Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T19:41:30.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition behaviour of weak turbulent fountains

Published online by Cambridge University Press:  11 May 2010

N. WILLIAMSON ,
S. W. ARMFIELD  and
WENXIAN LIN
N. WILLIAMSON*
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
S. W. ARMFIELD
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
WENXIAN LIN
Affiliation:
School of Engineering & Physical Sciences, James Cook University, Townsville, Queensland 4811, Australia
*
†Email address for correspondence: n.williamson@usyd.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical simulations of fully turbulent weak fountain flow are used to provide direct evidence for the scaling behaviour of fountain flow over the Froude number range Fr = 0.1–2.1 and Reynolds number range Re = 20–3494. For very weak flow at Fr < 0.4, the flow mean penetration height, Zm, scales with Zm/R0 = A1Fr2/3 + A2Fr2/3 where R0 is the source radius. A1 and A2 are constants which quantify the separate effects of the radial acceleration of fountain fluid from the source (A1) and the backpressure from the surrounding intrusion, if present, on the upflow (A2). The evidence presented in this work suggests that the mechanisms for the two parts in the scaling of Zm scale with Fr2/3. The intrusion behaviour varies with the Reynolds number (Re) but there is no Re affect on the fountain penetration height. For Re < 250 the radial intrusion flow is subcritical and has different behaviour. For Fr between 0.4 and 2.1 the effect of source momentum flux increases and the flow structure changes to one where there is a coherent upflow and a cap region where the flow stagnates and then reverses. The two regions have separate scaling behaviour such that the overall height, through this transition range of Froude numbers, can be described by Zm/R=C1Fr2/3 + C2Fr2, where C1 and C2 are constants. Over this transition range the effect of source velocity profile is more significant than the Reynolds number effects and the effect of inlet turbulence is minor.

JFM classification

Wakes/Jets: Jets Geophysical and Geological Flows: Gravity currents Waves/Free-surface Flows: Shear waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 655 , 25 July 2010 , pp. 306 - 326
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Armfield, S. W. & Street, R. 1999 The fractional step method for the Navier–Stokes equations on staggered grids: The accuracy of three variations. J. Comput. Phys. 153, 660–665.CrossRefGoogle Scholar
Armfield, S. W. & Street, R. 2002 An analysis and comparison of the time accuracy of fractional-step methods for the Navier–Stokes equations on staggered grids. Intl J. Numer. Meth. Fluids 38, 255–282.CrossRefGoogle Scholar
Baddour, R. E. & Zhang, H. 2009 Density effect on round turbulent hypersaline fountain. J. Hydraul. Engng 135, 57–59.CrossRefGoogle Scholar
Baines, W. D., Turner, J. S. & Campbell, I. H. 1990 Turbulent fountains in an open chamber. J. Fluid Mech. 212, 557–592.CrossRefGoogle Scholar
Bloomfield, L. J. & Kerr, R. C. 1998 Turbulent fountains in a stratified fluid. J. Fluid Mech. 358, 335–356.CrossRefGoogle Scholar
Bloomfield, L. J. & Kerr, R. C. 2000 A theoretical model of a turbulent fountain. J. Fluid Mech. 424, 197–216.CrossRefGoogle Scholar
Brandt, A. 1977 Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31 (138), 333–390.CrossRefGoogle Scholar
Campbell, I. H. & Turner, J. S. 1989 Fountains in magma chambers. J. Petrol. 30, 885–923.CrossRefGoogle Scholar
Cresswell, R. W. & Szczepura, R. T. 1993 Experimental investigation into a turbulent jet with negative buoyancy. Phys. Fluids A 5, 2865–2878.CrossRefGoogle Scholar
van der Vorst, H. A. 1992 Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (2), 631–644.CrossRefGoogle Scholar
Fukagata, K. & Kasagi, N. 2002 Highly energy-conservative finite difference method for the cylindrical coordinate system. J. Comput. Phys. 181, 478–498.CrossRefGoogle Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.CrossRefGoogle Scholar
Leonard, B. P. & Mokhtari, S. 1990 Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady state simulation of convection. Intl J. Numer. Meth. Engng 30, 729–766.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2000 a Direct simulation of weak axisymmetric fountains in a homogeneous fluid. J. Fluid Mech. 403, 67–88.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2000 b Very weak axisymmetric fountains in a homogeneous fluid. Numer. Heat Transfer A 38, 377–396.Google Scholar
Lin, W. & Armfield, S. W. 2003 The Reynolds and Prandtl number dependence of weak fountains. Comput. Mech. 31, 379–389.CrossRefGoogle Scholar
Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Wärme-und Stoffübertragung 16, 15–21.CrossRefGoogle Scholar
Pantzlaff, L. & Lueptow, R. M. 1999 Transient positively and negatively buoyant turbulent round jets. Exp. Fluids 27, 117–125.CrossRefGoogle Scholar
Snyder, D. & Tait, S. 1995 Replenishment of magma chambers: comparison of fluid-mechanic experiments with field relations. Contrib. Mineral Petrol. 122, 230–240.CrossRefGoogle Scholar
Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26, 779–792.CrossRefGoogle Scholar
Williamson, N., Armfield, S. W. & Lin, W. 2008 a Direct numerical simulation of turbulent intermediate Froude number fountain flow. ANZIAM J. 50, C16–C30.CrossRefGoogle Scholar
Williamson, N., Srinarayana, N., Armfield, S. W., McBain, G. D. & Lin, W. 2008 b Low-Reynolds-number fountain behaviour. J. Fluid Mech. 608, 297–317.CrossRefGoogle Scholar
Zhang, H. & Baddour, R. E. 1997 Maximum vertical penetration of plane turbulent negatively buoyant jets. J. Engng Mech. 123, 973–977.Google Scholar
Zhang, H. & Baddour, R. E. 1998 Maximum penetration of vertical round dense jets at small and large Froude numbers. J. Hydraul. Engng 124, 550–553.CrossRefGoogle Scholar

Williamson et al. supplementary movie

Movie 1.Visualisation of Re=3494 and Fr=0.4 flow from startup through to quasi-steady flow. Shading indicates non-dimensional scalar concentration φ, from φ=0 (white) to φ=1.0 (black). The surrounding annular re-circulation region is unmixed with ambient fluid and the Kelvin Helmholtz structures flow from the cap region into the intrusion.

Download Williamson et al. supplementary movie(Video)
Video 2.1 MB

Williamson et al. supplementary movie

Movie 2. Visualisation of Re=3494 and Fr=0.97 flow from startup through to quasi-steady flow. Shading indicates non-dimensional scalar concentration φ, from φ=0 (white) to φ=1.0 (black). The surrounding annular re-circulation region is unmixed with ambient fluid and the Kelvin Helmholtz structures flow from the cap region into the intrusion.

Download Williamson et al. supplementary movie(Video)
Video 4.2 MB

Williamson et al. supplementary movie

Movie 3. Visualisation of Re=3494 and Fr=1.4 flow from startup through to quasi-steady flow. Shading indicates non-dimensional scalar concentration φ, from φ=0 (white) to φ=1.0 (black). Ambient fluid is entrained into the annular re-circulation region surrounding the upflow.

Download Williamson et al. supplementary movie(Video)
Video 4.2 MB