Lazy Plumes Research Articles

Richardson and Reynolds number effects on the near field of buoyant plumes: flow statistics and fluxes

The near-field characteristics of highly buoyant plumes, commonly referred to as lazy plumes, remain relatively poorly understood across a range of flow conditions, particularly compared with our understanding of far-field characteristics. Here, we perform fully resolved three-dimensional numerical simulations of round helium plumes to characterize the effects of different inlet Richardson, ${Ri}_0$ , and Reynolds, ${Re}_0$ , numbers on first- and second-order statistical moments as well as average vertical fluxes in the near field. For sufficiently high ${Re}_0$ at a particular ${Ri}_0$ , heavy air can penetrate the core of the plume, reminiscent of spikes in the classical Rayleigh–Taylor instability. In the most turbulent simulation, this penetration becomes so strong that a recirculation zone forms along the centreline of the plume. Vertical fluxes are found to scale linearly with vertical distance from the plume inlet, consistent with experimental and numerical observations (Jiang & Luo, Flow Turbul. Combust., vol. 64, 2000, pp. 43–69; Kaye & Hunt, Intl J. Heat Fluid Flow, vol. 30, 2009, pp. 1099–1105). We analytically derive this linear scaling from the governing equations by making a radial entrainment hypothesis whereby ambient fluid is entrained, on average, only in the radial direction at a finite distance from the inlet. Through this derivation, we identify physical mechanisms that can cause these relationships to remain only approximately valid for the present simulations. Lastly, we identify near-field power-law scaling relations for the flux magnitudes based on ${Ri}_0$ , and also examine vertical profiles of the non-dimensional Richardson number flux. Ultimately, insights from the present simulations are used to define near-, intermediate- and far-field regions in buoyant plumes.

Two-dimensional buoyant plumes in a uniform co-flow

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

Downstream stratification of the buoyant contaminants produced by a highly lazy plume in a longitudinally ventilated tunnel

In an underground tunnel with longitudinal ventilation, the stratification of the buoyant contaminants is crucial for the safety of the workers and passengers. A series of experiments are conducted in a water tank to investigate the density stratification in the downstream of the buoyancy source. Highly lazy plumes are produced by releasing coloured brine solution into a longitudinally ventilated tunnel which is placed in a large reservoir filled with fresh water. The transient distribution of the reduced gravity is measured using a light attenuation technique. The results suggest that the vertical distribution of the reduced gravity in the buoyant layer is approximately linear. The gradient Richardson number demonstrates that the downstream stratification can be very unstable under the effect of forced longitudinal ventilation. The difference between the reduced gravity at the ceiling and that at the floor Δg′, the average reduced gravity g′‾ and the thickness of the buoyant layer h are chosen to quantify the stratification. With the increase in the longitudinal ventilation velocity, both Δg′ and g′‾ decrease but h tends to increase. As the source buoyancy flux increases, all the three parameters become larger. Based on the experimental data and dimensional analyses, quantitative models are proposed to estimate the above three parameters. It is hoped that the models may provide useful references for the downstream distribution of the buoyant contaminants produced by highly lazy plumes in a longitudinally ventilated tunnel.

Analytical solutions and virtual origin corrections for forced, pure and lazy turbulent plumes based on a universal entrainment function

Previous measurements and numerical simulations of buoyant turbulent plumes that develop from area sources provide convincing evidence that entrainment varies locally in response to an imbalance from the asymptotic state of equilibrium, a state referred to as a pure plume. Across the wide spectrum of possible source conditions, that span forced and lazy plume releases, this variation of entrainment has been successfully captured by a single, or universal, description in which the entrainment function $\unicode[STIX]{x1D6FC}$ varies linearly with the local Richardson number. Herein, an analytical solution for the virtual origin of forced, pure and lazy turbulent plumes from circular sources in unstratified environments is derived based on this universal description of entrainment. Prior to this, the analytical solutions reported were limited to those based on the simplifying assumption of invariant entrainment, so-called constant- $\unicode[STIX]{x1D6FC}$ solutions of the plume conservation equations. Analytical solutions for the fluxes of volume and specific momentum are first developed. These solutions highlight the deficit in near-field entrainment in forced plumes and enable the general imbalance from the equilibrium state to be predicted via the streamwise variation of the local Richardson number. Focus then turns to the virtual origin due to the practical benefits that a knowledge of this location offers experimentalists (e.g. in comparing measurement with theory) and theoretical modellers (e.g. in incorporating a turbulent plume within a broader modelling framework).

