Coexistence of multiple long-time solutions for two-dimensional laminar flow past a linearly sprung circular cylinder with a rotational nonlinear energy sink

Antoine B. Blanchard, Lawrence A. Bergman, Alexander F. Vakakis, and Arne J. Pearlstein
Phys. Rev. Fluids 4, 054401 – Published 30 May 2019
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Abstract

For two-dimensional flow past a linearly sprung circular cylinder to which has been attached an internal “nonlinear energy sink” consisting of a mass allowed to rotate about the cylinder axis and a viscous damper that linearly retards the rotational motion of that mass, we show, for a given set of parameters, that as many as three distinct unsteady long-time solutions (two periodic and one chaotic), in addition to one or more steady solutions, can coexist. For other combinations of the parameters, two unsteady solutions (both periodic, one periodic and one quasiperiodic, one periodic and one chaotic, one quasiperiodic and one chaotic, or both chaotic) can coexist with one or more steady solutions. These multiple solutions, all of which appear to be linearly stable, are found in a range of Reynolds number (15<Re<50) in which the flow is expected to be two dimensional. The discovery of this unprecedented degree of solution multiplicity establishes the potential of this system to serve, at low Re, as a test bed for detailed investigation of basins of attraction of the Navier-Stokes equations, and in studies of noise- and disturbance-induced transitions between different long-time solutions.

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  • Received 5 June 2018

DOI:https://doi.org/10.1103/PhysRevFluids.4.054401

©2019 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsFluid Dynamics

Authors & Affiliations

Antoine B. Blanchard1,*, Lawrence A. Bergman1, Alexander F. Vakakis2, and Arne J. Pearlstein2,†

  • 1Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
  • 2Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

  • *Present address: Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 5-318, Cambridge, Massachusetts 02139, USA.
  • Corresponding author: ajp@illinois.edu

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Issue

Vol. 4, Iss. 5 — May 2019

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