3.1 Bubble shape regime

From the image data, we measured the median bubble sizes to be 3.8 and 4.1 mm for $Q_{1}$ and $Q_{2}$ , respectively (see table 1). These bubbles appear to be ellipsoidal and wobbling in the video data. The quantitative presentation of bubble shape regime can be illustrated by three non-dimensional parameters: Reynolds number $Re=\unicode[STIX]{x1D70C}dU_{s}/\unicode[STIX]{x1D707}$ (where $d$ is bubble diameter and $\unicode[STIX]{x1D707}$ is the dynamic viscosity of water), Morton number $M=g\unicode[STIX]{x1D707}^{4}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\overline{\unicode[STIX]{x1D70C}}^{2}\unicode[STIX]{x1D70E}^{3}$ (where $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{p})$ and $\unicode[STIX]{x1D70E}$ is the interfacial tension), and Eötvös number $Eo=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}d^{2}/\unicode[STIX]{x1D70E}$ . Values for our experiments are shown in figure 4 along with the shape regime classification in Clift et al. (Reference Clift, Grace and Weber1978). Figure 4 demonstrates that the observed bubbles are in the ellipsoidal wobbling regime of the bubble theory. Bubbles at natural seeps have also been observed to have similar shapes and sizes (Wang et al. Reference Wang, Socolofsky, Breier and Seewald2016).

Figure 4.

Shape regime of the observed bubbles. The curves are reproduced after figure 2.5 in Clift et al. (Reference Clift, Grace and Weber1978).

3.2 Velocity and void fraction profiles

In figure 5, we examine the shapes of the mean velocity and void fraction profiles in the weak bubble plume. The horizontal coordinate for the velocity and void fraction is normalized at each height by $b_{g}$ and $b_{\unicode[STIX]{x1D712}}$ , respectively. Likewise, the velocity and void fractions are normalized by their centreline values $U_{m}$ and $\unicode[STIX]{x1D712}_{m}$ , respectively. The irregular spacing of the velocity data is due to the fact that the velocity profile is computed from the bubbles, and the bubbles occupy discrete points and are not evenly distributed in the plume. Because the bubbles are mainly located inside the plume, the velocity data for $|r/b_{g}|>1.5$ had few data points and were not statistically converged; therefore, they are not included in the figures or analysis. Despite the scatter, all data appear consistent with the Gaussian distribution, with the goodness of fit $R^{2}=0.73$ and 0.95 for velocity and void fraction, respectively.

Figure 5.

Self-similarity of the weak bubble plume: (a) mean velocity profile (error bars show standard deviation of the data); (b) mean void fraction profile.

The data in figure 5 confirm the Gaussian profile shapes assumed in (2.2) and (2.3) for continuous-phase velocity and void fraction in a weak bubble plume. Although the profile shapes are preserved with height, the scales used in the non-dimensionalization ( $b_{g}$ , $b_{\unicode[STIX]{x1D712}}$ , $U_{m}$ and  $\unicode[STIX]{x1D712}_{m}$ ) are not constant with height; hence, this result is not a confirmation of self-similar behaviour. Rather, these results give an analytical equation summarizing the results at each measurement height. Using the Gaussian distribution, the net upward volume flux $Q$ in the plume is obtained as

(3.1) $$\begin{eqnarray}Q=\int _{0}^{\infty }U(r)2\unicode[STIX]{x03C0}r\,\text{d}r=\unicode[STIX]{x03C0}b_{g}^{2}U_{m}.\end{eqnarray}$$

The validity of this relation is built upon the assumption of negligible gas void fraction in the plume (e.g. 0.1 % and 0.16 % in the lowest measurement location for $Q_{1}$ and $Q_{2}$ , respectively). With the spreading of the bubbles in the water column, the values of void fraction decrease substantially at high locations. Equation (3.1) was used in the analysis by Leitch & Baines (Reference Leitch and Baines1989) for a similar weak bubble plume, but they did not obtain the velocity profiles to validate the Gaussian distribution. We analyse the volume flux profile $Q(z)$ in § 4.2, where we compare entrainment models.

3.3 Plume width and spreading rate

The spreading rate in the lateral direction of a classic bubble plume has been shown to be similar to that for single-phase round plumes (Ditmars & Cederwall Reference Ditmars and Cederwall1974), following a linear spreading behaviour given by

(3.2) $$\begin{eqnarray}\frac{\text{d}b_{g}}{\text{d}z}=\unicode[STIX]{x1D6FD},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}\approx 0.1$ (Fisher et al. Reference Fisher, List, Koh, Imberger and Brooks1979).

Figure 6.

Half-widths of the plume at different heights above the orifice. The two solid lines without symbols represent the best fit to the data of $b_{\unicode[STIX]{x1D712}}$ (red line, $b_{\unicode[STIX]{x1D712}}=0.021z^{1/2}-0.002$ ) and $b_{g}$ (blue line, $b_{g}=0.028z^{1/2}+0.006$ ). (a) Linear scale plot, where the dashed line represents the linear spreading rate in (3.2). (b) Log–log scale plot, where the dashed line represents the slope of a diffusive spreading process.

Figure 6 shows the measured relationship between the plume half-widths $b_{g}$ and $b_{\unicode[STIX]{x1D712}}$ with the height $z$ above the orifice in our experiments. In the linear scale plot (figure 6 a), a reference line for linear spreading with $\unicode[STIX]{x1D6FD}=0.1$ is shown as the dashed line. Comparing the measured data with the linear growth prediction, the spreading of the weak plume is much slower than $b_{g}=0.1z$ in the current experiment. This indicates that the physical mechanism causing the spreading of the weak bubble plume differs from that for the single-phase plume and from that of the classic bubble plume. Leitch & Baines (Reference Leitch and Baines1989) also observed a much weaker growth rate than $\unicode[STIX]{x1D6FD}=0.1$ in their experiment for the cases of gas flow rate ${\geqslant}0.36~\text{NL}~\text{min}^{-1}$ . Leitch & Baines (Reference Leitch and Baines1989) suggested that the liquid volume flux is mostly contributed by the entrainment into the near wake of individual bubbles in the weak bubble plume, instead of the entrainment process by eddies in the turbulent shear layer of a classic bubble plume. While this may explain the flow rate, this mechanism does not directly explain the spreading rate of the plume. Also, for the lowest measurement point in our measurements ( $z=0.5$  m), the data seem to track the $b_{g}=0.1z$ line, suggesting that a linear spreading region may still exist close to the bubble source. Hugi (Reference Hugi1993) also showed a linear spreading rate of a bubble plume close to the source following $b_{g}=0.1z$ that transitioned to a spreading rate that decreases further away from the source.

If we plot the plume half-width as a function of height in log–log space (figure 6 b), the weak plume spreading follows $b_{g}\sim z^{1/2}$ , albeit with considerable scatter of the data deviating from the best-fitting lines (especially for $Q_{2}$ ). The scatter of the data is almost inevitable due to the experiments being conducted at similar to field scale and due to the small values of the state variables under these weak gas flow rate conditions. The deviation of the data from the $1/2$ power law growth lines could have been improved by increasing measurement locations in the experiment, which was not feasible due to our experimental constraints. The $z^{1/2}$ growth rate we argued for here has also been observed in Leitch & Baines (Reference Leitch and Baines1989) ( $b_{g}=0.2z^{1/2}$ ), but the mechanism responsible for the $1/2$ power law relationship has not been postulated. This observation suggests that the spreading of weak bubble plumes cannot be explained by the traditional bubble plume theory; hence, a new theory is needed to explain the physics of weak bubble plumes.

4.1 Lateral spreading rates

As shown in figure 6(b), the spreading of the weak bubble plume above $z=0.5$  m height follows $b_{g}\sim z^{1/2}$ . This region of the bubble plume behaviour is in the range of $z/D\geqslant 29$ (at low gas flow rate) and $z/D\geqslant 6.0$ (at high gas flow rate), which are both in the asymptotic regime defined by Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007). In this region, the measured centreline water velocities are between 3 and $7~\text{cm}~\text{s}^{-1}$ and the bubble rise velocities are nearly constant at $22~\text{cm}~\text{s}^{-1}$ . This allows us to substitute time and height by $z=U_{b}t$ , where $U_{b}$ is the mean net rise velocity $U(r)+U_{s}$ of bubbles, taken as approximately constant. Then, we have the plume spreading as $b_{g}\sim t^{1/2}$ , which is consistent with a diffusive process at constant diffusivity.

