2.1. Plume theory

Consider a turbulent plume that develops from an infinitely long slender source of buoyancy flux per unit length $B_0$ (dimension L$^3$T$^{-3}$, where L is length and T is time), in an otherwise quiescent and miscible environment of uniform density $\rho _e$, subject to the gravitational acceleration $g$. Let the vertical coordinate $z$ denote the distance from the source in the streamwise direction, $x$ the cross-stream coordinate where $x=0$ is the plume centreline and $y$ the spanwise coordinate. The effect of the buoyancy force, characterised by the buoyancy of the plume relative to the environment $g'=g(\rho _e-\rho )/\rho _e$, where $\rho$ is the local density of the plume, is to induce a vertical velocity $w$. Here, $g'$ and $w$ are the time- and spanwise-averaged quantities at a point ($x,z)$.

As is standard, the integral fluxes of volume $Q$ (L$^2$T$^{-1}$), specific momentum $M$ (L$^3$T$^{-2}$) and buoyancy $B$ through a horizontal cross-section are then defined as

(2.1a–c)\begin{equation} Q=\int_{-\infty}^{\infty} w\,\mathrm{d} x , \quad M=\int_{-\infty}^{\infty} w^2\,\mathrm{d} x, \quad B=\int_{-\infty}^{\infty} wg'\, \mathrm{d} x.\end{equation}

Following the assumptions made to the Navier–Stokes equations by Morton et al. (Reference Morton, Taylor and Turner1956) and Lee & Emmons (Reference Lee and Emmons1961), the plume is modelled by conservation equations for these integral quantities. For an unstratified environment these are

(2.2a–c)\begin{equation} \frac{\mathrm{d} Q}{\mathrm{d} z} =2u_e, \quad \frac{\mathrm{d} M}{\mathrm{d} z} =\int_{-\infty}^{\infty}g'\, \mathrm{d} x, \quad \frac{\mathrm{d} B}{\mathrm{d} z} =0.\end{equation}

To close (2.2a–c), we must specify the cross-plume variation of $w$ and $g'$ and relate $u_e$ to the integral quantities. It is at this stage that various conventions for the entrainment coefficient are introduced. One convention assumes ‘top-hat’ profiles whereby the plume is modelled as having a uniform average velocity $w_T=M/Q$ and buoyancy $g_T'=B_0/Q$ across a finite width $2b_T=Q^2/M$ and zero vertical velocity and buoyancy outside. The top-hat entrainment coefficient $\alpha _T$ is then defined such that $u_e=\alpha _T M/Q$. A second convention, based on observations that the time-averaged cross-stream profiles of velocity and buoyancy are self-similar and Gaussian-like (Rouse et al. Reference Rouse, Yih and Humphreys1952; Kotsovinos Reference Kotsovinos1975; Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989; Paillat & Kaminski Reference Paillat and Kaminski2014a; Parker et al. Reference Parker, Burridge, Partridge and Linden2020), assumes that the profiles are the Gaussians

(2.3a,b)\begin{equation} w(x,z)=w_c(z)\exp{\left(-\frac{x^2}{b^2}\right)}, \quad g'(x,z)=g'_c(z)\exp{\left(-\frac{x^2}{(\lambda b)^2}\right)}. \end{equation}

In (2.3a,b), $g'_c$ is the centreline buoyancy and $b$ and $\lambda b$ are the characteristic half-widths of the profiles where the velocity and buoyancy, respectively, drop to $1/e$ (where $\ln {e}=1$) of the centreline value. Experimental measurements show that the buoyancy profile is wider than the velocity profile, i.e. the profile coefficient $\lambda >1$ (e.g. Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989). Via (2.1a–c), these profiles correspond to the integral quantities

(2.4a–c)\begin{equation} Q=\sqrt{\rm \pi}\,w_cb, \quad M=\sqrt{\frac{\rm \pi}{2}}\,w_c^2b, \quad B=\frac{\lambda\sqrt{\rm \pi}}{\sqrt{\lambda^2+1}} w_c b g'_c. \end{equation}

For the idealised case of a true line plume source, the source volume and momentum fluxes are zero and the plume is in an asymptotic state whereby $\alpha$ (or $\alpha _T)$ and $\lambda$ are constants for any $z>0$. Solution of (2.2a–c) for this idealised case, based on the top-hat convention, yields

(2.5a–c)\begin{equation} Q=(2\alpha_T)^{2/3}B_0^{1/3}z , \quad M=(2\alpha_T)^{1/3}B_0^{2/3}z, \quad B=B_0,\end{equation}

and based on the Gaussian convention

(2.6a–c)\begin{equation} Q=2^{5/6}(1+\lambda^2)^{1/6}\alpha^{2/3}B_0^{1/3}z, \quad M=2^{1/6}(1+\lambda^2)^{1/3}\alpha^{1/3}B_0^{2/3}z, \quad B=B_0.\end{equation}

Clearly, there is a requirement that $Q, M$ and $u_e$ be identical irrespective of the form of the profile assumed and, thus, the values of $\alpha _T$ and $\alpha$ are linked. For example, $\alpha _T=\sqrt {2}\alpha$ on taking ${\lambda =1}$ as is a typical simplification in the top-hat model (van den Bremer & Hunt Reference van den Bremer and Hunt2014a; Paillat & Kaminski Reference Paillat and Kaminski2014a). This link is outlined here given top-hat profiles are widely adopted in applications of classic plume theory. However, for interpreting experimental measurements, it is convenient to work using the Gaussian convention as the majority of the reported values for $\alpha$ (table 1) were calculated from fits to measured Gaussian-like time-averaged velocity and buoyancy profiles.

On dimensional grounds, the characteristic velocity profile half-width, centreline velocity and centreline buoyancy of a turbulent plume from a line source can be expressed as

(2.7a–c)\begin{equation} b=C_bz, \quad w_c=C_w B_0^{1/3}, \quad g_c'=C_g B_0^{2/3} z^{{-}1}, \end{equation}

where $C_b, C_w$ and $C_g$ are dimensionless coefficients. Equating (2.4a–c) with (2.6a–c) and substituting for (2.7a–c) shows that these coefficients are related to $\alpha$ and $\lambda$ as follows:

(2.8)$$\begin{gather} \sqrt{\rm \pi} C_wC_b =2^{5/6}(1+\lambda^2)^{1/6}\alpha^{2/3}, \end{gather}$$

(2.9)$$\begin{gather}\sqrt{\rm \pi} C_w^2C_b =2^{2/3}(1+\lambda^2)^{1/3}\alpha^{1/3}, \end{gather}$$

(2.10)$$\begin{gather}\sqrt{\rm \pi} C_w C_g C_b =\lambda^{{-}1} (1+\lambda^2)^{1/2} . \end{gather}$$

These three coupled algebraic equations, expressed in similar forms by Lee & Emmons (Reference Lee and Emmons1961) and Yuan & Cox (Reference Yuan and Cox1996), have five unknowns. Thus, the measurement of a pair of coefficients is necessary before $\lambda$, the remaining coefficients and our ultimate target, $\alpha$, can be deduced. The pairings are not limited to the coefficients appearing in (2.8)–(2.10), and at this stage it is convenient to define the following four coefficients:

(2.11a–d)\begin{equation} \lambda b=C_{\lambda b}z, \quad Q=C_Q B_0^{1/3}z, \quad M=C_M B_0^{2/3}z, \quad u_e=C_e B_0^{1/3}. \end{equation}

These coefficients are readily linked to $\alpha$ and $\lambda$ following substitution into (2.8)–(2.10) via (1.1), (2.6a–c) and (2.7a–c).

