2. Experimental set-up and parameters

We measured velocities and temperatures using PIV and thermocouples, respectively, in thermal air plumes in a windowless, thermally insulated enclosure.

The experimental set-up is shown in figure 1. Air was fed from a compressor to a mass flow-rate meter, where the mass flux was monitored and controlled by an electronic feedback system. A small fraction of the air ( ${\approx}5\,\%$ by volume) passed via a chamber where it was seeded with incense particles. The seeded air rejoined the unseeded air immediately upstream of the heating element. The heated air then passed through a diverging–converging section, over a turbulence grid and finally through a diaphragm opening. Air temperature was monitored continuously at the diaphragm to ensure steady source conditions. The ambient air was seeded using a stage smoke generator, which filled the whole enclosure with approximately spherical $1~{\rm\mu}\text{m}$ polyethylene glycol particles. Over a 10 min period, the release of smoke spread to fill the enclosure uniformly. When the ambient air motion induced by the initial jet of smoke dissipated, acquisition of data could begin. Approximately one smoke injection per hour seeded the ambient sufficiently. Seeding both plume and ambient air was necessary in order to obtain proper velocity statistics, statistics which would otherwise be biased when solely seeding the plume.

Temperature measurements were made with a horizontal rake of thermocouples spaced at 10 mm intervals. The uncertainty associated with these measurements was estimated as $\pm 0.5~\text{K}$ . The rake was sequentially displaced vertically at increments of 10 mm, from $z=12$ to $z=512~\text{mm}$ . Temperatures were measured for 3 min at 10 Hz.

We acquired and processed PIV data using LaVision’s DaVis 7.2 software. Circular interrogation areas with a 50 % overlap were employed, resulting in a spatial resolution of 0.7 mm. To achieve this resolution, the measurement plane was split into various fields of view with 3000 image pairs acquired sequentially for each field. In order to record sufficiently large particle images to allow for PIV processing at $16\times 16~\text{pixels}$ and to minimise peak-locking, we split the acquisition into 8 adjacent fields of view measuring approximately 150 mm (horizontal) $\times$ 100 mm (vertical) with a 15 mm vertical overlap between successive fields to ensure continuity of data. The camera could not be moved further than 1.5 m from the laser plane to prevent peak-locking from becoming significant. Subpixel accuracy of the processing algorithm was thus maintained. Measurements on plumes at a spatial resolution of less than 1 mm have seldom been undertaken and those presented herein represent a resource for numerical practitioners, experimentalists and theoreticians alike. Velocity measurements were made at a frequency of 4 Hz, and the duration of each acquisition was 12.5 min. Dynamic statistics were obtained from the 3000 instantaneous measurements.

In order to compare the spatial and temporal resolutions of our measurements with characteristic (turbulent) time and length scales of the flows, estimates of a typical length scale $\ell$ and velocity scale ${\it\upsilon}$ , the former approximately equal to the plume radius and the latter to the standard deviation of the vertical velocities, were computed. In the flows analysed here, $\ell$ varied from a minimum of approximately 10 mm in the near field of release J to a maximum of 75 mm in the far field of release P (see table 1), whereas ${\it\upsilon}$ is of the order $1~\text{m}~\text{s}^{-1}$ in release J and $0.1~\text{m}~\text{s}^{-1}$ in release P. From these we obtained typical time scales ${\it\tau}\sim \ell /{\it\upsilon}$ of the order $10^{-2}~\text{s}$ for release J and $10^{-1}~\text{s}$ for release P. The temporal resolution of the PIV and thermocouple measurements are therefore slightly larger than ${\it\tau}$ and insufficient to allow the computation of turbulent velocity and temperature spectra. The spatial resolution of the PIV measurements is at least one order of magnitude lower than $\ell$ allowing for a detailed description of the velocity field structure. Conversely, for thermocouple measurements the spatial resolution is coarser and approaches $\ell$ in the near field of release J.

Table 1.

