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Evolution of localized disturbances in the elliptic instability

Published online by Cambridge University Press:  22 August 2014

Yuji Hattori  and
Mohd Syafiq bin Marzuki
Yuji Hattori*
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
Mohd Syafiq bin Marzuki
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
*
†Email address for correspondence: hattori@fmail.ifs.tohoku.ac.jp
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Abstract

The time evolution of localized disturbances in an elliptical flow confined in an elliptical cylinder is studied by direct numerical simulation (DNS). The base flow is subject to the elliptic instability. The unstable growth of localized disturbances predicted by the short-wavelength stability analysis is captured. The time evolution can be divided into four stages: linear, weakly nonlinear, nonlinear and turbulent. In the linear stage a single wavepacket grows exponentially without changing its shape. The exponential growth is accompanied by large oscillations which have time period half that of the fluid particles in the elliptical flow. An averaged wavepacket, which is a train of bending waves that has a finite spatial extent, also grows exponentially, while the oscillations of the growth rate are small. The averaged growth rate increases as the kinematic viscosity decreases; the inviscid limit is close to the value predicted by the short-wavelength stability analysis. In the weakly nonlinear stage the energy stops growing. The vortical structure of the initial disturbances is deformed into wavy patterns. The energy spectrum loses the peak at the initial wavenumber, developing a broad spectrum, and the flow goes into the next stage. In the nonlinear stage weak vorticity is scattered in the whole domain although strong vorticity is still localized. The probability density functions (p.d.f.) of a velocity component and its longitudinal derivative are similar to those of isotropic turbulence; however, the energy spectrum does not have an inertial range showing the Kolmogorov spectrum. Finally in the turbulent stage fine-scale structures appear in the vorticity field. The p.d.f. of the longitudinal derivative of velocity shows the strong intermittency known for isotropic turbulence. The energy spectrum attains an inertial range showing the Kolmogorov spectrum. The turbulence is not symmetric because of rotation and strain; the component of vorticity in the compressing direction is smaller than the other two components. The energy of the mean flow as well as the total energy decreases. The ratio of the lost energy to the initial energy of the mean flow is large in the core region.

JFM classification

Instability: Parametric instability Instability: Transition to turbulence Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , pp. 603 - 627
Copyright
© 2014 Cambridge University Press 

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References

Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81, 497–520.CrossRefGoogle Scholar
Arendt, S., Fritts, D. C. & Andreassen, Ø. 1997 The initial value problem for Kelvin vortex waves. J. Fluid Mech. 344, 181–212.CrossRefGoogle Scholar
Arquis, E. & Caltagirone, J. P. 1984 On the hydrodynamical boundary-conditions along a fluid layer porous-medium interface – application to the case of free-convection. C. R. Acad. Sci. Paris II 299, 1–4.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Cohen, J., Shukhman, I. G., Karp, M. & Philip, J. 2010 An analytical-based method for studying the nonlinear evolution of localized vortices in planar homogeneous shear flows. J. Comput. Phys. 229, 7765–7773.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence – An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Eloy, C. & Le Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660–676.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 3400–3403.CrossRefGoogle ScholarPubMed
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.CrossRefGoogle Scholar
Faddy, J. M. & Pullin, D. I. 2005 Flow structure in a model of aircraft trailing vortices. Phys. Fluids 17, 085106.CrossRefGoogle Scholar
Friedlander, S. & Vishik, M. M. 1991 Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206.CrossRefGoogle ScholarPubMed
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.CrossRefGoogle Scholar
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77–115.CrossRefGoogle Scholar
Gledzer, E. B. & Ponomarev, V. M. 1992 Instability of bounded flows with elliptical streamlines. J. Fluid Mech. 240, 1–30.CrossRefGoogle Scholar
Godeferd, F. S., Cambon, C. & Leblanc, S. 2001 Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with external rotation. J. Fluid Mech. 449, 1–37.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2003 Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15, 3151–3163.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2009 Short-wavelength stability analysis of a helical vortex tube. Phys. Fluids 21, 014104.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2010 Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability. Theor. Comput. Fluid Dyn. 24, 363–368.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2012 Effects of axial flow on the stability of a helical vortex tube. Phys. Fluids 24, 054102.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222–249.CrossRefGoogle Scholar
Hattori, Y. & Hijiya, K. 2010 Short-wavelength stability analysis of Hill’s vortex with/without swirl. Phys. Fluids 22, 074104.CrossRefGoogle Scholar
Hattori, Y. & Ishihara, T.2012 4D visualization of isotropic turbulence and dynamics of high-enstrophy structures. In Proceedings JSST 2012 (International Conference on Simulation Technology), pp. 170–174.Google Scholar
Hattori, Y., Ishizawa, A. & Rubinstein, R. 2004 Shell model for rotating turbulence. Phys. Rev. E 70, 046311.CrossRefGoogle ScholarPubMed
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.CrossRefGoogle Scholar
Kerr, R. 2013 Swirling, turbulent vortex rings formed from a chain reaction of reconnection events. Phys. Fluids 25, 065101.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Landman, M. J. & Saffman, P. G. 1987 The 3-dimensional instability of strained vortices in a viscous-fluid. Phys. Fluids 30, 2339–2342.CrossRefGoogle Scholar
Laporte, F. & Corjon, A. 2000 Direct numerical simulations of the elliptic instability of a vortex pair. Phys. Fluids 12, 1016–1031.CrossRefGoogle Scholar
Le Dizès, S. & Eloy, C. 1999 Short-wavelength instability of a vortex in a multipolar strain field. Phys. Fluids 11, 500–502.CrossRefGoogle Scholar
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptical instability in a two-vortex flow. J. Fluid Mech. 396, 73–108.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.CrossRefGoogle Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651.CrossRefGoogle Scholar
Lundgren, T. S. & Mansour, N. N. 1996 Transition to turbulence in an elliptic vortex. J. Fluid Mech. 307, 43–62.CrossRefGoogle Scholar
Malkus, W. V. R. 1989 An experimental study of the global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.CrossRefGoogle Scholar
Mason, D. M. & Kerswell, R. R. 1999 Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown. J. Fluid Mech. 396, 73–108.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159.CrossRefGoogle Scholar
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. In Proceedings of the International Colloquium on Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 139–156. Nauka.Google Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.CrossRefGoogle Scholar
Schaeffer, N. & Le Dizès, S. 2010 Nonlinear dynamics of the elliptic instability. J. Fluid Mech. 646, 471–480.CrossRefGoogle Scholar
Schneider, K. & Farge, M. 2005 Decaying two-dimensional turbulence in a circular container. Phys. Rev. Lett. 95, 244502.CrossRefGoogle Scholar
Shukhman, I. G. 2007 Evolution of a localized vortex in plane nonparallel viscous flows with constant velocity shear. II. Elliptic flow. Phys. Fluids 19, 017106.CrossRefGoogle Scholar
Sreenivasan, K. R. 1995 On the universality of the Kolmogorov constant. Phys. Fluids 7, 2778–2784.CrossRefGoogle Scholar
Terakado, D. & Hattori, Y. 2014 Density distribution in two-dimensional weakly compressible turbulence. Phys. Fluids 26, 085105.CrossRefGoogle Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.CrossRefGoogle Scholar
Waleffe, F. A.1989 The 3D instability of a strained vortex and its relation to turbulence. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar