OVERVIEW

Having discussed finite-difference methods for computing numerical solutions to the convection diffusion equation in its general form, we proceed to develop corresponding methods for solving the equations of steady and unsteady incompressible Newtonian flow. The set of governing equations includes the Navier Stokes equation and the continuity equation, and the primary unknowns are the velocity and the pressure. We recall, however, that a general rotational flow may also be described and therefore computed in terms of the secondary variables discussed in Chapter 2, including the vorticity, the stream functions, and the vector potential.

Considering the evolution of an unsteady flow, we regard the Navier Stokes equation as an evolution equation for the velocity, providing us with the rate of change of the velocity at a particular point in the flow in terms of the instantaneous velocity and pressure. We then note that if the pressure gradient were absent, the simplified evolution equation would be identical to the nonlinear convection diffusion equation, and could therefore be integrated in time using the finite-difference methods discussed in Chapter 12. Unfortunately, as discussed in Section 9.1, an evolution equation for the pressure is not available in an explicit form. In its place we have the restriction of incompressibility, which requires that the pressure evolve so as to ensure that the rate of expansion vanish and the velocity field remain solenoidal at all times. As we saw in Section 9.1, the restriction of incompressibility may be expressed in terms of a Poisson equation either for the pressure or...