Entrainment in pulsing plumes

Turbulent axisymmetric lazy plumes produced by discharging saline fluid downwards into a less dense uniform environment from a round pipe are examined experimentally. The plume development is controlled by the source flow rate $$Q_0$$ , momentum $$M_0$$ and buoyancy $$F_0$$ . This study investigated plumes where the source flow rate, momentum and buoyancy are sinusoidal functions of time. The pulsing flow is generated by a programmable ISMATEC gear pump. The maximum frequency f of this pulsing plume is of $$\mathcal {O}(U_0/D)$$ , where $$D/U_0$$ is the eddy turnover time scale at the source, D is the source diameter, and $$U_0$$ is the average velocity at the source. Experiments with pulsing plumes were carried out with a flow rate amplitude A of up to $$80\%$$ and with Strouhal number $$St=fD/U_0$$ ranging from 0.012 to 1.2. The bulk dilution and entrainment measurements were made using the experimental approach of Hunt and Kaye (J Fluid Mech 435:377–396, 2001). Average entrainment is obtained via the integral relationship for Q(z) and M(z) from the model established by Morton et al. (Proc R Soc Lond A 234(1196):1–23, 1956) for continuous sources. The influence of the amplitude and Strouhal number St on the entrainment coefficient $$\alpha$$ is examined, which was found to be very small over the entire range of source conditions considered.

On the effects of buoyancy on higher order moments in lazy plumes

ABSTRACTJets and plumes are shear flows created by momentum and buoyant sources. They can be classified as either pure jets, forced plumes, pure plumes, or lazy plumes. Lazy plumes are characterised as having a high source buoyancy flux relative to the momentum flux. We use direct numerical simulations to simulate lazy plumes to ascertain the effects of increasing plume ‘laziness’ on the higher order moments, such as Reynolds stresses, third order moments and turbulent kinetic energy budgets. The mildly lazy plumes behaved like most plumes in terms of higher order statistics: a self-similar collapse some distance downstream and peak values comparable to previous experiments and simulations. The highly lazy plumes did not show a self-similar collapse and, in most cases, had higher peak values in all moments. The highly lazy plumes had higher levels of turbulence intensities due to the high buoyancy flux, leading to significant buoyancy-induced turbulence production downstream.

On the evolution of the plume function and entrainment in the near-source region of lazy plumes

Plumes occur in many natural and industrial settings, such as chimney smoke, volcanic eruptions and deep water oil spills. A plume function, $\unicode[STIX]{x1D6E4}$, is used to characterize plumes and jets. The far-field behaviour of these flows has been studied in great detail while the near-field behaviour has not quite received the same attention. We examine near-field phenomena such as radial constriction, termed necking, and vortex structure formations with new high resolution direct numerical simulations. Four lazy plumes with increasing values of the source plume parameter, $\unicode[STIX]{x1D6E4}_{0}$, are simulated. We study the evolution of entrainment and the plume function. The original assumptions, that Reynolds stresses dominate viscous shear stresses, do not hold for lazy plumes in the near field. Due to this, a deviation from self-similarity occurs initially and is corrected by a large entrainment coefficient caused by vortex stretching and compression. After correcting for the virtual origin, comparison between theory and simulations shows a monotonic decay of $\unicode[STIX]{x1D6E4}$ towards pure plume behaviour. The entrainment coefficient asymptotes to a widely accepted constant value for plumes.

