3.1. Velocity and buoyancy profiles

Figures 4 and 5 show mean velocity and scalar concentration (buoyancy) profiles for fountains and NBJs with $Fr_o=30$, along with the best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) as a reference. Dimensionless scalar concentration ($0\leqslant c\leqslant 1$) and buoyancy (${\rm mm}\ {\rm s}^{-2}$) are related by a constant such that $b=c(\rho _o-\rho _a)g/\rho _a$. In Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), these profiles were shown to be Gaussian over a range of axial locations ($20\lesssim z/D\lesssim 40$), or equivalently, local Froude numbers, $Fr$ ($1.9\lesssim Fr\lesssim 5.9$), in the NBJ, which are given in figures 4(b) and 5(b). When normalised by the centreline quantities, $\bar {w}_c$ and $\bar {c}_c$, and respective half-widths, $r_{1/2,w}$ and $r_{1/2,c}$, these velocity and scalar profiles in the NBJ collapse onto a single curve.

Figure 4. Mean velocity profiles, $\bar {w}$, for a fully developed fountain, (a), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b), both with $Fr_o=30$, at a range of axial locations. The profiles are normalised by the local centreline value, $\bar {w}_c$, and half-width, $r_{1/2,w}$. The best-fit profile for a neutral jet (J) by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

Figure 5. Mean scalar profiles, $\bar {c}$, for a fully developed fountain, (a), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b), both with $Fr_o=30$, at a range of axial locations. The profiles are normalised by the local centreline value, $\bar {c}_c$, and half-width, $r_{1/2,c}$. The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

The profiles for a fully developed fountain obtained from the present experiments, also with $Fr_o=30$, are shown in figures 4(a) and 5(a), where it is clear immediately that the profiles do not collapse in the same way as the NBJ. Although the velocity and scalar profiles are reasonably similar for $r/r_{1/2,w}\lesssim 1$ and $r/r_{1/2,c}\lesssim 1$, respectively, they diverge for radial locations beyond this. For the velocity profile, this is particularly evident in the OF where $\bar {w}<0$. The minimum (most negative) value of the velocity profile will be denoted $\bar {w}_d$, with the radial location of this point denoted $r_d$. Figure 4(a) shows that $\bar {w}_d/\bar {w}_c$ becomes increasingly negative with increasing axial distance, while the location of the local minima, $r_d/r_{1/2,w}$, moves towards the centre. This is primarily a consequence of the normalisation, where $\bar {w}_c$ decreases strongly as the IF is decelerated under negative buoyancy, increasing the magnitude of $\bar {w}_d/\bar {w}_c$. The velocity half-width, $r_{1/2,w}$, also increases relative to $r_d$, reducing the value of $r_d/r_{1/2,w}$. The actual (non-normalised) magnitude of $\bar {w}_d$, however, decreases towards zero at the top of the fountain, and $r_d$ moves outwards as the IF expands.

The scalar profiles, normalised by local quantities, are given in figures 5(a) and 5(b) for the fountain and NBJ. Unlike the NBJ profiles in figure 5(b), the fountain velocity profiles in figure 5(a) do not collapse onto the same curve for the full radial width, although they are reasonably similar in the inner profile for $r/r_{1/2,c}\lesssim 1$. This means that, particularly in the outer profile ($r/r_{1/2,c}\gtrsim 1$), they are no longer well described by a similar Gaussian shape along the flow, appearing to get narrower with axial distance. Neither the velocity or scalar (buoyancy) profiles in a fully developed fountain can therefore be considered self-similar in the sense that they have similar shapes along the fountain. Despite this, the most significant shape change of the profiles occurs in the return flow, with the inner profiles collapsing approximately when normalised by their centreline values and respective half-widths.

3.2. Turbulence statistics

Profiles for the axial turbulence fluctuations and Reynolds stress, $\overline {w'^{2}}$ and $\overline {w'u'}$, are given in figures 6 and 7, normalised by both local and source quantities. The development of these turbulence profiles in an NBJ was discussed in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), where they were found to increase with axial distance due to the strongly decelerating mean flow, characterised here by $\bar {w}_c$. This differs from neutral jets and plumes, where $\overline {w'^{2}}$ and $\overline {w'u'}$ decrease at the same rate as the mean flow, so the normalised profiles, $\overline {w'^{2}}/\bar {w}_c^{2}$ and $\overline {w'u'}/\bar {w}_c^{2}$, are approximately constant along the flow (at least in the fully developed self-similar region) (Panchapakesan & Lumley Reference Panchapakesan and Lumley1993; Hussein, Capp & George Reference Hussein, Capp and George1994; Wang & Law Reference Wang and Law2002).

Figure 6. Profiles of the mean axial turbulence fluctuations, $\overline {w'^{2}}$, for a fully developed fountain, (a,c), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b,d), both with $Fr_o=30$. The profiles are normalised by the local centreline value, $\bar {w}_c^{2}$, and half-width, $r_{1/2,w}$, in (a,b), and by source velocity, $w_o^{2}$, and radius, $r_o$, in (c,d). The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

Figure 7. Profiles of the mean Reynolds stress, $\overline {w'u'}$, for a fully developed fountain, (a,c), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b,d), both with $Fr_o=30$. The profiles are normalised by the local centreline value, $\bar {w}_c^{2}$, and half-width, $r_{1/2,w}$, in (a,b), and by source velocity, $w_o^{2}$, and radius, $r_o$, in (c,d). The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

For the fully developed fountain, the behaviour of $\overline {w'^{2}}$ and $\overline {w'u'}$ remaining high relative to $\bar {w}_c^{2}$ is more prominent than in the NBJ. The normalised profiles in figures 6 and 7 increase more rapidly in the fountain than in the NBJ, with large peak values $\overline {w'^{2}}/\bar {w}_c^{2}\simeq 0.2$ and $\overline {w'u'}/\bar {w}_c^{2}\simeq 0.04$ for the $Fr=2.23$ profile. These are considerably higher than the NBJ peaks at a similar $Fr=2.21$ of $\overline {w'^{2}}/\bar {w}_c^{2}\simeq 0.09$ and $\overline {w'u'}/\bar {w}_c^{2}\simeq 0.02$. Figures 6(c), 6(d), 7(c) and  7(d), which show the $\overline {w'^{2}}$ and $\overline {w'u'}$ profiles instead normalised by source conditions, illustrate the fact that the turbulence intensities are driven primarily by the decreasing $\bar {w}_c$, rather than by an increase in turbulence production. From these it is clear that $\overline {w'^{2}}$ and $\overline {w'u'}$ decrease with axial distance for both the NBJ and the fountain, and are higher in the fountain than the NBJ at similar local $Fr$. This is particularly true of the $\overline {w'^{2}}/w_o^{2}$ profile, where the peak values in the fountain profiles are approximately twice that of the NBJ at similar $Fr$. This is likely due to interactions between the IF and OF that result in increased turbulence in the IF, and so also contributes to the greater $\overline {w'^{2}}/\bar {w}_c^{2}$ and $\overline {w'u'}/\bar {w}_c^{2}$ observed in the fountain. Cresswell & Szczepura (Reference Cresswell and Szczepura1993) also gathered turbulence data for a fully developed fountain, although one with a much lower $Fr_o\simeq 3.2$ ($z_{ss}/D\simeq 3.9$). When normalised by the local centreline value, their peak axial turbulence intensity was $\overline {w'^{2}}/\bar {w}_c^{2}\simeq 0.046$ near the source at $z/D=0.3$, which increased to $\overline {w'^{2}}/\bar {w}_c^{2}\simeq 1.8$ near the cap at $z/D=3$. Although the magnitudes at a given axial location differ from the present data due to the significantly different $Fr_o$, the observation of $\overline {w'^{2}}/\bar {w}_c^{2}$ increasing as the flow decelerates is consistent with our higher $Fr_o$ fountains.

Figures 8(a) and 8(b) show profiles of the mean turbulence scalar fluctuations, $\sqrt {\overline {c'^{2}}}/\bar {c}_c$, for the fountain and NBJ, respectively. The NBJ profiles are reasonably well collapsed, while the fountain profiles are not self-similar and also do not appear to follow a clear increasing or decreasing trend with axial distance. Profiles of the axial and radial fluxes, $\overline {w'c'}$ and $\overline {u'c'}$, for the fountain and NBJ are given in figures 9 and 10. When normalised by local quantities, these profiles clearly increase with axial distance. A moderate increase can also be observed between the NBJ profiles nearest and furthest from the source ($z/D=19.77$ and $z/D=38.86$), although the trend is less significant and there is some scatter in the profiles between these locations. As with the velocity fluctuations, the growth in the turbulent scalar profiles for the fountain is primarily a result of normalising by the decreasing $\bar {w}_c$, rather than an increase in axial/radial scalar transport. It is also notable that the $\overline {w'c'}$ profiles for the fountain drop considerably below zero in the outer profile. This corresponds to the OF region where the direction of axial mean flow is reversed, and so a sign reversal of the axial scalar flux is expected.

Figure 8. Profiles of the mean scalar fluctuations, $\sqrt {\overline {c'^{2}}}$, for a fully developed fountain, (a), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b), both with $Fr_o=30$. The profiles are normalised by the local centreline value, $\bar {c}_c$, and half-width, $r_{1/2,c}$. The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

Figure 9. Profiles of the mean axial turbulent flux, $\overline {w'c'}$, for a fully developed fountain, (a), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b), both with $Fr_o=30$. The profiles are normalised by the local product, $\bar {w}_c\bar {c}_c$, and half-width, $r_{1/2,c}$. The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

Figure 10. Profiles of the mean radial turbulent flux, $\overline {u'c'}$, for a fully developed fountain, (a), and NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), (b), both with $Fr_o=30$. The profiles are normalised by the local product, $\bar {w}_c\bar {c}_c$, and half-width, $r_{1/2,c}$. The best-fit profile for a neutral jet by Wang & Law (Reference Wang and Law2002) is also shown as a reference.

Mean and turbulence statistics have been presented for a fully developed fountain and an NBJ, both with $Fr_o=30$, at axial locations where they have a similar local $Fr$. Presenting these fountain profiles alongside those for an NBJ with the same $Fr_o$ allows the effect of a return flow to be revealed. In the fountain, although the $\bar {w}/\bar {w}_c$ and $\bar {c}/\bar {c}_c$ profiles collapse approximately in the IF, they do not maintain similar shapes in the OF and are therefore not generally self-similar. This is in contrast to the NBJ, which does not have an OF, and has Gaussian velocity and scalar profiles over the full range of observed $Fr$ locations. The presence of a return flow tends to increase turbulence in the IF of fountains, which can be seen by the significantly larger $\overline {w'^{2}}/\bar {w}_c^{2}$ and $\overline {w'u'}/\bar {w}_c^{2}$ compared to an NBJ at similar $Fr$ locations. In both fountains and NBJs, the turbulence profiles involving velocity fluctuations, namely $\overline {w'u'}$, $\overline {w'^{2}}$, $\overline {u'c'}$ and $\overline {w'c'}$, all increase with axial distance when normalised by centreline values. This is primarily a result of the strongly decelerating mean flow, characterised here by $\bar {w}_c$, rather than an increase in turbulence along the flow. This observation was first noted in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) for NBJs, with the present study identifying a similar behaviour in fountains.

Figures 4–10 have all shown profiles normalised by the local mean centreline values. Another potential scaling option would be to include the OF velocity, which could be characterised by the velocity at the location of ‘maximum outer flow’, ${\bar {w}(r=r_d)=\bar {w}_d}$. The mean velocity profile could then be normalised by a characteristic relative IF/OF velocity, $\bar {w}_c-\bar {w}_d$. However, since $\bar {w}_c\gg |\bar {w}_d|$ for the majority of the fountain height, this gives results similar to those for the $\bar {w}_c$ normalisation, and the profiles do not collapse across the full fountain width. Another approach would be to consider the quantity ${(\bar {w}-\bar {w}_d)/(\bar {w}_c-\bar {w}_d)}$, which is the fountain velocity relative to the moving reference frame of the OF. This is always positive and has centreline value 1. Although this could potentially be useful from a modelling point of view, it also did not collapse the present results over the full fountain width.

7.1. Jet and plume models

Morton et al. (Reference Morton, Taylor and Turner1956) proposed an integral model, referred to here as the MTT model, based on the conservation of mass, momentum and buoyancy, to describe plumes originating from a point source. These may also be applied to neutral and buoyant jets, as well as NBJs prior to the return flow forming (Morton Reference Morton1959; Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021). The MTT model invokes the ‘entrainment assumption’, which relates the radial inflow of ambient fluid into the jet to a characteristic vertical velocity at that height by an ‘entrainment coefficient’, $\alpha$. Simple applications of the model, such as to pure plumes (zero initial momentum) and neutral jets (zero buoyancy), typically use a constant value of $\alpha$ in the range $0.10\lesssim \alpha _p\lesssim 0.16$ and $0.065\lesssim \alpha _j\lesssim 0.080$ for plumes and jets, respectively (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Carazzo, Kaminski & Tait Reference Carazzo, Kaminski and Tait2006). Priestley & Ball (Reference Priestley and Ball1955) also developed an integral model for plumes/jets, referred to here as the PB model, but instead derived from the conservation of momentum, buoyancy and kinetic energy. Although the PB model did not make use of the entrainment assumption originally, Fox (Reference Fox1970) showed how when combined with the continuity equation, it implies that $\alpha$ is a function of the local $Ri$. For flows with constant $Ri$, such as pure jets and plumes, $\alpha$ is then constant and the MTT model is obtained (Fox Reference Fox1970; Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979). Analytical solutions to these models can be obtained by assuming fully self-similar velocity and buoyancy profiles in the flow, where the shapes of the profiles are assumed to be Gaussian or ‘top-hat’ (constant inside the plume/jet, zero outside it). More recent studies have developed these models further, by omitting assumptions about the shapes or self-similarity of the profiles (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015).

These more recent formulations of the models may be derived from the conservation of volume, momentum, buoyancy and kinetic energy equations for a high-$Re_o$ axisymmetric flow in a homogeneous environment (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015):

(7.1)$$\begin{gather} \frac{\partial}{\partial r}\left(r\bar{u}\right) + \frac{\partial}{\partial z}\left(r\bar{w}\right) = 0, \end{gather}$$

(7.2)$$\begin{gather}\frac{\partial}{\partial r}(r\bar{u}\,\bar{w}+r\overline{u'w'}) + \frac{\partial}{\partial z}(r\bar{w}^{2}) = r\bar{b}, \end{gather}$$

(7.3)$$\begin{gather}\frac{\partial}{\partial r}(r\bar{u}\bar{b} + r\overline{u'b'}) + \frac{\partial}{\partial z}(r\bar{w}\bar{b}) = 0, \end{gather}$$

(7.4)$$\begin{gather}\frac{\partial}{\partial r}(r\bar{u}\,\bar{w}^{2}+2r\overline{u'w'}\bar{w}) + \frac{\partial}{\partial z}(r\bar{w}^{3}) = 2r\overline{u'w'}\,\frac{\partial\bar{w}}{\partial r} + 2r\bar{w}\bar{b}. \end{gather}$$

Pressure contributions and second-order turbulence terms, namely those involving $\overline {w'^{2}}$ and $\overline {w'b'}$, are neglected here. Neglecting these higher-order terms is common in the jet and plume literature, where they are typically found to be negligible (Wang & Law Reference Wang and Law2002; Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015; van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016). For the case of fountains and NBJs, these terms have been estimated as they appear after integrating (7.1)–(7.4), and are also found to be small compared to the mean components for the majority of the flow. The terms become larger near the top of an NBJ/fountain, where the mean velocity goes to zero and there is non-zero turbulence, but this region is not considered in the present analysis.

For neutral or positively/negatively buoyant jets in a quiescent homogeneous ambient, these equations may be integrated to infinity, with respect to $r$, to obtain a set of ordinary differential equations (ODEs) consistent with the MTT and PB models (Priestley & Ball Reference Priestley and Ball1955; Morton et al. Reference Morton, Taylor and Turner1956; Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015):

(7.5)$$\begin{gather} \frac{\mathrm{d}Q}{\mathrm{d}z} = 2\alpha M^{1/2} , \end{gather}$$

(7.6)$$\begin{gather}\frac{\mathrm{d}M}{\mathrm{d}z} = B = \frac{FQ}{\theta_mM} , \end{gather}$$

(7.7)$$\begin{gather}\frac{\mathrm{d}F}{\mathrm{d}z} = 0 , \end{gather}$$

(7.8)$$\begin{gather}\frac{\mathrm{d}}{\mathrm{d}z}\left(\gamma_m\frac{M^{2}}{Q}\right) = \delta_m\frac{M^{5/2}}{Q^{2}} + 2F, \end{gather}$$

where the entrainment coefficient, $\alpha$, is defined as

(7.9)\begin{equation} \left(r\bar{u}\right)_{r=\tilde{r}} ={-}\alpha r_m w_m. \end{equation}

The quantities $\theta _m$, $\gamma _m$ and $\delta _m$ are ‘profile coefficients’ corresponding to the dimensionless buoyancy flux, mean kinetic energy and turbulence production, and are constant if the flow is fully self-similar:

(7.10)\begin{equation} \left. \begin{array}{l@{}} \theta_m=\dfrac{F}{w_mb_mr_m^{2}}, \\ \gamma_m=\dfrac{2}{w_m^{3}r_m^{2}}\displaystyle\int_{0}^{\tilde{r}}\bar{w}^{3} r\,\mathrm{d}r, \\ \delta_m=\dfrac{4}{w_m^{3}r_m}\displaystyle\int_{0}^{\tilde{r}}\overline{w'u'}\,\dfrac{\partial\bar{w}}{\partial r}\,r\,\mathrm{d}r. \end{array}\right\}\end{equation}

In the present context of neutral or positively/negatively buoyant jets and plumes, the integration limit is set to $\tilde {r}=\infty$ in all definitions.

For a constant $\alpha$, (7.5)–(7.7) then represent the MTT model in the case of self-similar flow, with the value of $\theta _m$ depending on the assumed profile shapes (e.g. $\theta _m=1$ for top-hat profiles). van Reeuwijk & Craske (Reference van Reeuwijk and Craske2015) used (7.5)–(7.8) to derive an analytical expression for $\alpha$ that does not make any assumptions about the shape of the profiles or whether they are self-similar:

(7.11)\begin{equation} \alpha ={-}\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}(\ln\gamma_m). \end{equation}

For a self-similar neutral jet, which has $Ri=0$ and constant profile coefficients, the second two terms are zero and a constant entrainment coefficient is obtained, while for self-similar buoyant jets, $\alpha$ follows a linear relationship with $Ri$. In a previous study, the present authors calculated the terms of (7.11) for an $Fr_o=30$ NBJ after assuming Gaussian velocity and buoyancy profiles (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021). It was found that $\alpha$ could be approximated reasonably by a constant $\theta _m\simeq 0.64$ and linear fit of $\delta _m$ with $Ri$, giving a linear relationship between $\alpha$ and $Ri$. Since $Ri\rightarrow -\infty$ as $w_m\rightarrow 0$ at the top of an NBJ, $\alpha$ decreases with axial distance and eventually becomes negative for sufficiently negative $Ri$. This was found to occur for $Ri\lesssim -0.25$, where there is a mean radial outflow of fluid from the jet to the ambient.

7.2. Application to fountains

The boundary between the IF and OF of a fully developed fountain is defined as the location where the mean axial velocity is first equal to zero – that is, $\bar {w}(r=r_{io})=0$ where $r_{io}$ is the radial location of the boundary. The IF of a fountain could then be considered an NBJ that is surrounded by an opposing shear flow, namely the OF, rather than a quiescent ambient. By integrating the conservation equations given in (7.1)–(7.4) to $r_{io}$ instead of infinity, a system of ODEs similar to (7.5)–(7.8) can be derived that describe the IF of a fountain. In this case, we let $\tilde {r}=r_{io}$ in (3.2), (7.9) and (7.10), and all integral quantities and profile coefficients correspond to the IF:

(7.12)$$\begin{gather} \frac{\mathrm{d}Q}{\mathrm{d}z} = 2\alpha M^{1/2} , \end{gather}$$

(7.13)$$\begin{gather}\frac{\mathrm{d}M}{\mathrm{d}z} = \frac{FQ}{\theta_mM} - 2(r\overline{u'w'})_{r=r_{io}} , \end{gather}$$

(7.14)$$\begin{gather}\frac{\mathrm{d}F}{\mathrm{d}z} ={-}2(r\bar{u}\bar{b})_{r=r_{io}} -2(r\overline{u'b'})_{r=r_{io}}, \end{gather}$$

(7.15)$$\begin{gather}\frac{\mathrm{d}}{\mathrm{d}z}\left(\gamma_m\frac{M^{2}}{Q}\right) = \delta_m\frac{M^{5/2}}{Q^{2}} + 2F. \end{gather}$$

The higher-order turbulence terms, i.e. turbulence components of the profile coefficients, have been neglected here. The continuity and kinetic energy equations, (7.12) and (7.15), are in the same form as (7.5) and (7.8) since $\bar {w}(r=r_{io})=0$, but the momentum and buoyancy equations have non-zero boundary conditions arising because $\overline {u'w'}$, $\overline {u'b'}$ and $\bar {b}$ do not go to zero at $r_{io}$. In the present formulation, $\alpha$ is related to the integral scales of the IF only, and describes the exchange of fluid between the IF and OF, where $\alpha >0$ corresponds to fluid moving from the OF to the IF. An alternative approach may be to define the velocity scale in the definition of $\alpha$ in (7.9) so that it includes information about the OF velocity. However, since no assumptions are being made regarding the value or behaviour of $\alpha$, any substitution here can be made valid provided that it is consistent with the conservation equations.

Similarly to the derivation of (7.11), an expression for $\alpha$ for the IF of a fountain may be derived using (7.12), (7.13) and (7.15) (van Reeuwijk & Craske Reference van Reeuwijk and Craske2015):

(7.16)\begin{equation} \alpha ={-}\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}(\ln\gamma_m) - \frac{2Q}{M^{3/2}}(r\overline{u'w'})_{r=r_{io}}. \end{equation}

This expression differs from (7.11) only in that it includes a non-zero shear stress term at the boundary, and all integral quantities and profile coefficients are defined up to $r_{io}$.

Carazzo et al. (Reference Carazzo, Kaminski and Tait2010) derived a set of equations similar to (7.12)–(7.15) for the IF of a fountain, but went on to construct additional conservation equations for $\bar {w}^{3}$ and $\bar {w}\bar {b}$ in order to replace the boundary conditions with integral profiles, and obtained a ‘confined top-hat’ model for the flow. For the purposes of the present investigation, it is sufficient to derive only the $\bar {w}^{3}$ equation, obtained by multiplying the momentum equation, (7.2), by $3\bar {w}^{2}$:

(7.17)\begin{equation} \frac{\partial}{\partial r}(r\bar{u}\,\bar{w}^{3}+3r\overline{u'w'}\bar{w}^{2}) + \frac{\partial}{\partial z}(r\bar{w}^{4}) = 3r\bar{w}^{2}\bar{b} + 6r\bar{w}\overline{u'w'}\,\frac{\partial \bar{w}}{\partial r}. \end{equation}

This may then be integrated with respect to $r$ from zero to $r_{io}$, giving

(7.18)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}z}\left(\varGamma_m\frac{M^{3}}{Q^{2}}\right) = 6\varDelta_m\frac{M^{7/2}}{Q^{3}} + 3\mu_m\frac{MF}{\theta_m Q}, \end{equation}

which is written in terms of the new profile coefficients

(7.19)\begin{equation} \left. \begin{array}{l@{}} \mu_m=\dfrac{1}{w_m^{2}b_mr_m^{2}}\displaystyle\int_{0}^{\tilde{r}}\bar{w}^{2}\bar{b}r\,\mathrm{d}r, \\ \varGamma_m=\dfrac{1}{w_m^{4}r_m^{2}}\displaystyle\int_{0}^{\tilde{r}}\bar{w}^{4}r\,\mathrm{d}r, \\ \varDelta_m=\dfrac{1}{w_m^{4}r_m}\displaystyle\int_{0}^{\tilde{r}}\bar{w}\overline{w'u'}\, \dfrac{\partial\bar{w}}{\partial r}\,r\,\mathrm{d}r, \end{array}\right\} \end{equation}

with $\tilde {r}=r_{io}$. Note that this approach may also be applied to jets and plumes in a homogeneous environment by setting $\tilde {r}=\infty$ in the profile coefficient definitions, and integrating the conservation equations to infinity. In this case, the same expression as (7.18) is obtained.

