Figure 4. 



significance.) This may be in the nature of things and it is quite possible that such 

 solutions are simply unattainable. 



It is therefore natural to fall back on numerical methods, and with the increas- 

 ing emergence of high speed computing machines it is appropriate that methods be 

 developed for the calculation of cavity flows. At the present time, the labor in these 

 calculations is formidable, and most questions of convergence and accuracy remain 

 unanswered. 



An interesting and valuable illustration is provided by the problem of the 

 vena contracta, which is perhaps the best studied of the axially symmetric free surface 

 problems. The experimental value for the contraction coefficient of a jet issuing from a 

 circular orifice in a plane wall is in the neighborhood of .61. This figure is of par- 

 ticular interest because it coincides with the known theoretical value tt/{tt + 2) ~ .611 

 of the contraction coefficient of a plane jet issuing from a slot. This agreement has 

 stimulated the conjecture that the theoretical values of corresponding plane and axially 

 symmetric contraction coefficients are indeed the same. Trefftz [9], in calculations we 

 shall not discuss here, first gave strong support to the conjecture by arriving at the value 

 .61 for the axially symmetric jet, and later several calculations by the relaxation 

 method yielded the same result [10]. However, Garabedian [11], using an altogether 

 different method, recently obtained the value .58. This difference is large enough to 

 raise serious doubts concerning the conjecture and to spur inquiry as to the sources of 

 the discrepancy. 



Garabedian's point of departure is the concept of axially symmetric flow in 

 a space of £ + 2 dimensions. The stream function of such a flow obeys the equation 



■fyxx + i>yy i>y = 0, 



y 



287 



(19) 



