Hirt 

 Now I have two questions. 



1. In most aeronautical and marine applications the surfaces are curved. 

 Have you any comments about methods for handling such boundaries? 



2. I understand that Professor Roache at Notre Dame has developed a finite- 

 difference method for the same problems as yours that is stable. Do you know 

 about it and do you have any comments about it? 



REPLY TO DISCUSSION 



C. W. Hirt 



REPLY TO COMMENTS BY M. T. MURRAY 



The computer used was the IBM 7030 (STRETCH) machine. This is a 98,000 

 (base 10) word machine, which was programed for the MAC method directly in 

 machine language. 



The 7030 machine can handle a maximum of 4440 cells with 12,500 marker 

 particles. If fewer cells are used more particles can be included because there 

 is some tradeoff in storage space. Typical problems usually involve on the or- 

 der of 2000 cells and several thousand particles. For example, the hydraulic 

 jump calculation used 100 x25 or 2500 cells and approximately 2700 particles 

 (not all cells contain particles), and the sluice gate calculation used 47 x 32 or 

 1504 cells with approximately 6200 particles. 



The computer time needed for a problem depends crucially on the number 

 of cells and particles used. The sluice gate calculations, as a typical example, 

 required approximately 12 seconds per cycle. This is not unreasonable consid- 

 ering that one cycle consists of the solution of a Poisson equation for pressure, 

 advancement of all velocity components, and the movement of particles. The 

 loss and gain of surface cells must also be recorded during each cycle. 



REPLY TO COMMENTS BY A. M. O. SMITH 



Curved boundaries are generally a problem, but not an insurmountable one. 

 It is certainly possible to think of a MAC technique that uses a mesh of irregu- 

 lar polyagonal cells. Assuming that a curved boundary can be approximated by 

 a set of short-line segments, the fluid region could be covered by a mesh of 

 polyagonal cells extending out from the boundary. In this way problems with 

 curved boundaries could be solved. We have not yet attempted this approach 

 with the MAC method, but we are investigating a similar method for compressi- 

 ble flow calculations. 



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