The three source conditions of the releases investigated giving the plume Richardson number ${\it\Gamma}_{0}=\{0.001,0.034,0.96\}$ ; note that the value ${\it\alpha}_{ref}=0.1$ has been assumed when calculating ${\it\Gamma}_{0}$ from (1.1). The source buoyancy is $g_{0}^{\prime }=g({\rm\Delta}T_{0}/T_{0})$ . Here J refers to jet-like, F to forced plume and P to pure plume. We use $z_{B}$ to denote the non-Boussinesq length scale (2.1),  $L_{M}=\mathscr{C}\cdot M_{0}^{3/4}/B_{0}^{1/2}$ to denote the jet-length, with $\mathscr{C}=(5/9{\it\alpha}_{ref})^{1/2}(2/{\rm\pi})^{1/4}$ . The final two columns report, respectively, the limiting measurement heights, $z_{max}$ , scaled on the source radius, $b_{0}$ , and on the jet length, $L_{M}$ .

The injected air velocity was kept as high as practicable in order to maximise the Reynolds number and reduce the influence of background disturbances. A nozzle was selected with a maximum radius of 2.5 cm. With the 2 kW coiled heating element, a maximum temperature difference of ${\rm\Delta}T_{0}=T_{0}-T_{e}\approx 90{-}100~\text{K}$ ( $T_{e}$ is the ambient air temperature) was possible and this enabled us to achieve a relatively high source buoyancy flux at the lower velocities we considered. Table 1 summarises the three different releases studied, where the letters J, F and P refer to jet-like, forced and pure plume releases, respectively. Measurements were made over a vertical extent ranging from 1 to 70 cm above the source. The corresponding non-dimensional heights are listed in the final two columns of table 1.

For our set-up, the Reynolds number at the source ( $\mathit{Re}_{0}=(w_{0}b_{0})/{\it\nu}$ , $w_{0}$ denoting the velocity at the source and ${\it\nu}$ the air kinematic viscosity) decreases with increasing ${\it\Gamma}_{0}$ . The higher source Richardson number (table 1) of the pure plume required relatively low flow velocities so that $\mathit{Re}_{0}\approx 1000$ , implying that the near-source flow field is not fully turbulent (we return to this in § 3). The temperature at the source reached 393 K, producing flow conditions that are nominally beyond the limits of the Boussinesq approximation. Woods (Reference Woods1997) suggested a length scale over which non-Boussinesq effects are significant as

An estimate of this length scale for our experiments leads to $z_{B}\sim 15~\text{cm}$ . A non-negligible part of the domain could therefore be affected by non-Boussinesq effects on the flow dynamics.

The experimental apparatus was conceived to produce buoyant turbulent plumes within an unstratified quiescent environment. We however observed two main features inducing non-negligible departures from these ideal conditions. The first concerned the diffusion of heat along the horizontal rigid wooden base plate within which the nozzle was mounted (figure 1). Heat transferred from this plate (at $z=0$ ) resulted in a region, of approximately 5 % of the vertical extent of the domain, where the ambient air temperatures could not be considered to be uniform. Figure 2 shows the mean temperatures in the ambient measured over the course of the three experiments using a rack of thermocouples. From these we estimate the temperature gradient to be $1~\text{K}~\text{cm}^{-1}$ over a layer approximately 3 cm thick immediately above the plate. The second concerned the ambient air which was not perfectly still. Background air motion was unavoidable for a number of reasons: the release induces a flow within the confines of the test room as it was necessary to seed the environment with smoke. Mean air velocities in the background, as estimated by PIV measurements, never exceeded 10 % (for releases J and F this percentage was far lower) of the mean centreline plume velocity $w_{m}$ and the standard deviation of this fluctuation about this mean was always less than $0.1w_{m}$ : equivalent to an actual velocity of $7~\text{cm}~\text{s}^{-1}$ . These intensities of background motion are directly comparable with those in previous plume studies.

Figure 2.

