ABSTRACT
Explicit and implicit methods for computing solutions to the Navier–Stokes equations are presented. For the compressible case, those included and described in detail are the MacCormack, Beam–Warming, and upwind methods. Information on computing solutions at low speeds is also presented and the discussion topics include preconditioning. A number of ways are shown to solve the incompressible Navier–Stokes equations. Both the vorticity–stream function and primitive-variable approach are included. The primitive variable methods presented are the method of artificial viscosity, coupled space marching, marker-and-cell, and the method of fractional steps. The SIMPLE family of methods, pressure correction approach, is discussed and applied on nonstaggered grids. The pressure-implicit approach with operator splitting (PISO) is also included. Detailed calculations for several example problems are shown and form a valuable reference for use in code validation. The driven cavity and channel flow in two dimensions, two-dimensional backward facing step and two-dimensional square cavity with natural convection are included. Detailed information on free-surface flows in given and several topics pertinent to this class of flows are included in sections on numerical methods for the level-set equations, the reinitialization equation, the issue of mass conservation, and time integration schemes. Detailed calculations on the flow of a two-dimensional jet and the numerical solution of the broken dam problem are given.
