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Growth and instability of a laminar plume in a strongly stratified environment

Published online by Cambridge University Press:  03 February 2011

M. LOMBARDI ,
C. P. CAULFIELD ,
C. COSSU ,
A. I. PESCI  and
R. E. GOLDSTEIN
M. LOMBARDI
Affiliation:
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
C. P. CAULFIELD
Affiliation:
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. COSSU
Affiliation:
IMFT, 1 Allée du Pr. Camille Soula, 31400 Toulouse, France
A. I. PESCI
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
R. E. GOLDSTEIN*
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: r.e.goldstein@damtp.cam.ac.uk
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Abstract

Experimental studies of laminar plumes descending under gravity into stably stratified environments have shown the existence of a critical injection velocity beyond which the plume exhibits a bifurcation to a coiling instability in three dimensions or a sinuous instability in a Hele-Shaw flow. In addition, flow visualization has shown that, prior to the onset of the instability, a stable base flow is established in which the plume penetrates to a depth significantly smaller than the neutral buoyancy depth. Moreover, the fresh water that is viscously entrained by the plume recirculates within a ‘conduit’ whose boundary with the background stratification appears sharp. Beyond the bifurcation, the buckling plume takes the form of a travelling wave of varying amplitude, confined within the conduit, which disappears at the penetration depth. To determine the mechanisms underlying these complex phenomena, which take place at a strikingly low Reynolds number but a high Schmidt number, we study here a two-dimensional arrangement, as it is perhaps the simplest system which possesses all the key experimental features. Through a combination of numerical and analytical approaches, a scaling law is found for the plume's penetration depth within the base flow (i.e. the flow where the instability is either absent or artificially suppressed), and the horizontal cross-stream velocity and concentration profile outside the plume are determined from an asymptotic analysis of a simplified model. Direct numerical simulations show that, with increasing flow rate, a sinuous global mode is destabilized giving rise to the self-sustained oscillations as in the experiment. The sinuous instability is shown to be a consequence of the baroclinic generation of vorticity, due to the strong horizontal gradients at the edge of the conduit, a mechanism that is relevant even at very low Reynolds numbers. Despite the strength of this instability, the penetration depth is not significantly affected by it, instead being determined by the properties of the plume in the vicinity of the source. This scenario is confirmed by a local stability analysis. A finite region of local absolute instability is found near the source for sinuous modes prior to the onset of the global instability. Sufficiently far from the source the flow is locally stable. Near the onset of the global instability, varicose modes are also found to be locally, but only convectively, unstable.

