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Spontaneous generation of a swirling plume in a stratified ambient

Published online by Cambridge University Press:  26 November 2014

Francisco Marques  and
Juan M. Lopez
Francisco Marques
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: [email protected]
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Abstract

The transition from laminar to complex spatio-temporal dynamics of plumes due to a localized buoyancy source is studied numerically. Several experiments have reported that this transition is sensitive to external perturbations. Therefore, a well-controlled set-up has been chosen for our numerical study, consisting of a localized heat source at the bottom of an enclosed cylinder whose sidewall is maintained at a fixed temperature which varies linearly up the wall. Restricting the dynamics to the axisymmetric subspace, the first instability is to a puffing state. However, for smaller Grashof numbers, the plume becomes unstable to three-dimensional perturbations and a swirling plume spontaneously appears. The next bifurcation, viewed in the rotating frame where the plume is stationary, also exhibits puffing and suggests a connection between the unstable axisymmetric puffing solution and the swirling plume. Further bifurcations result in quasi-periodic states with a very low-frequency modulation, and these eventually become spatio-temporally complex.

JFM classification

Instability: Nonlinear instability Convection: Plumes/thermals Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , pp. 443 - 463
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Copyright
© 2014 Cambridge University Press 

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References

Altmeyer, S., Do, Y., Marques, F. & Lopez, J. M. 2012 Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor–Couette flow. Phys. Rev. E 86, 046316.Google Scholar
Avila, M., Marques, F., Lopez, J. M. & Meseguer, A. 2007 Stability control and catastrophic transition in a forced Taylor–Couette system. J. Fluid Mech. 590, 471–496.Google Scholar
Battaglia, F., Rehm, R. G. & Baum, H. R. 2000 The fluid mechanics of fire whirls: an inviscid model. Phys. Fluids 12, 2859–2867.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.CrossRefGoogle Scholar
Carroll, J. J. & Ryan, J. A. 1970 Atmospheric vorticity and dust devil rotation. J. Geophys. Res. 75, 5179–5184.Google Scholar
Chossat, P. & Lauterbach, R. 2000 Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific.CrossRefGoogle Scholar
Cortese, T. & Balachandar, S. 1993 Vortical nature of thermal plumes in turbulent convection. Phys. Fluids A 5, 3226–3232.Google Scholar
Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341–387.CrossRefGoogle Scholar
Emmons, H. W. & Ying, S.-J. 1967 The fire whirl. In Proceedings 11th International Symposium on Combustion, pp. 475–488. Combustion Institute.Google Scholar
Fiedler, B. H. & Kanak, K. M. 2001 Rayleigh-Bénard convection as a tool for studying dust devils. Atmos. Sci. Lett.; doi:10.1006/asle.2001.0043.Google Scholar
Gray, D. D. & Giorgini, A. 1976 The validity of the Boussinesq approximation for liquids and gases. Intl J. Heat Mass Transfer 19, 545–551.CrossRefGoogle Scholar
Knobloch, E. 1996 Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows. Phys. Fluids 8, 1446–1454.Google Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.Google Scholar
Lopez, J. M. 2006 Rotating and modulated rotating waves in transitions of an enclosed swirling flow. J. Fluid Mech. 553, 323–346.Google Scholar
Lopez, J. M. & Marques, F. 2000 Dynamics of 3-tori in a periodically forced Navier–Stokes flow. Phys. Rev. Lett. 85, 972–975.Google Scholar
Lopez, J. M. & Marques, F. 2003 Small aspect ratio Taylor–Couette flow: onset of a very-low-frequency three-torus state. Phys. Rev. E 68, 036302.Google Scholar
Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.Google Scholar
Lopez, J. M. & Marques, F. 2013 Instability of plumes driven by localized heating. J. Fluid Mech. 736, 616–640.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 2006 Onset of three-dimensional unsteady states in small aspect-ratio Taylor–Couette flow. J. Fluid Mech. 561, 255–277.Google Scholar
Marques, F., Lopez, J. M. & Iranzo, V. 2002 Imperfect gluing bifurcation in a temporal glide-reflection symmetric Taylor–Couette flow. Phys. Fluids 14, L33–L36.Google Scholar
Marques, F., Lopez, J. M. & Shen, J. 2001 A periodically forced flow displaying symmetry breaking via a three-tori gluing bifurcation and two-tori resonances. Physica D 156, 81–97.CrossRefGoogle Scholar
Massaguer, J. M. & Mercader, I. 1988 Instability of swirl in low-Prandtl number thermal convection. J. Fluid Mech. 189, 367–395.Google Scholar
Massaguer, J. M., Mercader, I. & Net, M. 1990 Nonlinear dynamics of vertical vorticity in low-Prandtl number thermal convection. J. Fluid Mech. 214, 579–597.Google Scholar
Maxworthy, T. 1973 A vorticity source for large-scale dust devils and other comments on naturally occurring columnar vortices. J. Atmos. Sci. 30, 1717–1722.Google Scholar
Mercader, I., Batiste, O. & Alonso, A. 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224.Google Scholar
Morton, B. R. 1966 Geophysical vortices. Prog. Aerosp. Sci. 7, 145–194.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Munroe, J. R. & Sutherland, B. R. 2014 Internal wave energy radiated from a turbulent mixed layer. Phys. Fluids 26, 096604.Google Scholar
Muraszew, A., Fedele, J. B. & Kuby, W. C. 1979 The fire whirl phenomenon. Combust. Flame 34, 29–45.CrossRefGoogle Scholar
Murphy, J. O. & Lopez, J. M. 1984 The influence of vertical vorticity on thermal convection. Austral J. Phys. 37, 41–62.Google Scholar
Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom-A attractors near quasi-periodic flows on $T^{m}$ , $m\geqslant 3$ . Commun. Math. Phys. 64, 35–40.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.Google Scholar
Snow, J. T. 1987 Atmospheric columnar vortices. Rev. Geophys. 25, 371–385.CrossRefGoogle Scholar
Speer, K. G. & Marshall, J. 1995 The growth of covective plumes at seafloor hot springs. J. Mar. Res. 53, 1025–1057.Google Scholar
Strogatz, S. 1994 Nonlinear Dynamics and Chaos. Addison-Wesley.Google Scholar
Sutherland, B. R. & Linden, P. F. 1998 Internal wave excitation from stratified flow over a thin barrier. J. Fluid Mech. 377, 223–252.CrossRefGoogle Scholar
Torrance, K. E. 1979 Natural convection in thermally stratified enclosures with localized heating from below. J. Fluid Mech. 95, 477–495.CrossRefGoogle Scholar
Torrance, K. E., Orloff, L. & Rockett, J. A. 1969 Experiments on natural convection in enclosures with localized heating from below. J. Fluid Mech. 36, 21–31.Google Scholar
Torrance, K. E. & Rockett, J. A. 1969 Numerical study of natural convection in an enclosure with localized heating from below – creeping flow to the onset of laminar instability. J. Fluid Mech. 36, 33–54.CrossRefGoogle Scholar
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391–412.Google Scholar

