Brennen and Whitney 



IDacI = Id^eI = IDagI = |DaJ « |]§ = ^ + V2 



where d| , 62 are the lengths of the cell diagonals. For square cells, 

 "Y = -y = ^^^ *^® situation Is stable. However difficulties may 

 arise when the cells are very skewed or elongated and It Is In such 

 situations, In general, that care has to be taken with the relsixatlon 

 technique. 



C, Observation on the Cell Equation 



One feature of the basic cell equation, (18), Itself demands 

 attention. Note that without the higher order spatial correction, the 

 residuals, R In all of the cells (of Fig. 3) remain unaltered when 

 the W or Z values at alternating points (say the odd numbered 

 points of Fig, 3) are changed by the same amount. Such alternating 

 "errors" must be suppressed. Some damping is provided by the 

 higher order spatial correction since It Is not Insensitive to these 

 changes. But experience showed this to be Insufficient unless all 

 the boundary conditions also Inhibited such alternating "errors." 

 Solid boundaries usually provide adquate damping. For Instance, 

 in Fig. 4(a) fluctuations In U on BC, DA and In V on BC are 

 obviously barred. But the free surface provides little or no such 

 suppression and as will be seen In the next section this can lead to 

 difficulties. It Is of Interest to note that some of the solutions of 

 Hlrt, Cook and Butler [ 1970] exhibit the same kind of alternating 

 errors. 



In the MAC technique, neglected higher order derivatives of 

 the diffusion type and with negative coefficients (a "numerical" 

 viscosity) can lead to a numerical Instability If not counteracted by 

 the Introduction of sufficient real viscosity. In the present method, 

 as with that of Hlrt, Cook and Butler [ 1970] , the convection terms 

 which cause that problem are not present. The higher order spatial 

 correction does contain terms of diffusion order, but it cannot be 

 directly correlated with a viscosity since viscous terms are of a 



different form (I.e. , like J vVxy F dt). Also, the higher order spatial 



correction has a beneficial rather than a destabilizing effect. 



D. The Free Surface 



By Including previously neglected derivatives, the numerical 

 free surface condition (without surface tension) Is found to correspond 

 more precisely to: 



2 

 {X,X,, + Ya(Ytt + i)} +-^ {XoooXtt + Yaoa(Ytt + 1)) 



+ ^{X,X,,,, +Y,Yt,„} =0 (26) 



130 



