This book presents different formulations of the equations governing incompressible viscous flows, in the form needed for developing numerical solution procedures. The conditions required to satisfy the no-slip boundary conditions in the various formulations are discussed in detail. Rather than focussing on a particular spatial discretization method, the text provides a unitary view of several methods currently in use for the numerical solution of incompressible Navier-Stokes equations, using either finite differences, finite elements or spectral approximations. For each formulation, a complete statement of the mathematical problem is provided, comprising the various boundary, possibly integral, and initial conditions, suitable for any theoretical and/or computational development of the governing equations. The text is suitable for courses in fluid mechanics and computational fluid dynamics. It covers that part of the subject matter dealing with the equations for incompressible viscous flows and their determination by means of numerical methods. A substantial portion of the book contains new results and unpublished material.

The numerical resolution of the incompressible Navier-Stokes equations with the Fractional Step Method, based on the Helmholtz-Hodge theorem, is studied. Basic benchmark problems are solved previously, such as a generic transient 2D heat conduction problem, potential flow around a rotating and non-rotating cylinder and a generic convection-diffusion equation; with excellent agreement with the results obtained and the ones on the literature. The code for the incompressible Navier-Stokes equation is verified using the benchmark results of the Lid-driven cavity problem with really good agreement as well. Finally, laminar flow around a confined square cylinder is studied and compared with the results from Breuer et. al. The drag coefficient and Strouhal number are computed finding good agreement for Reynolds numbers lower than 100 but important discrepancies for higher Reynolds.

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases. Rosenfeld, Moshe and Kwak, Dochan and Vinokur, Marcel Ames Research Center...