2.2. Determining $\alpha$ and $\lambda$

Only a small fraction of all the possible pairings of coefficients have been used previously to determine $\alpha$ and $\lambda$. Kotsovinos (Reference Kotsovinos1975) and Paillat & Kaminski (Reference Paillat and Kaminski2014a) used the pair $(C_{b},C_{\lambda b})$, i.e. measurements of the widths of the velocity and buoyancy profiles. Rearrangement of (2.8) and (2.9), and the definition of $\lambda$, leads to

(2.12a,b)\begin{equation} \alpha=\frac{\sqrt{\rm \pi}}{2}C_b, \quad \lambda =\frac{C_{\lambda b}}{C_b}. \end{equation}

The straightforward forms of (2.12a,b) plausibly explains why the pair $(C_{b},C_{\lambda b})$ is one of the most commonly selected to determine $\alpha$ and $\lambda$. Notably, only measurement of $C_b$ is needed to estimate $\alpha$.

In contrast, Yuan & Cox (Reference Yuan and Cox1996) designed their experiment to determine $\alpha$ and $\lambda$ (Yuan & Cox (Reference Yuan and Cox1996) use $\beta =1/\lambda ^2$) from measurements of the centreline velocity and buoyancy, i.e. the pair $(C_w,C_g)$. For this pairing, (2.8) and (2.10) lead to the more complex relationships

(2.13a,b)\begin{equation} \alpha=\frac{\sqrt{2C_g^2+C_w^4}}{2C_w^3C_g}, \quad \lambda =\frac{C_w^2}{\sqrt{2}C_g}. \end{equation}

Yuan & Cox (Reference Yuan and Cox1996) used this pair to calculate a value of $\alpha$ (and $\lambda$) from measurements made by themselves and others – note the entries in table 1 linked to their analysis. In doing so, they appear to have been the first authors to reexamine past data by using a different coefficient pair than the original experimentalist(s).

There are $\binom{5}{2}=10$ possible pairings of the five most commonly measured coefficients $C_b, C_{\lambda b}, C_w, C_g$ and $C_Q$; the expressions for these pairings are listed in Appendix A. In addition to $(C_{b},C_{\lambda b})$ and ${(C_{w},C_{g})}$, only two more of the possible pairings have been chosen previously to determine $\alpha$ and $\lambda$, namely, ${(C_{\lambda b},C_{g})}$ by Lee & Emmons (Reference Lee and Emmons1961) and ${(C_{w},C_{Q})}$ by Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989). Our measurements of $C_e$ (§ 5) can be related to $\alpha$ and $\lambda$ via relationships involving $C_Q$: the change in volume flux being the result of entrainment into the two sides of the plume ($C_Q=2C_e$). We additionally include the relationships for two pairs involving the coefficients $C_M$ and $C_B=w_cg'_cz/B_0$, although do not include every possible combination. The pair ${(C_{M},C_{Q})}$ is closely linked to the top-hat plume width measured by Parker et al. (Reference Parker, Burridge, Partridge and Linden2020). The pair ${(C_{g},C_{B})}$ uses centreline measurements of the buoyancy and the product of the buoyancy and velocity, and highlights how experimental capabilities could motivate the study of additional pairs; in this case, possibly using a combined temperature and heat-flux sensor.

4.1. Effects of experimental conditions

It is natural to enquire if the reported variations in $\alpha$ can be attributed to differences in how ‘line like’ the experimental set-ups were. In an ideal scenario, measurements would be made from a long and slender source ($L/s\gg 1$) at downstream distances $z_m$ that are small relative to the source length ($L/z_m \gg 1$) and large relative to the source width ($z_m/s \gg 1$). Practically, it is infeasible to insist that $s \ll z_m \ll L$ and, consequently, experimentalists are forced to compromise on what conditions are appropriate. In general, an experiment can be deemed increasingly ‘line like’ if the ratios $L/s, L/z_m$ and $z_m/s$ are larger, and if there are end walls to encourage a two-dimensional induced-flow field.

The variation in the reported values for $\alpha$ with the experimental geometry is shown in figure 1. There is no clear evidence in these plots for a systematic relationship between how line-like experiments are and the value of $\alpha$. Further support for this claim is provided by Yuan & Cox (Reference Yuan and Cox1996) and Paillat & Kaminski (Reference Paillat and Kaminski2014a) whose measurements of plumes from sources of two different widths show no systematic difference in the centreline properties.

Figure 1. Variation of the reported values for $\alpha$ with experimental geometry. (a) $\alpha$ vs the source aspect ratio $L/s$; both values for $L/s$ are shown where two aspect ratios were studied, (b) $\alpha$ vs the source length $L$ scaled on the largest downstream distance that measurements are reported $z_{max}$, (c) $\alpha$ vs $z_{max}$ scaled on the source width; both values are shown where two different source widths were used, (d) The presence of end walls; note that three ‘No’ data points overlap (Yokoi Reference Yokoi1960; Yuan & Cox Reference Yuan and Cox1996; Paillat & Kaminski Reference Paillat and Kaminski2014a).

In addition to plumes from a large aspect ratio line-like source, Yokoi (Reference Yokoi1960) studied plumes from sources with $L/s=1.4, 2.7$ and $5.5$. His measurements of $g'_c$ above these small aspect ratio sources show that the departure from the two-dimensional scaling ($g'_c \sim z^{-1}$) toward the axisymmetric scaling ($g'_c \sim z^{-5/3}$) occurs for $z/L\approx 6 - 7$. As all of the experiments reported in table 3 have sources with $L/s > 5.5$ and only report measurements for $z/L \lesssim 3$, the apparent absence of systematic effects due to the source geometry is not surprising.

While in principle there should be no difference between measurements of dynamically equivalent plumes in water or air, possible systematic differences can be identified. For example, plumes above a fire source are subject to non-Boussinesq effects (van den Bremer & Hunt Reference van den Bremer and Hunt2014b) while those in water tanks will be affected by the confining walls and filling-box effects (Baines & Turner Reference Baines and Turner1969; Barnett Reference Barnett1991). A brief analysis suggests that the fire plume measurements reported in table 3 were unaffected by non-Boussinesq effects (Appendix B). In contrast, confinement effects could not be ruled out, and we discuss their possible impact on $\alpha$ in § 6; however, we did not identify a systematic variation due to confinement. Other possible effects, such as similarities in the design of the source, or measurement instrumentation and techniques, were not considered (and because of the small number of measurements, it is not clear that such an analysis would be fruitful).

