Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T07:48:13.851Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of sudden source buoyancy flux increases on turbulent plumes

Published online by Cambridge University Press:  10 September 2009

M. M. SCASE ,
A. J. ASPDEN  and
C. P. CAULFIELD
M. M. SCASE*
Affiliation:
Division of Process and Environmental Engineering, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK
A. J. ASPDEN
Affiliation:
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
C. P. CAULFIELD
Affiliation:
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Building upon the recent experimentally verified modelling of turbulent plumes which are subject to decreases in their source strength (Scase et al., J. Fluid Mech., vol. 563, 2006b, p. 443), we consider the complementary case where the plume's source strength is increased. We consider the effect of increasing the source strength of an established plume and we also compare time-dependent plume model predictions for the behaviour of a starting plume to those of Turner (J. Fluid Mech., vol. 13, 1962, p. 356).

Unlike the decreasing source strength problems considered previously, the relevant solution to the time-dependent plume equations is not a simple similarity solution. However, scaling laws are demonstrated which are shown to be applicable across a large number of orders of magnitude of source strength increase. It is shown that an established plume that is subjected to an increase in its source strength supports a self-similar ‘pulse’ structure propagating upwards. For a point source plume, in pure plume balance, subjected to an increase in the source buoyancy flux F0, the rise height of this pulse in terms of time t scales as t3/4 while the vertical extent of the pulse scales as t1/4. The volume of the pulse is shown to scale as t9/4. For plumes in pure plume balance that emanate from a distributed source it is shown that the same scaling laws apply far from the source, demonstrating an analogous convergence to pure plume balance as that which is well known in steady plumes. These scaling law predictions are compared to implicit large eddy simulations of the buoyancy increase problem and are shown to be in good agreement.

We also compare the predictions of the time-dependent model to a starting plume in the limit where the source buoyancy flux is discontinuously increased from zero. The conventional model for a starting plume is well approximated by a rising turbulent, entraining, buoyant vortex ring which is fed from below by a ‘steady’ plume. However, the time-dependent plume equations have been defined for top-hat profiles assuming only horizontal entrainment. Therefore, this system cannot model either the internal dynamics of the starting plume's head or the extra entrainment of ambient fluid into the head due to the turbulent boundary of the vortex ring-like cap. We show that the lack of entrainment of ambient fluid through the head of the starting plume means that the time-dependent plume equations overestimate the rise height of a starting plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 137 - 169
Copyright
Copyright © Cambridge University Press 2009

