Viscous Incompressible Fluid Flow
The finite element discretization concepts introduced in the previous chapters will now be applied to the solution of the Navier-Stokes equations for viscous incompressible flows in two dimensions. We will discuss the three types of algorithms most commonly used in the solution of the equations, namely, the mixed formulation, the fractional step method, and the penalty formulation. The latter will be treated in more detail, as it is the basis for very practical procedures for the solution of the Navier-Stokes equations. We will face a new difficulty because the incompressibility condition is a constraint on the space of possible solutions and is related in a nonexplicit form to the pressure field, which in this case, is a dynamic variable. Mathematically, this will be expressed as a consistency condition known as the Ladyzhenkaya, Babuska, and Brezzi (LBB) condition, which established the compatibility of the velocity and pressure spaces when the Navier-Stokes equations are discretized directly.