source 

 distribution over the wing and the cavity with a density q(x,y), defined by 



B0_^ ^ 3^2 - ii! - i^ . ■ . •■ ■■ 



Bn Bn 3x 3x ■ "' ' 



Far downstream, q is reduced to a function which depends only on y . However, 

 for the inverse problem we have a certain latitude in the choice of the source 

 distribution. In fact, the boundary conditions (a), (b), and (c) allow that on each 

 line parallel to the x-axis, within the limits defined by the wingspan, the potential 

 (p is fixed at an arbitrary level. If we indicate by k(y) the mean value of (t>^^ and 

 0t , we have 



k(y) = 0,(y) 4- • ■ ■''■ 



It is evident that the distribution of q over the surface of the wing and the 

 cavity, i.e., the cavitation shape, is directly influenced by the choice of the law 

 attributed to k(y). 



In order to facilitate analysis of the problem, the potential is subdivided 

 into two parts, defined by the following boundary conditions: 



(a') 0j = , 02 = , on z = 



(b') <P\ = 0ij(y) , 02 = •^2(y) • °n z+ = -d , xj < X < 00 



(c') 0- = 01 (y) , 0- = k2(y) , on z" = - d , x^ < x < 00 



(12) 



<^"i = ^ij(y) + <^t(y) g- <^2 = ^2(y) • °" z- = -d , x^ < x < x^ (cont) 



^ /1)4>\ B0-\ /f /B0+2 ^^2\ 



(d') lim + dx -» , lim + dx - . 



^ x^co J V Bn Bn / x^co J \ Bn Bn / 



Xj Xf 



391 