To model this diffusion process for plume spreading, we apply the far-field Taylor spreading theory (Taylor Reference Taylor1921) with an effective lateral diffusion coefficient $E_{t}$ and solve for a point source:

(4.1) $$\begin{eqnarray}\frac{\text{d}b_{g}^{2}}{\text{d}t}=\unicode[STIX]{x1D6FD}_{t}=2E_{t}\quad \Rightarrow \quad b_{g}=(2E_{t}t)^{1/2}.\end{eqnarray}$$

If we use the bubble slip velocity $U_{s}$ to approximate $U_{b}$ since the vertical liquid velocities are relatively small, we can convert the Lagrangian equation (4.1) to an Eulerian expression, given by

(4.2) $$\begin{eqnarray}\frac{\text{d}b_{g}^{2}}{\text{d}z}=\unicode[STIX]{x1D6FD}_{t}=2E_{t}U_{s}^{-1}\quad \Rightarrow \quad b_{g}=\sqrt{2}E_{t}^{1/2}z^{1/2}U_{s}^{-1/2}.\end{eqnarray}$$

The measured plume width can be used with this equation to estimate $E_{t}$ . We perform a regression to the measured data on the width of the plumes, which follow a linear spreading process with $b_{g}=0.1z$ for $z/D<5$ from a point source, followed by a diffusive process with $\text{d}b_{g}^{2}/\text{d}z=2E_{t}/U_{s}$ for $z/D\geqslant 5$ . From the regression to the data for  $b_{g}$ , we obtain $E_{t}=8.65\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ for $Q_{1}$ and $1.07\times 10^{-4}~\text{m}^{2}~\text{s}^{-1}$ for $Q_{2}$ ; for fitting to data for $b_{\unicode[STIX]{x1D712}}$ , we obtain $E_{t}=3.98\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ at the lower gas flow rate and $5.02\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ at the higher release rate (see also table 2).

Table 2.

Effective lateral diffusion coefficient from the data fit and estimated from two different mechanisms. For $D_{T}$ and $D_{E}$ , mean $\pm$ standard deviation calculated from data for all heights are shown.

The differences in $E_{t}$ estimated from $b_{g}$ and $b_{\unicode[STIX]{x1D712}}$ relate to the different spreading rates for velocity and concentration. This effect is normally quantified by the spreading ratio  $\unicode[STIX]{x1D706}_{b}$ , given by

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{b}=\frac{b_{\unicode[STIX]{x1D712}}}{b_{g}}.\end{eqnarray}$$

Here, we can estimate $\unicode[STIX]{x1D706}_{b}$ by $\sqrt{E_{\unicode[STIX]{x1D712}}/E_{g}}$ , as well as from the direct calculation from the definition; the values for our data are reported in table 2. These values can be compared to those reported by Socolofsky & Adams (Reference Socolofsky and Adams2005), who computed $\unicode[STIX]{x1D706}_{b}$ between $b_{\unicode[STIX]{x1D712}}$ and  $b_{c}$ , the half-width of the concentration distribution. If we use the single-phase relationship $b_{c}=1.2b_{g}$ , our values of $\unicode[STIX]{x1D706}_{b}$ reported here are similar to those in Socolofsky & Adams (Reference Socolofsky and Adams2005) for bubble plumes with very weak entrained fluid velocity. Hence, the differences in the observed values of $E_{t}$ for fitting to $b_{g}$ and $b_{\unicode[STIX]{x1D712}}$ also match expectations for weak bubble plumes.

The turbulent dispersion in natural or engineered water systems can be described by an analogy to Fickian or molecular diffusion, and the diffusion coefficient is termed ‘turbulent diffusivity’ in turbulent flows. Since the ambient water is stagnant in the current experiment, the turbulent diffusion coefficient in the background is likely to be very low. Therefore, the diffusive process of the bubble spreading may be the result of two mechanisms: (1) the turbulent wake flow behind the leading bubbles; and (2) the lateral excursion of the bubbles, considering the zigzag or helical paths of these ellipsoidal wobbling bubbles (Wang et al. Reference Wang, Socolofsky, Breier and Seewald2016).

