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AXISYMMETRIC PLUMES IN VISCOUS FLUIDS

Part of:Basic methods in fluid mechanicsHydrodynamic stability

Published online by Cambridge University Press:  17 May 2019

EMMA J. ALLWRIGHT,
L. K. FORBESOpen the ORCID record for L. K. FORBES[Opens in a new window] and
S. J. WALTERSOpen the ORCID record for S. J. WALTERS[Opens in a new window]
EMMA J. ALLWRIGHT
Affiliation:
Mathematics Department, T. U. München, Germany email Emma.Allwright@tum.deDepartment of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email Larry.Forbes@utas.edu.au, stephen.walters@utas.edu.au
L. K. FORBES*
Affiliation:
Department of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email Larry.Forbes@utas.edu.au, stephen.walters@utas.edu.au
S. J. WALTERS
Affiliation:
Department of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email Larry.Forbes@utas.edu.au, stephen.walters@utas.edu.au
*
✉Larry.Forbes@utas.edu.au
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Abstract

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We consider fluid in a channel of finite height. There is a circular hole in the channel bottom, through which fluid of a lower density is injected and rises to form a plume. Viscous boundary layers close to the top and bottom of the channel are assumed to be so thin that the viscous fluid effectively slips along each of these boundaries. The problem is solved using a novel spectral method, in which Hankel transforms are first used to create a steady-state axisymmetric (inviscid) background flow that exactly satisfies the boundary conditions. A viscous correction is then added, so as to satisfy the time-dependent Boussinesq Navier–Stokes equations within the fluid, leaving the boundary conditions intact. Results are presented for the “lazy” plume, in which the fluid rises due only to its own buoyancy, and we study in detail its evolution with time to form an overturning structure. Some results for momentum-driven plumes are also presented, and the effect of the upper wall of the channel on the evolution of the axisymmetric plume is discussed.

Keywords

Boussinesq fluidoverturning plumeunsteady flow

MSC classification

Primary:76E20: Stability and instability of geophysical and astrophysical flows
Secondary:76E17: Interfacial stability and instability76M22: Spectral methods