Vertical profiles of ambient temperature $T$ (degrees centigrade) registered in experiments for releases (a) J, (b) F and (c) P.

3. Mean flow and temperature field

First, we focus on the evolution of mean velocity and buoyancy (temperature). We examine the radial evolution of the vertical component of the mean velocity and of the mean temperature. Whilst over 800 velocity profiles were gathered for each release, plotting all of the data collected was unhelpful in explaining the trends; for this reason each plot shows 13 radial profiles equally spaced in height and spanning the entire vertical extent of each experiment. See table 2 for heights and symbols used in these plots. Mean temperature profiles are given at these same distances from the source. We discuss the reliability of the assumption of Gaussian profiles (§§ 3.1 and 3.2), profiles which allow us to readily identify characteristic scales for the plume width, velocity and buoyancy with height. The analysis of the variation with height of these local scales (§§ 3.3.1 and 3.3.2) provides first evidence of the dynamical behaviour of the three releases. To discuss this further, we compute the vertical evolution of the plume Richardson number ${\it\Gamma}(z)$ by integrating the radial profiles of mean vertical velocity and temperature (§ 3.3.3). Finally, we discuss the implication of self-similarity of radial profiles in light of the concept of ‘local self-similarity’ proposed by George et al. (Reference George, Alpert and Tamanini1977).

Table 2.

Non-dimensional profile heights $z_{b}=z/b_{0}$ plotted in figures 3, 4, 9, 10 and 15.

4. Turbulence

The aim here is to use turbulent statistics of the velocity data to illuminate the discussion that follows in § 5, particularly with regards to the (non-constant) entrainment coefficient.

5. Entrainment coefficient

Finally, we focus on the rate of entrainment. Our aim is twofold. First, we aim to quantify the apparent differential entrainment coefficient evidenced by the vertical profiles of centreline buoyancy examined in § 3.3.2. Second, in light of the analysis of the flow structure performed so far, we aim to shed light on the dynamical causes of this variation.

The entrainment coefficient ${\it\alpha}_{G}=u_{e}/w_{m}$ is defined as the ratio of two velocities, the entrainment velocity $u_{e}$ and the mean centreline vertical velocity, and represents the simplest way to close the volume flux conservation equation:

Although widely and successfully applied to many problems of practical interest (Turner Reference Turner1986), this form of turbulence closure that attempts to capture the turbulent process, by which ambient fluid is entrained across the shear layers forming the plume boundary and into the plume, is nonetheless a rather crude model of a complex physical phenomenon. Many researchers have attempted to infer the entrainment coefficient from experimental observations (Turner Reference Turner1986) as it is central to plume theory. A major problem in the experimental estimates of ${\it\alpha}_{G}$ reported in the literature, for both jets and plumes, is the consistent scatter in the data. A comprehensive review of the estimates of the entrainment coefficient provided by previous authors is presented by Linden (Reference Linden, Batchelor, Moffatt and Worster2000) and by Carazzo, Kaminski & Tait (Reference Carazzo, Kaminski and Tait2006). For Gaussian profiles, the entrainment coefficient is in the range $0.045<{\it\alpha}_{j}<0.056$ in pure momentum-dominated jets and $0.07<{\it\alpha}_{p}<0.11$ in pure buoyancy-dominated plumes (Carazzo et al. Reference Carazzo, Kaminski and Tait2006). Whilst this variability remains only partially explained it can, in part, be attributed to different conditions at the source, the nozzle geometry, or to the different experimental techniques deployed. In addition to this uncertainty, however, we can expect that buoyant plumes produce different entrainment rates according to their local dynamical condition. Assuming fully-turbulent plumes with relatively low density differences, we may therefore expect the dynamics of the entrainment process to depend on the plume Richardson number ${\it\Gamma}$ . Previous authors have tried to define the functional dependence of ${\it\alpha}_{G}$ on ${\it\Gamma}$ , or to equivalent non-dimensional parameters (the Froude number). Among these, we cite the semi-empirical model derived from the theoretical analysis of Priestley & Ball (Reference Priestley and Ball1955)