Here, we estimate the effective lateral diffusion coefficient for each of the above mechanisms. First, we estimate the turbulent diffusivity ( $D_{T}$ ) in the wake flow in three directions (i.e. bubble rising direction  $z$ , seeding flow direction  $x$ , and binormal direction  $y$ ). We applied an approach derived from Taylor’s theory following Holtappels & Lorke (Reference Holtappels and Lorke2011), giving turbulent diffusivity $D_{Ti}=u_{i}^{\prime }L_{i}=u_{i}^{\prime 2}T_{i}$ , where $u^{\prime }$ is the turbulent velocity scale, $L$ is the integral length scale, $T$ is the integral time scale and $i$ indicates the direction. In this work, $u_{i}^{\prime }$ is the standard deviation of the velocity measured by the ADV and $T_{i}$ is calculated by taking the integral of the autocorrelation function of the ADV data on the centreline (Tennekes & Lumley Reference Tennekes and Lumley1972; Holtappels & Lorke Reference Holtappels and Lorke2011). The estimated turbulent diffusivity is shown in figure 7. It is seen that $D_{T}$ for $Q_{2}$ is higher than that for $Q_{1}$ in all three directions, and $D_{T,z}$ is slightly larger than those in the other two directions. Overall, on the horizontal directions, $D_{T}$ is in the range of $10^{-6}~\text{m}^{2}~\text{s}^{-1}$ , and is approximately an order of magnitude smaller than $E_{t}$ estimated from the plume spreading data.

Figure 7.

Turbulent diffusivity at different heights for $Q_{1}$ and $Q_{2}$ , estimated using Taylor’s theory (Holtappels & Lorke Reference Holtappels and Lorke2011). Symbols present the data in the bubble plumes; the solid and dashed black lines represent the mean value and standard deviation of the diffusivity in a test run without the bubble plume but with the seeding flow.

The effective diffusion coefficient due to the second mechanism of bubble wobbling can be estimated from the product of the distance and velocity scales of the lateral excursion of the bubble motion:

(4.4) $$\begin{eqnarray}D_{E}=L_{E}U_{E},\end{eqnarray}$$

where $L_{E}$ is the mean of bubble lateral excursion distance and $U_{E}$ is the mean lateral velocity of the bubble excursion. The three-dimensional measurements of bubble location from the stereo camera data provide the information to compute these scales, and the calculated effective diffusion coefficients are reported in table 2. The $E_{t}$ calculated by fitting the void fraction is closer to $D_{E}$ than $D_{T}$ . In addition, the similar $E_{t}$ for both gas flow rates shown in the data may be well explained by the bubble excursion, which is determined by the bubble sizes, and is independent of the initial gas flow rate at these low flow rates. Here, we conclude that the bubble lateral excursion is probably a major contributing mechanism that is responsible for the spreading of the weak bubble plume in stagnant water, and the effective diffusion coefficient is determined by the extent and velocity of the lateral bubble motion. In more turbulent water systems, the turbulent diffusion may have a more profound contribution to the bubble plume spreading, but this is subject to further studies beyond our present scope. In any case, by these analyses, we conclude that a weak bubble plume spreads out by a diffusion process acting on the dispersed-phase particles.

4.2 Plume centreline velocity

The evolution of the centreline velocity with distance along the plume axis in a classic single-phase plume has been well documented. Fisher et al. (Reference Fisher, List, Koh, Imberger and Brooks1979) show that the centreline velocity scales with $z^{-1/3}$ in a continuous-phase plume. For classic bubble plumes, Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007) showed that

(4.5) $$\begin{eqnarray}\frac{U_{m}}{U_{s}}\sim \left(\frac{z}{D}\right)^{-1/3},\end{eqnarray}$$

for an integral bubble plume model for $z/D<5$ . This scaling is usually assumed to be valid in a classic bubble plume (Lemckert & Imberger Reference Lemckert and Imberger1993). The data of Fannelop & Sjoen (Reference Fannelop and Sjoen1980), Milgram & Van Houten (Reference Milgram and Van Houten1982) and Milgram (Reference Milgram1983) are close to this scaling, although they do not collapse onto it (see figure 6a in Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007) and figure 8 in the present paper).

Figure 8.

Normalized centreline water velocity $U_{m}$ (with respect to the bubble slip velocity, $U_{s}$ ) versus height $z$ above the orifice (with respect to the length scale  $D$ ), where $D$ is the length scale defined in Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007). The $Q_{1}$ and $Q_{2}$ in the legend are the data of the present study. Milgram83, MV82 and FS80 represent the data summarized in Milgram (Reference Milgram1983), including those from Fannelop & Sjoen (Reference Fannelop and Sjoen1980) (shown as FS80) and Milgram & Van Houten (Reference Milgram and Van Houten1982) (shown as MV82). The dashed lines show the slopes of $-1/3$ and $-1/2$ for classic and weak plumes, respectively.

Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007) obtained an analytical solution for the centreline velocity for $z/D\gg 1$ in a classical integral bubble plume model, namely

(4.6) $$\begin{eqnarray}\frac{U_{m}}{U_{s}}=3\left(\frac{z}{D}\right)^{-1/2},\quad \text{for}~z/D\gg 1.\end{eqnarray}$$

Bombardelli et al. (Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007) also used numerous simulation results to obtain the asymptotic equation:

(4.7) $$\begin{eqnarray}\frac{U_{m}}{U_{s}}=2\left(\frac{1.9(z/D)^{-1}}{1+0.563(z/D)^{1/2}}\right)^{1/3},\quad \text{for}~z/D>5.\end{eqnarray}$$

Figure 8 shows that the non-dimensional relationship between the centreline velocity in the plume and the height above the orifice for our data ( $z/D>5$ ) and the literature data ( $z/D<5$ ) falls in the vicinity of the prediction line given by (4.7). Our data and the data from the literature span the value of $z/D$ over four orders of magnitude. Although the discrepancy from the prediction line (4.7) grows for $z/D<0.5$ , it does pass through our weak bubble plume data and matches the observations over nearly the entire range of $z/D$ . Thus, we will consider (4.7) valid for classic and weak bubble plumes.

This result also seems to suggest that a classic integral model can quantitatively predict the centreline velocity in both classic and weak bubble plumes. However, this asymptotic solution is solved on the basis of a linear spreading hypothesis or constant entrainment coefficient hypothesis (see also later discussion in § 4.4). As a result, it cannot predict the correct plume spreading for the weak bubble plume (see figure 6 a), because the integral model with constant $\unicode[STIX]{x1D6FC}$ will follow $b_{g}=\unicode[STIX]{x1D6FD}z$ . Hence, the classic integral theory for a bubble plume is incapable of predicting the correct evolution of the liquid volume flux $Q=\unicode[STIX]{x03C0}b_{g}^{2}U_{m}$ in weak bubble plumes ( $z/D>5$ ) since it will estimate a correct $U_{m}$ while significantly overestimating  $b_{g}$ .

4.3 Liquid volume flux

In a general case, the relationship between $Q$ and $z$ in a plume is expected to follow a power law $Q\sim z^{m}$ (Leitch & Baines Reference Leitch and Baines1989). Because $\text{d}Q/\text{d}z\sim z^{m-1}$ represents the increase of liquid volume flux over height, which is due to the entrainment of ambient water, the value of $m$ can be understood as follows:

1. (i) $m>1$ , $Q$ increases faster than linearly (i.e. increasing entrainment);

2. (ii) $m=1$ , $Q$ increases linearly (i.e. constant entrainment);

3. (iii) $m<1$ , $Q$ increases slower than linearly (i.e. decreasing entrainment);

4. (iv) $m=0$ , $Q$ does not change (i.e. no entrainment); and

5. (v) $m<0$ , $Q$ decreases (i.e. detrainment).

From (4.5) and (3.2), the liquid volume flux in a classic integral model of a bubble plume can be derived as

(4.8) $$\begin{eqnarray}Q=\unicode[STIX]{x03C0}b_{g}^{2}U_{m}\sim \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}^{2}U_{s}D^{1/3}z^{5/3}.\end{eqnarray}$$

Equation (4.8) suggests that in a classic plume $Q$ increases faster than linearly as $m=5/3$ . Most of the literature data for bubble plumes (Fannelop & Sjoen Reference Fannelop and Sjoen1980; Milgram & Van Houten Reference Milgram and Van Houten1982; Milgram Reference Milgram1983) have $m$ values slightly smaller than $5/3$ , with the majority of these data observed in the classic bubble plume regime, below the asymptotic region (i.e.  $z/D<5$ ). This is a result of the presence of  $D$ , which violates the requirements for self-similarity, and expresses itself through a non-constant entrainment coefficient (or spreading rate  $\unicode[STIX]{x1D6FD}$ ). Milgram (Reference Milgram1983) developed a detailed theory for the entrainment coefficient in classic bubble plumes (see § 4.4), but most of the variability is near the source, and many successful models employ constant entrainment coefficients (Asaeda & Imberger Reference Asaeda and Imberger1993; Bombardelli et al. Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007; Socolofsky, Bhaumik & Seol Reference Socolofsky, Bhaumik and Seol2008).