By applying the product rule to (7.18), and using (7.12) and (7.15), a new expression for $\alpha$ without any boundary conditions may be derived:

(7.20)\begin{equation} \alpha = \frac{3}{2}\left(\frac{\delta_m}{\gamma_m}-\frac{4\varDelta_m}{\varGamma_m}\right) + 3\left(\frac{\theta_m}{\gamma_m}-\frac{\mu_m}{\varGamma_m}\right)Ri + \frac{Q}{M^{1/2}}\,\frac{\mathrm{d}}{\mathrm{d}z}\left(\ln\frac{\varGamma_m}{\gamma_m^{3/2}}\right), \end{equation}

which is valid for arbitrary velocity and buoyancy profiles and does not make any assumptions about the self-similarity of the flow. By setting $\tilde {r}=r_{io}$ in the profile coefficient and $\alpha$ definition, this describes entrainment between the IF and OF of a fully developed fountain. As with (7.19), it may also be applied to jets of arbitrary buoyancy in a homogeneous environment by setting $\tilde {r}=\infty$. This is somewhat similar to the expression for $\alpha$ in Carazzo et al. (Reference Carazzo, Kaminski and Tait2010) (their (4.38)), which is valid for self-similar profiles and corresponds to entrainment in their top-hat model of the flow. Although both formulations are equally valid with respect to their given framework, (7.20) may be applied immediately to both the IF of a fountain or an NBJ with arbitrary profiles, using the classical integral velocity, width and buoyancy scales defined in (3.3). If the IF of the fountain were self-similar, then all profile coefficients in (7.20) would be constant, and a linear expression for $\alpha$ with $Ri$ would be obtained. A linear relationship for $\alpha$ was also derived by Mehaddi et al. (Reference Mehaddi, Vaux, Candelier and Vauquelin2015) in their model of self-similar turbulent fountains. In their case, $\alpha$ and $Ri$ were defined in terms of centreline values of the mean profiles and $r_{io}$, rather than integral quantities as in the present case. Although this means that the $\alpha$ expressions are not immediately equivalent without making additional assumptions about the shapes of the mean profiles, the linear forms of the relationships are consistent.

Both (7.16) and (7.20) are valid expressions for the entrainment coefficient of the IF of a fountain ($\tilde r=r_{io}$), as well for an NBJ ($\tilde {r}=\infty$). Using data from the present experiments, it is possible to calculate each of the profile coefficients present in these expressions, as well as to measure directly the boundary condition $(r\overline {u'w'})_{r=r_{io}}$ in (7.16). Figure 16 shows $\alpha$ calculated using both (7.16) and (7.20), for two fully developed fountains with $Fr_o=30$ and $Fr_o=15$, as well as for an NBJ with $Fr_o=30$. Gaussian velocity/buoyancy profiles were assumed for the NBJ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), which results in $\gamma _m=4/3$, $\varGamma _m=1$, and the derivate in (7.16) and (7.20) equalling zero. For the fountain, Gaussian profiles were not assumed, though the derivative terms in these two equations were found to be negligible and so have been neglected.

Figure 16. The entrainment coefficient, $\alpha$, plotted with local $Ri$ for the IF of an $Fr_o=30$ and $Fr_o=15$ fountain, and of an $Fr_o=30$ NBJ, as calculated using (7.16) and (7.20).

As discussed in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) and shown in figure 16, $\alpha <0$ for $Ri\lesssim -0.25$ for an NBJ, indicating that there is a mean radial outflow of fluid from the jet. For the $Fr_o=30$ fountain, $\alpha$ becomes negative near $Ri\simeq -0.13$, reflecting a mean outflow of fluid from the IF to the OF. For the $Fr_o=15$ fountain, $\alpha <0$ for the full range shown, $Ri\lesssim -0.04$. This is consistent with studies by Williamson et al. (Reference Williamson, Armfield and Lin2011) ($Fr_o=7$) and Cresswell & Szczepura (Reference Cresswell and Szczepura1993) ($Fr_o\simeq 3.2$), who observed predominately a mean radial flow from IF to OF, other than in a small region near the source. More generally, figure 16 shows good agreement between the $\alpha$ estimated from (7.16), which uses a direct measurement of the IF/OF boundary condition, and that from (7.20), which uses the new profile coefficients defined in (7.19).

8.1. Decomposed top-hat model

Consider a control volume of a thin horizontal slice of an axisymmetric fully developed fountain, such as one taken from the fountain schematic shown in figure 1. The slice should be taken before the cap, $z< z_c$, where there are distinct IF and OF regions (e.g. $Ri\geqslant -0.5$) (Shrinivas & Hunt Reference Shrinivas and Hunt2014; Hunt & Debugne Reference Hunt and Debugne2016). For simplicity, the fountain is presumed to be orientated vertically upwards with a denser IF than OF (the opposite but equivalent case to the present experiments), and to have top-hat vertical velocity and buoyancy profiles. Here we have, for the IF and OF, $\hat {w}_{if}$ and $\hat {w}_{of}$ denoting the vertical velocities, $\hat {r}_{if}$ and $\hat {r}_{of}$ the widths, and $\hat {g}_{if}=g(\rho _a-\hat {\rho }_{if})/\rho _a$ and $\hat {g}_{of}=g(\rho _a-\hat {\rho }_{of})/\rho _a$ the buoyancies. Here, $\hat {\rho }_{if}$, $\hat {\rho }_{of}$ and $\rho _a$ are the densities of the IF, OF and ambient, so we have $\hat {g}_{if}<0$ and $\hat {g}_{of}<0$. There is also a radial exchange of fluid between IF and OF that, as will be argued in the following sections, is comprised of a component resulting from turbulent entrainment and a component resulting from the buoyancy-induced spreading of the IF. These two components can, simultaneously, be in opposite directions: the turbulent component, in isolation, producing a flow from OF to IF, and the buoyant spreading component, in isolation, producing a flow from IF to OF. The combination of the two components will, depending on their balance, produce either a positive or negative net flow across the IF/OF boundary at any time. The velocity component associated with the turbulent entrainment is denoted $\hat {u}_e$, while the velocity component associated with the buoyant spreading is denoted $\hat {u}_{out}$. The equations for conservation of volume, momentum and buoyancy flux for the IF of this system are then

(8.1)$$\begin{gather} \frac{\mathrm{d}\hat{Q}_{if}}{\mathrm{d}z} = 2\hat{r}_{if}\hat{u}_{e} - 2\hat{r}_{if}\hat{u}_{out} , \end{gather}$$

(8.2)$$\begin{gather}\frac{\mathrm{d}\hat{M}_{if}}{\mathrm{d}z} ={-}2\hat{r}_{if}\hat{u}_{e}\hat{w}_{of} - 2\hat{r}_{if}\hat{u}_{out}\hat{w}_{if} + \hat{r}_{if}^{2}\hat{g}_{if} , \end{gather}$$

(8.3)$$\begin{gather}\frac{\mathrm{d}\hat{F}_{if}}{\mathrm{d}z} = 2\hat{r}_{if}\hat{u}_{e}\hat{g}_{of} - 2\hat{r}_{if}\hat{u}_{out}\hat{g}_{if}, \end{gather}$$

where $\hat {Q}_{if}=\hat {r}_{if}^{2}\hat {w}_{if}$, $\hat {M}_{if}=\hat {r}_{if}^{2}\hat {w}_{if}^{2}$ and $\hat {F}_{if}=\hat {r}_{if}^{2}\hat {w}_{if}\hat {g}_{if}$ are the (scaled) volume, momentum and buoyancy fluxes.