As the variation in $\alpha$ cannot be explained by systematic differences between the experiments, we proceed by analysing the experimental campaigns in turn (§§ 4.2–4.7). We summarise the main findings in § 4.8 where we present a new curated table of values for $\alpha$ (table 4). Although some of the changes are relatively minor, the analysis results in several significant modifications. In particular, we identify an alternative (and more appropriate) value for $\alpha$ using the measurements of Lee & Emmons (Reference Lee and Emmons1961) and we reject the value that Yuan & Cox (Reference Yuan and Cox1996) attribute to Kotsovinos (Reference Kotsovinos1975). As a result, we show that $0.095\lesssim \alpha \lesssim 0.13$ is a better representation of the variation in past measurements.

Table 4. Updated list of values for the entrainment coefficient and profile coefficient of a turbulent line plume following analysis of the literature. Summary of differences from table 1: erroneous values for Lee & Emmons (Reference Lee and Emmons1961) and Kotsovinos (Reference Kotsovinos1975) have been removed; entries for Rouse et al. (Reference Rouse, Yih and Humphreys1952) and Yokoi (Reference Yokoi1960) have been removed because of concerns regarding the interpretation of their data; based on consideration of multiple coefficient pairs, the entries for Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989), Paillat & Kaminski (Reference Paillat and Kaminski2014a) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) are now a single representative value.

4.2. Rouse et al. (Reference Rouse, Yih and Humphreys1952)

Rouse et al. (Reference Rouse, Yih and Humphreys1952) measured $w(x)$ and $g'(x)$ at different heights in a thermal plume created by a line of gas burners. Although they do not calculate $\alpha$, the coefficients $C_b, C_{\lambda b}, C_w$ and $C_g$ can be determined from their reported profiles. These profiles were chosen to fit their measurements subject to constraints imposed by the conservation of momentum and buoyancy flux. The appropriateness of their fits was, however, challenged by Brooks (Reference Brooks1973) who claimed that redrawing the curves to better fit the data showed that $\alpha =0.14$ and $\lambda =1.0$ (the redrawn curves are not reproduced in Brooks Reference Brooks1973). Further challenge comes from Chen & Rodi (Reference Chen and Rodi1980) who propose new fits on the basis that the original fits implied $\lambda <1$, despite the measurements showing $\lambda >1$. The coefficients for these revised fits, subject to the conservation constraints, are reported in table 3. Despite the small apparent difference between the Rouse et al. (Reference Rouse, Yih and Humphreys1952) and Chen & Rodi (Reference Chen and Rodi1980) fits relative to the scatter in the measurements (figure 2), the effect on $\alpha$ and $\lambda$ is significant: $\alpha =0.160$ decreases to $\alpha =0.144$ and $\lambda =0.88$ increases to $\lambda =1.04$. Here, $\alpha$ and $\lambda$ were calculated using the pair $(C_{b},C_{\lambda b})$, although, because these fits were constrained by conservation of momentum and buoyancy, calculations using any pairing of $C_b, C_{\lambda b}, C_w$ and $C_g$ result in the same values (subject to rounding errors).

Figure 2. A redrawing of the comparison that Chen & Rodi (Reference Chen and Rodi1980) made between the original fit of Rouse et al. (Reference Rouse, Yih and Humphreys1952) and the revised profiles proposed by Chen & Rodi (Reference Chen and Rodi1980). A least-squares Gaussian fit to the data points is included for comparison.

To assess the appropriateness of either reported fit we have added an (unconstrained) Gaussian least-squares fit to figure 2, which yields the following coefficients: $C_b=0.167, C_{\lambda b}=0.175, C_w=1.72$ and $C_g=2.63$. While a comparison with the least-squares fit does not definitively show that Chen & Rodi (Reference Chen and Rodi1980) achieved a better fit to the data than Rouse et al. (Reference Rouse, Yih and Humphreys1952), the least-squares fit does support the Chen & Rodi (Reference Chen and Rodi1980) claim that the measurements show $\lambda >1$ ($C_{\lambda _b}/C_b=1.05$). While the least-squares fit might be expected to be more appropriate than the two constructed fits, this is not true for the purpose of calculating $\alpha$. There are four possible values for $\alpha$ that can be calculated from the least-squares coefficients: $0.148$ ($C_b$), $0.182$ ${(C_{\lambda b},C_{w})}$, $0.178$ ${(C_{w},C_{g})}$ and $0.079$ ${(C_{\lambda b},C_{g})}$. This poor agreement between values calculated with different pairs is not inevitable – in §§ 4.5 and 4.7 we show that unconstrained Gaussian fits to data can result in a set of values for $\alpha$ in much closer agreement.

We do not include values of $\alpha$ determined from the Rouse et al. (Reference Rouse, Yih and Humphreys1952) measurements in our curated list (table 4) as there is too much uncertainty regarding which coefficients best represent their measurements. However, the Rouse et al. (Reference Rouse, Yih and Humphreys1952) measurements are important in the historical context, not only as the first of their kind and as revealing the self-similar nature of the profiles, but also because the reported coefficient values influenced the analysis of later measurements, particularly those of Lee & Emmons (Reference Lee and Emmons1961).

4.3. Yokoi (Reference Yokoi1960)

Yokoi (Reference Yokoi1960) measured $w(x)$ and $g'(x)$ at different heights above a line fire. In his analysis, Yokoi adopts Prandtl's momentum transfer theory (Prandtl Reference Prandtl1925) and Taylor's vorticity transfer theory (Taylor Reference Taylor1932), rather than the framework of the Morton et al. (Reference Morton, Taylor and Turner1956) plume model. With modifications to these theories based on his measurements, Yokoi (Reference Yokoi1960) shows that

(4.1a,b)\begin{equation} C_w=1.040 c^{{-}2/9} \quad \mbox{and} \quad C_g=0.663 c^{{-}4/9} , \end{equation}

and obtains $c^{2/3}=0.13$ from experiment. It would appear that the values $C_w=2.05$ and $C_g=2.6$, and the corresponding values of $\alpha =0.125$ and $\lambda =1.15$, that Yuan & Cox (Reference Yuan and Cox1996) attribute to Yokoi (Reference Yokoi1960) result from (4.1a,b). Yuan & Cox (Reference Yuan and Cox1996) also report that Yokoi (Reference Yokoi1960) ‘explicitly’ measured $\lambda =1.01$. We could not find reference to this, but Yokoi (Reference Yokoi1960) does present plots of velocity and buoyancy profiles. As these plots do not clearly indicate Gaussian-like profiles, we have not attempted to use them to determine $C_b$ or $C_{\lambda b}$, but they do appear to support $\lambda \approx 1.0$ (and not $\lambda =1.15$).

