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On the role of vortex stretching in energy optimal growth of three-dimensional perturbations on plane parallel shear flows

Published online by Cambridge University Press:  19 July 2012

H. Vitoshkin ,
E. Heifetz ,
A. Yu. Gelfgat  and
N. Harnik
H. Vitoshkin*
Affiliation:
School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
E. Heifetz
Affiliation:
Department of Geophysics, Atmospheric and Planetary Sciences, Tel Aviv University, Tel Aviv 69978, Israel
A. Yu. Gelfgat
Affiliation:
School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
N. Harnik
Affiliation:
Department of Geophysics, Atmospheric and Planetary Sciences, Tel Aviv University, Tel Aviv 69978, Israel
*
†Email address for correspondence: vitosh@eng.tau.ac.il
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Abstract

The three-dimensional linearized optimal energy growth mechanism, in plane parallel shear flows, is re-examined in terms of the role of vortex stretching and the interplay between the spanwise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille and mixing-layer shear profiles is robust and resembles localized plane waves in regions where the background shear is large. The waves are tilted with the shear when the spanwise vorticity and the planar divergence fields are in (out of) phase when the background shear is positive (negative). A minimal model is derived to explain how this configuration enables simultaneous growth of the two fields, and how this mutual amplification affects the optimal energy growth. This perspective provides an understanding of the three-dimensional growth solely from the two-dimensional dynamics on the shear plane.

JFM classification

Waves/Free-surface Flows: Shear waves Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 369 - 380
Copyright
Copyright © Cambridge University Press 2012

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References

1. Bakas, N. A. 2009 Mechanisms underlying transient growth of planar perturbations in unbounded compressible shear flow. J. Fluid Mech. 639, 479–507.CrossRefGoogle Scholar
2. Butler, R. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids A 4, 1367–1654.CrossRefGoogle Scholar
3. Chagelishvili, G. D., Khujadze, G. R., Lominadze, J. G. & Rogava, A. D. 1997 Acoustic waves in unbounded shear flows. Phys. Fluids 9, 1955.Google Scholar
4. Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487.CrossRefGoogle Scholar
5. Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390–1400.Google Scholar
6. Farrell, B. F. & Ioannou, P. J. 2000 Transient and asymptotic growth of two-dimensional perturbations in viscous compressible shear flow. Phys. Fluids 12, 3021.CrossRefGoogle Scholar
7. Gelfgat, A. Yu. & Kit, E. 2006 Spatial versus temporal instabilities in a parametrically forced stratified mixing layer. J. Fluid Mech. 552, 189–227.CrossRefGoogle Scholar
8. Heifetz, E. & Methven, J. 2005 Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids 17, 064107.CrossRefGoogle Scholar
9. Hussain, F., Pradeep, D. S. & Stout, E. 2011 Nonlinear transient growth in a vortex column. J. Fluid Mech. 682, 304–331.Google Scholar
10. Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
11. Mao, X., Sherwin, S. J. & Blackburn, H. M. 2012 Non-normal dynamics of time-evolving co-rotating vortex pairs, J. Fluid Mech. (submitted).CrossRefGoogle Scholar
12. Methven, J., Hoskins, B. J., Heifetz, E. & Bishop, C. H. 2005 The counter-propagating Rossby wave perspective on baroclinic instability. Part IV: nonlinear life cycles. Q. J. R. Meteorol. Soc. 131, 1393–1424.CrossRefGoogle Scholar
13. Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. A 27, 69–138.Google Scholar
14. Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251–288.CrossRefGoogle Scholar
15. Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
16. Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar