2.1. Morton's (Reference Morton1959) analytical solution

Morton (Reference Morton1959) further assumed self-similar Gaussian velocity and buoyancy profiles, a constant entrainment coefficient and considered only the mean components of the profile coefficients (subscript $m$), to obtain (2.12a–c),

(2.12a–c)\begin{equation} \frac{\mathrm{d}Q}{\mathrm{d}z} = 2\alpha M^{1/2},\quad \frac{\mathrm{d}M}{\mathrm{d}z} = \frac{FQ}{\theta_mM},\quad \frac{\mathrm{d}F}{\mathrm{d}z} = 0. \end{equation}

In this case, all of the profile coefficients in (2.8) are constants, and an analytical solution was derived for a positively or negatively buoyant jet originating from a point source. The solution to this system of equations is plotted here in figure 1 for the case of a negatively buoyant jet ($F<0$). The following new variables are used to interpret the solution,

(2.13)\begin{gather} q =\alpha^{-1/2}\theta_m^{-1/2}|F_o|^{1/2} |M_o|^{-5/4}Q, \end{gather}

(2.14)\begin{gather}m = \frac{M}{|M_o|}, \end{gather}

(2.15)\begin{gather}\zeta = 2\alpha^{1/2}\theta_m^{-1/2}|F_o|^{1/2} |M_o|^{-3/4}z, \end{gather}

where the constant $M_o$ is the initial momentum flux at the point source and $F_o=F$ is the buoyancy flux. Integral velocity, width and buoyancy scales can be defined based on the above variables (Morton Reference Morton1959),

(2.16a–c)\begin{equation} \hat{w}_m = \frac{m}{q}, \quad \hat{r}_m = \frac{q}{m^{1/2}}, \quad \hat{b}_m = \frac{1}{q}. \end{equation}

These variables allow new ‘scaled’ local Froude and Richardson numbers to be defined,

(2.17)\begin{equation} \widehat{Fr}=\frac{\hat{w}_m}{(\hat{r}_m\hat{b}_m)^{1/2}} = \frac{1}{\widehat{Ri}^{1/2}}. \end{equation}

The solution to (2.12) for a negatively buoyant jet originating from a point source is plotted against $\zeta$ in figures 1(a) and 1(b) in terms of the integral scales defined in (2.16) (Morton Reference Morton1959). An additional term, $\hat H_2 = -{\hat r_m}/({2\hat w_m})({\mathrm {d}\hat w_m}/{\mathrm {d}\zeta })$, is plotted in figure 1(c) and is discussed in more detail in § 7. The velocity scale, $\hat {w}_m$, decreases with $\zeta$ until it reaches its maximum height at $\zeta _i=1.454$ where $\hat {w}_m=0$ and $\widehat {Fr}=0$. Here, the width scale, $\hat {r}_m$, approaches infinity and the buoyancy scale decreases to a finite value, $\hat {b}_m=0.791$. Figure 1(b) reveals that $1/\hat {w}_m$, $1/\hat {b}_m$ and $\hat {r}_m$ increase approximately linearly with $\zeta$ for $\zeta \lesssim 0.5$ ($\widehat {Fr}\gtrsim 2$), implying that the velocity and buoyancy scales are nearly proportional to $\zeta ^{-1}\sim z^{-1}$, and radial scale to $\zeta \sim z$, in the lower portion of the jet. Although $\hat w_m$, $\hat b_m$ and $\hat r_m$ all change continuously along the negatively buoyant jet, the flow may still be characterised as consisting of two separate regimes. A ‘forced regime’ where the scales may be approximated as linear with $\zeta$ (for $\zeta \lesssim 0.5$, $\widehat {Fr}\gtrsim 2$), and a ‘buoyancy dominated’ regime where this approximation is no longer suitable. The $\widehat {Fr}\lesssim 2$ regime is characterised by the strong deceleration of the flow, which is captured by the non-dimensional term $\hat H_2$. This is plotted in figure 1(c) against $\widehat {Fr}$ for both a negatively buoyant jet (NBJ) and neutral jet (J). For $\widehat {Fr}\gtrsim 2$ in the NBJ, $\hat H_2$ is small and similar to the neutral jet value of $\hat H_2=0.5$, corresponding to the ‘forced’ regime. After this, for $\widehat {Fr}\lesssim 2$, $\hat H_2$ rapidly increases with decreasing $\widehat {Fr}$ as the flow is strongly decelerated, corresponding to the ‘buoyancy dominated’ regime.

Figure 1.

The solution to the system of ordinary differential equations in (2.12) for the case when $F<0$ (i.e. a NBJ). The solution assumes fully self-similar velocity and buoyancy profiles and a constant entrainment coefficient, $\alpha$, and is presented in terms of the integral buoyancy, velocity and width scales defined in (2.16). Plots of $\hat {b}_m$, $\hat {r}_m$, $\hat {w}_m$, $\widehat {Fr}$ and $\hat H_2 = -{\hat r_m}/({2\hat w_m})({\mathrm {d}\hat w_m}/{\mathrm {d}\zeta })$ against $\zeta$ are shown in (a). The quantities $1/\hat {b}_m$ and $1/\hat {w}_m$ are additionally plotted in (b), which, for relatively small $\zeta$, are approximately proportional to $\zeta$. The quantities $\hat {b}_m$, $1/\hat {r}_m$, $\hat {w}_m$ and $\hat H_2$ are plotted against $\widehat {Fr}$ in (c). For a neutral jet (J), $\hat H_2=0.5$ everywhere, which is also plotted in (c) as a horizontal line.