In a fully developed fountain, the mean radial velocity is from the IF to the OF for the majority of its height. A mean radial outflow is also seen in NBJs that do not have a return flow, where it is driven by the decelerating mean flow as characterised by $Ri\rightarrow -\infty$. Despite this, there may still be instantaneous ‘entrainment’ from the OF to the IF of a fountain across the IF/OF interface (or from the ambient to the jet in the case of an NBJ). The present analysis will treat $\hat {u}_e$ as the ‘entrainment velocity’, and we will assume that it is proportional to the velocity of the IF by an entrainment coefficient, $\alpha _e$:

(8.4)\begin{equation} \hat{u}_e = \alpha_e\hat{w}_{if}. \end{equation}

An alternative approach would be to treat $\alpha _e$ as instead proportional to the relative velocity difference between the IF and OF, which was also considered by Bloomfield & Kerr (Reference Bloomfield and Kerr2000) alongside (8.4). This approach, as well as an alternative treatment of the body-force that controls the buoyancy of the IF, is discussed in Appendix A.

No assumptions will be made about the velocity of the outflowing fluid, $\hat {u}_{out}$, which in practice is likely dependent on both IF and OF. In this formulation, even if $\hat {w}_{of}=0$ and the flow is effectively an NBJ without a return flow, there may still be a radial outflow of fluid occurring simultaneously to entrainment (i.e. $\hat {u}_e>0$ and $\hat {u}_{out}>0$). This is a key difference when compared to previously proposed fountain models, such as by McDougall (Reference McDougall1981) or Bloomfield & Kerr (Reference Bloomfield and Kerr2000), who modelled flow from the IF to the OF as proportional to $\hat {w}_{of}$, and related them by a second constant entrainment coefficient. The conservation equations for the present system may then be written as

(8.5)$$\begin{gather} \frac{\mathrm{d}\hat{Q}_{if}}{\mathrm{d}z} = 2\alpha_e\hat{r}_{if}\hat{w}_{if} - 2\hat{r}_{if}\hat{u}_{out} , \end{gather}$$

(8.6)$$\begin{gather}\frac{\mathrm{d}\hat{M}_{if}}{\mathrm{d}z} ={-}2\alpha_e\hat{r}_{if}\hat{w}_{if}\hat{w}_{of} - 2\hat{r}_{if}\hat{u}_{out}\hat{w}_{if} + \hat{r}_{if}^{2}\hat{g}_{if} , \end{gather}$$

(8.7)$$\begin{gather}\frac{\mathrm{d}\hat{F}_{if}}{\mathrm{d}z} = 2\alpha_e\hat{r}_{if}\hat{w}_{if}\hat{g}_{of} - 2\hat{r}_{if}\hat{u}_{out}\hat{g}_{if}, \end{gather}$$

which will be referred to as the ‘decomposed top-hat’ model. It was mentioned previously that the system of equations could be applied to an NBJ by considering $\hat {w}_{of}=0$, since there is no return flow. As is clear from the $\bar {w}/\bar {w}_c$ profiles for NBJs in figure 4(b), this is a reasonable description since the profiles go to approximately zero by $r/r_{1/2,w}\simeq 3$. However, since NBJs are subject to a net radial outflow, characterised by $\alpha <0$ in figure 16, we should expect that $\hat {u}_{out}>\hat {u}_e$ for a portion of the flow. This ejected fluid will be negatively buoyant, implying that $\hat {g}_{of}\neq 0$ in this top-hat description of the flow. This may be counter-intuitive since NBJs do not have a fully formed OF like a fountain, so invoking a non-zero $\hat {g}_{of}$ term potentially complicates the physical interpretation of the model. For this reason, the original model described by (7.5)–(7.7) and solved in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) may be more appropriate for NBJs than the top-hat description, which is presented primarily for the application to fountains. Despite this, characterising entrainment in NBJs as the balance of inflowing and outflowing fluid is still a physically meaningful description, and it is useful to compare this formulation to the original model. For this reason, we will now proceed to estimate the inflow and outflow components of entrainment with respect to the top-hat model in both fountains and NBJs using the present experimental data.

If the top-hat variables in (8.5)–(8.7) are replaced with integral quantities calculated from the experimental data, then the only unknowns in the volume and momentum flux equations are $\alpha _e$ and $\hat {u}_{out}$. For the IF velocity and radius, we set $\hat {w}_{if}=w_m$ and $\hat {r}_{if}=r_m$ as defined using (3.2) and (3.3a,b) with $\tilde r=r_{io}$ for a fountain and $\tilde {r}=\infty$ for an NBJ. We will define the OF velocity, $\hat {w}_{of}$, similarly to $w_m$ except that the integrals in (3.2) are evaluated from $r_{io}$ to infinity. For the IF buoyancy scale, $\hat {g}_{if}$, one choice might be $b_m$ from (3.3c), which is based on integrals of the buoyancy and velocity profiles. However, we choose instead $\hat {g}_{if}=g_m$, where $g_m$ is defined in (8.8) (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Armfield and Kirkpatrick2020). In this definition, $g_m$ is based on the buoyancy profile only, so is more closely related to the definition of $\hat {g}_{if}$, which is dependent only on the local density difference and gravity. If the real $\bar {w}$ and $\bar {b}$ profiles were exactly top-hat and had the same width, then $b_m$ and $g_m$ would be equal, where we have,

(8.8a,b)\begin{equation} G = 2\int_{0}^{\tilde{r}}\bar{b}^{2}r\,\mathrm{d}r, \quad g_m = \frac{G}{B}. \end{equation}

The OF buoyancy for a fountain, $\hat {g}_{of}$, may be defined similarly to $\hat {g}_{if}$ except with the integration limits in (8.8) set from $r=r_{io}$ to infinity. The derivatives $\mathrm {d}\hat {Q}_{if}/\mathrm {d}z$ and $\mathrm {d}\hat {M}_{if}/\mathrm {d}z$ are estimated using a second-order accurate finite difference stencil. When referring exclusively to NBJs, which do not have an OF, the subscript ‘$if$’ is omitted from the notation.

With all top-hat variables in (8.5) and (8.6) defined based on local integral scales, the pair of equations may then be solved simultaneously to obtain estimates for $\alpha _e$ and $\hat {u}_{out}$ along the fountain and NBJ. Figure 17 shows $\alpha _e$ plotted with local $Ri$ for $Fr_o=30$ and $Fr_o=15$ fountains, and an $Fr_o=30$ NBJ. Here, only results prior to the cap region are shown, since this is where the present top-hat model is expected to be valid. The entrainment coefficient for a neutral jet, which has $\hat {u}_{out}=0$, is also shown as a horizontal line as a reference. This corresponds to $\alpha =0.071$, as calculated using the same experimental set-up (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021). For the fully developed fountains, we have a slightly higher $\alpha _e$ in the $Fr_o=30$ case than in the $Fr_o=15$ case, with average values $\alpha _e=0.062$ and $\alpha _e=0.052$, respectively. Both fountains have $\alpha _e>0$ over the full range shown, which contrasts to the original definition of $\alpha$ discussed in § 7 and shown in figure 16. This had $\alpha <0$ for the majority of the $Ri$ range shown ($Ri\lesssim -0.13$ for $Fr_o=30$, and $Ri\lesssim -0.04$ for $Fr_o=15$), and also had $\alpha$ generally lower in the $Fr_o=15$ case. This previous formulation of $\alpha$, which was expressed by (7.16) and (7.20), will henceforth be referred to as the ‘full model’.

Figure 17. The entrainment coefficient, $\alpha _e$, plotted with local $Ri$ for the IF of an $Fr_o=30$ and $Fr_o=15$ fountain, and of an $Fr_o=30$ NBJ. Here, $\alpha _e=\hat {u}_e/\hat {w}_{if}$, corresponding to the decomposed top-hat model described by (8.5)–(8.7). The ‘standard’ entrainment coefficient for the neutral jet (J), $\alpha$, as reported in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Armfield and Kirkpatrick2020) using the same experimental set-up, is also shown as a reference. For neutral jets, $Ri=0$ everywhere so $\alpha$ is shown as a horizontal line for clarity.

