Projection Methods for the Calculation of Incompressible Unsteady Flows

A general formula for the second-order projection method for solution of unsteady incompressible Navier-Stokes equations is presented. It includes the fourand three-step projection methods. Also, RKCN (Runge-Kutta=Crank-Nicholson) three-step and four-step projection methods are presented, in which the three-stage Runge-Kutta and semi-implicit Crank-Nicholson techniques are employed to update the convective and diffusion terms, respectively. The RKCN projection method is further simplified. The pressure Poisson equation (PPE) is solved only at the final substage for the simplified RKCN projection method, which greatly reduces the computation time. The high-order boundary conditions for the intermediate velocities have also been given for the four-step RKCN projection method and its simplified version. A 2-D vortex flow, a 2-D oscillating cavity flow, and a 3-D lid-driven cavity flow are simulated to validate the analysis. The projection method is also used to do the direct numerical simulation (DNS) of a fully developed channel flow.