The solution of the Navier-Stokes equations in velocity-pressure variables is the subject of this chapter. This set of equations is of broader application than the vorticity-streamfunction equations which are restricted to two-dimensional flows. First, the Fourier method for computing fully periodic flows is discussed. Then the major part of the chapter is devoted to the case of one or more nonperiodic directions. In such a situation, the classical difficulty is the determination of the pressure field ensuring that the velocity field is solenoidal. This problem is discussed according to the kind of method used for the advancement in time. Time-discretization methods leading, at each time-cycle, either to a Stokes problem or to a combination of Helmholtz and Darcy problems will be considered. Various solution methods will be discussed and compared in typical examples. Finally, an application to the calculation of a three-dimensional rotating flow will be presented.
KeywordsProjection Method Algebraic System Helmholtz Equation Collocation Point Stokes Problem
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