The results in figure 17 for the ‘decomposed model’ are more scattered in the NBJ since there are fewer data available, but we clearly have $\alpha _e>0$ over the $Ri$ range shown (with average value $\alpha _e=0.077$). This again contrasts to the full model in figure 16, where $\alpha <0$ for $Ri\lesssim -0.25$. Since there is no clear systematic trend for $\alpha _e$ with $Ri$ in the NBJ or fountains, $\alpha _e$ could be approximated by taking the average value as constant. This returns values of $\alpha _e$ in the NBJ similar to $\alpha$ in a neutral jet, and fountain values somewhat lower.

In the decomposed formulation, $\alpha _e$ can be interpreted as a measure of the rate of ‘turbulent entrainment’ in the flow. For fountains, this represents ‘entrainment’ from the OF to the IF across the IF/OF interface, and in NBJs it is entrainment from the ambient into the jet across the TNTI. This contrasts to $\alpha$ in the full model shown in figure 16, which simply reflects the direction and magnitude of the mean radial velocity.

Solving (8.5) and (8.6) simultaneously also provides values for $\hat {u}_{out}$, the outflow velocity. Figure 18 shows $\hat {u}_{out}/\hat {w}_{if}$ plotted with $Ri$ for the $Fr_o=30$ and $Fr_o=15$ fountains, and $Fr_o=30$ NBJ. We see that, for all three flows, $\hat {u}_{out}$ increases with $-Ri$, and that $\hat {u}_{out}>0$ for all $Ri<0$. Near the source, where the IF is much stronger than the OF, $\hat {u}_{out}/\hat {w}_{if}\rightarrow 0$ and $Ri\rightarrow 0$. This is consistent with the neutral jet case, which has $Ri=0$ and does not have a mean outflow. Regions where there is a net radial outflow of fluid from the IF correspond to where $\hat {u}_{out}>\hat {u}_e$. Since $\alpha _e=\hat {u}_e/\hat {w}_{if}$ from (8.4), this is equivalent to where $\hat {u}_{out}/\hat {w}_{if}>\alpha _e$. Horizontal lines showing average values of $\alpha _e$ for the fountains and NBJ are also shown in figure 18, which therefore indicate the start of the region where there is a net radial outflow of fluid from the IF. Figure 18 therefore implies that the IFs of both $Fr_o$ fountains and the NBJ have a net radial outflow of fluid for a large portion of their height, with the local $Ri$ at the start of this region broadly consistent with figure 16 – that is, where $\alpha <0$ in the original formulation given by (7.16) and (7.20).

Figure 18. The normalised radial outflow velocity, $\hat {u}_{out}/\hat {w}_{if}$, with respect to the decomposed top-hat model for $Fr_o=30$ and $Fr_o=15$ fountains, and an $Fr_o=30$ NBJ. Horizontal lines indicating average values of $\alpha _e$ from figure 17 are also shown.

In summary, the decomposed formulation separates radial flow between the IF and OF of a fountain into two components that are constrained by both the conservation of mass and momentum. The inflow term is described by the approximately constant $\alpha _e$, which represents ‘turbulent entrainment’ from the OF to the IF. The outflow term, $\hat {u}_{out}/\hat {w}_{if}$, represents fluid ejected from the IF to the OF and depends on the local $Ri$. This term approaches zero as $Ri\rightarrow 0$ where the IF is much stronger than the OF, and increases for more negative $Ri$.

8.2. Connection to the full model

Figures 17 and 18 show that in the present decomposed top-hat formulation, the radial flow across the IF/OF boundary of a fountain (or between an NBJ and the ambient) can be described by an approximately constant turbulent entrainment coefficient, $\alpha _e$, and a local $Ri$-dependent outflow component, $\hat {u}_{out}/\hat {w}_{if}$. If the net radial flow from the IF to the OF is denoted $\hat {u}$, so that $\hat {u}=\hat {u}_{out}-\hat {u}_e$, then the ratio $\hat {\alpha }=-\hat {u}/\hat {w}_{if}$ is a non-dimensional measure of the direction and magnitude of the mean radial flow at a given height. Using (8.4), this may then be written as

(8.9)\begin{equation} \hat{\alpha} = \alpha_e - \frac{\hat{u}_{out}}{\hat{w}_{if}}, \end{equation}

which expresses $\hat {\alpha }$ as the sum of a constant and an $Ri$-dependent component, $\alpha _e$ and $-\hat {u}_{out}/\hat {w}_{if}$, respectively. This is then the present top-hat model's equivalent to $\alpha$ in the full model, defined by (7.11) or (7.20), which is also a measure of the mean radial velocity relative to the IF velocity scale. This equivalence may be seen by substituting (8.9) into (8.5), and recalling that the same integral definitions for $\hat {Q}_{if}$ and $\hat {M}_{if}$ are used as for $Q$ and $M$ in (7.5) and (7.12). If the profile coefficients in (7.20) are assumed constant, then $\alpha$ in the full model may be written as

(8.10)\begin{equation} \alpha = \frac{3}{2}\left(\frac{\delta_m}{\gamma_m}-\frac{4\varDelta_m}{\varGamma_m}\right) + 3\left(\frac{\theta_m}{\gamma_m}-\frac{\mu_m}{\varGamma_m}\right)Ri, \end{equation}

which is also expressed as the sum of a constant and an $Ri$-dependent component, namely the first and second terms on the right-hand side of (8.10), respectively.

For Gaussian velocity and buoyancy profiles, shown in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) to be a reasonable assumption in NBJs, we have that $\gamma _m=4/3$, $\varGamma _m=1$, $\theta _m=2/(\lambda ^{2}+1)$ and $\mu _m=2/(2\lambda ^{2}+2)$. Here, $\lambda$ is defined as

(8.11)\begin{equation} \lambda=\frac{r_b}{r_w}, \end{equation}

where $r_b$ and $r_w$ are the $1/\mathrm {e}$ widths of the buoyancy and velocity profiles, respectively. In Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), it was shown that $\lambda$, and hence $\theta _m$ and $\mu _m$, are not constant in NBJs. However, their variation has little effect on $\alpha$ once multiplied by $Ri$, such as in (7.11) or (8.10), so they may be approximated as constant, with $\lambda \simeq 1.46$ for this purpose. For the IF of a fountain, the profiles cannot be described simply by Gaussian functions, so these profile coefficients must be estimated numerically from their integral definitions, (7.10) and (7.19). By assuming that these are constant we obtain, by taking averages, $\theta _m\simeq 0.68$ and $\mu _m\simeq 0.36$ for $Fr_o=30$, and $\theta _m\simeq 0.65$ and $\mu _m\simeq 0.34$ for $Fr_o=15$. The data suggest that $\gamma _m$ and $\varGamma _m$ have a weak $Ri$ dependence in fountains, with $1.27\gtrsim \gamma _m\gtrsim 1.18$ and $0.90\gtrsim \varGamma _m\gtrsim 0.76$ over $0\lesssim -Ri\lesssim 0.5$. Since this variation is not very significant, these were approximated as constant using average values $\gamma _m\simeq 1.23$ and $\varGamma _m\simeq 0.85$ for $Fr_o=30$, and $\gamma _m\simeq 1.23$ and $\varGamma _m\simeq 0.84$ for $Fr_o=15$. All four of these profile coefficients, $\theta _m$, $\mu _m$, $\gamma _m$ and $\varGamma _m$, therefore have similar values for both fountains, with a negligible $Fr_o$ dependence.