Using the operational definition $Q=\unicode[STIX]{x03C0}b_{g}^{2}U_{m}$ , we can derive an equivalent expression in the weak bubble plume regime from our data. We select the diffusion growth process for plume spreading and substitute (4.2) for  $b_{g}$ . For the centreline velocity, we use the asymptotic solution at large $z/D\gg 1$ (4.6). This yields

(4.9) $$\begin{eqnarray}Q=6\unicode[STIX]{x03C0}E_{t}z^{1/2}D^{1/2}.\end{eqnarray}$$

From these fitted equations, $Q$ scales with $z^{1/2}$ in the weak plume, which demonstrates a decreasing entrainment of ambient water (i.e.  $m<1$ ). Therefore, the weak and classic bubble plumes have different entrainment processes (increasing and decreasing) and follow different scaling laws ( $m=1/2$ and $m=5/3$ ). In the intermediate range of $z/D\sim O(1)$ , $m$ is expected to be in the range between $1/2$ and $5/3$ .

Here, we also consider the similar laboratory measurements in weak bubble plumes by Leitch & Baines (Reference Leitch and Baines1989). They observed the same bubble spreading rate ( $b_{g}\sim z^{1/2}$ ) but report a different $m$ ( $m=1$ in their work instead of $m=1/2$ in this study). This discrepancy follows from their observation of $U_{m}(z)\sim z^{0}$ instead of $U_{m}(z)\sim z^{-1/2}$ . Their observation may represent an accurate measurement, which the scatter in figure 8 allows, or may result from their shallow experimental water depth (i.e. 0.5 m), which may not allow adequate development of the velocity gradient to observe the $-1/2$ power law. In our experiments, the observations were conducted over 16 m of water depth, and the $-1/2$ power law is consistent across this whole water depth, linking our experiments to results reported in Milgram & Van Houten (Reference Milgram and Van Houten1982). Hence, we conclude that $m=1/2$ is the more accurate estimate of the entrainment process in a weak bubble plume.

4.4 Apparent entrainment coefficient

In single-phase jets and plumes, an integral model is usually closed using an entrainment hypothesis or a spreading hypothesis, and the two hypotheses are equivalent (Lee & Chu Reference Lee and Chu2003) (see also appendix A). A similar spreading hypothesis was recently adapted to a particle plume (Lai et al. Reference Lai, Chan, Law and Adams2016). The equivalence of these two hypotheses can be summarized as follows:

(4.10) $$\begin{eqnarray}entrainment~hypothesis~(for~Gaussian~profile)\quad \frac{\text{d}Q}{\text{d}z}=2\unicode[STIX]{x03C0}b_{g}\unicode[STIX]{x1D6FC}U_{m},\end{eqnarray}$$

is equivalent to

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle jet\left\{\begin{array}{@{}ll@{}}linear~spreading\quad & \unicode[STIX]{x1D6FD}_{j}={\displaystyle \frac{\text{d}b_{g}}{\text{d}z}},\\ dimensional~analysis\quad & U_{m}=\left({\displaystyle \frac{M_{0}}{z^{2}}}\right)^{1/2},\end{array}\right. & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle plume\left\{\begin{array}{@{}ll@{}}linear~spreading\quad & \unicode[STIX]{x1D6FD}_{p}={\displaystyle \frac{\text{d}b_{g}}{\text{d}z}},\\ dimensional~analysis\quad & U_{m}=\left({\displaystyle \frac{B_{0}}{z}}\right)^{1/3},\end{array}\right. & \displaystyle\end{eqnarray}$$

where $M_{0}$ is the initial kinematic momentum flux of the jet. Thus, linear spreading of the single-phase jet and plume is mathematically equivalent to the entrainment hypothesis with a constant entrainment coefficient  $\unicode[STIX]{x1D6FC}$ . The entrainment hypothesis states that the growth of the liquid volume flux is due to the shear entrainment of ambient liquid at the edge of the jet or plume with the entrainment velocity  $U_{e}$ , where $U_{e}$ is proportional to the centreline velocity multiplied by the entrainment coefficient $\unicode[STIX]{x1D6FC}$ (Turner Reference Turner1986). Because the classic bubble plume spreads out linearly, similar to single-phase jets and plumes, the entrainment hypothesis in single-phase jets and plumes is adopted for many integral models of the classic bubble plume (Asaeda & Imberger Reference Asaeda and Imberger1993; Bombardelli et al. Reference Bombardelli, Buscaglia, Rehmann, Rincon and Garcia2007; Socolofsky et al. Reference Socolofsky, Bhaumik and Seol2008; Dissanayake, Gros & Socolofsky Reference Dissanayake, Gros and Socolofsky2018).

