Mathematics Colloquium
and
Computational and Applied Math Proseminar

Department of Mathematics, Arizona State University

Thursday, February 10, 2000, 4:00 p.m. in PSA Room 103

Francisco Marques Truyol

Applied Physics Polytechnic University of Catalunia, Barcelona/Spain

Dynamics and bifurcations in a periodically forced Navier-Stokes flow

Abstract Recent experiments with the flow contained in the annular gap between two concentric cylinders, the outer being stationary and the inner rotating steadily and oscillating harmonically in the axial direction, have demonstrated that the onset of the centrifugal instability leading to Taylor vortex flow can be delayed by the axial oscillations. Floquet analysis has described the observed control of the instability over a wide range of frequencies and amplitudes of this oscillation; the dynamics remain axisymmetric and the response to the applied periodic control mechanism is synchronous over an extensive range of parameters. Here we implement an accurate and efficient spectral-projection scheme for solving the fully nonlinear axisymmetric Navier-Stokes equations to examine the effects of endwalls on the flow dynamics and the breaking of space-time symmetries. By varying a single parameter we have detected a route to chaos. As the parameter is increased the system undergoes a quasiperiodic Naimark-Sacker bifurcation, goes through a 1:9 resonance horn (Arnold's tongue), and then chaos. A separate branch of 3-tori solutions has also been detected. The 3-tori solutions exist in a finite region of parameter space, and undergo various global bifurcations with parameter variation. We have observed global bifurcations associated with homoclinic and heteroclinic connections to other unstable solutions (2-tori). These unstable 2-tori act as organizing centers for the 3-tori dynamics. The system has a discrete space-time symmetry which influences the dynamics and bifurcations.

For further information please contact: mittelmann@asu.edu