Time-dependent plumes and jets with decreasing source strengths

Abstract

The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) is generalized to allow for time-dependence in the various fluxes driving the flow. This new system models the spatio-temporal evolution of jets in a homogeneous ambient fluid and Boussinesq and non-Boussinesq plumes in stratified and unstratified ambient fluids.Separable time-dependent similarity solutions for plumes and jets are found in an unstratified ambient fluid, and proved to be linearly stable to perturbations propagating at the velocity of the ascending plume fluid. These similarity solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles ($\tan^{-1}(2\alpha/3)$) than either constant-source-buoyancy-flux pure plumes (with spreading angle $\tan^{-1}(6\alpha/5)$) or constant-source-momentum-flux pure jets (with spreading angle $\tan^{-1}(2\alpha)$), where $\alpha$ is the conventional entrainment coefficient. These new similarity solutions are closely related to the similarity solutions identified by Batchelor (Q. J. R. Met. Soc., vol. 80, 1954, p. 339) in a statically unstable ambient, in particular those associated with a linear increase in ambient density with height.If the source buoyancy flux (for a rising plume) or source momentum flux (for a rising jet) is decreased generically from an initial to a final value, numerical solutions of the governing equations exhibit three qualitatively different regions of behaviour. The upper region, furthest from the source, remains largely unaffected by the change in buoyancy flux or momentum flux at the source. The lower region, closest to the source, is an effectively steady plume or jet based on the final (lower) buoyancy flux or momentum flux. The transitional region, in which the plume or jet adjusts between the states in the lower and upper regions, appears to converge very closely to the newly identified stable similarity solutions. Significantly, the predicted narrowing of the plume or jet is observed. The size of the narrowing region can be determined from the source conditions of the plume or jet. Minimum narrowing widths are considered with a view to predicting pinch-off into rising thermals or puffs.

Generalised unsteady plume theory

“…Note that the radius of the plume is independent of the dimensionless buoyancy flux θ g (inclusive of turbulent transport) when θ m = 1. When θ g = 1 (A 6b) and (A 7) are identical to the 'top-hat' solutions obtained by Scase et al (2006b). Consistent with the analysis of § 5.2, (A 8) indicates that the spreading rate of the unsteady Gaussian plume is identical to that associated with the steady state.…”

“…First is the factor 1/γ g preceding ∂ t (Q 2 /M), which, by implying an effective area Q 2 /(Mγ g ), provides a definition of the plume edge for continuous velocity profiles. Reassuringly, for top-hat profiles γ g = 1, which makes the left-hand side of (2.20) consistent with the top-hat model of Scase et al (2006b). Second, is the entrainment coefficient α, which is not necessarily constant.…”

“…The top-hat unsteady plume model (see, e.g. Scase et al 2006b) is a degenerate case of the generalised unsteady plume equations, which, in general, describe a hyperbolic system with three distinct characteristic curves. For physically realistic assumptions about the underlying velocity field, buoyancy distribution and turbulence, the generalised unsteady plume equations are well posed.…”

“…The solutions show that for Gaussian plumes the entrainment coefficient increases, relative to its steady-state value α 0 , in such a way that the plume's radius retains its steady-state dependence on z and is independent of time. Conversely, in plumes with a top-hat velocity profile, the solutions predict that the entrainment coefficient is always equal to α 0 and that the spreading rate of the plume decreases relative to that associated with the steady state due to the time dependence of the source conditions, the latter result being first obtained by Scase et al (2006b).…”

“…In all but the simplest cases (e.g. the rigorously derived top-hat model of Scase et al (2006b)), such models have questionable foundations, because typically, ensemble-averaged quantities in plumes do not have a well-defined edge and depend continuously on the radial coordinate. The use of a generalised plume theory, resulting in (2.9)-(2.11), circumvents these issues altogether and allows one to understand unsteady plume models in a broader context.…”