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INVISCID AND VISCOUS MODELS OF AXISYMMETRIC FLUID JETS OR PLUMES

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  12 November 2012

NICHOLAS A. LETCHFORD ,
LAWRENCE K. FORBES  and
GRAEME C. HOCKING
NICHOLAS A. LETCHFORD
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia (email: Larry.Forbes@utas.edu.au)
LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia (email: Larry.Forbes@utas.edu.au)
GRAEME C. HOCKING
Affiliation:
School of Chemical and Mathematical Sciences, Murdoch University, 90 South Street, Murdoch, Western Australia 6150, Australia (email: G.Hocking@murdoch.edu.au)
*
✉For correspondence; e-mail: Larry.Forbes@utas.edu.au
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Abstract

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The vertical rise of a round plume of light fluid through a surrounding heavier fluid is considered. An inviscid model is analysed in which the boundary of the plume is taken to be a sharp interface. An efficient spectral method is used to solve this nonlinear free-boundary problem, and shows that the plume narrows as it rises. A generalized condition is also introduced at the boundary, and allows the ambient fluid to be entrained into the rising plume. In this case, the fluid plume first narrows then widens as it rises. These features are confirmed by an asymptotic analysis. A viscous model of the same situation is also proposed, based on a Boussinesq approximation. It qualitatively confirms the widening of the plume due to entrainment of the ambient fluid, but also shows the presence of vortex rings around the interface of the rising plume.

Keywords

fluid jetaxisymmetric flowinviscid theoryspectral methodsplumeBoussinesq approximationentrainment

MSC classification

Secondary: 76D25: Wakes and jets
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 3 , January 2012 , pp. 228 - 250
Copyright
Copyright © Australian Mathematical Society 2012

