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On the spectral solution of the three-dimensional Navier-Stokes equations in spherical and cylindrical regions. (English) Zbl 0923.76206
This paper investigates the application of spectral methods to the simulation of three-dimensional incompressible viscous flows within spherical or cylindrical boundaries. The Navier-Stokes equations for the primitive variables are considered, and a generalized unsteady Stokes problem is derived, using an explicit time discretization of the nonlinear term. A split formulation of the linearized problem is then chosen by introducing a separate Poisson equation for the pressure supplemented by conditions of an integral character which assure that the incompressibility and the velocity boundary condition are simultaneously and exactly satisfied. A Chebyshev spectral approximation is then considered to resolve the radial structure of the flow field. Numerical results are given to illustrate the convergence properties of the discrete equations obtained by the tau projection method. Finally, the treatment of coordinate singularity in regions bounded by a single spherical or cylindrical surface is discussed.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
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