Projection methods for incompressible unsteady flows

A general formula of the 2^nd-order projection method for solution of unsteady incompressible Navier-Stokes equations has been presented. The four- and three-step projection methods have been developed. Several existed famous projection methods have been analyzed. The pressure corrected SIMPLE method for unsteady flows has also been discussed. The three- and four-step RKCN (Runge-Kutta/Crank-Nicholson) projection methods have also been presented. The explicit three-stage Runge-Kutta and semi-implicit Crank-Nicholson techniques are employed to update the convective and diffusion terms respectively. The four-step RKCN projection method has been simplified furthermore. In the simplified version, the pressure Poisson equation (PPE) is solved only at the final sub-stage, which will save the computing time greatly. The high-order boundary conditions for the intermediate velocities have also been given for the four-step RKCN projection method and its simplified version. Numerical examples for 2-D vortex flows and oscillating cavity driven flows have been simulated to validate the effectiveness of the above developed methods.