In the analytical solution for high $Re_o$ self-similar neutral jets, which have $Fr=\infty$, $w_m$ scales with $z^{-1}$ exactly (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979), and so it may be expected that the velocity in self-similar, constant $\alpha$, NBJs scales in approximately the same way for sufficiently high local $Fr$ (i.e. in the forced regime). Similarly, the scalar concentration and width scales in a neutral jet are also proportional to $z^{-1}$ and $z$ respectively, just as is approximately true for $\hat {b}_m$ and $\hat {r}_m$ in the forced regime. Figure 1(c) also shows how these scales change with $\widehat {Fr}$, with $\hat {b}_m$, $1/\hat {r}_m$, $\hat {w}_m$ all scaling approximately linearly for high $\widehat {Fr}$.

The present experimental results show that, even in the forced regime ($\zeta \lesssim 0.5$, $\widehat {Fr}\gtrsim 2$), there are significant differences between the behaviour of NBJs and what is captured in this simplified model. These include turbulence intensities and shear stress that scale differently to the mean flow, discussed in §§ 4.3 and 5, a non-constant entrainment coefficient discussed in § 6, and the spreading of the velocity and buoyancy profiles discussed in § 7. Although these effects can be distinguished from each other, as are shown in the following sections they are also linked. For example, the turbulence intensities and shear stress have an effect on both entrainment and the spreading rate, as is discussed in §§ 6 and 7.

4.1. Centreline decay

For a self-similar neutral jet with constant $\alpha$, the decay of the centreline velocity along the jet axis follows the relation,

(4.1)\begin{equation} \frac{w_o}{\bar{w}_c} = K\left(\frac{z}{D} - \frac{z_o}{D}\right), \end{equation}

where $w_o$ is the velocity at the source, $\bar {w}_c$ is the velocity at the jet centreline, $z_o$ is a virtual origin and $K$ is a constant (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Papanicolaou & List Reference Papanicolaou and List1988; Hussein, Capp & George Reference Hussein, Capp and George1994). For a neutral jet, the centreline velocity therefore scales with $z^{-1}$. There is some variation in the literature for the value of the constant $K$, such as $K=0.17$ by Hussein et al. (Reference Hussein, Capp and George1994) or $K=0.149$ by Papanicolaou & List (Reference Papanicolaou and List1988), but a robust finding is the linear $w_o/\bar {w}_c$ relation with $z$. Figure 6 shows $w_o/\bar {w}_c$ plotted against $z/D$ for neutral and negatively buoyant jets. A linear fit of our neutral jet data gives $K=0.142$, in good agreement with Papanicolaou & List (Reference Papanicolaou and List1988). The virtual origin for the present experiments is found to be close to the actual source at $z_o/D=0.707$.

Figure 6.

The quantity $w_o$/$\bar {w}_c$ plotted against axial distance $z/D$, showing the decay of the centreline velocity for a neutral and negatively buoyant jet. Each cluster of points was obtained from a separate experiment with the same source Froude number ($Fr_o=30$ for the NBJ) and similar Reynolds numbers ($5500\lesssim Re_o\lesssim 5900$). The prediction of Morton's (Reference Morton1959) model for a constant $\alpha =0.0714$ and $\lambda =1.2$ is shown, as well as ‘model 2’, a linear model for $\alpha$ with $Ri$ using coefficients based on values found by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005). Additionally, ‘model 1’, which assumes a linear relationship for $\delta _m$ with $Ri$, is also shown. These models are discussed further in § 6.

For the NBJ, the decay is not linear over the $z/D$ range shown, and can be seen diverging from the neutral jet results for $z/D\gtrsim 18$. As $z/D$ increases the local $Fr$ decreases towards zero (and $Ri$ asymptotes to negative infinity), which may be interpreted as negative buoyancy playing an increasingly important role in decelerating the flow. However, for the points closest to the source, e.g. for $z/D\lesssim 26$ ($Fr\gtrsim 3.0$, $Ri\gtrsim -0.11$), $w_o/\bar {w}_c$ could be approximated as linear with $z$, although with a different slope to a neutral jet. This may be considered the ‘forced’ regime where the flow is more similar to a neutral jet, and is consistent with the arguments made in § 2 regarding the solution to Morton (Reference Morton1959) simplified model. Although this is a local regime based on local $Fr$, it may be compared to the classification of fountains by source Froude number, $Fr_o$, such as those suggested by Burridge & Hunt (Reference Burridge and Hunt2012). They classified $2.8\lesssim Fr_o\lesssim 5.5$ and $Fr_o\gtrsim 5.5$ as ‘forced’ and ‘highly forced’ fountains, which is consistent with the presently suggested local ‘forced’ regime of $Fr\gtrsim 3.0$.

4.2. Velocity and buoyancy profiles

Time-averaged profiles for axial velocity and buoyancy, where buoyancy is presented in terms of scalar concentration $\bar {c}$, are given in figure 7(a,b), normalised by their centreline values and respective half-widths. Dimensionless concentration ($0\leq c\leq 1$) and buoyancy (mm s$^{-2}$) are related by a constant such that $b=c(\rho _o-\rho _e)g/\rho _e$. All NBJ profiles are close to Gaussian for the full range of local $Fr$ investigated, $1.85\lesssim Fr\lesssim 5.91$, and are similar to the profiles for neutral jets. This is despite the considerable deceleration of the mean flow in the NBJ compared to the neutral jet, as demonstrated in figure 6, showing that the profiles maintain a Gaussian shape even outside of the forced regime ($Fr\lesssim 3.0$).

Figure 7.

