Bvennen and Whitney 



+ TZ ^^^'6 + ^9 - W| - ^2)^2, - Zg) Higher 



- (W|5 + W,o - W3 - W4)(Z4 - Z3) Order 

 + (W7 + W,2 - Wg - W3)(Z2 - Z3) Correction 



- (^6 + ^13 - ^1 - ^4^<2| - ^4^^ ^^ required 



= = Rj + iRg = R, the cell residual . (18) 



The higher order correction, included for completeness, cillows the 

 shape of the cell sides and the variations in velocity along them to be 

 of cubic form. Without it the neglected terms are of order ZgW(j|j^, 

 ZflbWrth, etc. , with it they are.pf order ZnWhhbhh» etc. Values 

 refe] 



,bWab, etc. , witj it they are of oi^er Z,Wb5bbb» el 

 f erred to are Z and W^ ,\s .V^ 



Though this derivation of the cell equation is instructive, it 

 can be obtained more directly (except for the continuity correction) 

 by integration of (8) over the area of the cell in the (a,b) plane 

 (using Taylor expansions about the center of the cell). 



The cell equations must now be solved for W = (U - iV) , 

 Z being known, in order to proceed in time. 



In a recently published paper, Hirt, Cook and Butler [ 1970] 

 take a rather different approach in which the (a,b) plane is employed 

 merely as a tagging space. The equations are written in essentially 

 Eulerian terms, no derivatives with respect to a,b appearing. The 

 numericcil method (LINC) is similar to that of the MAC technique 

 (Fromm and Harlow [ 1963] , Welch, et al. [ 1966] , Chan, Street 

 and Strelkoff [ 1969] , etc.) and involves solving for the pressure at 

 the center of a cell as well as for the vertex velocities. Advantages 

 of the nnethod described in the present paper are: the pressure has 

 been eliminated (though this may be disadvantageous in compressible 

 flows); no special treatment is required for cells adjacent to bound- 

 aries; inhomogeneous density terms are relatively easily included. 

 However, since the LINC system is based on the Eulerian equations 

 of motion, the inclusion of viscous terms is more easily accomplished 

 than in the present method where such an attennpt leads to horrendous 

 difficulties. 



C. Boundary Conditions 



p+i/2 

 To complete the specifications , a condition upon W is 



required at each of the boundary nodes. This usually takes the form 



of an expression connecting U and V . For example, solid 



boundaries, whether fixed or moving in time, may be prescribed by 



a function, F(X,Y,t) = 0. Then the required relation is 



124 