For an NBJ, it was shown in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) that $\delta _m$ is not constant, and instead varies linearly with $Ri$. The present data show that an approximately linear relation also occurs in fountains. Additionally, the profile coefficient $\varDelta _m$, defined in (7.19) and related to the turbulent production, is also well described by a linear $Ri$ relation. These can be expressed as

(8.12)\begin{equation} \left. \begin{array}{l@{}} \delta_m=\delta_o+\tilde\delta\,Ri, \\ \varDelta_m=\varDelta_o+\tilde\varDelta\,Ri, \end{array} \right\} \end{equation}

where $\delta _o$ and $\varDelta _o$ represent the values at the source, and $\tilde {\delta }$ and $\tilde {\varDelta }$ are additional ad hoc terms that capture the observed $Ri$ dependence. The increasing of $-\delta _m$ and $-\varDelta _m$ with negative $Ri$ is related to the behaviour of the $\overline {w'u'}$ profiles discussed in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) and § 3.2, where the turbulence did not decrease at the same rate as the mean flow in NBJs and fountains. A detailed analysis of the governing equations may shed light on the precise reason for this behaviour, which is left for potential future research. For the purposes of the present investigation, the observation of a linear trend and subsequent empirical expressions in (8.12) will be used.

By using these linear relations, the expression for $\alpha$ in (8.10) becomes

(8.13)\begin{equation} \alpha = \underbrace{\frac{3}{2}\left(\frac{\delta_o}{\gamma_m}-\frac{4\varDelta_o}{\varGamma_m}\right)}_{P_1} + \underbrace{3\left(\frac{\theta_m}{\gamma_m}-\frac{\mu_m}{\varGamma_m} +\frac{\tilde{\delta}}{2\gamma_m}-\frac{2\tilde{\varDelta}}{\varGamma_m} \right)Ri}_{P_2}. \end{equation}

This expresses $\alpha$ as the sum of a constant and an $Ri$-dependent component, $P_1$ and $P_2$, and is valid for both NBJs and fountains. The following expression, obtained by combining the linear $\delta _m$ relation with (7.20), is valid for NBJs only:

(8.14)\begin{equation} \alpha = \underbrace{-\frac{\delta_o}{2\gamma_m}}_{L_1} + \underbrace{\left(1-\frac{\theta_m}{\gamma_m}-\frac{\tilde{\delta}}{2\gamma_m}\right)Ri}_{L_2}. \end{equation}

Although both (8.13) and (8.14) are equally valid for NBJs, (8.14) is simpler since it does not depend on $\varDelta _m$ or $\varGamma _m$. Equation (8.14) will therefore be used for calculating $\alpha$ in NBJs, while (8.13) will be used for fountains. It is then useful to attempt to unify the decomposed top-hat model with the full model by considering $\hat {\alpha }=\alpha$. Both (8.13) and (8.14) express $\alpha$ in terms of constant and $Ri$-dependent components, which may then be compared individually to the components of $\hat {\alpha }$ defined in (8.9). We would then have

(8.15)$$\begin{gather} \left. \begin{array}{l@{}} L_1 = \alpha_e, \\ \displaystyle L_2 ={-}\dfrac{\hat{u}_{out}}{\hat{w}}, \end{array} \right\} \text{NBJ} \end{gather}$$

(8.16)$$\begin{gather}\left. \begin{array}{l@{}} P_1 = \alpha_e, \\ \displaystyle P_2 ={-}\dfrac{\hat{u}_{out}}{\hat{w}_{if}}, \end{array} \right\} \text{Fountain} \end{gather}$$

for the NBJ and fountains.

These different components and their summation are given in figure 19 for the $Fr_o=30$ NBJ, and in figure 20 for the $Fr_o=30$ and $Fr_o=15$ fountains. For the NBJ, figure 19(a) shows $L_1=0.075$ and $L_2=0.292\,Ri$ corresponding to (8.14), with the values obtained by assuming Gaussian velocity and buoyancy profiles, a constant $\lambda \simeq 1.46$, and a linear fit of $\delta _m$ (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021). Although very good agreement is observed between $\alpha _e$ and $L_1$, the constant component of the entrainment coefficient, there is a discrepancy between the $Ri$-dependent terms, $L_2$ and $-\hat {u}_{out}/\hat {w}$. It was found that better agreement could be obtained by including the turbulence components of the profile coefficients $\delta _f$ and $\gamma _f$ in the expression for $\alpha$ (van Reeuwijk & Craske Reference van Reeuwijk and Craske2015):

(8.17a,b)\begin{equation} \gamma_f=\frac{4}{w_m^{3}r_m^{2}}\int_{0}^{\tilde{r}}\bar{w}\overline{w'^{2}}r\,\mathrm{d}r,\quad \delta_f=\frac{4}{w_m^{3}r_m}\int_{0}^{\tilde{r}}\overline{w'^{2}}\,\frac{\partial\bar{w}}{\partial z}\,r\,\mathrm{d}r. \end{equation}

These may be included by replacing $\delta _m$ and $\gamma _m$ with $\delta _g=\delta _m+\delta _f$ and $\gamma _g=\gamma _m+\gamma _f$, wherever they appear. By initially neglecting higher-order turbulence terms in the conservation equations, (7.1)–(7.4), $\delta _f$ and $\gamma _f$ were assumed negligible prior to this point. From the present NBJ data, these terms are estimated to be approximately $18\,\%$ and $35\,\%$ of the corresponding mean components, $\delta _m$ and $\gamma _m$, respectively. As a comparison, these are approximately $1\,\%$ and $25\,\%$ of the mean values for a neutral jet in the present data, and are commonly neglected for this flow (Kaminski et al. Reference Kaminski, Tait and Carazzo2005; van Reeuwijk & Craske Reference van Reeuwijk and Craske2015). Now, average values $\delta _f\simeq -0.044$ and $\gamma _f\simeq 0.47$, as estimated from the present experimental data for the NBJ, will be included and assumed constant. This results in the following expression for $\alpha$ that can be used in place of (8.14):

(8.18)\begin{equation} \alpha = \underbrace{-\frac{\delta_o+\delta_f}{2\gamma_g}}_{L_1} + \underbrace{\left(1-\frac{\theta_m}{\gamma_g}-\frac{\tilde\delta}{2\gamma_g}\right)Ri}_{L_2}. \end{equation}

Figure 19(b) then shows the same $\hat {\alpha }$ decomposition, but now with $L_1$ and $L_2$ as calculated by (8.18). Although the constant terms, $\alpha _e$ and $L_1$, are not as close as in figure 19(a), the agreement between the $Ri$ terms $-\hat {u}_{out}/\hat {w}$ and $L_2$, and the overall summations $\hat {\alpha }$ and $\alpha$, have been improved significantly. Although this supports the validity of the decomposed top-hat formulation, it raises questions as to whether the assumption made up to this point, that the turbulence components of the profile coefficients may be ignored, is appropriate. Although the turbulence components do have an effect on the entrainment rate, the modelling of NBJs in Milton-McGurk et al. (Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021) showed that reasonable $\bar {w}_c$ agreement could nevertheless be achieved without them. It is argued therefore that neglecting these higher-order contributions is a reasonable simplification for the purposes of the present investigation. The effect of including additional turbulence components is left for potential future research.

Figure 19. The constant and $Ri$-dependent components of the entrainment coefficient of an $Fr_o=30$ NBJ from both the decomposed top-hat model ($\hat {\alpha }=\alpha _e-\hat {u}_{out}/\hat {w}$) and the full model ($\alpha =L_1+L_2$). Panel (a) shows the terms $L_1$ and $L_2$ as calculated using (8.14), while (b) uses (8.18). That is, (a) neglects the turbulence component of profile coefficients, $\delta _f$ and $\gamma _f$, while (b) includes them.

Figure 20. The constant and $Ri$-dependent components of the entrainment coefficient of a fully developed fountain from both the decomposed top-hat model ($\hat {\alpha }=\alpha _e-\hat {u}_{out}/\hat {w}$) and the full model ($\alpha =L_1+L_2$). Results for the $Fr_o=30$ fountain are shown in (a), and for the $Fr_o=15$ case in (b).

