Description
- Abstract:
- Statistical descriptions are well-suited for chaotic systems and turbulent flows. A common strategy for studying the statistics of such systems is Direct Numerical Simulation (DNS). However accumulating statistics via DNS can be inefficient because convergence to a statistically steady-state is slow, especially in systems with rare but large deviations as are often found in chaotic systems and turbulent flows. Another drawback of DNS is that it is difficult to extract the physics from the computed numbers. In this thesis I present two alternative strategies to directly solve for the statistics. In the first part, a Fokker-Planck description is used to study the equal-time statistics of the stochastically-forced Lorenz attractor. In particular, the steady-state probability distribution of the attractor is solved for directly by computing the zero mode of a Fokker-Planck operator. I also investigate a perturbative expansion in the equal-time cumulants of the system, tested on the Lorenz attractor and on two idealized barotropic models. Since low-order statistics tend to be spatially smoother than the corresponding dynamical fields, this method can capture the macroscopic features of turbulent flows using fewer degrees of freedom. An added benefit of such a cumulant expansion scheme is that the small-scale modes are integrated out, leaving only large-scale modes containing information about the coherent structures of interest. These modes are associated with the low-order statistics of the system and might be described by a fixed point or slow manifold, allowing for quicker convergence to a statistically steady state. Since this perturbative expansion suffers from the "curse of dimensionality", I detail my efforts to resolve this issue via a dimensional reduction scheme that uses a type of unsupervised machine learning technique known as Proper Orthogonal Decomposition. This involves rotating into a new coordinate system spanned by the eigenvectors of the vorticity second cumulants of the system and retaining only the most energetic modes. A substantial basis reduction with order of magnitude computational gains is demonstrated, providing an accurate alternative to directly accessing the low-order statistics of turbulent flows.
- Notes:
- Thesis (Ph. D.)--Brown University, 2017
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Citation
Allawala, Altan Turowicz,
"Direct Statistical Simulation of Chaos and Turbulence"
(2017).
Physics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/73qk-vp54