There are difficulties with the interpretation of Yokoi's results, principally, as no clear statement is given as to how $c$ was measured, or inferred, so judgements concerning its accuracy or precision cannot be readily made. Moreover, the constant $c$ is a component in his analysis using modified transfer theories and non-Gaussian profiles, and it is unclear how this analysis can be reconciled with the Morton et al. (Reference Morton, Taylor and Turner1956) entrainment hypothesis and Gaussian-profile framework adopted here. Although it is unclear how Yokoi (Reference Yokoi1960) interpreted his measurements, we follow Yuan & Cox (Reference Yuan and Cox1996) and use (4.1a,b) to calculate the entries reported for Yokoi (Reference Yokoi1960) in table 3, which are in broad agreement with others. However, we do not include entries for Yokoi's measurements in table 4.

4.4. Lee & Emmons (Reference Lee and Emmons1961)

Lee & Emmons (Reference Lee and Emmons1961) also studied the thermal plume above a line fire. Unlike Rouse et al. (Reference Rouse, Yih and Humphreys1952) and Yokoi (Reference Yokoi1960) they only measured temperature profiles and, consequently, could only determine $C_{\lambda b}$ and $C_g$. Lee & Emmons (Reference Lee and Emmons1961) present the inverse relations, i.e. $C_g$ and $C_{\lambda b}$ in terms of $\alpha$ and $\lambda$, which can be rearranged to show

(4.2a,b)\begin{equation} \alpha =\frac{\sqrt{\rm \pi}C_{\lambda b}}{2}\left(A\pm\sqrt{A^2-1}\right), \quad \lambda =A\mp\sqrt{A^2-1}, \quad \mbox{where } A=\frac{\rm \pi}{\sqrt{2}} C_g^3C_{\lambda b}^2. \end{equation}

The solutions for $\alpha$ and $\lambda$ in (4.2a,b) share the same basic form and, for a given value of $A>1$, there are clearly two pairs of real-valued solutions. Lee & Emmons (Reference Lee and Emmons1961) report $\alpha =0.16$ and $\lambda =0.9$, which closely align with the values determined from the fits reported by Rouse et al. (Reference Rouse, Yih and Humphreys1952). However, they do not comment on the existence of multiple solutions or the possibility that measurement error could result in non-real-valued solutions.

The solution pair $\alpha =0.16$ and $\lambda =0.9$ can be calculated to 2 s.f. using $C_{\lambda b}=0.16248$ and $C_g=2.5786$, the latter requiring 5 s.f. in order to achieve the aforementioned precision. This extreme sensitivity is reflected by the large values of the uncertainty multipliers for this pair (table 2). These same coefficient values can be used to calculate a previously unreported second solution pair for which $\lambda >1$, namely, $\alpha =0.13$ and $\lambda =1.1$. It is impossible to select one pair over the other from measurements of the buoyancy profile alone: two different velocity profiles, one ‘faster’ and narrower than the other, each implying different values of $\alpha$ and $\lambda$, could be paired with the buoyancy profile and satisfy the conservation equations. In contrast, a coefficient pair with at least one measurement of the velocity profile does not lead to such an ambiguity. It is unclear if Lee & Emmons (Reference Lee and Emmons1961) deliberately chose the solution branch which agreed more closely with the fits originally reported by Rouse et al. (Reference Rouse, Yih and Humphreys1952) (which erroneously suggested $\lambda <1$) or if they were only aware of the branch chosen. Regardless, the revised fits to the Rouse et al. (Reference Rouse, Yih and Humphreys1952) data (Chen & Rodi Reference Chen and Rodi1980) and all subsequent measurements of $(C_{b},C_{\lambda b})$ (Kotsovinos Reference Kotsovinos1975; Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989; Paillat & Kaminski Reference Paillat and Kaminski2014a; Parker et al. Reference Parker, Burridge, Partridge and Linden2020) show $\lambda >1$. Consequently, we consider $\alpha =0.13$ to be the correct interpretation of the Lee & Emmons (Reference Lee and Emmons1961) measurements.

A similar analysis error has propagated in the fire plume literature (e.g. Thomas Reference Thomas1987; Poreh et al. Reference Poreh, Morgan, Marshall and Harrison1998) where, because the transport of smoke is of particular concern, there is emphasis on the coefficient for the variation of volume flux, $C_Q$, rather than on $\alpha$. Using the framework set out in § 2

(4.3)\begin{equation} C_Q=2^{5/6}\alpha^{2/3}(1+\lambda^2)^{1/6}. \end{equation}

Thomas (Reference Thomas1987) used (4.3) to calculate $C_Q=0.58$ using the original, but seemingly erroneous, values for $\alpha$ and $\lambda$ reported by Lee & Emmons (Reference Lee and Emmons1961). With the revised values for $\alpha$ and $\lambda$ $(>1)$, we obtain $C_Q=0.52$, which is in closer agreement with the other measurements (table 3). Given the potential for ambiguity, we recommend $C_Q$ also be related to the measured quantities directly, viz.

(4.4)\begin{equation} C_Q=\frac{1}{C_g}\sqrt{\frac{2A}{A\pm\sqrt{A^2-1}}}, \end{equation}

a form that does not obscure the fact that two solutions are possible. Thomas et al. (Reference Thomas, Morgan and Marshall1998) describe how the excess plume volume flux implied by the values of $\alpha$ and $\lambda$ originally reported by Lee & Emmons (Reference Lee and Emmons1961) resulted in unnecessarily complex spill plume models used by the Building Research Establishment (BRE) and state that this ‘complexity can be removed by decoupling early BRE work from the experiments of Lee and Emmons’.

A final conclusion concerning the use of the Lee & Emmons (Reference Lee and Emmons1961) data for estimating $\alpha$ is worthy of mention. The magnitudes of their uncertainty multipliers are $18.1$ ($C_{\lambda b}$) and $28.7$ ($C_g$), values that (i) are considerably larger than typical multipliers (table 2) and (ii) render the use of (3.2a) inappropriate. Using (4.2a), we note that if both values of ${(C_{\lambda b},C_{g})}$ were $1\,\%$ smaller then $A<1$ and $\alpha$ could not be determined, but if both values were $1\,\%$ larger then $\alpha$ would reduce from $0.13$ to $0.10$ (both changes being more significant than would be predicted using (3.2a)). Given such sensitivity, the accuracy and precision of their measurements is particularly important but information on this is not provided.

4.5. Kotsovinos (Reference Kotsovinos1975) and Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989)

Both Kotsovinos (Reference Kotsovinos1975) and Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) measured $w(x)$ and $g'(x)$ at a number of downstream locations in a thermal plume in water.

It is evident from table 1 that there is a peculiarity regarding the data of Kotsovinos (Reference Kotsovinos1975): Kotsovinos (Reference Kotsovinos1975) reports $\alpha =0.11$ based on his own measurements of the velocity profile width (2.12a), while Yuan & Cox (Reference Yuan and Cox1996) report $\alpha =0.20$ using (2.13a) and the Kotsovinos (Reference Kotsovinos1975) data for ${(C_{w},C_{g})}$. By contrast, independent assessments of the data of Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) yield only marginal variations in $\alpha$: Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) determined $\alpha =0.113$ from their own measurements of ${(C_{w},C_{Q})}$, while calculations by Yuan & Cox (Reference Yuan and Cox1996), using ${(C_{w},C_{g})}$, and Paillat & Kaminski (Reference Paillat and Kaminski2014a), using $C_b$, both concluded $\alpha =0.117$.

