1.5 Fluid Flow Applications

The three types of algorithms most commonly used in the solution of the equations for viscous incompressible flow are the mixed formulation, the fractional step method, and the penalty formulation. The latter is the basis for very practical procedures for the solution of the Navier-Stokes equations. Incompressibility is related in a nonexplicit form to the pressure field, which in this case, is a dynamic variable. Mathematically, this is expressed as a consistency condition known as the Ladyzhenkaya, Babuska, and Brezzi (LBB) condition, which establishes the compatibility of the velocity and pressure spaces when the Navier-Stokes equations are discretized directly. A very large number of algorithms have been proposed for the solution of these equations.