Time-averaged axial velocity and scalar concentration (buoyancy) profiles, in (a) and (b) respectively, of a NBJ with different local Froude numbers. Neutral jet (J) data from the present experimental set-up are also shown ($Re_o=5900$), as well as by Webster, Roberts & Ra'ad (Reference Webster, Roberts and Ra'ad2001), Wang & Law (Reference Wang and Law2002) and Darisse, Lemay & Benaïssa (Reference Darisse, Lemay and Benaïssa2015). The NBJ profiles were obtained using data from multiple experiments using the same pipe, $D=10$ mm, at the same source Froude number, $Fr_o=30$, and similar Reynolds number, $5500\lesssim Re_o\lesssim 5900$, while varying the location of the region of interest relative to the source. All velocity and scalar concentration points have been normalised by their respective centreline values, $\bar {w}_c$ and $\bar {c}_c$, and half-widths, $r_{1/2,w}$ and $r_{1/2,c}$.

The ratio of widths between the buoyancy/scalar and velocity profiles is given by $\lambda$ and defined in (2.9). For neutral jets, $r_b$ corresponds to the $1/e$ width of the scalar profile, with values estimated in the literature ranging $1.15\lesssim \lambda \lesssim 1.30$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Wang & Law Reference Wang and Law2002; Ezzamel, Salizzoni & Hunt Reference Ezzamel, Salizzoni and Hunt2015), and is assumed to be constant in the far field where the flow is self-similar. As pointed out by Ezzamel et al. (Reference Ezzamel, Salizzoni and Hunt2015), discrepancies in the literature may be attributed to the distance from the source where the profiles were measured (e.g. if the jet has not fully developed), and that it is likely that source conditions play a role.

Figures 8(a) and 8(b) show $\lambda$ plotted with axial distance and local $Ri$ for both the neutral and NBJs from the present experiments. The values for the neutral jet are reasonably constant, and have an average value of $\lambda =1.181$, in good agreement with $\lambda _j=1.189$ from Fischer et al. (Reference Fischer, List, Koh, Imberger and Brooks1979). The slight decreasing trend may be attributed to the jet still developing at this axial distance. For the NBJ, $\lambda$ is higher than the neutral jet and increases with axial distance over the range shown. From figure 8(b), $\lambda$ can be seen to be increasing almost immediately from $Ri\cong -0.04$ ($Fr\cong 5.0$) with more negative $Ri$, which is within the previously suggested ‘forced’ regime of $Ri\gtrsim -0.11$ ($Fr\gtrsim 3.0$). So even for relatively high local $Fr$, the velocity and buoyancy/scalar profile widths grow at different rates compared with a neutral jet.

Figure 8.

The $1/e$ width ratio between the buoyancy (or scalar) and velocity profiles, $\lambda =r_b/r_w$, for a NBJ and neutral jet (J). The data are plotted against axial distance normalised by source diameter $z/D$ in (a), and against the local $Ri$ in (b). At the source, $Fr_o=30$ ($Ri_o=-0.0011$), which decreases towards $Fr=0$ ($Ri\rightarrow -\infty$) at the top of the NBJ. In (b), the values for a neutral jet, which have $Ri=0$ everywhere, are shown as horizontal lines for clarity.

A varying $\lambda$ can be interpreted as a type of ‘similarity drift’ of the velocity and buoyancy profiles, which has also been reported in jets and plumes that have not yet reached a state of full self-similarity (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; Carazzo et al. Reference Carazzo, Kaminski and Tait2006; Ezzamel et al. Reference Ezzamel, Salizzoni and Hunt2015). The mechanism causing the increasing $\lambda$ with $z$ for negatively buoyant jets is discussed in § 7.

4.3. Turbulence statistics

Figures 9 and 10 show the profiles of the axial turbulence intensity, $\overline {w'^2}/\bar {w}_c^2$, and the normalised Reynolds stress, $\overline {w'u'}/\bar {w}_c^2$. The profiles at $z/D=73$ for the neutral jet in figures 9(a) and 9(b) are in good agreement with both the best fit curve from Wang & Law (Reference Wang and Law2002) and data from Darisse et al. (Reference Darisse, Lemay and Benaïssa2015). This experiment used a smaller, $D=5$ mm, pipe so that measurements could be taken at a larger downstream distance relative to the source diameter, and a high $Re_o=10\ 700$ could be achieved. The flow in this experiment could therefore be expected to be fully developed and self-similar and so could be compared to similar experiments in the literature, such as Wang & Law (Reference Wang and Law2002) who had $Re_o=12\ 700$ and $40<z/D<80$. The remaining J profiles in figure 9 correspond to experiments using a $D=10$ mm pipe and have $Re_o=5900$ in order to closely match the source conditions of the NBJ experiments. These $18<z/D<27$ profiles are generally a little lower than the $z/D=73$ case since they may not be completely developed at this distance, but are nevertheless reasonably close and can be compared to a negatively buoyant jet at the same axial location.

Figure 9.

Mean turbulent axial velocity fluctuations (a), and Reynolds stress (b), for the neutral jet (J) at three different axial locations, normalised by the local mean centreline velocity, $\bar {w}_c^2$. The $z/D=18$, $23$ and $27$ profiles correspond to an experiment with $D=10$ mm and $Re_o=5900$. The $z/D=73$ profiles were obtained used $D=5$ mm and $Re_o=10\ 700$. The best fit curve from Wang & Law (Reference Wang and Law2002) ($40<z/D<80$) and the data from Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) ($z/D=30$) for a neutral jet are also shown.

Figure 10.

