On the Generalization of Shadowing to Fluid Turbulence: Practical Methods For Quantifying Dynamical Similarity
Author(s)
Pughe-Sanford, Joshua L.
Advisor(s)
Editor(s)
Collections
Supplementary to:
Permanent Link
Abstract
Chaos is an intrinsic property of many real world systems, impacting a number of today's open research questions. While many chaotic systems have known governing equations and are deterministically “solved,” we still lack a comprehensive framework for predicting, controlling, and simply making sense of such systems. And while recent advances in technology allow us to explore these systems through direct numerical simulation better than ever before, the need for an insightful theoretical framework is still very much alive.
Such a framework exists in a subset of chaotic systems, known as Axiom A chaotic systems. As a result, Axiom A systems are understood quite well. However, the requirements for a system to be Axiom A are quite strict, and the overlap between systems that are Axiom A and those that are physically significant is quite small.
A very important concept in Axiom A systems is the notion of shadowing, which allows the chaotic dynamics to be decomposed piecewise-in-time in terms of much easier to analyze solutions known as periodic orbits. Periodic orbits are solutions to the governing equations that, unlike chaos, repeat in time. Their compactness make periodic orbits very simple objects to manipulate, both numerically and theoretically. This decomposition ultimately results in a predictive theory of Axiom A systems both deterministically and statistically.
In this dissertation, we seek to generalize the concept of shadowing to a broader class of (non Axiom A) chaotic systems, specifically, fluid turbulence. Although recent studies suggest that Exact Coherent Structures—e.g., repeating solutions to the Navier-Stokes equation—are descriptive of turbulence, it is an open question whether they are to turbulence what periodic orbits are to Axiom A chaos. Here, we propose a generalized method for quantifying shadowing and discuss the generalized nature of shadowing in turbulence. Our results suggest that an axiom A framework for chaos may be more generalizable than previously thought.
Sponsor
Date
2023-07-31
Extent
Resource Type
Text
Resource Subtype
Dissertation