An entrainment model for lazy turbulent plumes

An entrainment model for lazy turbulent plumes is proposed, the resulting solutions of the plume conservation equations are developed and the implications for plume behaviour are considered and compared with the available experimental data. Indeed, the applicability of the classic solutions of the conservation equations subject to source conditions that produce lazy plumes, i.e. those with suitably high source Richardson number, contains an essential weakness: the underlying assumption of a constant entrainment coefficient. While entrainment models prescribing the dependence of the entrainment coefficient on the local Richardson number have been proposed for forced plumes, corresponding formulations for lazy plumes have not until now been considered. In the context of saline plumes, the model is applied directly. For hot gaseous plumes, we use a modified definition of buoyancy flux to recover a constant buoyancy flux in a non-stratified environment, despite the specific heat varying with the temperature. After a brief review of existing forced-plume formulations of entrainment, a power-law variation is adopted for the lazy plume. The plume equations are solved for the parameter $0\leqslant \unicode[STIX]{x1D714}<1$, where $\unicode[STIX]{x1D714}$ denotes the exponent of the power law. The cases of pure plumes and lazy plumes are then analysed in more detail; to the best of our knowledge this represents the first modelling of variable entrainment for lazy plumes. Specifically, it is shown that classic plume theory is recovered for $\unicode[STIX]{x1D714}=0$, while for $\unicode[STIX]{x1D714}=1/5$ the plume equations may be solved using usual functions (notably polynomials) only. The results of the models for these cases are very similar, which advocates the idea of selecting systematically $\unicode[STIX]{x1D714}=1/5$, instead of $\unicode[STIX]{x1D714}=0$, for cases where the effect of variation of entrainment is weak, since the new model leads to simple calculations. In the case of very lazy plumes, it is shown that, provided that a relevant value of $\unicode[STIX]{x1D714}$ is chosen, the new model reproduces the available experimental results well.

An examination of the high-order statistics of developing jets, lazy and forced plumes at various axial distances from their source

In this study, direct numerical simulations of a turbulent free jet (Re = 2000), a lazy plume (), and a forced plume (Re = 1684, Ri = 0.025) are compared. The evolution of the various fluxes and the so-called source parameter, Γ, are examined as a function of distance from the source. The first-, second-, and third-order statistics of the flows are calculated and discussed. The radial profiles of such statistics, as well as that of the turbulent kinetic energy balance and other second-order transport equations are examined at two axial distances, one axial distance before the flows have adjusted to their similarity solution, and the other beyond the similarity adjustment length scale. Vortical structures are visualised and discussed along with entrainment. The source term Γ was not found to monotonically decrease with axial distance from the source as predicted by past researchers. While the mean flow and turbulent velocity statistics of the simulated lazy and forced plumes took on similar behaviour far from the sources turbulent statistics which involve buoyancy did not.

Two-dimensional planar plumes: non-Boussinesq effects

AbstractIn an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

Large-eddy simulation of starting buoyant jets

A series of Large Eddy Simulations (LES) are performed to investigate the penetration of starting buoyant jets. The LES code is first validated by comparing simulation results with existing experimental data for both steady and starting pure jets and lazy plumes. The centerline decay and the growth rate of the velocity and concentration fields for steady jets and plumes, as well as the simulated transient penetration rate of a starting pure jet and a starting lazy plume, are found to compare well with the experiments. After validation, the LES code is used to study the penetration of starting buoyant jets with three different Reynolds numbers from 2000 to 3000, and with a wide range of buoyancy fluxes from pure jets to lazy plumes. The penetration rate is found to increase with an increasing buoyancy flux. It is also observed that, in the initial Period of Flow Development, the two penetrative mechanisms driven by the initial buoyancy and momentum fluxes are uncoupled; therefore the total penetration rate can be resolved as the linear addition of these two effects. A fitting equation is proposed to predict the penetration rate by combining the two independent mechanisms.

Universal solutions for Boussinesq and non-Boussinesq plumes

Closed-form solutions describing the behaviour of buoyant axisymmetric turbulent rising plumes and fountains, emitted vertically from area sources in unconfined quiescent environments of uniform density, are proposed in a form that is universally applicable to Boussinesq and non-Boussinesq plumes. This paper, thereby, generalizes the results obtained separately for steady Boussinesq and non-Boussinesq plumes, including asymptotic virtual source corrections. The flux balance parameter Γ = Γ(z), a local Richardson number, is instrumental in describing the behaviour of steady plumes and the initial rise behaviour of fountains with heightz. Non-dimensional graphs (cf. the ‘scale diagrams’ of Morton & Middleton,J. Fluid Mech., vol. 58, 1973, pp. 165–176) are plotted, showing certain characteristic heights for different source conditions, characterized by one single source flux balance parameter, giving a unique representation of the behaviour of Boussinesq fountains and both Boussinesq and non-Boussinesq plumes. Finally, a length scale has been identified that characterizes the height over which non-Boussinesq effects are important for lazy plumes rising from area sources.

