Hostname: page-component-f7d5f74f5-g2fc4 Total loading time: 0 Render date: 2023-10-03T03:37:24.899Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Straight-sided solutions to classical and modified plume flux equations

Published online by Cambridge University Press:  20 June 2011

Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA
Division of Process and Environmental Engineering, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK
Email address for correspondence:


The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23) is re-cast in terms of the non-dimensional plume radius, the plume ‘laziness’ defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339–358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248). We are able to explain the unstable behaviour previously found when the power lies in the range (−4, −8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.

Copyright © Cambridge University Press 2011

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.)



Baines, W. D. 1983 A technique for the direct measurement of volume flux of a plume. J. Fluid Mech. 132, 247256.CrossRefGoogle Scholar
Baines, W. D., Turner, J. S. & Campbell, I. H. 1990 Turbulent fountains in an open chamber. J. Fluid Mech. 212, 557592.CrossRefGoogle Scholar
Basu, A. J. & Narasimha, R. 1999 Direct numerical simulation of turbulent flows with cloud-like off-source heating. J. Fluid Mech. 385, 199228.CrossRefGoogle Scholar
Batchelor, G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Meteorol. Soc. 80, 339358.CrossRefGoogle Scholar
Bhat, G. S. & Narasimha, R. 1996 A volumetrically heated jet: large-eddy structure and entrainment characteristics. J. Fluid Mech. 325, 303330.CrossRefGoogle Scholar
Campbell, A. & Cardoso, S. 2010 Turbulent plumes with internal generation of buoyancy by chemical reaction. J. Fluid Mech. 655, 122151.CrossRefGoogle Scholar
Cardoso, S. S. & McHugh, S. T. 2010 Turbulent plumes with heterogeneous chemical reaction on the surface of small buoyant droplets. J. Fluid Mech. 642, 4977.CrossRefGoogle Scholar
Caulfield, C. P. & Woods, A. W. 1995 Plumes with non-monotonic mixing behaviour. Geophys. Astrophys. Fluid Dyn. 79, 173199.CrossRefGoogle Scholar
Caulfield, C. P. & Woods, A. W. 1998 Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech. 360, 229248.CrossRefGoogle Scholar
Conroy, D. T. & Llewellyn Smith, S. G. 2008 Endothermic and exothermic chemically reacting plumes. J. Fluid Mech. 612, 291310.CrossRefGoogle Scholar
Conroy, D. T., Smith, S. G. L. & Caulfield, C. P. 2005 Evolution of a chemically reacting plume in a ventilated room. J. Fluid Mech. 537, 221253.CrossRefGoogle Scholar
Diez, F. J. & Dahm, W. J. A. 2007 Effects of heat release on turbulent shear flows. Part 3. Buoyancy effects due to heat release in jets and plumes. J. Fluid Mech. 575, 221255.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329338.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. G. 2001 Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435, 377396.CrossRefGoogle Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319328.CrossRefGoogle Scholar
Mather, T. A., Harrison, R. G., Tsanev, V. I., Pyle, D. M., Karumudi, M. L., Bennett, A. J., Sawyer, G. M. & Highwood, E. J. 2007 Observations of the plume generated by the December 2005 oil depot explosions and prolonged fire at Buncefield (Hertfordshire, UK) and associated atmospheric changes. Proc. R. Soc. Lond. A 463 (2081), 11531177.CrossRefGoogle Scholar
Morton, B. R. 1959 Forced plumes. J. Fluid Mech. 5, 151163.CrossRefGoogle Scholar
Morton, B. R. & Middleton, J. 1973 Scale diagrams for forced plumes. J. Fluid Mech. 58, 165176.CrossRefGoogle 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, 123.CrossRefGoogle Scholar
Priestly, C. H. B. & Ball, F. K. 1955 Continuous convection from an isolated source of heat. Q. J. R. Meteorol. Soc. 81, 144157.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2006 a Boussinesq plumes and jets with decreasing source strengths in stratified environments. J. Fluid Mech. 563, 463472.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, 181199.CrossRefGoogle 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, 443461.CrossRefGoogle Scholar