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Plumes with time dependent source conditions in a uniformly<br />
stratified environment<br />
M. M. Scase ∗ , C. P. Caulfield †∗ ,S. B. Dalziel ∗ and J. C. R. Hunt ‡<br />
Motivated by Hunt 1 et al., the classical bulk models for isolated jets and plumes<br />
due to Morton, Taylor & Turner 2 are generalised to allow for time dependence in the<br />
various fluxes driving the flow. These new systems model the spatio-temporal evolution<br />
of both Boussinesq and non-Boussinesq jets and plumes in uniformly stratified<br />
fluids. Separable time-dependent similarity solutions for plumes and jets are found.<br />
These similarity solutions are characterized by having time-independent plume orjet<br />
radii, with appreciably smaller spreading angles than either constant source buoyancy<br />
flux pure plumes or constant source momentum flux pure jets.<br />
If the source buoyancy flux (for a plume) or source momentum flux (for a jet)<br />
is decreased generically from an initial to a final value, numerical solutions of the<br />
governing equations exhibit three qualitatively different regions of behaviour. The<br />
upper region remains largely unaffected by the change in buoyancy flux or momentum<br />
flux at the source. The lower region is an effectively steady plume or jet based on the<br />
final (lower) buoyancy flux or momentum flux. The intermediate transition region,<br />
in which the plume or jet adjusts between the states in the lower and upper regions<br />
appears to converge very closely to the newly identified stable similarity solutions. In<br />
figure 1, we consider an unstratified ambient fluid. Figure 1(a) shows the well-known<br />
conical plume shape established by Morton et al. 2 . In figure 1(b) we show the shape<br />
of a plume which has been convecting steadily for some time and then has its source<br />
buoyancy flux rapidly reduced. The spreading angles and velocities of the plumes are<br />
considered with a view to predicting pinch-off.<br />
∗DAMTP, University of Cambridge, CMS, Wilberforce Road, Cambridge CB3 0WA, UK.<br />
† BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK.<br />
‡ CPOM, University College London, London WC1E 6BT, UK.<br />
1Hunt, J. C. R., Vrieling, A. J., Nieuwstadt, F. T.M. &Fernando, H. J.S.,J. Fluid Mech. 491,<br />
183–205 (2003).<br />
2Morton, B. R., Taylor, G. I. &Turner,J.S.,Proc. Roy. Soc. Lon. A 234, 1–32 (1956).<br />
(a) (b)<br />
Figure 1: (a) The ‘well-known’ steady Boussinesq plume shape. (b) Boussinesq plume<br />
that at some time t0 has had its source buoyancy flux reduced.<br />
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