which assumes a linear variation between two asymptotic values (determined experimentally) for a pure jet ${\it\alpha}_{j}$ and a pure plume ${\it\alpha}_{p}$ . Then 50 years later Kaminski et al. (Reference Kaminski, Tait and Carazzo2005), following Priestley & Ball (Reference Priestley and Ball1955), proposed a formulation of the entrainment coefficient based on a mean kinetic energy budget, and that includes explicitly the direct contribution of the variation of ${\it\Gamma}$ on ${\it\alpha}_{G}$ .

The entrainment coefficient is computed herein from the PIV velocity estimates of the mean vertical volume flux $Q(z)$ . To avoid scatter in the data due to spatial discretisation, prior to determining its derivative, the volume flux variation with height was fitted by means of a sixth-order polynomial, and thereafter ${\it\alpha}_{G}$ estimated from (5.1). Results are shown in figure 16, where we plot the entrainment coefficient for the three releases as a function of ${\it\Gamma}$ . We have excluded only the data of release P for $z/b_{0}\leqslant 5$ , data that was highly affected by the non-fully turbulent condition of the flow, as widely discussed in §§ 3.3.1, 3.3.3 and 4.

Figure 16.

Entrainment coefficient ${\it\alpha}_{G}$ plotted against local plume Richardson number ${\it\Gamma}$ . Experimental data of the present study refers to release J (○), F ( $+$ ) and P ( $\Box$ ). The grey shaded area represents estimates based on the Priestley & Ball model (5.2) assuming an upper and lower bound provided by maximal and minimal entrainment values for jets and plumes from the literature data (see the text). Experimental data from Wang & Law (Reference Wang and Law2002) is referred to as W&L.

Results are compared with the model of Priestley & Ball (Reference Priestley and Ball1955) discussed above and with experimental results reported by Wang & Law (Reference Wang and Law2002). To the best of the authors’ knowledge, Wang & Law (Reference Wang and Law2002) provide the only experimental estimate of the entrainment coefficient as a function of the local Richardson number in the literature. The data of Wang & Law (Reference Wang and Law2002) was originally plotted against a non-dimensional parameter, denoted here as $\mathit{Fr}$ , which is related to the plume Richardson number via ${\it\Gamma}=\mathit{Fr}^{2}(5/(2^{7/2}\sqrt{{\rm\pi}}{\it\alpha}_{ref}))$ . Wang & Law (Reference Wang and Law2002) estimated $\mathit{Fr}$ , taking also into account the turbulent fluxes of buoyancy and momentum, which are neglected in (1.1). Since the turbulent fluxes constitute approximately 20 % of the total buoyancy flux and 10 % of the total momentum flux, neglecting these leads to overestimates of ${\it\Gamma}$ of approximately 5 %. The outcome of the Priestley & Ball (Reference Priestley and Ball1955) model is plotted assuming the two limits for both asymptotic values of ${\it\alpha}_{G}$ identified by Carazzo et al. (Reference Carazzo, Kaminski and Tait2006), i.e.  ${\it\alpha}_{j}=0.045$ and ${\it\alpha}_{j}=0.056$ for ${\it\Gamma}\rightarrow 0$ , and ${\it\alpha}_{p}=0.07$ and ${\it\alpha}_{p}=0.11$ for ${\it\Gamma}\rightarrow 1$ .

As a general remark we note that our estimates show a clear tendency for ${\it\alpha}_{G}$ to increase with ${\it\Gamma}$ . For the condition of a ‘highly forced plume’, ${\it\Gamma}\rightarrow 0$ , our ${\it\alpha}_{G}$ tends to be slightly lower than that in the literature for jets. For increasing ${\it\Gamma}$ , our results show generally good agreement with the lower bound defined by the semi-empirical model of Priestley & Ball (Reference Priestley and Ball1955). Across the whole range of forced plume conditions $0<{\it\Gamma}<1$ , our estimates are systematically lower than those provided by Wang & Law (Reference Wang and Law2002). This discrepancy is significantly reduced for pure plume conditions. In addition, our estimates show good agreement with the other literature on pure plumes, in particular the PIV estimates by Pham et al. (Reference Pham, Plourde and Kim2005) within a thermal plume rising above a heated plate.

