2. The entrainment flow field and plume boundary

The complex potential of a line sink of strength $-m(z)$ at the origin is

where $Z=x+\text{i}y$ . This differs from the complex potential of a line vortex by only a factor of $\text{i}$ . Kaye & Linden (Reference Kaye and Linden2004) used a line sink to represent the converging, horizontal and irrotational flow exterior to a plume which is induced by plume entrainment (see also Taylor Reference Taylor1958).

The real part of the complex potential corresponds to the velocity potential. For a fluid of density ${\it\rho}$ , Batchelor (Reference Batchelor1967, § 6.10) proposes a physical interpretation of the velocity potential as $(1/{\it\rho})$ times the pressure impulse required to set up, or bring to rest, the irrotational velocity arising from the potential. In this sense, contours of equal velocity potential may be interpreted as connecting points at which an equal pressure impulse must be applied to bring about a given irrotational motion. Furthermore, the equations for mean plume motion are consistent with the assumption that the horizontal pressure gradient within the plume is neglected (Turner Reference Turner1973, § 6.1.2), and thus the plume body is approximately isobaric at any given height.

Hence, contours of equal velocity potential may be used to approximate the mean plume boundary at different stages (i.e. heights) in the evolution of single or multiple plumes. An additional feature of velocity-potential contours is that the flow is normal to the contours at all points, which would be expected of plume entrainment flow.

The total complex potential of an array of plumes, represented by line sinks, inherently incorporates the interaction of the flow fields induced exterior to the plumes.

3. Infinite row of line sinks

For an infinite array of parallel line sinks at positions $na~(n\in \mathbb{Z})$ , i.e. at intervals of length $a$ along the real line, the total complex potential is

derived as in the vortex-street summation, e.g. Acheson (Reference Acheson1990, § 5.7), where ${\it\Pi}$ is an arbitrary constant. The velocity potential is

where $x^{\prime }+\text{i}y^{\prime }={\rm\pi}Z/a$ and hence $x^{\prime }/{\rm\pi}=x/a$ , etc. The velocity components $u$ and $v$ may be obtained from the complex derivative, $\text{d}{\it\Omega}/\text{d}Z=u-\text{i}v$ . In this case,

The flow speed is given by $q=|\text{d}{\it\Omega}/\text{d}Z|$ , hence

From (3.2), the contours of equal velocity potential are given by

where $p$ is constant, or in terms of $x^{\prime }$ and $y^{\prime }$ ,

Hence,

Figure 2 shows this solution plotted for a range of $p$ . It is periodic in $x^{\prime }$ with period ${\rm\pi}$ , has roots at $x^{\prime }=\pm (1/2)\cos ^{-1}(1-2p^{2})$ for $0\leqslant p\leqslant 1$ , and tends to a constant value of $\ln 2p$ as $p$ tends to infinity. Its maximum value in the domain $-{\rm\pi}/2\leqslant x^{\prime }\leqslant {\rm\pi}/2$ is $y_{0}^{\prime }=(1/2)\ln (2p^{2}+1+2[p^{2}(p^{2}+1)]^{1/2})$ at $x^{\prime }=0$ . The area under (3.7) is

where

$A^{\prime }$ is a function of $p$ only, and is related to the actual area $A$ by $A^{\prime }={\rm\pi}^{2}A/a^{2}$ .

Figure 2.

Contours of velocity potential. The corresponding values of $p$ are those in table 1 in the range $0.05\leqslant p\leqslant 5$ . The domain spans $-{\rm\pi}/2\leqslant x^{\prime }\leqslant {\rm\pi}/2$ . The contour $p=0.05$ is dotted, and $p=5$ is dot-dashed. The contour $p=1$ is the thicker solid line which extends into the lower corners of the plot.

From (3.4) and (3.6), the flow speed at points along a particular velocity-potential contour satisfies

The flow is directed towards the origin.