Mean turbulent axial velocity fluctuations (a), and Reynolds stress (b), for a NBJ at several axial locations, normalised by the local mean centreline velocity, $\bar {w}_c^2$. The source conditions were $Fr_o=30$ and $5500\lesssim Re_o\lesssim 5900$ using $D=10$ mm, with data gathered across multiple experiments.

Figures 10(a) and 10(b) shows the normalised $\overline {w'^2}$ and $\overline {w'u'}$ profiles of a NBJ at some of the same axial distances as the neutral jet in figure 9, as well as two additional further downstream locations. Despite some scatter in the data, there is a clear upwards trend for both the axial turbulence intensity and Reynolds stress relative to the centreline velocity with increasing axial distance, or equivalently, decreasing local $Fr$. This is most evident for the $Fr\lesssim 3.38$ profiles as the flow exits the ‘forced’ regime, and is particularly strong in the $\overline {w'^2}/\bar {w}_c^2$ plot shown in (a). This does not imply that the magnitude of $\overline {w'^2}$ or $\overline {w'u'}$ is increasing with distance, but is instead revealing that $\overline {w'^2}$ and $\overline {w'u'}$ do not decrease at the same rate as the mean flow. This is consistent with the qualitative description of a NBJ, where the mean velocity is reduced to zero at the top of the jet, $z_i$, but where we can still expect non-zero turbulence.

Cresswell & Szczepura (Reference Cresswell and Szczepura1993) also obtained $\overline {w'^2}$ and $\overline {w'u'}$ profiles, but for a fully developed fountain with $Fr_o\cong 3.2$. When their data (originally presented normalised by source conditions) are normalised by the centreline velocity, the peak values also increase with axial distance from the source as the mean flow decelerates. Near to the source, $0.03\lesssim z/D\lesssim 1.7$, their peak values for the IF cover the range $0.05\lesssim \overline {w'^2}/\bar {w}_c^2\lesssim 0.2$ and $0.01\lesssim \overline {w'u'}/\bar {w}_c^2\lesssim 0.03$, which is broadly similar to the present values. However, the flow of Cresswell & Szczepura (Reference Cresswell and Szczepura1993) was at a much lower $Fr_o\cong 3.2$ and also included a return flow, and so is notably different to the present case of $Fr_o=30$ NBJs. The phenomenon of increasing turbulence intensities in a decelerating mean flow is not exclusive to NBJs/fountains. In a flow through a conical diffuser, for example, increasing turbulence fluctuations and shear stresses, relative to the local centreline velocity, can be seen with increasing axial distance as the flow expands and decelerates (Okwuobi & Azad Reference Okwuobi and Azad1973; Singh & Azad Reference Singh and Azad1995). Although such a flow is significantly different to the present case, since the evolution of turbulence with axial distance is affected by the velocity shear near the wall (even at the centreline Singh & Azad Reference Singh and Azad1995), a decelerating mean flow will still work to increase the turbulence intensity if it is normalised in this way.

Figures 11 and 12 show the turbulent scalar fluctuation and axial and radial fluxes, $\sqrt {\overline {c'^2}}$, $\overline {w'c'}$ and $\overline {u'c'}$, normalised by the centreline values and scalar half-widths for the neutral and NBJ at different axial distances. The neutral jet profiles in figure 11 generally all agree with Wang & Law (Reference Wang and Law2002) and Webster et al. (Reference Webster, Roberts and Ra'ad2001). The $\sqrt {\overline {c'^2}}/\overline {c_c}$ data for $z/D\geq 23$ are in very close agreement with Webster et al. (Reference Webster, Roberts and Ra'ad2001), although the shortest axial location $z/D=18$ is slightly higher near the centreline. The $\sqrt {\overline {c'^2}}/\overline {c_c}$ centreline value of the best fit by Wang & Law (Reference Wang and Law2002) ($Re_o=12\ 700$, $40<z/D<80$) is slightly lower than the present jet data ($Re_o\cong 5700$, $18<z/D<27$). However, this difference is small ($\lesssim 0.5\,\%$) when compared to the furthest jet experiment ($Re_o=10\ 700$, $z/D=73$), and so the difference may be attributed to the larger $z/D$ and $Re_o$ in the Wang & Law (Reference Wang and Law2002) experiments. The present axial flux data, $\overline {w'c'}/\bar {w}_c\overline {c_c}$, are close to both Wang & Law (Reference Wang and Law2002) and Webster et al. (Reference Webster, Roberts and Ra'ad2001) for $18\leq z/D\leq 27$ in the neutral jet, but here the $z/D=73$ profile is a little higher. The radial flux, $\overline {u'c'}/\bar {w}_c\overline {c_c}$, is in reasonable agreement with both studies at all locations. The negatively buoyant profiles, given in figure 12, are of similar shape and order to the neutral jet data in figure 11, with no discernible trend with axial location evident. Although one might expect to see an increasing trend in $\overline {w'c'}/\bar {w}_c\bar {c}_c$ or $\overline {u'c'}/\bar {w}_c\bar {c}_c$ with axial distance, due to the decelerating mean flow, $\bar {w}_c$, this effect is not noticeable within the experimental scatter. This is likely due to the fact that, unlike with the velocity fluctuation, there is no clear relative increase of the scalar fluctuations compared to $\bar {c}_c$ (which does not go to zero at the top of the jet), and so the effect of a decreasing $\bar {w}_c$ is less significant. Cresswell & Szczepura (Reference Cresswell and Szczepura1993) also obtained these quantities for their $Fr_o=3.2$ fountain using temperature measurements. When normalised by centreline quantities, and treating temperature as a passive scalar, their peak values in the IF for $0.3\lesssim z/D\lesssim 2.3$ covered the range $0.12\lesssim \sqrt {\overline {c'^2}}/\bar {c}_c\lesssim 0.3$, $0.01\lesssim \overline {w'c'}/\bar {w}_c\bar {c}_c\lesssim 0.02$ and $0.004\lesssim \overline {u'c'}/\bar {w}_c\bar {c}_c\lesssim 0.02$. As with the turbulent velocity fluctuations, these are broadly similar to the present range of NBJ values, despite the differences in the flow.