To help explain the differences between our estimates of ${\it\alpha}_{G}$ and the estimates of Wang & Law (Reference Wang and Law2002), as well as the differences in ${\it\alpha}_{G}$ in our own data at a given value of ${\it\Gamma}$ (i.e. for releases J and F for $0.06<{\it\Gamma}<0.2$ ) we turn to the expression developed by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) for the entrainment coefficient. Following recent developments proposed by Craske & van Reeuwijk (Reference Craske and van Reeuwijk2014) for the analysis of unsteady jets, we adopt a formulation that makes no assumption about the slenderness of the flow. In particular, this allows us to quantify the behaviour of the entrainment in the very near field, where the usual assumption of ‘thin plume’ (negligible vertical gradients of second-order statistics compared with radial gradients) does not necessarily hold. Assuming that the radial profiles of velocity and buoyancy are well approximated by a Gaussian, even close to the source (as we verified in §§ 3.1 and 3.2), and adopting a simple gradient-law closure to model the Reynolds stress, the entrainment coefficient can be expressed as

where

and where $I_{6}$ and $I_{7}$ are related to integrals of radial profiles of second-order velocity statistics. Details on the derivation of (5.3) are provided in the appendix A.

The formulation of the entrainment coefficient (5.3) helps to clarify the role of the different terms in the entrainment process and their physical meaning. The first term $(2{\it\varphi}^{2}-1)(2{\it\alpha}_{ref}/5){\it\Gamma}$ reflects the effect of the radial gradient of hydrostatic pressure, induced by the presence of a column of warm air, in drawing ambient air into the plume. This term is therefore related to the mean radial velocity field. The second term $(3/2)\langle \widehat{{\it\nu}_{T}}\rangle$ is directly linked to the local production of t.k.e. by inertial instabilities (Kaminski et al. Reference Kaminski, Tait and Carazzo2005), i.e. to the product of Reynolds stress and mean vertical velocity radial gradient. It is therefore related to the fluctuating component of the velocity field. The remaining two terms, originally neglected by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005), are both related to the t.k.e. production and to the vertical mean kinetic energy transfer (and thereby related to the vertical gradient of the first- and second-order velocity statistics). The term ${\it\alpha}_{nf}$ tends to zero in the far field, as the second-order velocity statistics attain self-similarity and therefore plays a role only in the very near field. The term ${\it\alpha}_{m}$ tends to zero in the far field only in pure jets, ${\it\Gamma}=0$ , as the mean momentum flux maintains a constant value. Its contribution however is non-null for buoyant releases. It is instructive to evaluate the magnitude of ${\it\alpha}_{m}$ in the case of a nominal pure plume, i.e.  ${\it\Gamma}\approx 1$ . Considering simple scaling relations for a pure plume, i.e.  $b_{w}^{2}w_{m}^{2}\propto z^{4/3}$ and $b_{w}\propto (6/5){\it\alpha}_{ref}z$ , and estimating the integral $I_{6}\simeq 0.05$ from our experimental data, we can estimate ${\it\alpha}_{m}\simeq 1.6\times 10^{-2}$ , which represents a contribution of approximately 10 % to the total entrainment.

As pointed out in appendix A, it is worth noting that, under the assumption of Gaussian radial profiles of mean vertical velocity and temperature, the contribution to ${\it\alpha}_{G}$ given by the ‘drift term’ and pointed out by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) vanishes, even in the case of a varying ${\it\varphi}$ .