For the weak bubble plume, a formulation of the integral plume equations with a constant entrainment breaks down. Instead, we derive the gradient of liquid volume flux from a spreading hypothesis, following the same procedure as in single-phase flow (see appendix A) to obtain

(4.13) $$\begin{eqnarray}\frac{\text{d}Q}{\text{d}z}=\frac{1}{2}\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}_{b}U_{m},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FD}_{b}=\text{d}b_{g}^{2}/\text{d}z$ . This equation differs fundamentally from the constant entrainment hypothesis (i.e. equation (4.10)). The form of (4.13) also suggests that the characteristic length scale where entrainment occurs is not  $b_{g}$ . We suspect that there is no entrainment at the edge of the weak bubble plume, giving $U_{e}=0$ . Instead, the growth of the liquid volume flux in the weak bubble plume is due to the increasing width of the plume as the result of diffusive spreading of the bubbles and the changing wake flow below the leading bubbles as they spread out. This hypothesis is also consistent with ideas in Leitch & Baines (Reference Leitch and Baines1989), who attributed the flow rate in a weak bubble plume to the individual wakes behind bubbles.

Although we conclude for a weak bubble plume that there is no entrainment velocity in the normal sense of a plume, the equation for the growth rate of the liquid volume flux can be treated mathematically in the same manner as a classic plume, but with a different interpretation of the entrainment coefficient. If we combine (4.10) and (4.13), we can obtain the relationship

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{\unicode[STIX]{x1D6FD}_{b}}{4b_{g}}.\end{eqnarray}$$

Here, we name $\unicode[STIX]{x1D6FC}$ the ‘apparent entrainment coefficient’, as it represents the mathematical formulation of entrainment, but in a system that does not obey the entrainment hypothesis. This equation predicts that $\unicode[STIX]{x1D6FC}$ is not a constant and is proportional to the turbulent diffusivity causing the spread of the bubbles ( $\unicode[STIX]{x1D6FD}_{b}$ ) and inversely proportional to the width of the plume ( $b_{g}(z)$ ). Using the definitions of $\unicode[STIX]{x1D6FD}_{b}$ and $b_{g}(z)$ in (4.2), we obtain

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=2^{-3/2}E_{t}^{1/2}U_{s}^{-1/2}z^{-1/2},\end{eqnarray}$$

and (4.13) becomes (this is also the direct derivative of (4.9))

(4.16) $$\begin{eqnarray}\frac{\text{d}Q}{\text{d}z}=3\unicode[STIX]{x03C0}E_{t}z^{-1/2}D^{1/2}.\end{eqnarray}$$

Equation (4.15) predicts that the apparent entrainment coefficient $\unicode[STIX]{x1D6FC}$ decreases with height following $\unicode[STIX]{x1D6FC}\sim z^{-1/2}$ in the weak bubble plume. In a single-phase round jet or plume, the constant Gaussian entrainment coefficients are $\unicode[STIX]{x1D6FC}=0.057$ and 0.083, respectively (Lee & Chu Reference Lee and Chu2003). Although integral models for classic bubble plumes also usually assume constant entrainment, many experimentalists have found that $\unicode[STIX]{x1D6FC}$ is not a constant in bubble plumes, and this arises because the classic bubble plume is not strictly self-similar. Milgram (Reference Milgram1983) used a saturation growth curve to describe the relationship between the entrainment coefficient and a local bubble Froude number (defined by $F_{B}=Q_{b}^{2/5}\unicode[STIX]{x1D712}^{1/3}g^{3/10}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}^{1/2}\unicode[STIX]{x1D70E}^{-1/2}$ , where $Q_{b}$ is gas volume flux). In his empirical fit to data, the entrainment coefficient approaches a plateau at 0.16 when the local bubble Froude number increases, which occurs close to the release (e.g.  $z/D\rightarrow 0$ ). Figure 9 compares our data with this empirical curve, where we computed the apparent entrainment coefficient $\unicode[STIX]{x1D6FC}$ at mid-points between measurements using (4.10). The apparent entrainment coefficients in our experiment are approximately an order of magnitude smaller than entrainment coefficients reported by previous authors for larger $z/D$ and show remarkable correspondence with the empirical fit of Milgram (Reference Milgram1983).

