T 



Y 



i 



T 



Y 



i 



Figure 5. 



where .r is the axial and y the radial coordinate in the meridian plane. One can now 



formulate the problem of jet flow in terms of e', namely, a curve T(e) (Fig. 5) and a 



stream function \b{x, y; £ ) are sought such that \b — on the axis of symmetry, 



1 dip 

 if/ — const, on the bounding streamline C + T, and — — = 1 on r. Of course, 



y E dn 



£ = 1 corresponds to axially symmetric flow and is the case of immediate physical 

 interest. The contraction coefficient as a function of f is C c (e) = (Y( e )/Y) 1+e where 

 Y and Y are the orifice and jet radii (cf. Fig. 5). If it were indeed true that C c (l) 

 = C c (0) z= Tr/i-rr + 2), one might conjecture that C c ( f ) remains constant, and hence 

 that C' c (e) e=0 = 0. However, this quantity can be accurately computed and is equal 

 to — .057. Garabedian arrives at his estimate C c (l) = .58 by cubic interpolation, 

 using the exact values of r — Y/Y at £ — — 1,0, and oo where r = 0, tt/{tt + 2), 



d - 

 and 1, respectively, and the computed quantity — (Y/Y) e=0 — .243. Even without 



de 

 the latter figure, interpolation gives C c (l) = .586, an indication that .58 is probably 

 accurate. 



Turning to the relaxation method, we recall that the procedure here is to cover 

 the flow region with a rectangular mesh and to replace the differential equation by an 

 appropriate difference equation. In the problem at hand, the free streamline must be 

 chosen so that the velocity computed by finite differences is constant on the curve. 

 This requires a trial and error procedure which finally yields a curve on which the 

 computed speed variation is considered sufficiently small. Unfortunately, it is not 

 entirely clear what should be considered "sufficiently small" in a free boundary problem, 

 especially since the computed velocity fluctuation can be insensitive to relatively large 

 displacements of the free streamline. That this should be so can be anticipated for 

 theoretical reasons, but is also confirmed by specific calculations which compare an 

 exact solution for plane Riabouchinsky flow with one computed by the finite difference 

 method [12]. The speed variations on the free streamline in the latter calculation were 



288 



