Hostname: page-component-5cf477f64f-fcbfl Total loading time: 0 Render date: 2025-04-07T13:45:11.733Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteady turbulent line plumes

Published online by Cambridge University Press:  28 September 2018

Andrew J. Hogg Open the ORCID record for Andrew J. Hogg [Opens in a new window] ,
Edward J. Goldsmith  and
Mark J. Woodhouse
Andrew J. Hogg*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Edward J. Goldsmith
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
Mark J. Woodhouse
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK School of Earth Science, University of Bristol, Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon ‘top-hat’ distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.

JFM classification

Convection: Plumes/thermals Turbulent Flows: Turbulent convection Turbulent Flows: Turbulent Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 856 , 10 December 2018 , pp. 103 - 134
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. 1964 A Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55. National Bureau of Standards.Google Scholar
Anwar, H. O. 1969 Experiment on an effluent discharging from a slot into stationary or slow moving fluid of greater density. J. Hydraul. Res. 7, 411–431.Google Scholar
Baker, E. T., Massoth, G. J., Feely, R. A., Embley, R. W., Thomson, R. E. & Burd, B. J. 1995 Hydrothermal event plumes from the CoAxial seafloor eruption site, Juan de Fuca Ridge. Geophys. Res. Lett. 22, 147–150.Google Scholar
Bradbury, L. J. S. 1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23, 31–64.Google Scholar
van den Bremer, T. S. & Hunt, G. R. 2014 Two-dimensional planar plumes and fountains. J. Fluid Mech. 750, 210–244.Google Scholar
Ching, C. Y., Fernando, H. J. S. & Robles, A. 1995 Breakdown of line plumes in turbulent environments. J. Geophys. Res. 100, 4707–4713.Google Scholar
Craske, J. 2017 The properties of integral models for planar and axisymmetric unsteady jets. IMA J. Appl. Maths 82, 305–333.Google Scholar
Craske, J. & van Reeuwijk, M. 2016 Generalised unsteady plume theory. J. Fluid Mech. 792, 1013–1052.Google Scholar
Glaze, L. S., Baloga, S. M. & Wimert, J. 2011 Explosive volcanic eruptions from linear vents on Earth, Venus, and Mars: comparisons with circular vent eruptions. J. Geophys. Res. 116, E01011.Google Scholar
Kaminski, E. L., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–376.Google Scholar
Kurganov, A., Noelle, S. & Petrova, G. 2001 Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707–740.Google Scholar
Lee, S. & Emmons, H. W. 1961 A study of natural convection above a line fire. J. Fluid Mech. 11, 353–368.Google Scholar
Linden, P. F. 2000 Convection in the environment. In Perspectives in Fluid Dynamics (ed. Moffatt, H. K., Batchelor, G. K. & Worster, M. G.), pp. 289–345. Cambridge University Press.Google Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous source. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Priestley, C. H. B. & Ball, F. K. 1955 Continuous convection from an isolated source of heat. Q. J. R. Meteorol. Soc. 81, 144–157.Google Scholar
Rouse, H., Yih, C. S. & Humphreys, H. W. 1952 Gravitational convection from a boundary source. Tellus 4, 201–210.Google Scholar
Scase, M. M., Caulfield, C. P., Dalziel, S. B. & Hunt, J. C. R. 2006 Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech. 563, 443–461.Google Scholar
Scase, M. M. & Hewitt, R. E. 2012 Unsteady turbulent plume models. J. Fluid Mech. 697, 455–480.Google Scholar
Stothers, R. B. 1989 Turbulent atmospheric plumes above line sources with an application to volcanic fissure eruptions on the terrestrial planets. J. Atmos. Sci. 46, 2662–2670.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption and its application to geophysical flows. J. Fluid Mech. 173, 431–471.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Woodhouse, M. J., Phillips, J. C. & Hogg, A. J. 2016 Unsteady turbulent buoyant plumes. J. Fluid Mech. 794, 595–639.Google Scholar
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391–412.Google Scholar