There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D. / There are still many open questions regarding chaos. Rayleigh-Bènard convection is a type of natural convection which occurs when a fluid is placed between a hot bottom plate and a cold top plate. Rayleigh-Bènard convection is a classical model to explore chaos in space and time. The major application of Rayleigh-Bènard convection is weather prediction which is an extremely difficult problem of intense interest. The governing equations can only be solved using supercomputing resources. The main reason for this difficulty is the presence of a very large number of degrees of freedom that may influence the weather. To reduce the number of degrees of freedom by only including ones that contribute significantly is a difficult problem. In this thesis, vectors describing the growth of disturbances have been calculated for Rayleigh-Bènard convection. These vectors give us information about which regions in space are more important than others. For weather example, the knowledge of these vectors would tell us which regions are important. With this information, scientists and engineers can focus on the important regions and possibly improve their long term predictions. These vectors also yield the number of degrees of freedom to characterize a chaotic system, on average. In this thesis, this number is also explored for Rayleigh-Bènard convection. This thesis extends the calculation of these vectors to a realistic fluid model which gives us new insights into fundamental questions about chaos in space and time.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/79499 |
| Date | 04 October 2017 |
| Creators | Xu, Mu |
| Contributors | Mechanical Engineering, Paul, Mark R., Tafti, Danesh K., Ross, Shane D., Leonessa, Alexander, Dancey, Clinton L. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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