Figure 11.

The turbulent scalar fluctuations and axial/radial flux profiles, $\sqrt {\overline {c'^2}}$, $\overline {w'c'}$ and $\overline {u'c'}$ are given in (a), (b) and (c), respectively, normalised by the centreline values, $\bar {w}_c$ and $\bar {c}_c$, and scalar half-width, $r_{1/2,c}$ for a neutral jet at various axial locations. The data were obtained from the same experiments as figure 9. The best fit curve from Wang & Law (Reference Wang and Law2002) ($30<z/D<80$) and data from Webster et al. (Reference Webster, Roberts and Ra'ad2001) ($50<z/D<90$) are also shown.

Figure 12.

The normalised turbulent scalar fluctuations and axial/radial flux profiles, $\sqrt {\overline {c'^2}}$, $\overline {w'c'}$ and $\overline {u'c'}$ are given in in (a), (b) and (c), respectively, for a NBJ at different axial locations. The data were obtained from the same experiments as figure 10.

6.1. Estimating the entrainment coefficient

For jets and plumes with arbitrary buoyancy, using the conservation of volume, momentum and mean kinetic energy, and making no assumptions about the self-similarity of the profiles, the entrainment coefficient can be expressed as,

(6.1)\begin{equation} \alpha = -\frac{\delta_g}{2\gamma_g} + \left(\frac{1}{\beta_g}-\frac{\theta_m}{\gamma_g}\right)Ri + \frac{Q}{2M^{1/2}}\frac{\mathrm{d}}{\mathrm{d}z} \left(\log\frac{\gamma_g}{\beta_g^2}\right), \end{equation}

where $\log (\cdot )$ is the natural logarithm (van Reeuwijk & Craske Reference van Reeuwijk and Craske2015). In a simplified form, neglecting the turbulence components of the profile coefficients (subscript $f$ in (2.8)), this can be written as (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015),

(6.2)\begin{equation} \alpha_{M} = -\frac{\delta_m}{2\gamma_m} + \left(1-\frac{\theta_m}{\gamma_m}\right)Ri + \frac{Q}{2M^{1/2}}\frac{\mathrm{d}}{\mathrm{d}z}(\log\gamma_m). \end{equation}

The first term, $-\delta _g/2\gamma _g$, is the ratio of turbulence production to mean energy flux, and is the only non-zero term in a self-similar neutral jet and is constant. The second term shows the effect of buoyancy on entrainment through the local Richardson number, and provides a mechanism for buoyancy-driven entrainment associated with the mean flow, rather than by directly affecting turbulence (van Reeuwijk & Craske Reference van Reeuwijk and Craske2015). The third term reflects how the profile coefficients $\gamma _g$ and $\beta _g$ change along the jet, and is zero if the flow is fully self-similar.

By assuming self-similar profiles, (6.2) becomes,

(6.3)\begin{equation} \alpha_{MS} = -\frac{\delta_m}{2\gamma_m} + \left(1-\frac{\theta_m}{\gamma_m}\right)Ri, \end{equation}

which can be further simplified by assuming Gaussian profiles and a constant $\delta _m$ to obtain the entrainment relation given in (2.11) (Priestley & Ball Reference Priestley and Ball1955; Fox Reference Fox1970; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015). Alternatively, if $\alpha$ is assumed constant then the MTT model is obtained (Morton et al. Reference Morton, Taylor and Turner1956; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015). For the case of negatively buoyant jets, although we have shown an assumption of Gaussian profiles is realistic over the $Fr$ range observed, the non-constant $\lambda$ indicates that the full self-similarity assumption is not. It is therefore useful to invoke the assumption of Gaussian velocity and buoyancy profiles, yet allowing for a variable $\lambda$, to (6.2). The resulting expression is,

(6.4)\begin{equation} \alpha_{MG} = \underbrace{-\frac{3}{8}\delta_m}_{A_1} + \underbrace{\left(1-\frac{3}{2(1+\lambda^2)}\right)Ri}_{A_2}, \end{equation}

where $\theta _m=2/(\lambda ^2 +1)$, and the third term from (6.2) vanishes since $\gamma _m=4/3$ is constant, for Gaussian profiles.

To calculate $\delta _m$ from (2.8), the derivative $\partial \bar {w}/\partial r$ must be estimated from the experimental data. To avoid scatter in the derivative due to the spatially discrete velocity data, a two-dimensional Gaussian filter of width $100$ pixels (approximately $4$ times the width of a PIV interrogation window) was first applied to $\bar {w}$, and then the derivative estimated using a second-order accurate finite difference stencil. A similar procedure was used to calculate the other derivatives present in the profile coefficient definitions. In calculating $\delta _m$, as well as other quantities such as $Q$, it is also necessary to approximate an integral defined from $r=0$ to infinity using data from a finite region. In obtaining $Q$ for the NBJ, for example, the integral was first calculated using the trapezoidal rule over the full $r$ range captured in the region of interest. This was compared to the values obtained if the integral was calculated only up to the point where the mean axial velocity first equals zero. The latter gives $Q$ typically around $3\,\%$ lower than using the full range, since there are small $\bar {w}\lesssim 0$ values in the outer region. This difference is considered negligible, and given these negative velocities are much lower than inside the jet ($\lesssim 0.01\bar {w}_c$), this region is regarded as part of an approximately quiescent ambient. Integrating from $r=0$ to the edge of the region of interest was therefore used in computing all integral quantities.

