Simulation of the Navier-Stokes Equations in Three Dimensions with a Spectral Collocation Method

dc.contributor.authorSubich, Christopher
dc.date.accessioned2011-05-13T17:57:23Z
dc.date.available2011-05-13T17:57:23Z
dc.date.issued2011-05-13T17:57:23Z
dc.date.submitted2011
dc.description.abstractThis work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables, so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.en
dc.identifier.urihttp://hdl.handle.net/10012/5926
dc.language.isoenen
dc.pendingfalseen
dc.publisherUniversity of Waterlooen
dc.subjectfluid dynamicsen
dc.subjectspectral methodsen
dc.subjectnavier-stokes equationsen
dc.subjectinternal wavesen
dc.subject.programApplied Mathematicsen
dc.titleSimulation of the Navier-Stokes Equations in Three Dimensions with a Spectral Collocation Methoden
dc.typeDoctoral Thesisen
uws-etd.degreeDoctor of Philosophyen
uws-etd.degree.departmentApplied Mathematicsen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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