In general, it is not surprising that different coefficient pairs should lead to marginally different values for $\alpha$ on account of small errors in measurement or in the fitting of Gaussian profiles to determine $C_{b}, C_{\lambda b}, C_w$ and $C_g$. The least-squares fit to the Rouse et al. (Reference Rouse, Yih and Humphreys1952) data highlighted the challenge in achieving a consistent set of values for $\alpha$ determined from different pairs. However, the difference between $\alpha =0.11$ and $\alpha =0.20$ cannot be simply explained by a small error and suggests that the measurements made by Kotsovinos (Reference Kotsovinos1975) are not self-consistent.

Chen & Rodi (Reference Chen and Rodi1980), Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) have all cast doubt on some aspects of the Kotsovinos (Reference Kotsovinos1975) measurements, and it seems likely that the Kotsovinos (Reference Kotsovinos1975) value of $C_w$ is erroneous. Consequently, we reject the value $\alpha =0.20$ calculated by Yuan & Cox (Reference Yuan and Cox1996). Looking beyond $C_w$, we could not identify any specific reason that the widths of the velocity and scalar profiles would be incorrect, and the values of $\alpha =0.10$ and $\lambda =1.28$ that result from the pair $(C_{b},C_{\lambda b})$ are not unreasonable. These values are slightly different to those usually attributed to Kotsovinos (Reference Kotsovinos1975), as explained in Appendix C.

The values of the measurements reported by Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) are in good agreement with others (table 3), and are seemingly self-consistent given the close similarities in the independently calculated values of $\alpha$. We further demonstrate the self-consistency in § 4.7 where we attribute the value $\alpha =0.115$ to their dataset. However, some aspects of their dataset raise concern. They report on only four experiments, including one where the plume buoyancy flux drops by over $50\,\%$ in their measurement region. The small number of experiments is concerning as the (unreported) variation of the value of $C_Q$ in individual experiments exceeds $10\,\%$.

4.6. Yuan & Cox (Reference Yuan and Cox1996)

The approach of Yuan & Cox (Reference Yuan and Cox1996) is an exemplar of the application of plume theory in the design of experiments. They showed that measuring only the centreline properties, rather than entire plume profile(s), is sufficient to deduce $\alpha$. Although they could not then show the self-similarity of cross-stream profiles, they confirm that their measurements were of a fully developed line plume by showing that their centreline measurements followed the power laws $g_c'\propto B_0^{2/3}z^{-1}$ and $w_c\propto B_0^{1/3}z^0$. Their measurements of volume flux using the hood method (Zukoski et al. Reference Zukoski, Kubota and Cetegen1981), while not as comprehensive, were complementary and the measured value of $C_Q=0.51$ is in good agreement with the value $C_Q=0.48$ which they calculated using their values of ${(C_{w},C_{g})}$ (table 3). Yuan & Cox (Reference Yuan and Cox1996) do not report any uncertainties for their measurements, but their well-populated plots of centreline temperature and velocity show minimal scatter.

4.7. Paillat & Kaminski (Reference Paillat and Kaminski2014a) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020)

Paillat & Kaminski (Reference Paillat and Kaminski2014a) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) measured $w(x,z)$ and $g'(x,z)$ in the central plane of a saline plume using particle image velocimetry and laser induced fluorescence. Paillat & Kaminski (Reference Paillat and Kaminski2014a) report values of $\alpha$ and $\lambda$ calculated using $(C_{b},C_{\lambda b})$. While values for $(C_w,C_g)$ are not reported, $C_w\approx 2.1 - 2.2$ and $C_g\approx 2.8 - 2.9$ can be read from their plots and are listed in table 3 as the mid-range values: $C_w=2.15$ and $C_g=2.85$.

Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) report $\alpha _T=0.14$ for the top-hat entrainment coefficient, which corresponds to $\alpha =0.10$ (approximately). While they do not report values for the coefficients $C_b, C_{\lambda b}, C_w, C_g$ or $C_M$, our analysis of the data they provide allows these coefficients to be estimated (Appendix D). Values resulting from this analysis are included in table 3.

The three sets of entries in table 3 for Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) account for the three different representative buoyancy fluxes that we used to normalise their profile measurements: the source buoyancy flux $B_{0,S}$ (determined from the volume flux and buoyancy of the source fluid), the mean buoyancy flux $B_{0,M}$ measured in the plume and the total buoyancy flux $B_{0,T}$ measured in the plume (determined as the sum of the mean and turbulent components). Classic plume theory assumes that the turbulent component is relatively small, and, thus, $B_{0,S}\approx B_{0,M} \approx B_{0,T}$. However, Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) measured, on average, $B_{0,M}=0.84B_{0,S}$ and $B_{0,T}=0.94B_{0,S}$, and normalising their measurements by these buoyancy fluxes results in differences in coefficient values of similar magnitude to the differences between the other experimental campaigns (table 3). There is no obvious consensus from past campaigns on the ‘most’ representative buoyancy flux as all three have been used – Rouse et al. (Reference Rouse, Yih and Humphreys1952) normalise using the mean, Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) normalise with the total and Lee & Emmons (Reference Lee and Emmons1961) normalise with the source buoyancy flux – although these choices are, in part, a result of experimental capabilities.

Based on the coefficient values we have determined from the data of Paillat & Kaminski (Reference Paillat and Kaminski2014a) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020), $\alpha$ may now be estimated in multiple ways, a selection of which are plotted in figure 3. To aid wider comparisons, also plotted are the values for $\alpha$ determined from (i) the measurements of $\alpha _T$ by Parker et al. (Reference Parker, Burridge, Partridge and Linden2020), (ii) from the dataset of Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989), where we have normalised the measurements by $B_{0,M}$ in Appendix E, and (iii) from our own measurements (§ 5).

Figure 3. Values of $\alpha$ calculated from coefficient pairs or $\alpha _T$. Error bars show $95\,\%$ confidence intervals calculated with (3.2a), for pairs where the confidence interval for the measurements could be determined. Note: PBPL (Parker et al. Reference Parker, Burridge, Partridge and Linden2020), PK (Paillat & Kaminski Reference Paillat and Kaminski2014a), RC (Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989).

Amidst the range of values of $\alpha$ in figure 3, we can highlight four key results. First, for a dataset where at least three coefficients are measured it is potentially misleading to report a single value for $\alpha$ (i.e. as determined from a single coefficient pair) given a different coefficient pairing could lead to a significantly different value for $\alpha$. While the value $\alpha =0.113$ reported by Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) using the pair ${(C_{w},C_{Q})}$ is the median value of the seven values calculated for their data, the value $\alpha =0.126$ reported by Paillat & Kaminski (Reference Paillat and Kaminski2014a) using $C_b$ is the highest of the three values for theirs.

