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Nonlinear transient growth in a vortex column

Published online by Cambridge University Press:  19 July 2011

FAZLE HUSSAIN ,
DHOORJATY S. PRADEEP  and
ERIC STOUT
FAZLE HUSSAIN*
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA
DHOORJATY S. PRADEEP
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA
ERIC STOUT
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA
*
†Email address for correspondence: [email protected]
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Abstract

Growth of optimal transient perturbations to an Oseen vortex column into the nonlinear regime is studied via direct numerical simulation (DNS) for Reynolds number, Re (≡ circulation/viscosity), up to 10000. An optimal bending-wave transient mode is obtained from linear analysis and used as the initial condition. (DNS of a vortex column embedded in finer-scale turbulence reveals that optimal modes are preferentially excited during vortex–turbulence interaction.) Tilting of the optimal mode's radial vorticity perturbation into the azimuthal direction and its concomitant stretching by the column's strain field produces positive Reynolds stress, hence kinetic energy growth. Modes experiencing the largest growth are those with initial vorticity localized at a ‘critical radius’ outside the core, such that this perturbation vorticity resonantly induces core waves. Resonant forcing leads to growth of perturbation energy concentrated within the core. Moderate-amplitude (~5%) perturbations cause significant distortion of the core and generate secondary filament-like spiral structures (‘threads’) outside the core. As the mode evolves into the nonlinear regime, radially outward self-advection of thread dipoles accelerates growth arrest by removing the perturbation from the critical radius and disrupting resonant forcing. With increasing Re, the evolving vorticity patterns become more chaotic, more turbulent-like (finer scaled, contorted vorticity), and persist longer. This suggests that at typical Re (~106), nonlinear transient growth may indeed be able to break up, hence induce rapid decay of, column vortices – highly relevant for addressing the aircraft wake hazard crisis and the looming air traffic capacity crisis. In addition, we discover a regenerative transient growth scenario in which threads induce secondary perturbations closer to the vortex column. A parent–offspring regenerative mechanism is postulated and verified by DNS. There is a clear trend towards stronger regenerative growth with increasing Re. These results, showing an important role of transient growth in turbulent vortex decay, are highly relevant to the prediction and control of vortex-dominated flows.

JFM classification

Instability: Transition to turbulence Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 304 - 331
Copyright
Copyright © Cambridge University Press 2011

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