Marques and Lopez supplementary movie

Movie 1 (5a): Isotherms of the axisymmetric periodic puffing plume state at $Gr=3.5 \times 10^5$, $Ar=2$, $A_z=2$, $A_T=1$ and $\sigma=7$ over one puffing period $2\pi/\omega_0 \approx 3.9 \times 10^{−4}$.

Download Marques and Lopez supplementary movie(Video)
Video 860 KB

Marques and Lopez supplementary movie

Movie 2 (5b): Azimuthal vorticity contours of the axisymmetric periodic puffing plume state at $Gr=3.5 \times 10^5$, $Ar=2$, $A_z=2$, $A_T=1$ and $\sigma=7$ over one puffing period $2\pi/\omega_0 \approx 3.9 \times 10^{−4}$.

Download Marques and Lopez supplementary movie(Video)
Video 843.3 KB

Marques and Lopez supplementary movie

Movie 3(6a): Isotherms of the axisymmetric periodic puffing plume state at $Gr=10^6$, $Ar=2$, $A_z=2$, $A_T=1$ and $\sigma=7$ over one puffing period $2\pi/\omega_0 \approx 4.1 \times 10^{−4}$.

Download Marques and Lopez supplementary movie(Video)
Video 3.5 MB

Marques and Lopez supplementary movie

Movie 4 (6b): Azimuthal vorticity contours of the axisymmetric periodic puffing plume state at $Gr=10^6$, $Ar=2$, $A_z=2$, $A_T=1$ and $\sigma=7$ over one puffing period $2\pi/\omega_0 \approx 4.1 \times 10^{−4}$.

Download Marques and Lopez supplementary movie(Video)
Video 3 MB

Marques and Lopez supplementary movie

Movie 5 (12a): Three-dimensional isosurfaces of azimuthal vorticity $\eta$ of the rotating wave state RW at $Gr=2 \times 10^5$, $A_r=2$, $A_z=2$, $A_T=1$ and $\sigma=7$, over one precession period $2\pi/\omega_\text{pr} \approx 5 \times 10^{−4}; the isosurface levels are at $\eta = \pm 10^3$.

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Video 504.7 KB

Marques and Lopez supplementary movie

Movie 6 (16): Three-dimensional isosurfaces of azimuthal vorticity $\eta$ of the VLF state at $Gr=3.5 \times 10^5$, $A_r=2$, $A_z=2$, $A_T=1$ and $\sigma=7$; the isosurface levels are at $\eta = \pm 10^3$.

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Video 770 KB