Second, the choice of normalisation (by $B_{0,S}, B_{0,M}$ or $B_{0,T}$) only impacts certain coefficient pairs. As would be expected, there is no impact if $\alpha$ is determined from $C_b$, where only the velocity profile width matters, or from the pairs ${(C_{w},C_{Q})}$ and ${(C_{Q},C_{M})}$, where the buoyancy flux ‘cancels out’. However, for other pairings such as ${(C_{w},C_{g})}$ or ${(C_{g},C_{Q})}$ the difference can be significant. Intriguingly, normalising the Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) measurements by $B_{0,M}$ results in very consistent values for $\alpha$, with an absolute difference of only $0.005$ between the largest and smallest value. This consistency is perhaps unsurprising given that only the mean buoyancy flux is considered in classic plume theory and van Reeuwijk et al. (Reference van Reeuwijk, Salizzoni, Hunt and Craske2016) reached a similar conclusion with their direct numerical simulation data for axisymmetric plumes. Normalisation with $B_{0,M}$ did not result in a more consistent set using the Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) data, although there are concerns with the dataset (§ 4.5).

Third, while selecting a single value for $\alpha$ to represent each dataset is not necessarily trivial, the median of the plotted values yields the following representative values (rounded to the nearest $0.005$): $\alpha =0.095$ (Parker et al. Reference Parker, Burridge, Partridge and Linden2020), $\alpha =0.115$ (Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989) and $\alpha =0.120$ (Paillat & Kaminski Reference Paillat and Kaminski2014a). The rounded values for Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) are the same for each ‘set’ of coefficient measurements, irrespective of the normalising buoyancy flux.

Fourth, the $95\,\%$ confidence intervals for the Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) measurements span a considerable range of values for $\alpha$, and the size of the interval varies with the coefficient pairing. The smallest interval is $\pm 6\,\%$ for $(C_{\lambda b},C_{w})$ when the measurements are normalised by the mean or total buoyancy flux. From the standard deviation of $\pm 7\,\%$ that Paillat & Kaminski (Reference Paillat and Kaminski2014a) reported for $C_b$ (based on measurements from ten experiments), we calculate a confidence interval of $\pm 5\,\%$. The $\pm 2\,\%$ interval determined from our experiments (§ 5) compares favourably with these past measurements.

Figure 3 also shows the link between $\alpha$ calculated using the pair ${(C_{Q},C_{M})}$ and based on $\alpha _T=\mathrm {d} (Q^2/M)/ \mathrm {d} z$, as the values determined from these two approaches using the data of Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) are almost identical. While this agreement is not surprising given the dual reliance on the measurements of the volume and momentum flux, the difference in the confidence intervals calculated using either approach highlights that care should be taken when evaluating the uncertainty in an experiment or a particular pair. In this case, where $Q$ and $M$ are determined from the same velocity profile and thus correlated, the approach described in § 3 may not be the most appropriate; the narrower confidence intervals around the value determined from $\alpha _T$ are likely better reflections of the precision achieved when using measurements of the volume and momentum flux.

The range of values for $\lambda$ calculated from different coefficient pairings (figure 4) can be analysed in a similar manner, and we only highlight the following points herein. The variation in the individual values for $\lambda$, and the extent of the confidence intervals, is larger than for $\alpha$, although this is not surprising given the larger uncertainty multipliers for $\lambda$ (table 2). The representative values reported in table 4 for these datasets were determined in a similar manner to $\alpha$: determining the median values and rounding to the nearest $0.05$.

Figure 4. Values of $\lambda$ calculated using different coefficient pairs. Error bars show $95\,\%$ confidence intervals for $\lambda$ calculated using (3.2b), for pairs where the confidence interval for the measurements could be determined. Note: PBPL (Parker et al. Reference Parker, Burridge, Partridge and Linden2020), PK (Paillat & Kaminski Reference Paillat and Kaminski2014a), RC (Ramaprian & Chandrasekhara Reference Ramaprian and Chandrasekhara1989).

4.8. Summary of the analysis of the experimental literature

The primary objective of the preceding analysis was to determine the underlying reasons for the wide variation in the reported values for $\alpha$, which we achieved by reasoning why three of the largest reported values for $\alpha$ are not appropriate. While doubts about aspects of the experiments of Rouse et al. (Reference Rouse, Yih and Humphreys1952), Lee & Emmons (Reference Lee and Emmons1961) and Kotsovinos (Reference Kotsovinos1975) have been previously raised, we believe we are the first to identify that an alternate value of $\alpha$ can be calculated from the Lee & Emmons (Reference Lee and Emmons1961) measurements. In addition, using the measurements of Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) we showed that the theoretical extension to determine $\alpha$ from a multitude of coefficient pairings was appropriate, as the many calculated values were broadly in agreement, allowing for uncertainty (figure 3). By considering multiple pairs, we also determined a single representative value of $\alpha$ for each of the datasets reported by Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989), Paillat & Kaminski (Reference Paillat and Kaminski2014a) and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020). Table 4 is the culmination of our analysis and contains a curated list of values which we have reasoned represents the spread in the values of $\alpha$. The range $0.095\lesssim \alpha \lesssim 0.13$ is narrower than $0.10\lesssim \alpha \lesssim 0.20$, and more appropriately represents the uncertainty surrounding $\alpha$. Also summarised therein are the values of $\lambda$ determined from these datasets, and we consider $\lambda =1.2$ (average value, rounded to 2 s.f.) to be the representative value.

5.1. General set-up and approach

Experiments to determine both $C_e$ and $C_{\lambda b}$ were initiated by steadily releasing a sodium chloride solution from a slender rectangular source into a freshwater-filled clear acrylic visualisation tank of horizontal dimensions $2770~\textrm {mm} \times 250~\textrm {mm}$ and vertical dimension 500 mm. After accounting for residual temperature differences between the freshwater and saline solution, the source density difference was determined to within $1\,\%$ using an Anton Paar 5000M densitometer.

The source was a slot of length $L=220$ mm machined in the lower face of a bespoke constant head tank, centred in the visualisation tank and supported 400 mm above the base. Using digital callipers, the width of the slot ($s=2b_0$) was measured as $3.00\pm 0.15$ mm at ten points along its length, giving $L/s = 73.3$. Besides the $15$ mm wall thickness at either end, the source spanned the entire $250$ mm dimension of the visualisation tank. The constant head tank was supplied with saline solution using an Ismatec MCP-Z Process gear pump. Near uniform conditions were produced along the full length of the slot as shown in figure 5. While the majority of the experiments considered the central plane of the visualisation tank ($y=0$), ten experiments involved measurements at $y\approx 0.25 L$ to verify the two-dimensionality of the flow field. The source flow rate $Q_0$ (cm$^{2}$ s$^{-1}$) was determined by tracking the free surface in the constant head tank (horizontal dimensions $486 \pm 1$ mm by $200\pm 1$ mm). Recording the vertical position of the free surface at three frames per second (f.p.s.) and a resolution of 20 pixels per cm (approximately), $Q_0$ was determined to within $1\,\%$. Table 5 reports the mean flow rate during an experiment and the largest deviation from the mean (typically less than $2\,\%$).

