Numerical Simulation of Viscous Incompressible Fluid Flows 



THE MAC DIFFERENCE EQUATIONS 



The discussion in the previous section showed the futility of applying explicit 

 finite-difference methods, designed for compressible flows, to the solution of 

 incompressible flow problems. Time increments must be chosen very small to 

 limit the distance which sound waves travel in one time step to less than one 

 space increment. The incompressible limit assumes, however, that sound 

 speeds are much larger than fluid speeds, so that it would be necessary to cal- 

 culate an enormous number of time steps to see a significant flow change. Thus, 

 the wave propagation stability condition precludes the use of purely explicit 

 finite-difference methods. An implicit method is needed that will adjust the flow 

 field simultaneously at all space locations to maintain fluid incompressibility. 



The MAC technique accomplishes this task in a fast and novel way. 

 jin with the incompressible Navier-Stokes equations 



We be- 



3u Bv 



3x 3y 



(11a) 



Bu 



3t 



Bu 3uv 



Bx Bv 



3qp 



b7 



Bx2 



32u_ 

 By 2 



(lib) 



Bv 

 Bt 



Buv B V 2 



Bx By 



B^v B' 



(lie) 



where <? is the ratio of pressure to constant density, g^ and gy are the compo- 

 nents of a body acceleration, and v is the kinematic viscosity. 



Finite-difference approximations for Eqs. (11) require a finite set of points 

 on which to specify local values of the field variables. In MAC, this is accom- 

 plished by covering the flow region with a mesh of stationary rectangular cells. 

 The region actually occupied by fluid is further covered by a set of marker par- 

 ticles (Fig. 1). These particles move with the fluid and are used to locate free 

 surfaces, but they do not otherwise influence the flow dynamics. More is said 

 about marker particles in the section on Corrective Procedure. In each cell of 

 the stationary mesh, flow variables are specified at the positions indicated in 

 Fig. 2. By not recording all variables at the center of the cell it is possible to 

 obtain more compact finite-difference approximations. It also makes it easier 

 to satisfy boundary conditions at rigid walls, if the walls are assumed to coin- 

 cide with the cell boundaries. 



Fig. 1 - Schematic of a typical 

 cell and marker particle layout 



419 