JFM classification

Convection: Plumes/thermals Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 184 - 206
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Auteri, F., Guermond, J.-L. & Parolini, N. 2001 Role of the LBB condition in weak spectral projection methods. J. Comput. Phys. 174, 405–420.CrossRefGoogle Scholar
Batchelor, G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Meteorol. Soc. 80, 339–358.CrossRefGoogle Scholar
Blanchette, F., Peacock, T. & Cousin, R. 2008 Stability of a stratified fluid with a vertically moving sidewall. J. Fluid Mech. 609, 305–317.CrossRefGoogle Scholar
Brandt, L., Cossu, C., Chomaz, J.-M., Huerre, P. & Henningson, D. S. 2003 On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485, 221–242.CrossRefGoogle Scholar
Cartwright, J. H. E., Garcia-Ruíz, J. M., Novella, M. L. & Otálora, F. 2002 Formation of chemical gardens. J. Colloid Interface Sci. 256, 351–359.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Davis, T. A. & Duff, I. S. 1997 An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl. 18, 140–158.CrossRefGoogle Scholar
Delbende, I. & Chomaz, J.-M. 1998 Nonlinear convective/absolute instabilities of parallel two-dimensional wakes. Phys. Fluids 10, 2724–2736.CrossRefGoogle Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.CrossRefGoogle Scholar
Dombrowski, C., Lewellyn, B., Pesci, A. I., Restrepo, J. M., Kessler, J. O. & Goldstein, R. E. 2005 Coiling, entrainment, and hydrodynamic coupling of decelerated fluid jets. Phys. Rev. Lett. 95, 184501.CrossRefGoogle ScholarPubMed
Drazin, P. G. & Riley, N. 2006 The Navier–Stokes Equations: A Classification of Flows and Exact Solutions. Cambridge University Press.CrossRefGoogle Scholar
Helfrich, K. R. & Whitehead, J. A. 1990 Solitary waves on conduits of buoyant fluid in a more viscous fluid. Geophys. Astrophys. Fluid Dyn. 51, 35–52.CrossRefGoogle Scholar
Hill, R. I., Campbell, I. H., Davies, G. F. & Griffiths, R. W. 1992 Mantle plumes and continental tectonics. Science 256, 186–193.CrossRefGoogle ScholarPubMed
Huerre, P. 2001 Open shear flow instabilities. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 159–229. Cambridge University Press.Google Scholar
Katz, R. F. & Worster, M. G. 2008 Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell. J. Comput. Phys. 227, 9823–9840.CrossRefGoogle Scholar
Kessler, J. O. 1985 Hydrodynamic focusing of motile algal cells. Nature 313, 218–220.CrossRefGoogle Scholar
Kessler, J. O. 1986 Individual and collective fluid-dynamics of swimming cells. J. Fluid Mech. 173, 191–205.CrossRefGoogle Scholar
Lister, J. R. 1987 Long-wavelength instability of a line plume. J. Fluid Mech. 175, 413–428.CrossRefGoogle Scholar
Lombardi, M. 2008 Numerical analysis of a laminar plume in a stratified environment. MPhil thesis, BP Institute, University of Cambridge, Cambridge, UK.Google Scholar
Maxworthy, T. 1999 The flickering candle: transition to a global oscillation in a thermal plume. J. Fluid Mech. 390, 297–323.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. A 234, 1–23.Google Scholar
Peppin, S. S. L., Huppert, H. E. & Worster, M. G. 2008 Steady-state solidification of aqueous ammonium chloride J. Fluid Mech. 599, 465–476.CrossRefGoogle Scholar
Pesci, A. I., Porter, M. A. & Goldstein, R. E. 2003 Inertially driven buckling and overturning of jets in a Hele-Shaw cell. Phys. Rev. E 68, 056305.CrossRefGoogle Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines. J. Fluid Mech. 133, 133–145.CrossRefGoogle Scholar
Ribe, N. M. 2004 Coiling of viscous jets. Proc. R. Soc. A 460, 3223–3239.CrossRefGoogle Scholar
Schofield, S. P. & Restrepo, J. M. 2010 Stability of planar buoyant jets in stratified fluids. Phys. Fluids 22, 053602.CrossRefGoogle Scholar
Shraiman, B. I. 1987 Diffusive transport in a Rayleigh–Bénard convection cell. Phys. Rev. A 36, 261–267.CrossRefGoogle Scholar
Standing, R. G. 1971 The Rayleigh problem for a slightly diffusive density-stratified fluid. J. Fluid Mech. 48, 673–688.CrossRefGoogle Scholar
Stone, D. A. & Goldstein, R. E. 2004 Tubular precipitation and redox gradients on a bubbling template. Proc. Natl Acad. Sci. USA 101, 11537–11541.CrossRefGoogle ScholarPubMed
Stone, D. A., Lewellyn, B., Baygents, J. C. & Goldstein, R. E. 2005 Precipitative growth templated by a fluid jet. Langmuir 21, 10916–10919.CrossRefGoogle ScholarPubMed
Thouvenel-Romans, S. & Steinbock, O. 2003 Oscillatory growth of silica tubes in chemical gardens. J. Am. Chem. Soc. 125, 4338–4341.CrossRefGoogle ScholarPubMed
Thouvenel-Romans, S., van Saarloos, W. & Steinbock, O. 2004 Silica tubes in chemical gardens: radius selection and its hydrodynamic origin. Europhys. Lett. 67, 42–48.CrossRefGoogle Scholar
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies Proc. R. Soc. Lond. A 293, 310–328.Google Scholar
Weissburg, M. J. & Zimmer-Faust, R. K. 1994 Odor plumes and how blue crabs use them in finding prey. J. Exp. Biol. 197, 349–375.CrossRefGoogle ScholarPubMed