Title page for ETD etd-08262010-151429


Type of Document Dissertation
Author Kumar, Pankaj
Author's Email Address pkumar06@vt.edu
URN etd-08262010-151429
Title Chaos in Pulsed Laminar Flow
Degree PhD
Department Engineering Science and Mechanics
Advisory Committee
Advisor Name Title
Stremler, Mark A. Committee Chair
Aref, Hassan Committee Member
Cramer, Mark S. Committee Member
Ross, Shane D. Committee Member
Vlachos, Pavlos P. Committee Member
Keywords
  • Topological chaos
  • Kolmogorov entropy
  • Lyapunov exponent
  • Lid-driven cavity flow
  • Source-sink
  • Set oriented approach
  • Almost invariant set
Date of Defense 2010-08-09
Availability restricted
Abstract
Fluid mixing is a challenging problem in laminar flow systems. Chaotic advection can play an important role in enhancing mixing in such flow. In this thesis, different approaches are used to enhance fluid mixing in two laminar flow systems. In the first system, chaos is generated in a flow between two closely spaced parallel circular plates by pulsed operation of fluid extraction and reinjection through singularities in the domain. A singularity through which fluid is injected (or extracted) is called a source (or a sink). In a bounded domain, one source and one sink with equal strength operate together as a source-sink pair to conserve the fluid volume. Fluid flow between two closely spaced parallel plates is modeled as Hele-Shaw flow with the depth averaged velocity proportional to the gradient of the pressure. So, with the depth-averaged velocity, the flow between the parallel plates can effectively be modeled as two-dimensional potential flow.

This thesis discusses pulsed source-sink systems with two source-sink pairs operating alternately to generate zig-zag trajectories of fluid particles in the domain. For reinjection purpose, fluid extracted through a sink-type singularity can either be relocated to a source-type one, or the same sink-type singularity can be activated as a source to reinject it without relocation. Relocation

of fluid can be accomplished using either “first out first in” or “last out first in” scheme. Both relocation methods add delay to the pulse time of the system. This thesis analyzes mixing in pulsed source-sink systems both with and without fluid relocation. It is shown that a pulsed source-sink system with “first out first in” scheme generates comparatively complex fluid flow than pulsed source-sink systems with “last out first in” scheme. It is also shown that a pulsed source-sink system without fluid relocation can generate complex fluid flow.

In the second system, mixing and transport is analyzed in a two-dimensional Stokes flow system. Appropriate periodic motions of three rods or periodic points in a two-dimensional flow are determined using the Thurston-Nielsen Classification Theorem (TNCT), which also predicts a lower bound on the complexity generated in the fluid flow. This thesis extends the TNCT -based framework by demonstrating that, in a perturbed system with no lower order fixed points, almost invariant sets are natural objects on which to apply the TNCT. In addition, a method is presented to compute line stretching by tracking appropriate motion of finite size rods. This method accounts for the effect of the rod size in computing the complexity generated in the fluid flow. The last section verifies the existence of almost invariant sets in a two-dimensional flow at finite Reynolds number. The almost invariant set structures move with appropriate periodic motion validating the application of the TNCT to predict a lower bound on the complexity generated in the fluid flow.

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