The effect of sudden source buoyancy flux increases on turbulent plumes

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 F 0 , the rise height of this pulse in terms of time t scales as t 3/4 while the vertical extent of the pulse scales as t 1/4 . The volume of the pulse is shown to scale as t 9/4 . 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 † Email address for correspondence: matthew.scase@nottingham.ac.uk 138 M. M. Scase, A. J. Aspden and C. P. Caulfield plume with time. However, by modifying the entrainment coefficient appropriately, we see that realistic predictions consistent with experiment can be attained.

Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets

“…Indeed, their governing integral equations are consistent with the view that lateral slices of the jet or plume do not interact longitudinally (Scase, Aspden & Caulfield 2009). However, possible sources of longitudinal mixing such as turbulence and lateral gradients of mean velocity (causing dispersion) are evident in numerous experimental studies of steady jets and plumes (e.g.…”

“…In addition, numerical simulations allow one to control the source conditions arbitrarily and therefore investigate a broad range of problems or, alternatively, a canonical case that is difficult to realise in a laboratory. In Scase et al (2009) the unsteady plume theory of Scase et al (2006) was able to successfully predict the scaling of a 'pulse' in the plume's radius that was observed in implicit large-eddy simulations of a plume, whose buoyancy flux was suddenly increased. However, the observed longitudinal scaling of the pulse and effects relating to longitudinal mixing were not predicted by the theoretical model and these remain issues warranting further attention.…”

Generalised unsteady plume theory

“…The similarity scaling employed to derive (6.22), and the observation of self-similarity in figure 12, support the view that the length scales in the plume vary according to t 3/4 . This scaling applies to both the position of a disturbance and its longitudinal extent and therefore provides some resolution of the open question debated in Scase et al (2009), as to the longitudinal scaling of a propagating pulse structure in a plume. However, we note that our observations are based on results from a domain of relatively limited longitudinal extent.…”

“…To date, the assumption of top-hat velocity profiles and the omission of turbulence has resulted in the unsteady plume equations being regarded as a parabolic system (Scase et al 2009). However, use of the generalised framework described in § 2 reveals that, in general, the unsteady plume equations comprise a hyperbolic system, even when higher-order turbulent transport terms are neglected from (2.6) and (2.8) (resulting in β g = β m and γ g = γ m ).…”

Self-similar properties of decelerating turbulent jets

“…Validation of the various self-similarity-based statistically unsteady turbulent jet and plume models in Scase, Caulfield & Dalziel (2008), Musculus (2009), Scase, Aspden & Caulfield (2009) and Craske & van Reeuwijk (2015b) indicates that assumption of self-similar mean axial velocity profiles provides a useful basis for development of models in statistically unsteady jets; however, the underlying assumption that self-similarity persists as the jet decelerates has not been examined directly. On the contrary, the measurements by Borée et al (1996) indicate that the radial profiles of phase-averaged axial velocity and other velocity moments deviate from the self-similar profiles of a statistically steady jet as the jet decelerates, and that the profiles vary axially through the confined deceleration region (Borée et al 1996(Borée et al , 1997.…”