Figure 15 shows the average value of $\alpha _{MG}=A_1=0.0714$ as a horizontal line calculated from the present neutral jet data, assuming self-similar Gaussian profiles. This is in reasonable agreement with the ‘mean self-similar’ value calculated in van Reeuwijk & Craske (Reference van Reeuwijk and Craske2015) of $\alpha =0.073$, and their direct estimate (from the conservation of volume (2.4)) of $\alpha =0.069$.

Figure 15.

The entrainment coefficient in the neutral and NBJs from the present experiments, as calculated from (6.4), which assumes Gaussian velocity and buoyancy profiles, and for the NBJ case, allows for non-constant $\lambda$. For the neutral jet, which has $Ri=0$ everywhere, $\alpha$ is shown as a horizontal line for clarity. The prediction of $\alpha$ from (6.4), using the constant values $\delta _m=-0.216$ and $\lambda =1.05$ obtained from Kaminski et al. (Reference Kaminski, Tait and Carazzo2005), is also shown, as well as linear fits of the terms $A_1$, $A_2$, and $A_1+A_2$.

For the NBJ, neither term in (6.4) is necessarily constant, and both are plotted against local $Ri$ in figure 15. We see that the first term, $A_1$, which corresponds to the ratio $-\delta _m/2\gamma _m$, increases with more negative $Ri$. This is a consequence of the profile coefficient $\delta _m$, which increases in magnitude with axial distance primarily due to $\overline {w'u'}$ remaining high relative to $w_m$, as discussed in §§ 4.3 and 5. However, the overall value of $\alpha _{MG}$ decreases for more negative $Ri$ due to the second term, $A_2$, which reflects the effect of negative buoyancy on entrainment through the factor of $Ri$. This can also be seen by considering linear fits with $Ri$ of the terms $A_1$, $A_2$ and their summation to give $\alpha$,

(6.5)\begin{equation} \left. \begin{array}{c@{}} A_1 = 0.075 - 0.227 Ri \\ A_2 = 0.520 Ri \\ \alpha = 0.075 + 0.292 Ri \end{array} \right\}. \end{equation}

This empirical $\alpha$ relation can be expressed as,

(6.6)\begin{equation} \alpha = -\frac{\delta_j}{2\gamma_m} + \left(1-\frac{\theta_m}{\gamma_m}- \frac{\varDelta_m}{2\gamma_m}\right)Ri,\end{equation}

which is equivalent to (6.3) with $\delta _m=\delta _j + \varDelta _m Ri$. The coefficients of the empirical fits of $A_1$ and $A_2$ in (6.5) imply $\delta _j\cong -0.200$, $\varDelta _m\cong 0.604$ and a constant $\theta _m\cong 0.640$ ($\lambda \cong 1.46$ for Gaussian profiles). The value for $\delta _j$ can be interpreted as the ‘neutral jet value’ implied by the model, which agrees with the jet values reported in van Reeuwijk & Craske (Reference van Reeuwijk and Craske2015) of $0.19\lesssim -\delta _m\lesssim 0.21$ (Panchapakesan & Lumley Reference Panchapakesan and Lumley1993; Wang & Law Reference Wang and Law2002; Ezzamel et al. Reference Ezzamel, Salizzoni and Hunt2015). The entrainment relation in (6.4) is derived from the conservation equations (2.4), (2.5) and (2.7). In (6.6), we make the ad hoc addition of the $\varDelta _m$ term motivated by the approximately linear $\delta _m$ trend observed from the data in figure 15. The inclusion of $\varDelta _m$ is ad hoc since there has not yet been an analysis of the governing equations to show the necessity of this term. Nevertheless, its inclusion is supported by the data in figure 15, which raises the open question for potential future research to explain the apparent need for it. The linear $\alpha$ relation in (6.6) is then a semi-empirical description of the flow based on fits of the present experimental data, rather than a direct derivation from the conservation equations, and will be henceforth referred to as ‘model 1’.

The $\alpha$ relation given in (6.5) has a similar form to the linear relationship for positively buoyant jets/plumes, which have $Ri>0$ (Priestley & Ball Reference Priestley and Ball1955; Fox Reference Fox1970). However, (6.5) is only proposed valid for $Ri<0$, where we have observed the linear relationship between $\delta _m$ and $Ri$. It is not intended as a universal relation for both positively and NBJs. In the far field, buoyant jets approach a state of self-similarity where they become indistinguishable from pure plumes, approaching a constant $Ri=Ri_p>0$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Papanicolaou & List Reference Papanicolaou and List1988). For these self-preserving flows, $\delta _m$ is constant and hence the linear $\alpha$ relation with $Ri$ in the form of (6.4), where $A_1$ is constant, is obtained. It has been reported that $\delta _m$ is approximately the same for pure jets ($Ri=0$) and plumes ($Ri=Ri_p$) (Wang & Law Reference Wang and Law2002; Ezzamel et al. Reference Ezzamel, Salizzoni and Hunt2015; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015; van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016), and an approximately constant $\delta _m$ has been reported in buoyant jets for $0.25\lesssim Ri/Ri_p\lesssim 0.75$ (van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016). However, this buoyant jet value was slightly lower than the jet and plume values, and also varied near the source before the flow had developed (van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016).

Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) derived an equation for $\alpha$ equivalent to (6.2), but in terms of the coefficients $\tilde A$ and $\tilde C$ (denoted $A$ and $C$ in their (3.33)). These can be related to the profile coefficients in (6.2) by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) and van Reeuwijk & Craske (Reference van Reeuwijk and Craske2015),

(6.7a,b)\begin{equation} \tilde C=-\delta_m/(\sqrt{2}\theta_m\gamma_m), \quad \tilde A=\gamma_m/\theta_m. \end{equation}

In this formulation, $\tilde C$ is related to the ratio of turbulent production to the mean energy flux, but is also influenced by the shape of $\bar {w}$ and $\bar {b}$ through $\theta _m$. The parameter $\tilde A$ is related to the mean energy flux and is also influenced by the shape of the mean profiles. Using (6.7), (6.3) may then be written in terms of these parameters,

(6.8)\begin{equation} \alpha_{MS} = \frac{\gamma_m\tilde C\sqrt{2}}{2\tilde A} + \left(1-\frac{1}{\tilde A}\right)Ri. \end{equation}

Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) calculated these parameters based on literature for positively buoyant jets/plumes (see their table 3), with average values of $\tilde C=0.12$ and $\tilde A=1.4$. Although limited data were available, Carazzo et al. (Reference Carazzo, Kaminski and Tait2008) later found similar values in NBJs. Using (6.7), these correspond to $\delta _m\cong -0.216$ and $\lambda \cong 1.05$ ($\theta _m\cong 0.952$) for Gaussian velocity/buoyancy profiles. When substituted into (6.4), these give the linear relationship,

(6.9)\begin{equation} \alpha=0.081 + 0.286Ri. \end{equation}

This relationship will be referred to as ‘model 2’, and is a reformulation of the entrainment relation given in (2.11) based on the work by Fox (Reference Fox1970) and Priestley & Ball (Reference Priestley and Ball1955), and uses constants based on values reported in Kaminski et al. (Reference Kaminski, Tait and Carazzo2005). This is very similar to model 1 given in (6.5) and based on empirical fits for an NBJ, which can be seen in figure 15 where both models are shown.

Model 2 gives good predictions of $\alpha$ in NBJs, but assumes a constant $\delta _m\cong -0.216$ and $\lambda \cong 1.05$ that are not consistent with the increasing $A_1$ and $\lambda$ observed in figures 15 and 8, respectively. Model 2 can also be described by (6.6), but where $\delta _j=\delta _m\cong -0.216$ and $\varDelta _m=0$. It can then be seen that the similarity between the models is partly due to the $\delta _j\cong -0.200$ assumed by model 1 being similar to $\delta _j\cong -0.216$ in model 2 (i.e. the ‘neutral jet value’), which results in the first term of the linear $\alpha$ relations in (6.5) and (6.9) being similar. The second term (the $Ri$ coefficient) of the relations are also similar, but for different reasons. In model 2, the value is determined by $\lambda \cong 1.01$ ($\theta _m\cong 0.952$) only, since $\varDelta _m=0$ (and $\gamma _m=4/3$) with respect to (6.6). For model 1, we have $\varDelta _m\cong 0.604$ and $\lambda \cong 1.46$ ($\theta _m\cong 0.640$), which give a similar $Ri$ coefficient when inserted into (6.6). Model 1 appears to provide a representation of $\alpha$ in NBJs that is more consistent with the observed $\delta _m$ and $\lambda$ in the present data.

For $Ri\gtrsim -0.11$ ($Fr\gtrsim 3.0$), in the forced regime, $\alpha _{MG}$ in the NBJ is positive but generally lower than it is for the neutral jet. For $Ri\lesssim -0.25$ ($Fr\lesssim 2.0$), $\alpha _{MG}$ becomes negative, implying there is a mean radial outflow of fluid from the jet to the ambient. This phenomenon has also been reported in the literature on fully developed fountains, where a mean outflow is observed from the IF to OF (Cresswell & Szczepura Reference Cresswell and Szczepura1993; Williamson et al. Reference Williamson, Armfield and Lin2011). In a general sense, entrainment may be considered the process where fluid is transported from a non-turbulent to turbulent region across some interface (e.g. Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016). The $\alpha$ given by (6.1), however, is simply a consistency requirement for the conservation of mass, momentum, buoyancy and mean kinetic energy equations. This expression, and subsequent simplifications such as (6.4), are therefore not necessarily describing entrainment in this general sense, but instead reflecting what the radial mean flow must be in order to satisfy the conservation equations. An NBJ with a mean radial outflow could still be subject to instantaneous ‘entrainment’ (flow from a non-turbulent to turbulent region) at some times, while ejecting fluid into the ambient at others. The $\alpha <0$ observed in the present NBJ indicates this net radial outflow, and means that on average there is more fluid ejected outwards than flowing into the jet in this region. This can be observed directly by examining the mean radial velocity profiles across the jet, where the net outflow of fluid corresponds to $\bar {u}>0$ outside of the jet. This can be seen on the left axis of figure 16, which shows $\bar {u}/\bar {w}_c$ plotted for the NBJ at $Fr=3.00$ ($Ri=-0.11$) and $Fr=1.86$ ($Ri=-0.29$), within the $\alpha >0$ and $\alpha <0$ regions respectively. The right axis shows the axial velocity profiles, $\bar {w}/\bar {w}_c$, at the same locations as a reference. The $Fr=3.00$ curve has $\bar {u}/\bar {w}_c<0$ for $r/r_{w,1/2}\gtrsim 1.2$, indicating net entrainment of fluid into the edge of the jet where $\bar {w}/\bar {w}_c\rightarrow 0$. The $Fr=1.86$ curve, however, has $\bar {u}/\bar {w}_c>0$ in this region, indicating there is a net radial outflow of fluid into the ambient.