The components of the entrainment coefficient for the fully developed fountain are given in figures 20(a) and 20(b) for the $Fr_o=30$ and $15$ cases, respectively. For both $Fr_o$ cases, (8.13) has been used to calculate the $\alpha$ components, $P_1$ and $P_2$, with the turbulence profile coefficients neglected. Here, the linear fits used to obtain ${\delta _m=\delta _o+\tilde {\delta }\,Ri}$ and ${\varDelta _m=\varDelta _o+\tilde {\Delta }\,Ri}$ have been obtained from the experimental data for each $Fr_o$ fountain separately. The $Fr_o=30$ case in figure 20(a) shows good agreement between the constant and $Ri$-dependent components, with $\alpha _e\simeq P_1$ and $-\hat {u}_{out}/\hat {w}_{if}\simeq P_2$, and subsequently ${\alpha \simeq \hat {\alpha }}$. Although the agreement is more modest in the $Fr_o=15$ case, the components and their summation are still broadly consistent.

In the case of the NBJ, the constant component corresponds to turbulent entrainment from ambient fluid into the NBJ. Near the source where $Ri\rightarrow 0$, this fully describes entrainment and may be interpreted as the ‘neutral jet’ value. This is supported by the present experimental data, which give similar values of $\alpha =0.071$ and $\alpha _e=0.077$ for the neutral and negatively buoyant jets, respectively. For a fully developed fountain, the constant component can be interpreted as describing the ‘turbulent entrainment’ from the OF to the IF associated with the mean IF velocity. In this case, the values are slightly lower than the neutral/negatively buoyant jets, and also have a source $Fr_o$ dependence, with $\alpha _e\simeq 0.062$ and $\alpha _e\simeq 0.052$ for the $Fr_o=30$ and $15$ cases, respectively. This difference may be attributed to the existence of a return flow, which will depend on $Fr_o$ and is still present even at the source where $Ri=0$.

In describing the radial flow between the IF and OF of a fountain, the highly simplified decomposed top-hat description of a fountain is consistent with the the more general full model approach discussed in § 7.2. This is also applicable to an NBJ, which can be thought of as a fountain with a zero-velocity OF under the same top-hat formulation. This provides evidence that despite fountains generally having a mean radial flow from the IF to the OF (or into the ambient for an NBJ), they still ‘entrain’ fluid at a rate proportional to their characteristic axial velocity. This entrainment is captured by $\alpha _e$ in the top-hat model, and by terms $L_1$ and $P_1$ in the full model.

While buoyant jets and plumes are subject to an acceleration due to their positive buoyancy and $Ri>0$, NBJs are instead decelerated as characterised by their increasingly negative $Ri$. This has implications for entrainment as captured by the $\hat {u}_{out}$ term in the decomposed formulation, and by $L_2$ and $P_2$ in the full model. In NBJs under the decomposed formulation, this has the effect of encouraging a radial outflow of fluid. In a physical flow, there will be instantaneous entrainment at some times and radial outflow at others. A mean radial outflow of fluid then occurs when the $Ri$-dependent term is larger in magnitude than the constant entrainment term, resulting in $\alpha <0$. In addition to the present experiments, different regions of mean radial inflow and outflow have also been observed in other fountain investigations, such as by Williamson et al. (Reference Williamson, Armfield and Lin2011) and Cresswell & Szczepura (Reference Cresswell and Szczepura1993). The present decomposed top-hat model provides a consistent description of this observation, allowing it to be understood in a context similar to that for the classical entrainment relations applied originally to jets and plumes.

8.3. Special cases: pure jets/plumes

It is also useful to consider more closely the application of these models to neutral jets and plumes. Once fully developed, these idealised flows can be considered fully self-similar with constant profile coefficients. By assuming additionally that the velocity and buoyancy/scalar profiles are Gaussian, (7.11) and (7.20) become

(8.19)$$\begin{gather} \alpha ={-}\frac{3}{8}\delta_m + \left(1-\frac{3}{2(\lambda^{2}+1)}\right)Ri, \end{gather}$$

(8.20)$$\begin{gather}\alpha = \frac{3}{2}\left(\frac{3}{4}\delta_m-4\varDelta_m\right) + 3\left(\frac{3}{2(\lambda^{2}+1)}-\frac{2}{2\lambda^{2}+1}\right)Ri, \end{gather}$$

where $\lambda$ is the ratio of $1/\mathrm {e}$ widths of the scalar and velocity profiles. For a neutral jet, $Ri=0$, so by equating (8.20) and (8.19), we find that $\delta _m=4\varDelta _m$. This is supported by the experimental data for a neutral jet (Milton-McGurk et al. Reference Milton-McGurk, Williamson, Arfmfield, Kirkpatrick and Talluru2021), where we find $\delta _m\simeq 4\varDelta _m\simeq -0.20$.

If a self-similar buoyant jet or pure plume is now considered, with $0< Ri\leqslant Ri_p$ and constant $\delta _m$, then by equating the $Ri$ components of (8.19) and (8.20), we obtain

(8.21)\begin{equation} \frac{6}{\lambda^{2}+1}-\frac{6}{2\lambda^{2}+1} -1 = 0, \end{equation}

that is, a non-linear equation restricting the possible values of $\lambda$ to maintain consistency between (8.20) and (8.19), with positive solutions $\lambda =1$ and $\lambda =1/\sqrt {2}$. Under these assumptions, we then have that $\lambda$ can take only the values $1$ and $1/\sqrt {2}$ in order to maintain consistency with the conservation of mass, momentum and kinetic energy equations, (7.1)–(7.4), and the additionally derived $\bar {w}^{3}$ equation in (7.17). Although, to the best of the authors knowledge, there have been no studies to have reported a value as low as $\lambda =1/\sqrt {2}\simeq 0.707$ in buoyant jets or plumes, the prediction $\lambda =1$ is remarkably consistent with the existing plume literature (Papanicolaou & List Reference Papanicolaou and List1988; Shabbir & George Reference Shabbir and George1994; Wang & Law Reference Wang and Law2002; van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016). This then provides a new theoretical justification of the commonly reported $\lambda =1$ observation in pure plumes. It should be noted that although this consistency requirement exists for plumes/buoyant jets under these assumptions, there is no constraint here on $\lambda$ in neutral jets ($Ri=0$), which are typically reported to have $1.15\lesssim \lambda \lesssim 1.30$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Wang & Law Reference Wang and Law2002; Ezzamel, Salizzoni & Hunt Reference Ezzamel, Salizzoni and Hunt2015). If $\delta _m$ is not constant, such as in NBJs or near the source in positively buoyant jets (van Reeuwijk et al. Reference van Reeuwijk, Salizzoni, Hunt and Craske2016), then (8.21) is also not valid.

Consider now the application of the decomposed top-hat model to a Gaussian neutral jet by equating $\alpha =\hat {\alpha }$ from (8.19) and (8.9). Here, the $Ri$-dependent term is zero and we have $\alpha _e=-3\delta _m/8$ – that is, a constant turbulent entrainment coefficient $\alpha _e$, and no outflow term. This is consistent with the classical description of a turbulent jet, which entrains ambient fluid at a rate proportional to its mean axial velocity, and is not subject to any ‘mean radial outflow’.

For a buoyant jet with $\lambda =1$ and constant $\delta _m$, by setting $\alpha =\hat {\alpha }$ we obtain

(8.22)\begin{equation} \alpha = \alpha_e - \frac{\hat{u}_{out}}{\hat{w}} ={-}\tfrac{3}{8}\delta_m + \tfrac{1}{4}Ri. \end{equation}

Equating the constant and $Ri$-dependent terms would then give $\alpha _e=-3\delta _m/8$ and $-\hat {u}_{out}/\hat {w}=Ri/4$. We then have $-\hat {u}_{out}/\hat {w}>0$ and $\alpha _e>0$, that is, both terms promoting entrainment from the ambient into the plume, rather than a balance of inflow and outflow as is the case in an NBJ. If applying the decomposed top-hat formulation to positively buoyant jets or plumes, it is therefore potentially misleading to refer to the $Ri$-dependent term as the ‘outflow term’, $\hat {u}_{out}$. Despite this, it is still a consistent description of the flow, and distinguishes between radial inflow associated with turbulent entrainment and that associated with positive buoyancy. In these applications, it may simply be preferable to use an alternative nomenclature, such as replacing $\hat {u}_{out}$ with $\hat {u}_b$ to indicate that it is radial flow associated with buoyancy.