An experimental study of large area source turbulent plumes

A series of experiments was performed to measure the rate of entrainment of ambient fluid into a negatively buoyant turbulent plume emerging from a large area source with low initial momentum flux, a so called ‘lazy plume’. Immediately below the source, the plume contracts until it reaches a neck of diameter approximately half the source diameter. Beyond the neck it expands, eventually forming a classic pure plume with linear radial growth rate. The variation of volume flow rate, Q, with vertical distance from the source, z, was measured in the near source region, between the source and the neck for different source values of Γ 0, equivalent to the source Richardson number. Experiments were run for 10 5 < Γ 0 < 5 × 10 7. Our results indicate that the volume flux in the plume increases linearly from the source to the neck, that is, Q ∝ z. This scaling is in contrast to pure plumes where the flow rate increases as Q ∝ z 5/3. Our experimental results collapsed onto a single line when scaled, giving an empirical expression for the near source flow rate of the form Q ∝ Γ 0 1 / 3 z . We discuss this result in relation to the classic entrainment model for turbulent plumes and show that modifications to existing closure schemes are required for lazy plumes.

Analytical solutions for turbulent non-Boussinesq plumes

Analytical solutions are developed for non-Boussinesq turbulent plumes rising from horizontal area sources in unconfined quiescent environments of uniform density. The approach adopted follows and extends an earlier approach for Boussinesq plumes and replaces the non-Boussinesq area source of interest and located at z=0 with an idealized point source located at a virtual origin z=z v such that the flow above the idealized source approximates that from the actual source. Asymptotic analytical expressions are developed for the location of the virtual source that are valid for large vertical distances above the non-Boussinesq source. The non-Boussinesq source is characterized by a non-dimensional parameter Γ nb which is a measure of the relative strengths of the mass, momentum and density deficit fluxes at, or at a specified height above, the source. The vertical distance between the actual and virtual sources scales on the length scale l that characterizes the height over which the flow is non-Boussinesq and expressions for z v /l are developed for lazy (Γ nb > 1) and forced plume (Γ ub < 1) sources. For pure-plume source conditions Γ nb = 1, and the virtual source provides an exact representation of the actual plume above z =0. The limiting cases of a nearly pure lazy plume and of a highly lazy plume are also explored analytically

Lazy plumes

We examine the dynamics of turbulent lazy plumes rising from horizontal area sources and from vertically distributed line sources into a quiescent environment of uniform density. First, we consider plumes with internal buoyancy flux gain and, secondly, plumes from horizontal area sources that have significant momentum flux deficits. We re-cast the conservation equations of Morton et al. (1956) for a constant entrainment coefficient with height. We show that near the source there is a region of zero entrainment.

Virtual origin correction for lazy turbulent plumes

The location of the asymptotic virtual origin of positively buoyant turbulent plumes with a deficit of initial momentum flux when compared with equivalent pure plumes is investigated. These lazy plumes are generated by continuous steady releases of momentum, buoyancy and volume into a quiescent uniform environment from horizontal sources (at z = 0) of finite area, and are shown to be equivalent to the far-field flow above point source pure plumes, of buoyancy only, rising from the asymptotic virtual source located below the actual source at z = −zavs.An analytical expression for the location of the asymptotic virtual source relative to the actual source of the lazy plume is developed. The plume conservation equations are solved for the volume flow rate, and the position of the asymptotic virtual origin is deduced from the scaling for the volume flow rate at large distances from the source.The displacement zavs of the asymptotic virtual origin from the actual origin scales on the source diameter and is a function of the source parameter Γ ∝ Qˆ20Fˆ0/Mˆ5/20 which is a measure of the relative importance of the initial fluxes of buoyancy Fˆ0, momentum Mˆ0, and volume Qˆ0 in the plume. The virtual origin correction developed is valid for Γ &gt; 1/2 and is therefore applicable to lazy plumes for which Γ &gt; 1, pure plumes for which Γ = 1, and forced plumes in the range 1/2 &lt; Γ &lt; 1. The dimensionless correction z*avs decreases as Γ increases, and for Γ [Gt ] 1, z*avs → 0.853Γ−1/5. Comparisons made between the predicted location of the asymptotic virtual origin and the location inferred from measurements of lazy saline plumes in the laboratory show close agreement.