Hirt 

 2v§t < (26) 



The necessary diffusion coefficient conditions are derived by collecting all 

 truncation errors in the MAC equations that contribute to diffusions of u and v. 

 Keeping only these terms through order h t and S x ^^ 



dtoxdv ox \ 2 27 



,2 



t Sy'' 3v\ 92^ 



2 



4 3y/ 3 



'y 



By Buy By^ _ 3 (p ^ /'^^ _ 2 St Sx^ ^u 



Bt 3x By By ^ \ 2 4 



(27) 



2 li _ Sy^ By'\ 3^ V 

 ^2 2 By/By2 



Several remarks can be made about the diffusion coefficients appearing in Eq. 

 (27). First, the MAC equations are always unstable if v - 0. Second, neglecting 

 the Sx^ terms, the diffusion coefficients are positive if 



. " V >^. . > ^ . . ^^ (28) 



These conditions are approximately the stability conditions obtained from a lin- 

 ear Fourier analysis (7). However, the Sx^ terms cannot be ignored, which re- 

 quires 



2 



(29) 



where u' is a typical velocity derivative in the direction of flow. Instabilities 

 do occur when Eq. (29) is violated, and they are quite insidious, since a reduc- 

 tion in ot cannot cure them. Furthermore, they are more likely to occur when 

 V is small, i.e., in high Reynolds number flows. Therefore, it is extremely im- 

 portant to recognize the condition Eq. (29), and to distinguish these numerical 

 instabilities from physical instabilities. 



The relationship between diffusion-like truncation errors and computational 

 stability is applicable to all finite -difference approximations. It should now be 

 clear why it is so difficult to perform high Reynolds number calculations. Oc- 

 casionally an investigator claims to calculate at very high Reynolds numbers, 

 but a check on truncation errors most often reveals positive diffusion terms 

 much larger than the real viscosity. 



426 