Figure 5. Images of the span of the plume ($y$–$z$ plane). (a) An instantaneous snapshot showing that the flow appears uniform and turbulent along the length $L$ of the source. (b) A time-averaged image of the near-source region showing the horizontally integrated dye concentration, a proxy for the amount of dye introduced from the source. Averaged vertically between the indicated depths ($z/b_0\approx 1$ and $z/b_0\approx 10$), the concentration varies from the mean by $10\,\%$ over the central $90\,\%$ of the span and by only $5\,\%$ over the central $50\,\%$.

Table 5. Experimental parameters and measurements for the 32 entrainment coefficient experiments. The distance $z_p$ is an estimate of the distance required for the plume to adjust from its source condition, as characterised by the Richardson number $\varGamma _0$, to ‘pure-like’ behaviour. The values of $\alpha$ and $\lambda$ for an individual experiment are calculated via (5.3a,b). The percentage uncertainties are described in the main text: $Q_0, g'_0$ and $B_0$ (§ 5.1); $C_e$ (§ 5.2) and $C_{\lambda b}$ (§ 5.3). In the starred experiments, measurements of $C_e$ were conducted at $y\approx 0.25 L$ rather than in the central plane of the visualisation tank.

We conservatively report the uncertainty in the source buoyancy flux ($B_0=Q_0g'_0$) as the sum of the uncertainties in $Q_0$ and $g'_0$. As $C_e$ is calculated by normalising measurements of $u_e$ by $B_0^{1/3}$, the uncertainty in $B_0$ typically affects the values of $C_e$ by less than $1\,\%$, and by $2\,\%$ in the worst case. This variation resulting from the uncertainty in the source conditions is significantly smaller than the actual variation in the measurement of $u_e$ (§ 5.2).

The entrainment velocity and scalar width were determined from a set of 32 experiments, the details of which are summarised in table 5. The source flow rate $Q_0$ and reduced gravity $g'_0$ were selected to approximate the dynamical conditions of a pure turbulent plume. The source Reynolds number $Re_0$ is the ratio of $Q_0$ to the kinematic viscosity of the source fluid; typically $Re_0=200 - 300$. How ‘pure like’ the release was at source was characterised using the scaled source Richardson number $\varGamma _0$, where $\varGamma _0=1$ indicates a pure plume (van den Bremer & Hunt Reference van den Bremer and Hunt2014a). Assuming top-hat profiles at the source, as is reasonable for the high Reynolds number release conditions,

(5.1)\begin{equation} \varGamma_0=2^{3/2}\frac{b_0^3g'_0}{\alpha Q_0^2}. \end{equation}

The entries for $\varGamma _0$ in table 5, based on $\alpha =0.11$, confirm that the plumes were nominally pure at source. Moreover, using the van den Bremer & Hunt (Reference van den Bremer and Hunt2014a) solutions for an area source plume, the downstream distance $z_p$ over which the Richardson number adjusts from the value at source to within $10\,\%$ of $\varGamma =1$ was estimated to be $z_p/b_0\approx 0 - 10$, while the measurement region of $65 \lesssim z_m/b_0 \lesssim 130$ was considerably further downstream. Given $z_m \gg z_p$ our measurements were recorded within the pure-plume equilibrium region.

The visualisation tank was backlit with a white LED panel and images of the flow recorded at a frame rate of $12.5$ f.p.s. using a 5 mega-pixel monochrome camera (JAI SP-5000M-USB) with a $75$ mm fixed focal length lens (Kowa Optimed) and a $660$ nm bandpass filter (Midwest Optical Systems). The spatial resolution was $42$ pixels per cm (approximately). Figure 6 shows an image taken from one experiment, annotated to indicate the regions where the plume width and entrainment velocity were measured. The representative length scale $b_r$, defined as the average scalar half-width at the midpoint of the measurement region (estimated using our measured value $C_{\lambda b}=0.126$), and used as a reference for interpretation of the measurements of $C_e$, is also shown. Details specific to the measurement of $C_e$ and $C_{\lambda b}$ are discussed in §§ 5.2 and 5.3, respectively.

Figure 6. Instantaneous image of a saline plume and its induced-flow field. The entrainment coefficient was determined from combined measurements of: (i) the average velocity of dyeline segments for ${x \in -[16b_r,5b_r]}$ (approximately) and (ii) the plume scalar width in the region bounded by the dashed white outline. The representative length scale $b_r$ is the plume half-width evaluated at the midpoint of the measurement region ($z=97.5b_0$). Circles at $x=-23b_r$ indicate the sources of the dyelines.

5.2. Measurement of $C_e$

The velocity $u_e$ was determined by tracking six dyeline segments per experiment in the induced-flow field. The dyelines emerged at three depths from hypodermic needles (figure 6) connected to a syringe pump programmed to dispense laminar pulses of dyed freshwater (density within $0.01\,\%$ of the environment). Needles of different length were used to release dyelines at $y=0$ (Exps. 1–22) or at $y=56$ mm (Exps. 23–32). The horizontal position of the leading edge of each segment was tracked from $x/b_r=-16$ until the segment was distorted by turbulent fluctuations, which typically occurred $2b_r - 4b_r$ from the plume centreline; $90\,\%$ of the segments were tracked for at least $10b_r$, a distance large relative to the plume width. A typical segment covered this distance ($\sim 21$ cm, $\sim 900$ pixels) in 200 frames ($\sim 16$ s) and thus the velocity of each segment, determined from the gradient of a least-squares linear fit to the position-against-time data (figure 7a) was well resolved. Given the spatial uniformity of the predicted induced-flow field (Taylor Reference Taylor1958), the variation in the velocity of the segments was expected to be small. This was indeed the case, with the relative standard deviation being only $3\,\%$ of the average velocity for the six segments shown in figure 7. Similarly small variations were recorded for the segments tracked during Exp. 1 through to Exp. 32 (table 5).

Figure 7. (a) Dimensionless position $x/b_r$ of the leading edge of six dyeline segments against time $t_s B_0^{1/3}/b_r$ from Exp. 17. The time origin $t_s=0$ was taken as the first frame where the leading edge of a segment crossed the ‘start’ line at $x/b_r=-16$. (b) Entrainment velocity $u_e/B_0^{1/3}$ against measurement time $t_fB_0^{1/3}/W$, where $W$ ($=2770$ mm) is the tank width, for all 192 segments measured to determine $C_e$ (Pearson's correlation coefficient of $-0.18$. The $p$-value is $0.01$ and the null hypothesis of no correlation is rejected at the $5\,\%$ significance level). Ensemble average velocity and measurement time for the first ($\blacksquare$) and second ($\blacktriangle$) waves of 96 segments.

Tracking occurred during the quasi-steady period after the starting plume vortex had passed and before the measurement region filled with buoyant fluid. To assess whether the measured velocities were influenced by filling-box effects, the leading edge of two consecutive ‘waves’ of segments were tracked. Figure 7(b) plots the normalised velocity for each segment against the mean time that that measurement was made. The time origin $t_f=0$ corresponds to the instant the plume source was activated. Averaged over 96 segments released in the first wave, $C_e=0.231$ while $C_e=0.229$ over the second. The difference is small relative to the overall scatter in the measurements, and both values fall within the confidence interval we report for $C_e$. Thus, we conclude that the entrainment velocity was not significantly impacted by any transient filling behaviour.

