Numerical Simulation of Viscous Incompressible Fluid Flows 



An additional advantage is that initial conditions for a calculation do not 

 have to satisfy the incompressibility conditions. After one cycle of calculation 

 the condition is automatically satisfied. A complete discussion and generaliza- 

 tion of this corrective procedure can be found in Ref. 5. 



WALL BOUNDARY CONDITIONS 



Boundary conditions are needed to complete the basic MAC difference Eqs. 

 (12), (13), and (16). Free boundaries introduce special problems and will be 

 considered later. Here we are deriving conditions for rigid, input, and output 

 boundaries. 



Rigid boundaries are assumed to coincide with cell boundaries, otherwise 

 cells would have variable dimensions and this would require additional computer 

 storage. Boundary conditions are imposed by specifying the appropriate values 

 for field variables located in fictitious cells immediately outside the boundary. 



For the purposes of illustration, suppose that a vertical boundary coincides 

 with the right-hand side of cell (i , j ), and that cell (i + i, j ) is outside the 

 boundary. Rigid boundaries are further classified as free-slip or no-slip. At 

 a free-slip boundary, the normal component of velocity is zero, and no tangen- 

 tial shear is allowed: 



j n J± 1/2 _ j±l/2 (17) 



At a no-slip boundary, both the normal and tangential velocity vanish: 



ui../. = o, vi-/^.-vi-/^ . (18) 



The one velocity component of the fictitious cell still undetermined, u|^3/2> 

 is chosen to make Dj^j zero. If this D is not zero, it will diffuse into the flow 

 region through Eqs. (16). ^; 



The necessity for a condition on D in the fictitious cell arises from the form 

 chosen for the differential equations in Eqs. (11) and the cell layout in Fig. 2. 

 We might just as well have written the viscous terms in Eqs. (11), which are 

 components of the vector .-.,•. ■■■■■■. 



i^v-Vu , (19) 



as components of the equivalent vector 



u Vx Vy u . (20) 



When Eq. (20) is used as a starting point for making finite-difference ap- 

 proximations, the D diffusion terms will not appear in Eq. (16b) and the fictitious 

 cell velocity ^{ + 3^2 ^^H ^^^ otherwise be required. Equation (20), therefore, is 

 preferred, but we have chosen to describe Eq. (19), since it was used in the orig- 

 inal MAC development. 



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