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

Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comp. Phys. 142, 1–46.Google Scholar
Almgren, A. S., Bell, J. B. & Crutchfield, W. Y. 2000 Approximate projection methods. Part I. Inviscid analysis. SIAM J. Sci. Comp. 22, 1139–1159.CrossRefGoogle Scholar
Aspden, A. J., Nikiforakis, N., Dalziel, S. B. & Bell, J. B. 2008 Analysis of implicit LES methods. Comm. Appl. Math. Comput. Sci. 3, 103–126.Google Scholar
Bell, J. B., Colella, P. & Howell, L. H. 1991 An efficient second-order projection method for viscous incompressible flow. In 10th A.I.A.A. Computational Fluid Dynamics Conference, Honolulu, US.Google Scholar
Boris, J. P. 1990 On large eddy simulation using subgrid turbulence models. Comment 1. In Lecture notes in Physics (ed. Lumley, J. L.), vol. 357, pp. 344–353. Springer Verlag.Google Scholar
Boris, J. P., Grinstein, F. F., Oran, E. S. & Kolbe, R. L. 1992 New insights into large eddy simulation. Fluid Dyn. Res. 10, 199–229.Google Scholar
Caulfield, C. P. 1991 Stratification and buoyancy in geophysical flows. PhD thesis, University of Cambridge, UK.Google Scholar
Caulfield, C. P. & Woods, A. W. 1995 Plumes with non-monotonic mixing behaviour. Geophys. Astrophys. Fluid Dyn. 79, 173–199.Google Scholar
Colella, P. 1985 A direct Eulerian MUSCL scheme for gasdynamics. SIAM J. Sci. Stat. Comp. 6, 104–117.Google Scholar
Colella, P. 1990 A multidimensional second order Godunov scheme for conservation laws. J. Comp. Phys. 87, 171–200.Google Scholar
Delichatsios, M. A. 1979 Time similarity analysis of unsteady buoyant plumes in neutral surroundings. J. Fluid Mech. 93, 241–250.Google Scholar
Drikakis, D., Fuerby, C. Grinstein, F. F. & Youngs, D. L. 2007 Simulation of transition and turbulence decay in the Taylor–Green vortex. J. Turbul. 8, 1–12.Google Scholar
Fureby, C. & Grinstein, F. F. 1999 Monotonically integrated large eddy simulations of free shear flows. AIAA J. 37, 544–556.Google Scholar
Grinstein, F. F., Margolin, L. G. & Rider, W. J. 2007 Implicit Large Eddy Simulation. Cambridge University Press.Google Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. A 185, 213–245.Google Scholar
Hunt, G. R. & Kaye, N. B. 2001 Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435, 377–396.CrossRefGoogle Scholar
Hunt, J. C. R., Vrieling, A. J., Nieuwstadt, F. T. M. & Fernando, H. J. S. 2003 The influence of the thermal diffusivity of the lower boundary on eddy motion in convection. J. Fluid Mech. 491, 183–205.CrossRefGoogle Scholar
Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–376.Google Scholar
Levine, J. 1959 Spherical vortex theory of bubble-like motion in cumulus clouds. J. Meteor. 16, 653–662.Google Scholar
Margolin, L. G., Rider, W. J. & Grinstein, F. F. 2006 Modeling turbulent flow with implicit LES. J. Turbul. 7, 1–27.Google Scholar
Middleton, J. H. 1975 The asymptotic behaviour of a starting plume. J. Fluid Mech. 72, 753–771.Google Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–32.Google Scholar
Oran, E. S. & Boris, J. P. 1993 Computing turbulent shear flows – a convenient conspiracy. Comp. Phys. 7, 523–533.Google Scholar
Porter, D. H., Pouquet, A. & Woodward, P. R. 1992 Three-dimensional supersonic homogeneous turbulent: a numberical study. Phys. Rev. Lett. 68, 3156.Google Scholar
Ricou, F. P. & Spalding, D. B. 1961 Measurements of entrainment by axisymmetrical turbulent jets. J. Fluid Mech. 8, 21–32.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2006 a Boussinesq plumes with decreasing source strengths in stratified environments. J. Fluid Mech. 563, 463–472.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2008 Temporal variation of non-ideal plumes with sudden reductions in buoyancy flux. J. Fluid Mech. 600, 181–199.Google Scholar
Scase, M. M., Caulfield, C. P., Dalziel, S. B. & Hunt, J. C. R. 2006 b Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech. 563, 443–461.Google Scholar
Scase, M. M., Caulfield, C. P., Dalziel, S. B. & Hunt, J. C. R. 2006 c Plumes and jets with time-dependent sources in stratified and unstratified environments. In Proceedings of 6th International Symposium on Stratified Flows (ed. Ivey, G. N.), University of Western Australia, Perth, Australia, pp. 112–117.Google Scholar
Scase, M. M., Caulfield, C. P., Linden, P. F. & Dalziel, S. B. 2007 Local implications for self-similar turbulent plume models. J. Fluid Mech. 575, 257–265.Google Scholar
Scorer, R. S. 1954 The nature of convection as revealed by soaring birds and dragonflies. Q. J. R. Met. Soc. 80, 68–77.Google Scholar
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. A 239, 61–75.Google Scholar
Turner, J. S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356–368.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy effects in fluids. Cambridge University Press.Google Scholar
Youngs, D. L. 1991 Three-dimensional numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Phys. Fluids A 4, 1312–1320.Google Scholar
Zeldovich, Y. B. 1937 The asymptotic laws of freely-ascending convective flows. Zhur. Eksper. Teor. Fiz. 7, 1463–1465 (in Russian). English translation In Selected Works of Yakov Borisovich Zeldovich, 1992 (ed. J. P. Ostriker), vol. 1, pp. 82–85. Princeton University Press.Google Scholar