A systematic comparison of estimates of the entrainment coefficient obtained from (5.1) with those from (5.3) offers a means to explain the variability of ${\it\alpha}_{G}={\it\alpha}_{G}({\it\Gamma})$ shown in figure 16.

First, we analyse the Wang & Law (Reference Wang and Law2002) data. Their estimates of ${\it\alpha}_{G}$ are obtained for releases with $0.01<{\it\Gamma}_{0}<0.1$ and in the range $60<z/b_{0}<110$ . In contrast to us, Wang & Law (Reference Wang and Law2002) focus on the far-field region, i.e. where $\langle \widehat{{\it\nu}_{T}}\rangle$ is expected to have reached its asymptotic value, and where the second-order statistics have clearly already attained a condition of self-similarity, so that the contribution of the term ${\it\alpha}_{nf}$ is null. An examination of their Reynolds stress and mean velocity vertical profiles suggests $\langle \widehat{{\it\nu}_{T}}\rangle \simeq 0.35$ , which leads to values of the parameter $C=(3/2)\langle \widehat{{\it\nu}_{T}}\rangle ({\it\varphi}^{2}+1)$ (see appendix A) in the range $0.12<C<0.14$ (Kaminski et al. Reference Kaminski, Tait and Carazzo2005). This value of $\langle \widehat{{\it\nu}_{T}}\rangle$ corresponds approximately to the same value as observed in the far field of all three releases examined here (see figure 11). Over a similar range of distances from the source Wang & Law (Reference Wang and Law2002) data exhibits a decrease of ${\it\varphi}$ , from a near-field value of 1.25 down to 1.05 in the far field. As a first approximation, we neglect the contribution of ${\it\alpha}_{m}$ in (5.3), and estimate the entrainment coefficient as

On imposing $\langle \widehat{{\it\nu}_{T}}\rangle =0.35$ we plot ${\it\alpha}_{G}$ from (5.6) for ${\it\varphi}=$ 1.05, 1.15 and 1.25. As is clear from figure 17, the tendency of ${\it\alpha}_{G}={\it\alpha}_{G}({\it\Gamma})$ from Wang & Law (Reference Wang and Law2002) can be fully reproduced by (5.6) with ${\it\varphi}=1.05$ . For ${\it\Gamma}\geqslant 10^{-1}$ we note that entrainment responds sensitively to ${\it\varphi}$ , with variation of ${\it\varphi}$ less than 10 % inducing considerable variation in ${\it\alpha}_{G}$ . As a consequence of this sensitivity, the actual variation of ${\it\varphi}$ with height in the Wang & Law (Reference Wang and Law2002) experiments, that we have here neglected, is likely to alter the close agreement seen. However, a detailed estimate of all terms in (5.3) for the Wang & Law releases is clearly beyond the scope of the present study and requires further information on the velocity statistics, specifically those related to the term ${\it\alpha}_{m}$ . Our aim here is only to demonstrate that the estimates of ${\it\alpha}_{G}$ provided by Wang & Law (Reference Wang and Law2002) are consistent with the plume dynamics in the far field, where both ${\it\varphi}$ and $\langle \widehat{{\it\nu}_{T}}\rangle$ have reached their asymptotic values: values that closely match their experimental estimates.

Figure 17.

Entrainment coefficient ${\it\alpha}_{G}$ plotted against local plume Richardson number ${\it\Gamma}$ . Experimental data from Wang & Law (Reference Wang and Law2002) and estimates of the formulation of Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) given by (5.6) for a fixed $\langle \widehat{{\it\nu}_{T}}\rangle =0.035$ and three different values of ${\it\varphi}$ (see text).