5.3. Measurement of $C_{\lambda b}$

Measurements of the light attenuated by methylene blue dye mixed into the source fluid were used to infer the salt concentration in the plume (Cenedese & Dalziel Reference Cenedese and Dalziel1998; Allgayer & Hunt Reference Allgayer and Hunt2012). The time-averaged concentration field and the corresponding buoyancy profiles from a typical experiment are shown in figure 8(a,b), respectively. The concentration field has a resolution of 407 pixels (vertical) by 600 pixels (horizontal) and is an average of 200 frames recorded over $\tau _a=16$ s. The averaging time $\tau _a$ represents the available measuring time, which was limited primarily by the physical scale of the visualisation tank. Importantly, $\tau _a$ was large relative to two physical time scales $(\tau _e,\tau _t)$ that characterise the flow, namely, the time scale associated with a typical eddy $\tau _e$ (a turnover time) and the time scale required for a notional eddy to traverse the vertical extent ($\sim 65b_0$) of the measurement window $\tau _t$. Using the centreline velocity as the representative velocity scale, we define

(5.2a,b)\begin{equation} \tau_e=\frac{2C_{\lambda b} z_m}{C_wB_0^{1/3}} \quad \mbox{and} \quad \tau_t=\frac{65b_0}{C_wB_0^{1/3}}, \end{equation}

and take $C_w=2.157$ (i.e. the value used to estimate the uncertainty multipliers (§ 3)) and $C_{\lambda b}=0.126$ from our measurements. Based on the source conditions (table 5), ${\tau _e \approx 0.16 - 0.48 ~\mathrm {s}}$ and so we expect that 30 to 100 ‘eddies’ would form at a particular depth in the interrogation window during the averaging period. Moreover, $\tau _t \approx 0.65 - 0.96$ s and, thus, we expect between 16 and 25 ‘cycles’ where plume fluid traversing the measurement window is entirely refreshed. Given $\tau _a/\tau _e \gg 1$ and $\tau _a/\tau _t \gg 1$, and the Gaussian-like concentration profiles in figure 8(b), we conclude that our measurements appropriately capture a time-averaged line plume.

Figure 8. (a) Time-averaged image of the plume in the measurement window for Exp. 14 showing the expected ‘cigar-shaped’ contours of constant buoyancy. (b) In grey, time-averaged dimensionless buoyancy profiles recorded at ten different heights in the plume. In blue, a Gaussian fit calculated using the value $C_{\lambda b}=0.125$ measured in Exp. 14 shows the Gaussian model is appropriate. The virtual origin $z_0$ was determined by extrapolating the variation of the $1/e$ scalar width with depth to zero width. (c) The half-width of the buoyancy profile using the $g'/g'_c=1/e$ and $g'/g'_c=1/2$ thresholds.

Figure 8(c) shows the variation of $\lambda b$ and the half-maximum half-width $\lambda b_{1/2}$, i.e. the horizontal distance from the centreline to the point where $g'=0.5g'_c$. The gradient of the $\lambda b$ line, determined using a least-squares linear fit, gives $C_{\lambda b}$. For a Gaussian profile, the ratio of the half-widths $r_G=\lambda b_{1/2}/\lambda b=\sqrt {\ln 2}\approx 0.83$; for the buoyancy profile shown in figure 8(c), this measured ratio is $r_M=0.81$. Averaged across the 32 experiments, we measured $r_M=0.80$, which is within $4\,\%$ of the expected Gaussian value. The ratio of these widths is used to give an indication of the uncertainty in the measurements of $C_{\lambda b}$, and the reported uncertainty estimates in table 5 are values of $|1-r_M/r_G|$.

5.4. Overall results

Our measurements of $C_e$ and $C_{\lambda b}$ from the 32 experiments are plotted in figure 9. Figures 9(a) and 9(b) indicate the mean values and variation of the measurements. Figure 9(c) shows that the variation in $C_{\lambda b}$ is not correlated with the variation in $C_e$, and thus we conclude that it is reasonable to apply the analysis from § 3 to determine the uncertainty in $\alpha$ and $\lambda$. Following the framework established in § 2, our estimates for $\alpha$ and $\lambda$ are

(5.3a,b)\begin{equation} \alpha=C_e^{3/2}\sqrt{\frac{\sqrt{2A^2+1}-1}{A}}, \quad \lambda =\sqrt{\frac{2}{\sqrt{2A^2+1}-1}}, \end{equation}

where $A=8C_e^3/({\rm \pi} C_{\lambda b}^2)$. The associated uncertainty multipliers are given in Appendix A.

Figure 9. (a) Values of $C_e$. (b) Values of $C_{\lambda b}$. In (a), in grey, values for every individual dyeline segment ($6 \times 32$ dots). In (a,b), $\bullet$ average value for each of the 32 experiments; $\blacktriangle$ average value and standard deviation across the centred dyeline experiments; $\blacksquare$ average value and standard deviation across all 32 experiments. (c) Value of $C_e$ against $C_{\lambda b}$ for each of the 32 experiments, grey dots denote an off-centred dyeline experiment; the scatter indicates there is no clear correlation between the two quantities (Pearson's correlation coefficient of $0.19$. The $p$-value is $0.29$ so the null hypothesis of no correlation is not rejected at the $5\,\%$ significance level).

Table 6. The entrainment coefficient $\alpha$ and profile coefficient $\lambda$ determined in the present study from measurements of $C_e$ and $C_{\lambda b}$. The values of the standard deviation (SD) and the $95\,\%$ confidence interval (CI) for $\alpha$ and $\lambda$ were calculated using the corresponding values in the measurements and the uncertainty multipliers for the pair (A9a,b). Results are shown as a summary of all experiments and of the subsets categorised by dyeline position.

Table 6 summarises the experimental data and the calculated values for $\alpha$ and $\lambda$ for all the experiments and for the centred and off-centred subsets. There are small differences between measurements depending on the dyeline location. As $C_{\lambda b}$ is determined from measurements of light attenuation through the length of the plume, the mean value of $C_{\lambda b}$ should be unaffected by the dyeline position; the $2\,\%$ difference between the centred and off-centred subsets is attributed to random variation. The mean $C_e$ values measured at each dyeline needle position are within $3\,\%$ of each other, and this similarity supports the claim that the flow field is approximately two-dimensional. The small difference between the mean $C_e$ values may result from the effects of the end walls, small variations in the conditions along the plume source or random variation. The difference in the values of $\alpha$ and $\lambda$ determined from all experiments or from only the centred experiments is small compared with the variation between other reported values (table 4). In the absence of a detailed study of the variations along the span of a line plume, only the measurements from the experiments with centred dyelines were used to determine the key result: $\alpha =0.108$ with a $95\,\%$ confidence interval of $2\,\%$.