Hirt 



Sx2 



oy 



(16a) 



where 



R.J 



Ox by -^ 



{ulX - 2(u.^)^ + (u|.J 



{vi'r - 2{.,r + (v| 



j-1. 



)xdy 



/ nJ + ^/2 j- 1/2 



("V)i+l/2 + ("^)i-l/2 



/ sJ-1/2 

 (^^)i+l/2 



, , J + 1/ 2 

 ("^)i-l/2 



(16b) 



Equations (16) are the approximation for a Poisson equation for the pres- 

 sure. These equations must be solved by matrix inversion or an iteration proc- 

 ess. The necessity for solving a set of coupled equations arises because of the 

 fluid incompressibility. Each finite -difference cell containing fluid influences 

 every other cell containing fluid. This is the implicit part of the difference ap- 

 proximations needed for incompressible flows. . ^ , 



It is very important to note the presence of D/ terms in Eq. (16b). The 

 D.J -type term is the source term for the Poisson equation. These terms are so 

 important that we will digress here to consider them in more detail. 



CORRECTIVE PROCEDURE 



Let us assume Eqs. (16) are solved by an iteration method (matrix inver- 

 sions are too time-consuming). Every iterative solution of Eq. (16a) must be • 

 terminated with some error. Thus, "^^D-^ will not be exactly zero. An error in 

 the velocity divergence, however, means an error in mass conservation. To 

 maintain the accuracy of a MAC calculation, in which Eqs. (16) are repeatedly 

 solved at each time step, the accumulated value of D.J must be kept as small as 

 possible. This could be accomplished, for example, by iterating the pressure 

 equation to a high degree of accuracy at each time step. Unfortunately, the 

 computing time needed to solve sets of linear equations like Eq. (16a) increases 

 very rapidly as the convergence criterion is refined. Another means of pre- 

 venting a significant accumulation of D error is used in MAC. The D terms re- 

 tained in Eq. (16b) act as a self-correcting mechanism. An error in D at time 

 step n is automatically corrected at time step n+ l, since the difference equa- 

 tions are set up to make ""^^D zero, regardless of the value of "D. Therefore, a 

 relatively crude iteration solution of Eq. (16a) can be tolerated at each time step 

 without leading to a disastrous accumulation of error after many time steps. 

 Enormous savings in computer time are realized with this technique. 



422 



