Numerical Simulation of Viscous Incompressible Fluid Flows 



To correctly satisfy the conditions in Eqs. (22) and (23), and the conserva- 

 tion of mass requires a knowledge of the exact location and slope of the surface 

 within a surface cell. This information is not available in MAC. Instead, sev- 

 eral approximations are introduced which have been found to work quite well, 

 except at very low Reynolds numbers. 



The quantities to be determined in each surface cell by the boundary condi- 

 tions are the pressure and the velocities at each cell boundary adjacent to an 

 empty cell. Conservation of mass is approximated by choosing the velocities to 

 make D vanish in each surface cell. This is just an approximation, since D 

 should be zero only in that part of the cell which is filled with fluid. If the cell 

 has more than one side adjacent to an empty cell, the vanishing of D does not 

 uniquely determine all the velocities. In this case the finite-difference forms of 

 'du/dx and 3v/9y are individually required to vanish. Other possibilities can be 

 envisioned, but this particular choice seems to work well. 



Tangential velocities needed in empty cells at the surface are chosen to 

 make the normal derivative of the tangential velocity zero 3Uj^/3n = o. The tan- 

 gential stress condition in Eq. (23) is approximately satisfied by this choice. 



Once the surface velocities are determined, it is easy to satisfy the normal 

 stress condition in Eq. (22). Complete details are given in Ref. 6, where it is 

 shown that the tangential stress condition and the viscous contribution to the 

 normal stress condition are important only at Reynolds numbers less than about 

 10. 



STABILITY AND ACCURACY 



To use the MAC equations effectively, it is necessary to know their stability 

 properties. We have already seen that with an incorrect choice for 6t, the sim- 

 ple difference Eq. (2) has very misbehaved solutions. Similar solutions develop 

 in MAC. Analogous to the example in the Linear Equation section, there are two 

 kinds of stability conditions for MAC: wave propagation, and the positive diffu- 

 sion coefficient. 



Wave propagative instabilities occur for two reasons. First, in free-surface 

 flows, surface waves may develop with wave speed 



— tanh (hk) 

 k 



(24) 



where k is the wave number, h the depth of fluid, and g the downward accelera- 

 tion of gravity, gy = -g. The first wave propagation condition, therefore, limits 

 the distance surface waves travel in a single time step. Second, in two dimen- 

 sions the condition is 



C St < r f- • (25) 



ox + by 



The second wave condition is analogous to that needed for Eq. (2). For two 

 dimensions, 



425 



