give pretty good answers. The likelihood is, from what we know, you will get useful 

 information, which is not too accurate, whatever that happens to mean. 



C. J. Cohen 



I would just comment that one could do this matter by systematically reducing 

 the size of the mesh, so one would at least get a bound-under answer and you would 

 know where you stand. 



D. Gil bar g 



Yes, I think that should be done, definitely. 



C. J. Cohen and L. D. Gates, Jr. 



Professor Gilbarg stated in his paper that the relaxation solution of the 

 Riabouchinsky problem at Dahlgren indicated that the method was ill-suited for cavi- 

 tational flow problems because although the spread in the computed velocities along 

 the free boundary was of the order of one per cent, the error in the maximum cavity 

 width was of the order of 10 per cent. 



During the Symposium, one of us, (C. J. Cohen) commented from the floor 

 that to his recollection the 10 per cent error in maximum boundary width had been 

 mostly eliminated by applying corrections at the outer boundaries of the flow. (These 

 outer boundaries had been introduced to allow a difference system solution, and the 

 outer boundary values for the stream function were at first taken to be those for 

 undisturbed flow.) 



Dr. Cohen's recollection was in error although we feel that the large error in 

 cavity width noted by Professor Gilbarg was in fact due to the original crude condi- 

 tions imposed on the outer boundaries. 



Perhaps a more serious failure of the difference method was revealed when a 

 trial boundary for plane flow computed from the analytical plane flow solution was 

 tested using corrections to the outer boundary conditions. The observed velocity varia- 

 tion was of the order of 5 per cent, which was felt to be rather large and due to 

 truncation error. The average velocity was about 2 per cent above the true velocity. 

 This latter figure represented an improvement over previous data obtained without 

 the outer boundary correction. This was what prompted Dr. Cohen's remark. 



M. D. Van Dyke 



I should like to ask whether there is any doubt that locally, just at the edge of 

 the disk, the singularity has the same nature as in the two-dimensional free streamline 

 flow. 



D. Gilbarg 



No, there is no doubt about that. 



T. Y. Wu 



In Professor Gilbarg's lecture the re-entrant jet model seems to be preferred 

 as the "only non-artificial model" for the irrotational cavity flows. Actually there 

 are many other mathematical models, e.g., Riabouchinsky's image model and Roshko's 

 dissipation model; and they are all, at least in my opinion, more or less artificial. I 

 like to discuss these models by pointing out their physical basis and certain limitations, 

 hoping that a better understanding of these models may help to clarify some unsolved 

 problems. 



In the real fluid flow past a bluff body with a separated flow region in the 

 wake, experimental observations indicate that the discontinuous surfaces in the flow, 

 or free streamlines, are actually thin shear layers, into which the vorticity is fed from 

 the boundary layer in front of the separation point. The shear layers in general do not 



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