Finite-difference approximations of the Navier-Stokes equations for incompressible flows

Numerical solutions of the Navier-Stokes equations for incompressible unsteady two and three dimensional viscous flows are introduced. The differential formulations of the continuity equation and of the momentum equations are derived from the integral relations for mass conservation and momentum balance. The energy fluxes and the energy equation are described. The vorticity transport equation with the Poisson equation for the stream function and the pressure; the Poisson equations for the velocity components for two dimensional flow related to the gradients of the vorticity component; the fourth order equation for the stream function of two dimensional flow; and the divergence form of the equations of motion are obtained.