Figure 9.

The relationship between $\unicode[STIX]{x1D6FC}$ and local bubble Froude number, defined after Milgram (Reference Milgram1983), where $\unicode[STIX]{x1D6FC}$ is the entrainment coefficient in the classic bubble plume and is the apparent entrainment coefficient in the weak bubble plume. The dashed line represents the saturation curve proposed by Milgram (Reference Milgram1983).

Milgram (Reference Milgram1983) interpreted the local bubble Froude number $F_{B}$ as a ratio between the mixing length scale of turbulence in the plume and the characteristic length scale of bubble–bubble separation. The turbulence and the bubble–bubble separation change as the plume develops away from the source, and the resulting bubble Froude number is a complex nonlinear function of  $z$ . However, this scaling law is hidden within the $z$ dependence of the parameters that make up $F_{B}$ (e.g.  $\unicode[STIX]{x1D712}$ , $Q_{b}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ ). Our analysis also predicts that $\unicode[STIX]{x1D6FC}\sim z^{1/2}$ in a weak bubble plume. Hence, it is useful to evaluate the relationship between entrainment coefficient or apparent entrainment coefficient and the height above the source in the full range of the available bubble plume data. Figure 10 shows the relationship between $\unicode[STIX]{x1D6FC}$ and $z/D$ over a span of four orders of magnitude of $z/D$ , including both classic and weak bubble plumes. The constant entrainment coefficient in a continuous-phase plume, 0.083, and the plateau value in Milgram (Reference Milgram1983), 0.16, are shown for reference. A reference line of $\unicode[STIX]{x1D6FC}\sim z^{-1/2}$ is also shown, and our data agree well with this theoretical relationship in the region $z/D>5$ .

Figure 10.

The relationship between $\unicode[STIX]{x1D6FC}$ and the normalized height. The data key follows figure 8 with two additional datasets from Seol et al. (Reference Seol, Bhaumik, Bergmann and Socolofsky2007) (shown as SBBS07) and Hugi (Reference Hugi1993) (shown as Hugi93). Hugi’s data are directly digitized from the figure in Hugi (Reference Hugi1993) and are regrouped into two regimes: (1)  $z/D$ in the same regime as our data, and (2)  $z/D$ smaller than those of our data. The solid line is the reference of $\unicode[STIX]{x1D6FC}\sim z^{-1/2}$ . The constant entrainment coefficient in a continuous-phase plume (i.e. 0.083) and the upper bound of a classic bubble plume (i.e. 0.16) suggested by Milgram (Reference Milgram1983) are plotted as dashed lines.

These data have had different interpretations in past papers, each supported by part of the current complete dataset of observations. An asymptotic limit of entrainment coefficient is recognized in the classic bubble plume as $z\rightarrow 0$ but the value seems not to be universal. Ditmars & Cederwall (Reference Ditmars and Cederwall1974) suggested 0.082 (which does appear to be the correct bound as $D\rightarrow \infty$ , consistent with $U_{s}\rightarrow 0$ ) and Milgram (Reference Milgram1983) proposed 0.16 (which here appears as a peak near $z/D\sim O(0.1)$ ). In addition, Seol et al. (Reference Seol, Bhaumik, Bergmann and Socolofsky2007) suggested a lower limit of entrainment coefficient (i.e. 0.04) for bubble plumes, a conclusion dominated by a single dataset of Hugi (Reference Hugi1993). Our data show significantly smaller $\unicode[STIX]{x1D6FC}$ compared to the cases with orders of magnitude smaller $z/D$ in the literature and invalidate the asymptotic assumption in Seol et al. (Reference Seol, Bhaumik, Bergmann and Socolofsky2007). Our $\unicode[STIX]{x1D6FC}$ values are also smaller than Hugi (Reference Hugi1993) (except for one data point), although both datasets are in a similar range of $z/D$ . The scattering of data for $z/D>5$ might be due to the relatively large uncertainty associated with measuring $\text{d}Q/\text{d}z$ from a finite difference approximation using an integral quantity, an error which should be less in our deep tank (16 m) compared to Hugi (3 m). Hence, despite the scatter, the bulk of the data seem to support a decreasing trend of $\unicode[STIX]{x1D6FC}$ versus $z/D$ that is consistent with our scale relationship $\unicode[STIX]{x1D6FC}\sim z^{-1/2}$ at large $z/D$ .