Brennen and Whitney 



aP+"/2 ftp" 1/2 



^1234 " ''1234 



« - iln (1 - »i) [{(U, +U2 + U3 + U4r'^' - (U, +U2 + U3+U/"'^'} 



X {X, -Xg-Xj+xy 



D + l/? P-l/2 PT 



+ {V, +V2+V3+V^)' ""-{V, +V2+V3+V^r +4Trg}{Y, -Yg-Yj+Y^} J 

 + {1+ ^-\ ln(l-,x)} f(X,.x/{(U, +U,r'/'-(U, +U/""} 



P, p-H/2 P-1/2, 



- (X4-X3)^{(U3 + U4) - (U3 + U^) } 



p p+1/2 p-l/2 , 



+ (Y, - Yg) {(V, + Vg) - (V, - Vg) + 2Tg} 



p p*l/2 P-l/2 .I 



- (Y4-Y3) {(V3+V4) -(V3+V4) +2Tg}J 



where ft = Y^b/(1 + "ybj^) , b34 being the b value on side 34 of the 

 cell and Ab the difference across each and every cell. The first 

 term is of order p., the second order |i , The boundary conditions 

 are usually identical to the homogeneous case. 



V. ACCURACY, STABILITY, CONVERGENCE AND SINGULARITIES 



A, Accuracy 



If the cell equation, (18), is used without the higher order 

 spatial correction, an Indication of the errors due to neglected higher 

 order spatial derivatives can be obtained by assessing the value of 

 that correction and inferring its effect upon the final values of W. 

 Unfortunately, the mesh distribution and mesh size required for a 

 solution of given accuracy will not be known a priori and can only be 

 arrived at either by trial and error or by using some technique of 

 rezoning. The latter method in which cells are subdivided where and 

 when the violence of the motion demands It, can be difficult to pro- 

 gram satisfactorily and has not been attempted thus far. 



Errors due to higher order temporal derivatives are most 

 easily regulated by ensuring- that, for each cell, both 

 t|W, - W3I/IZ, - Z3I and t] Wg - W4 |/ | Zg- Zj are comfortably less 

 than unity. A workable rule of thumb can be devised in which a 

 suitable t for a particular time step is determined from the W and 

 Z values of the preceding step. 



128 