Second, we focus on our three releases J, F and P. The measurements gathered during our experimental campaign allow for a direct estimate of all terms comprising ${\it\alpha}_{G}$ in (5.3). However, an estimate of the terms in ${\it\alpha}_{nf}$ , involving vertical derivatives of second-order velocity statistics, is characterised by a large uncertainty due to an insufficient number of sampled velocity fields. Accurate estimates of these terms would require a number of samples of at least an order of magnitude larger. For these reasons, in estimating ${\it\alpha}_{G}$ by means of (5.3), we consider as a first approximation, ${\it\alpha}_{nf}=0$ which we expect to lead to underestimations of ${\it\alpha}_{G}$ in the near field.

Since release P exhibits almost no variation of ${\it\Gamma}$ , the evolution of the estimates of ${\it\alpha}_{G}$ provided by (5.1) and (5.3) shown in figure 18 is plotted against $z/b_{0}$ , rather than against ${\it\Gamma}$ . Estimates from (5.3) are shown for all three releases with 20 % error bars, evaluated taking into account the uncertainties related to ${\it\Gamma}$ (figure 8), $\langle \widehat{{\it\nu}_{T}}\rangle$ (figure 11) and ${\it\varphi}$ (figure 12).

Figure 18.

Variation of entrainment coefficient as a function of distance from source. Comparison of entrainment coefficient estimated from the volume flux balance equation (5.1) and from the formulation (5.3) for releases (a) J, (b) F and (c) P.

For releases J and F, estimates from (5.3) tend to systematically underestimate those from (5.1) in the near field. This can be explained by the neglected contribution of ${\it\alpha}_{nf}$ . Even in the case of plume P, the two estimates of ${\it\alpha}_{G}$ differ significantly in the near field. This difference can primarily be attributed to the non-fully turbulent condition of the plume for $z/b_{0}<5$ , conditions that invalidate the formulation of (5.3). Discrepancies in the near-field region can also be attributed to the two features discussed in § 2, i.e. eventual non-Boussinesq effects and the thermal stratification of the ambient very close to the source (in a region that extends up to $z/b_{0}\sim 6$ for release J, $z/b_{0}\sim 2$ for F and $z/b_{0}\sim 2$ for P). Both features are not accounted for in the formulation of (5.3). However, despite the non-negligible extent of the error bars, we observe a relatively good agreement between the two estimates for releases J, F and P. This allows us to interpret the physical variation of ${\it\alpha}_{G}$ with $z/b_{0}$ as given by the variations with $z/b_{0}$ of the terms on the right-hand side (r.h.s.) of (5.3) and to shed light on the role of ${\it\Gamma}$ in the intensity of the entrainment of ambient air. Focusing on the first two terms on the r.h.s. of (5.3), terms that represent the highest contribution to the total entrainment, we may assert that the role of ${\it\Gamma}$ is twofold: directly through term $(2{\it\varphi}^{2}-1)(2{\it\alpha}_{ref}/5){\it\Gamma}$ and indirectly through the vertical evolution of $\langle \widehat{{\it\nu}_{T}}\rangle$ .

The relatively low values of ${\it\alpha}_{G}$ in the near field of releases J and F can then be fully explained by the corresponding low values of $\langle \widehat{{\it\nu}_{T}}\rangle$ (figure 11). Moving away from the source, ${\it\Gamma}\rightarrow 1$ and the contribution of $\langle \widehat{{\it\nu}_{T}}\rangle$ to ${\it\alpha}_{G}$ increases (at a rate that depends on ${\it\Gamma}$ ) as does the direct contribution of the term including ${\it\Gamma}$ . As widely discussed in § 4, a buoyant release can exhibit different $\langle \widehat{{\it\nu}_{T}}\rangle$ and ${\it\varphi}$ for a given ${\it\Gamma}$ , depending on the release conditions and distance from the source. This explains why, in general, a plume can exhibit a different value of ${\it\alpha}_{G}$ for the same local ${\it\Gamma}$ and provides, in this particular case, a robust justification for the differences observed between our estimates of ${\it\alpha}_{G}$ and those reported by Wang & Law (Reference Wang and Law2002). This feature is also likely to explain the high variability of values of the entrainment coefficient reported in the literature.