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 2 , April 2019, pp. 119 - 147
Copyright
© 2019 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I. A. (eds), Handbook of mathematical functions (Dover, New York, 1972).Google Scholar
Andrews, M. J. and Dalziel, S. B., “Small Atwood number Rayleigh–Taylor experiments”, Philos. Trans. R. Soc. Lond. Ser. A 368 (2010) 1663–1679; doi:10.1098/rsta.2010.0007.Google Scholar
Atkinson, K. E., Elementary numerical analysis (Wiley, New York, 1985).Google Scholar
Baker, G., Caflisch, R. E. and Siegel, M., “Singularity formation during Rayleigh–Taylor instability”, J. Fluid Mech. 252 (1993) 51–78; doi:10.1017/S0022112093003660.Google Scholar
Baker, G. R. and Pham, L. D., “A comparison of blob methods for vortex sheet roll-up”, J. Fluid Mech. 547 (2006) 297–316; doi:10.1017/S0022112005007305.Google Scholar
Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Boyd, J. P., Chebyshev and Fourier spectral methods, 2nd edn (Dover, New York, 2001).Google Scholar
Craske, J. and van Reeuwijk, M., “Generalised unsteady plume theory”, J. Fluid Mech. 792 (2016) 1013–1052; doi:10.1017/jfm.2016.72.Google Scholar
Farrow, D. E. and Hocking, G. C., “A numerical model for withdrawal from a two-layer fluid”, J. Fluid Mech. 549 (2006) 141–157; doi:10.1017/S0022112005007561.Google Scholar
Forbes, L. K., “The Rayleigh–Taylor instability for inviscid and viscous fluids”, J. Engrg. Math. 65 (2009) 273–290; doi:10.1007/s10665-009-9288-9.Google Scholar
Forbes, L. K., “A cylindrical Rayleigh–Taylor instability: radial outflow from pipes or stars”, J. Engrg. Math. 70 (2011) 205–224; doi:10.1007/s10665-010-9374-z.Google Scholar
Forbes, L. K., “Rayleigh–Taylor instabilities in axi-symmetric outflow from a point source”, ANZIAM J. 53 (2011) 87–121; doi:10.1017/S1446181112000090.Google Scholar
Forbes, L. K., “How strain and spin may make a star bipolar”, J. Fluid Mech. 746 (2014) 332–367; doi:10.1017/jfm.2014.130.Google Scholar
Gómez, L., Rodríguez, L. F. and Loinard, L., “A one-sided knot ejection at the core of the HH 111 outflow”, Rev. Mex. Astron. Astrofís. 49 (2013) 79–85; http://www.scielo.org.mx/pdf/rmaa/v49n1/v49n1a9.pdf.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Tables of integrals, series and products, 6th edn (Academic Press, San Diego, CA, 2000).Google Scholar
Hocking, G. C. and Forbes, L. K., “Steady flow of a buoyant plume into a constant-density layer”, J. Engrg. Math. 67 (2010) 341–350; doi:10.1007/s10665-009-9324-9.Google Scholar
Horsley, D. E., “Bessel phase functions: calculation and application”, Numer. Math. 136 (2017) 679–702; doi:10.1007/s00211-016-0853-7.Google Scholar
Hunt, G. R. and Burridge, H. C., “Fountains in industry and nature”, Annu. Rev. Fluid Mech. 47 (2015) 195–220; doi:10.1146/annurev-fluid-010313-141311.Google Scholar
Hunt, G. R. and Kaye, N. B., “Lazy plumes”, J. Fluid Mech. 533 (2005) 329–338; doi:10.1017/S002211200500457X.Google Scholar
Krasny, R., “Desingularization of periodic vortex sheet roll-up”, J. Comput. Phys. 65 (1986) 292–313; doi:10.1016/0021-9991(86)90210-X.Google Scholar
Letchford, N. A., Forbes, L. K. and Hocking, G. C., “Inviscid and viscous models of axisymmetric fluid jets or plumes”, ANZIAM J. 53 (2012) 228–250; doi:10.1017/S1446181112000156.Google Scholar
Lovelace, R. V. E., Romanova, M. M., Ustyugova, G. V. and Koldoba, A. V., “One-sided outflows/jets from rotating stars with complex magnetic fields”, Mon. Not. R. Astron. Soc. 408 (2010) 2083–2091; doi:10.1111/j.1365-2966.2010.17284.x.Google Scholar
Matsuoka, C. and Nishihara, K., “Analytical and numerical study on a vortex sheet in incompressible Richtmyer–Meshkov instability in cylindrical geometry”, Phys. Rev. E 74 (2006); article 066303, 12 pages; doi:10.1103/PhysRevE.74.066303.Google Scholar
Mikaelian, K. O., “Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells”, Phys. Fluids 17 (2005); article 094105, 13 pages; doi:10.1063/1.2046712.Google Scholar
Moore, D. W., “The spontaneous appearance of a singularity in the shape of an evolving vortex sheet”, Proc. R. Soc. Lond. Ser. A 365 (1979) 105–119; doi:10.1098/rspa.1979.0009.Google Scholar
Morton, B. R., Taylor, G. I. and Turner, J. S., “Turbulent gravitational convection from maintained and instantaneous sources”, Proc. R. Soc. Lond. Ser. A 234 (1956) 1–23; doi:10.1098/rspa.1956.0011.Google Scholar
Proskurowski, G., Lilley, M. D., Kelley, D. S. and Olson, E. J., “Low temperature volatile production at the Lost City Hydrothermal Field, evidence from a hydrogen stable isotope geothermometer”, Chem. Geol. 229 (2006) 331–343; doi:10.1016/j.chemgeo.2005.11.005.Google Scholar
Rayleigh, Lord, “Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density”, Proc. Lond. Math. Soc. (3) 14 (1883) 170–177; doi:10.1112/plms/s1-14.1.170.Google Scholar
Russell, P. S., Forbes, L. K. and Hocking, G. C., “The initiation of a planar fluid plume beneath a rigid lid”, J. Engrg. Math. 106 (2017) 107–121; doi:10.1007/s10665-016-9895-1.Google Scholar
Sharp, D. H., “An overview of Rayleigh–Taylor instability”, Physica D 12 (1984) 3–18; doi:10.1016/0167-2789(84)90510-4.Google Scholar
Stahler, S. W. and Palla, F., The formation of stars (Wiley-VCH, Berlin, 2004).Google Scholar
Taylor, Sir G. I., “The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, I”, Proc. R. Soc. Lond. Ser. A 201 (1950) 192–196; doi:10.1098/rspa.1950.0052.Google Scholar
Trefethen, L. N., “Finite difference and spectral methods for ordinary and partial differential equations”, unpublished text, 1996; http://people.maths.ox.ac.uk/trefethen/pdetext.html.Google Scholar
Vadivukkarasan, M. and Panchagnula, M. V., “Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet”, J. Fluid Mech. 812 (2017) 152–177; doi:10.1017/jfm.2016.784.Google Scholar
von Winckel, G., lgwt.m, at: MATLAB file exchange website (2004); http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=4540&objectType=file.Google Scholar
Woods, A. W., “Turbulent plumes in nature”, Annu. Rev. Fluid Mech. 42 (2010) 391–412; doi:10.1146/annurev-fluid-121108-145430.Google Scholar