On the divergence-free (i.e., mass conservation, solenoidal) condition in computational fluid dynamics - How important is it?

Techniques for satisfying the incompressibility (divergence-freeness) in flows during numerical simulations are examined. Attention is focused on the incompressible Navier-Stokes equations for a two-dimensional flow. The equations are discretized, a Marker and Cell (MAC) mesh is defined in a staggered configuration, and boundary conditions are obtained through application of the momentum equations and the incompressibility condition at the boundary walls. A Helmholtz decomposition of the velocity field is demonstrated in generalized form to derive the incompressibility field and its uniqueness is proved. Additionally, the interaction between the vorticity and the boundary conditions are discussed.