Figure 16.

The left axis and solid lines show the normalised radial velocity profiles, $\bar {u}/\bar {w}_c$ of a NBJ at two different axial locations. The right axis and dashed lines show the corresponding axial velocity profiles, $\bar {w}/\bar {w}_c$. The right and left arrows are shown to indicate the axis each profile corresponds to.

Although this mean radial outflow does not typically occur in neutral or positively buoyant jets and plumes, which have $\alpha >0$ everywhere, in NBJs, a negative $\alpha$ towards the top of the jet is a natural consequence of (6.4), rather than any fundamentally different physics. This comes from the negative buoyancy and a decelerating mean flow resulting in $Ri\rightarrow -\infty$ at the top of the jet. This causes the second term, $A_2$, to become increasingly negative, dominating (6.4) until $\alpha <0$ and there is a net radial outflow of fluid from the jet.

Future work may seek to incorporate the traditional notion of entrainment (an inflow across a turbulent/non-turbulent interface), into the description of an NBJ with a net outflow. This could be achieved by splitting up the mean radial velocity into ‘entrainment’ and ‘outflow’ components, e.g. $\bar {u} = \bar {u}_e + \bar {u}_{out}$, where only the inflowing fluid is assumed to be proportional to the mean flow by an entrainment coefficient. This description could provide a clearer notion of ‘entrainment’ that is conceptually similar to that used in the traditional integral models (Morton et al. Reference Morton, Taylor and Turner1956; Morton Reference Morton1959), and that can occur simultaneously with a mean outflow.

6.2. Predictions of the simplified integral model

The system of equations defined in (2.12) is now solved numerically using models 1 and 2 discussed in § 6.1. Model 1 corresponds to using the $\alpha$ relation defined in (6.5), which assumes a linear $\delta _m$ relationship with $Ri$ and constant $\lambda \cong 1.46$. Although we observed from figure 8 that $\lambda$ is not constant in NBJs, $\lambda \cong 1.46$ lies within the range of observed values and so, when solving (2.12), we take it to be constant to simplify the model. When solving model 2, the $\alpha$ relation in (6.9) is used, which assumes a constant $\delta _m\cong -0.216$ and $\lambda \cong 1.05$ (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; Carazzo et al. Reference Carazzo, Kaminski and Tait2008). By assuming a constant $\delta _m$, model 2 is simply a reformulation of the PB model (Priestley & Ball Reference Priestley and Ball1955; Fox Reference Fox1970), and although these $\delta _m$ and $\lambda$ values are not consistent with those observed from our NBJ data, this model provides a useful reference case for comparison. The system of equations is also solved using the Morton (Reference Morton1959) model (constant $\alpha =0.0714$ and $\lambda =1.2$) as an additional comparison.

The predictions for $w_o/\bar {w}_c$ obtained by solving (2.12) with these models have been presented in figure 6 alongside the experimental data. The Morton (Reference Morton1959) model gives similar predictions to both model 1 and 2 for $z/D\lesssim 27$, where there is also reasonably good agreement with our NBJ data. Models 1 and 2 continue to agree with each other and the NBJ data until $z/D\cong 30$. For $30\lesssim z/D\lesssim 38$, model 2 actually gives better predictions than model 1 based on the empirical data, despite assuming values for $\delta _m$ and $\lambda$ that are not consistent with the present NBJ results. Part of the reason that two similar $\alpha$ models can give significantly different predictions for $\bar {w}_c$ is due to $\lambda$. Firstly, this is due to the fact that in addition to the $\alpha$ relation, $\lambda$ appears in the momentum equation (2.12b) through $\theta _m$. Additionally, $\lambda$ appears in the definition of $Ri$ through $\theta _m$, which further complicates its influence on the flow.

Neither model 1 nor 2 accurately predict the flow over the full $Ri$ range observed while simultaneously maintaining consistency with the present $\delta _m$ and $\lambda$ observations. This suggests that neither model is complete. The $\lambda \cong 1.46$ used in model 1 is closer to the observed experimental data in figure 8 than model 2, although has poorer agreement for $z/D\gtrsim 27$. Although the $w_o/\bar {w}_c$ agreement of model 2 is very good, it is not consistent with the observed $\lambda$ and $\delta _m$ in the present data, and we leave this apparent discrepancy as an open question.

Mizushina et al. (Reference Mizushina, Ogino, Takeuchi and Ikawa1982) found in their experimental study of fully developed fountains ($3\lesssim Fr_o\lesssim 258$), that the radius of the fountain was approximately constant and given by $r_{f}/D\cong 0.26Fr_o$. If the cap region of the fountain was hemispherical, then this would also be equal to the radius and vertical thickness of the cap. The cap region would then extend from $z=z_{ss}-r_{f}$ to the top of the fountain, $z=z_{ss}$. If the top of a NBJ resembles the cap of a fully developed fountain, then this region would not be well described by the present integral model, which was derived for jet-like flows. The present $Fr_o=30$ NBJ, which has $z_i/D\cong 53.5$, would have $r_f/D\cong 7.8$. The end of the ‘jet-like’ region where the models are applicable would then occur at $z/D\cong 45.7$. This is somewhat further than $z/D\cong 30$ where model 1 departs from the data, possibly due to the limitations of the model discussed above, but is nevertheless broadly consistent.