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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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References

REFERENCES

Antkowiak, A. & Brancher, P. 2004 Transient growth for the Lamb–Oseen vortex. Phys. Fluids 16 (1), L1–L4.Google Scholar
Antkowiak, A. & Brancher, P. 2007 On vortex rings around vortices: an optimal mechanism. J. Fluid Mech. 578, 295–304.CrossRefGoogle Scholar
Bailey, S. C. C. & Tavoularis, S. 2008 Measurements of the velocity field of a wing-tip vortex, wandering in grid turbulence. J. Fluid Mech. 601, 281–315.CrossRefGoogle Scholar
Crouch, J. D. 1997 Instability and transient growth for two trailing vortex-pairs. J. Fluid Mech. 350, 311–330.Google Scholar
Fabre, D., Jacquin, L. & Loof, A. 2002 Optimal perturbations in a four-vortex aircraft wake in counter-rotating configuration. J. Fluid Mech. 451, 319–328.Google Scholar
Heaton, C. J. & Peake, N. 2007 Transient growth in vortices with axial flow. J. Fluid Mech. 587, 271–301.Google Scholar
Kida, S. & Tanaka, M. 1994 Dynamics of vortical structures in homogeneous shear flow. J. Fluid Mech. 274, 43–68.Google Scholar
Mansour, N. N. & Wray, A. A. 1994 Decay of isotropic turbulence at low Reynolds number. Phys. Fluids 6 (2), 808–814.CrossRefGoogle Scholar
Marshall, J. S. & Beninati, M. L. 2000 Turbulence evolution in vortex dominated flows. In Advances in Fluid Mechanics 25 (Nonlinear Instability, Chaos and Turbulence II (ed. Debnath, L. & Riahi, D. N.), pp. 1–40. WIT Press.Google Scholar
Melander, M. V. & Hussain, F. 1993 Coupling between a coherent structure and fine-scale turbulence. Phys. Rev. E 48 (4), 2669–2689.Google ScholarPubMed
Pradeep, D. S. 2005 Mechanisms of perturbation growth and turbulence evolution in a columnar vortex. PhD thesis, University of Houston.Google Scholar
Pradeep, D. S. & Hussain, F. 2004 Effects of boundary condition in numerical simulations of vortex dynamics. J. Fluid Mech. 516, 115–124.Google Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in vortex column. J. Fluid Mech. 550, 251–288.Google Scholar
Pradeep, D. S. & Hussain, F. 2010 Vortex dynamics of turbulence-coherent structure interaction. Theor. Comput. Fluid Dyn. 24 (1–4), 265–282.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105 (15), 154502.Google Scholar
Rennich, S. C. & Lele, S. K. 1997 Numerical method for incompressible vortical flows with two unbounded directions. J. Comput. Phys. 137, 101–129.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Smith, C. R. & Walker, J. D. A. 1995 Turbulent wall-layer vortices. In Fluid Vortices (ed. Green, S. I.), pp. 235–290. Kluwer.CrossRefGoogle Scholar
Spalart, P. 1998 Aircraft trailing vortices. Annu. Rev. Fluid Mech. 30, 107–138.CrossRefGoogle Scholar
Wallin, S. & Girimaji, S. S. 2000 Evolution of an isolated turbulent trailing vortex. AIAA J. 38 (4), 657–665.Google Scholar
Zeman, O. 1995 The persistence of trailing vortices: a modeling study. Phys. Fluids 7 (1), 135–143.CrossRefGoogle Scholar
Zhou, J., Adrian, R. J., Balachander, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.CrossRefGoogle Scholar