1 Computational Fluid Dynamics Reading Group : Projection Methods

In the literature, discretization methods for solving the Navier-Stokes equation which are written in terms of the velocity and pressure are known as "primitive (or primary) variable formulation" methods (we actually solve for u and P; not for the vorticity, say). It is important to distinguish those which rely on a standard time discretization step, in which velocity and pressure terms evolve simultaneously, and those which are fractional-step projection methods. This last type of projection methods was …rst introduced by Chorin and Temam in 1969, and is characterized by a "fractional-step", followed by an orthogonal projection onto the space of solenoidal vector …elds (that is, incompressible vector …elds). We will see that in both methods, it is often the case that a Poisson equation for the pressure is solved. Key di¤erence: In the standard discretization methods, this will not ensure incompressibility, and an extra boundary condition (really painful) has to be ful…lled. However, in fractional-step projection methods this equation will indeed replace the incompressibility condition, rendering it a signi…cantly more attractive method.