References

[1] M. Abramowitz and I. A. Stegun (eds), Handbook of mathematical functions: with formulas, graphs and mathematical tables (Dover Publications, New York, 1972.Google Scholar
[2]Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
[3]Bhat, G. S. and Krothapalli, A., “Simulation of a round jet and a plume in a regional atmospheric model”, Monthly Weather Rev. 128 (2000) 4108–4117; doi:10.1175/1520-0493(2000)129<4108:SOARJA>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
[4]Borgas, M. S. and Tuck, E. O., “Thin slender water jets”, J. Fluid Mech. 118 (1982) 379–391; doi:10.1017/S0022112082001128.CrossRefGoogle Scholar
[5]Button, E. C., Davidson, J. F., Jameson, G. J. and Sader, J. E., “Water bells formed on the underside of a horizontal plate. Part 2. Theory”, J. Fluid Mech. 649 (2010) 45–68; doi:10.1017/S0022112009993363.CrossRefGoogle Scholar
[6]Chen, M. J. and Forbes, L. K., “Accurate methods for computing inviscid and viscous Kelvin–Helmholtz instability”, J. Comput. Phys. 230 (2011) 1499–1515; doi:10.1016/j.jcp.2010.11.017.CrossRefGoogle Scholar
[7]Christodoulides, P and Dias, F., “Impact of a rising stream on a horizontal plate of finite extent”, J. Fluid Mech. 621 (2009) 243–258; doi:10.1017/S0022112008004746.CrossRefGoogle Scholar
[8]Faltinsen, O. M., Rognebakke, O. F., Lukovsky, I. A. and Timokha, A. N., “Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth”, J. Fluid Mech. 407 (2000) 201–234; doi:10.1017/S0022112099007569.Google Scholar
[9]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
[10]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.CrossRefGoogle Scholar
[11]Forbes, L. K., Chen, M. J. and Trenham, C. E., “Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow”, J. Comput. Phys. 221 (2007) 269–287; doi:10.1016/j.jcp.2006.06.010.CrossRefGoogle Scholar
[12]Forbes, L. K. and Hocking, G. C., “Unsteady plumes in planar flow of viscous and inviscid fluids”, IMA J. Appl. Math. to appear; doi:10.1093/imamat/hxr045.CrossRefGoogle Scholar
[13]Geer, J. F. and Strikwerda, J. C., “Vertical slender jets with surface tension”, J. Fluid Mech. 135 (1983) 155–169; doi:10.1017/S0022112083003006.CrossRefGoogle Scholar
[14]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.CrossRefGoogle Scholar
[15]Hunt, G. R. and van den Bremer, T. S., “Classical plume theory: 1937–2010 and beyond”, IMA J. Appl. Math. 76 (2011) 424–448; doi:10.1093/imamat/hxq056.CrossRefGoogle Scholar
[16]Hunt, G. R. and Kaye, N. B., “Lazy plumes”, J. Fluid Mech. 533 (2005) 329–338; doi:10.1017/S002211200500457X.Google Scholar
[17]Kaminski, E., Tait, S. and Carazzo, G., “Turbulent entrainment in jets with arbitrary buoyancy”, J. Fluid Mech. 526 (2005) 361–376; doi:10.1017/S0022112004003209.CrossRefGoogle Scholar
[18]Lin, W. and Armfield, S. W., “Onset of entrainment in transitional round fountains”, Int. J. Heat Mass Transfer 51 (2008) 5226–5237; doi:10.1016/j.ijheatmasstransfer.2008.02.047.CrossRefGoogle Scholar
[19]List, E. J., “Turbulent jets and plumes”, Annu. Rev. Fluid Mech. 14 (1982) 189–212; doi:10.1146/annurev.fl.14.010182.001201.CrossRefGoogle Scholar
[20]Moore, D. W., “The spontaneous appearance of a singularity in the shape of an evolving vortex sheet”, Proc. R. Soc. Lond. A 365 (1979) 105–119; doi:10.1098/rspa.1979.0009.Google Scholar
[21]Morton, B. R., Taylor, G. and Turner, J. S., “Turbulent gravitational convection from maintained and instantaneous sources”, Proc. R. Soc. Lond. A 234 (1956) 1–23; doi:10.1098/rspa.1956.0011.Google Scholar
[22]Plourde, F., Pham, M. V., Kim, S. D. and Balachandar, S., “Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction”, J. Fluid Mech. 604 (2008) 99–123; doi:10.1017/S0022112008001006.CrossRefGoogle Scholar
[23]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.CrossRefGoogle Scholar
[24]Reynolds, C. S., Heinz, S. and Begelman, M. C., “The hydrodynamics of dead radio galaxies”, Mon. Not. R. Astron. Soc. 332 (2002) 271–282; doi:10.1046/j.1365-8711.2002.04724.x.CrossRefGoogle Scholar
[25]Scase, M. M., Aspden, A. J. and Caulfield, C. P., “The effect of sudden source buoyancy flux increases on turbulent plumes”, J. Fluid Mech. 635 (2009) 137–169; doi:10.1017/S002211200900740X.CrossRefGoogle Scholar
[26]Scase, M. M. and Hewitt, R. E., “Unsteady turbulent plume models”, J. Fluid Mech. 697 (2012) 455–480; doi:10.1017/jfm.2012.77.CrossRefGoogle Scholar
[27]Stoker, J. J., Water waves (Wiley Interscience, New York, 1957).Google Scholar
[28]Tuck, E. O., “Annular water jets”, IMA J. Appl. Math. 29 (1982) 45–58; doi:10.1093/imamat/29.1.45.CrossRefGoogle Scholar
[29]Turner, J. S., “Buoyant plumes and thermals”, Annu. Rev. Fluid Mech. 1 (1969) 29–44; doi:10.1146/annurev.fl.01.010169.000333.CrossRefGoogle Scholar
[30]Turner, J. S., “Turbulent entrainment: the development of the entrainment assumption, and its applications to geophysical flows”, J. Fluid Mech. 173 (1986) 431–471; doi:10.1017/S0022112086001222.CrossRefGoogle Scholar
[31]Vallis, G. K., Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
[32]Wilcock, W. S. D., “Cellular convection models of mid-ocean ridge hydrothermal circulation and the temperatures of black smoker fluids”, J. Geophys. Res. 103 (1998) 2585–2596; doi:10.1029/97JB03252.Google Scholar
[33]Woods, A. W., “Turbulent plumes in nature”, Annu. Rev. Fluid Mech. 42 (2010) 391–412; doi:10.1146/annurev-fluid-121108-145430.Google Scholar