The method also takes charge of reentrant jet problems such as the impact of 

 a stream on a wedge, Fig. 5. 



Applied to the impact of a stream on a lamina, Fig. 6, the image flow indicates 

 a layer of sources on the downstream face, and thus offers at least a suggestion for 

 tackling the corresponding problem of a stream impinging on a circular disc. 



Riabouchinsky [15] 30 years ago discovered a method of finding the drag on 

 a lamina exposed to a stream, Fig. 7. The method consists in placing an image lamina 

 downstream, the two being joined by free streamlines. 



This problem is capable of exact solution [16] in terms of Jacobian elliptic 

 functions, where PrandtFs cavitation number, CT = (p x — p c )/V2pu 2 , appears in the 

 parameter of the elliptic functions. The case of an unlimited cavity can then be 

 approximated by increasing the distance between image and plate. 



Quite recently Garabedian [17] has undertaken the numerical study of the 

 axisymmetrical problem, in particular that of the circular disc with an equal image 

 disc behind it on the Riabouchinsky model. The two discs are joined by a free stream 

 surface to enclose a region containing water vapor. 



The equation satisfied by the stream function \p is 



1 



^xx +$yy 1^ = (28) 



y 



with if, = on the entire boundary and 



1 ty 



=1 (29) 



y dn 



on the free stream surface, where x and y are coordinates in a meridian plane. 

 From these we derive 



d I 1 dip k \ 



— + -* =0 (30) 



dn\y dn y ) 



on the free stream surface where k is the curvature of the free streamline in the 

 meridian plane. 



The boundary conditions can be combined to give 



1 dxP K 



+ -*=1 (31) 



y dn y 



on the free streamline and this boundary condition is therefore stationary in the sense 

 that by (30) a normal displacement 8n of the free streamline leads to an error of order 

 (§h) 2 . A convergent iterative process can be founded on this observation. Garabedian 

 has carried this out, taking an initial position of the free streamline based on the curve 

 afforded by the plane flow solution of Riabouchinsky. 



Let us now turn to the problem of free surfaces when the liquid moves under 

 gravity. By a free surface we shall mean a surface which always consists of the same 

 fluid particles and on which the pressure is constant. The grand illustration in nature 

 is the surface of the ocean. 



But few simple complete solutions of this problem are known. 



The only non-trivial cases which occur to one are Gerstner's trochoidal wave 

 and Rankine's combined vortex. In the Gerstner wave the free surface is a trochoid 

 and the motion is rotational. In Rankine's combined vortex the motion is rotational 

 within a vertical cylindrical core and is irrotational outside the core, Fig. 8. 