6. Summary of findings

We have presented a highly resolved set of velocity and temperature measurements carried out on three turbulent plumes of source Richardson number (1.1) in the range $10^{-2}<{\it\Gamma}_{0}<1$ . These measurements have been used to assess local dynamical self-similarity, to confirm the evolution of key dynamic quantities, to investigate the large scale structure of the flow and the variability of the rate of entrainment with height from the source.

Despite a variable local dynamical condition, characterised by a varying plume Richardson number ${\it\Gamma}$ , the radial profiles of the mean and of the r.m.s. of the velocity components can be rescaled on local quantities, namely the mean centerline velocity $w_{m}$ and the mean plume radius $b_{w}$ , to yield self-similar profiles for the flow variables for a wide range of distances from the source. This local self-similarity applies also to the mean temperature (buoyancy) profiles and to the turbulent viscosity $\widehat{{\it\nu}_{T}}$ , showing that even the increased radial turbulent momentum transfer induced by an excess of buoyancy can be fully rescaled by local quantities. Even the Eulerian length scales $\mathscr{L}_{uu}$ and $\mathscr{L}_{ww}$ appear to rescale locally when normalised with the plume radius. In contrast to the other flow variables however, the local self-similar function describing the spatial evolution of the Eulerian length scales differs significantly depending on  ${\it\Gamma}$ .

The influence of ${\it\Gamma}$ on the turbulence dynamics within each release was studied by focusing on the evolution of radial profiles of second-order velocity statistics and estimating bulk quantities, including the spatially averaged turbulent viscosity $\langle \widehat{{\it\nu}_{T}}\rangle$ and ${\it\varphi}=b_{g^{\prime }}/b_{w}$ which is directly linked to the spatially averaged turbulent Prandtl number. Summarising, our results show that:

1. (a) the higher the value of ${\it\Gamma}$ , the lower is the fetch required for second-order statistics to attain local self-similarity;

2. (b) the higher the value of ${\it\Gamma}$ , the more rapid the rate of increase of $\langle \widehat{{\it\nu}_{T}}\rangle$ toward its asymptotic value;

3. (c) the influence of ${\it\Gamma}$ on ${\it\varphi}$ is likely to be similar to the influence of ${\it\Gamma}$ on $\langle \widehat{{\it\nu}_{T}}\rangle$ , but this effect is difficult to unequivocally confirm, given the vertical variation of ${\it\varphi}$ exhibited by a buoyant release is comparable with the uncertainty associated with its experimental estimate;

4. (d) neither $\langle \widehat{{\it\nu}_{T}}\rangle$ nor ${\it\varphi}$ do not show a one-to-one dependence with ${\it\Gamma}$ , since their magnitude depends, at least in a region of flow transition, also on the distance from the source.

The study culminated with an analysis of the influence of ${\it\Gamma}$ on the rate of entrainment of ambient air as quantified by the entrainment coefficient ${\it\alpha}_{G}$ . Two distinct experimental estimates of ${\it\alpha}_{G}$ were obtained. The first by estimating the vertical variation of the volume flux. The second adopting a formulation of ${\it\alpha}_{G}$ similar to that originally proposed by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005), and that makes explicit the role of the main non-dimensional parameters governing the dynamics of the plume, i.e.  ${\it\Gamma}$ , $\langle \widehat{{\it\nu}_{T}}\rangle$ and ${\it\varphi}$ . In this way we could fully explain the observed variations of ${\it\alpha}_{G}$ in the three releases studied as well as in the existing literature data, depending on ${\it\Gamma}$ and on the distance from the source.

Our analysis shows that it is not possible to identify a one-to-one dependence of ${\it\alpha}_{G}$ on ${\it\Gamma}$ , since its variations are due also to changes in $\langle \widehat{{\it\nu}_{T}}\rangle$ and ${\it\varphi}$ , whose magnitude, for a given ${\it\Gamma}$ depends on the distances from the source and the value of ${\it\Gamma}$ at the source.