Numerical Simulation of Viscous Incomipressible Fluid Flows 



ox 



■^y 



^(vr^'"- 



(13) 



the notation used here is, e.g., n^j = u( iSx, j5y,nSt ). All quantities on the right- 

 hand side of Eqs. (12) and (13) are evaluated at time net. Several quantities in 

 Eqs. (12) and (13) are located at positions other than those indicated in Fig. 2. 

 In each case a simple average is implied; for example, .. 



"i^ " 2 ("i+i/ 2 + ^1-1/2) • 

 or 



j + 1/ 2 1 / j + l/2 j + l/2\ i- -, - ^'i- "• ' f"-V' 



^i+1/2 = T (^i +^i+i ) • ... ■ ^, . . ,.,.••■■;■ 

 For a product, each factor is first averaged and then the product is formed, i.e., 



("^)i-l/2 = ("i-l/2)(Vi-l/2) 



C j + 1 j \ / i + 1/ 2 j+l/2-, . ' 



Equations (12) and (13) conserve momentum exactly. A sum over consecutive i 

 and j in either equation leads to a cancellation in pairs of terms on the right- 

 hand side. The only momentum changes occurring in a group of cells are caused 

 by surface fluxes. 



The momentum equations are completed by specifying the pressure. The 

 pressure must then be determined to make the velocity field satisfy the 

 conservation-of-mass condition, the first equation of Eqs. (11). For this pur- 

 pose we need the finite -difference expression for the velocity divergence in cell 

 (i-j), 



Di^ = ^(u!.V2-u!../.) . ^(vi-/^-vi-/^)- - (14) 



Then, the conservation-of-mass or incompressibility condition is 



"^^D/ ^ (15) 



for every ( i , j). Inserting Eqs. (12) and (13) for the values of u and v at 

 (n+ 1) St, into Eqs. (14) and (15) yields the following equations for cp;^ 



421 