The contour length is not used directly in the plume analysis; however, its calculation is a model for the calculation of the entrainment flux, which follows. It is also of interest to observe its evolution alongside the other derived quantities. First, (3.6) may be differentiated to obtain

and hence

The length of the contour in the upper half-plane is

where $\text{d}\ell ^{\prime 2}=\text{d}x^{\prime 2}+\text{d}y^{\prime 2}$ . For $p>1$ , the length is given by

For $p<1$ , numerical evaluation of $\ell ^{\prime }$ is complicated by the divergence of $\text{d}y^{\prime }/\text{d}x^{\prime }$ at $x_{\pm }^{\prime }$ , and it is convenient to split the integral into pieces. Defining

and $y_{+/2}^{\prime }$ as the value obtained by substituting for $x_{+/2}^{\prime }$ into (3.7), then

For $p=1$ , the length may be evaluated as

Given the orthogonality of streamlines and velocity-potential contours, the flux $E$ across the plume–ambient boundary in the upper half-plane is simply

where

Like $A^{\prime }$ , $I$ is a function of $p$ only. It may be evaluated in a similar manner to $\ell ^{\prime }$ using (3.10). Values of the main variables for a range of values of $p$ are shown in table 1.

Table 1.

Values of the main variables described in § 3, for different values of $p$ .

4. Generalised plume equations

To make use of the results from the previous section, it is convenient to write equations for flow in a half-plume. This will also simplify the later application. The half-plume (Boussinesq) equations may be written in a general form

where $w$ is the vertical velocity, $g^{\prime }$ is the reduced gravity and $N^{2}$ is the buoyancy frequency of the environment. The configuration is shown in figure 1. Equation (4.1a ) is a combination of the usual volume-flux and momentum-flux equations. In this form it emphasises that the plume acceleration is a combination of buoyancy acting on the plume body and drag from entrainment acting on the plume boundary. The plume shape determines the relative strengths of these (Cenedese & Linden Reference Cenedese and Linden2014).

Equations (4.1) revert to the (top-hat) axisymmetric (Morton, Taylor & Turner Reference Morton, Taylor and Turner1956) or two-dimensional (Lee & Emmons Reference Lee and Emmons1961; Stothers Reference Stothers1989) plume equations upon substituting the usual forms for the area and entrainment in those cases,

where ${\it\alpha}$ is an entrainment constant, $b$ is the plume radius in the axisymmetric case (4.2a,b ), $d$ is the plume half-width in the two-dimensional case (4.2c,d ) and $a$ in this limit becomes an arbitrary length along the line plume (which cancels out).

From (3.10), the speed of entrained fluid at the centreline of the domain ( $x^{\prime }=0$ ) is

One possible form of entrainment closure in the general case is to assume $q_{0}={\it\alpha}w$ . This is the same as the conventional entrainment assumption in the limiting axisymmetric and two-dimensional cases, where the entrainment flow speed is uniform along the plume perimeter. The strength of the line sink is then determined from the plume velocity $w$ and the value of $p$ as

Finally, it may be observed that, at any given height, if $A$ and $a$ are known, then the value of $p$ may be determined by inverting (3.8), e.g. by root finding. That is, the 1:1 relationship between $p$ and $A^{\prime }$ means that any function of $p$ only can be represented equivalently as a function of $A^{\prime }$ only. Thus, $m$ and $I$ can be obtained from the values of $A$ , $a$ , ${\it\alpha}$ and $w$ . The entrainment assumption is therefore sufficient to close the general equations (4.1).

5. Merging plumes in an unstratified environment

In an unstratified environment, (4.1c ) may be integrated to obtain the conserved half-plume buoyancy flux, $B=Awg^{\prime }$ . Using (3.18) and (4.4), the remaining equations become

where $V=Aw$ is the half-plume volume flux and

is a function of $p$ only. Non-dimensionalising (5.1) using

yields the equivalent equations in dimensionless variables, denoted by an overbar $\bar{\cdot }$ ,

Requiring that $\bar{A}=\bar{V}/\bar{w}$ then implies $\bar{A}=A/a^{2}=A^{\prime }/{\rm\pi}^{2}$ .

For small values of $p$ , the plume is expected to be approximately axisymmetric. In the axisymmetric case, the parameter describing the departure from the pure-plume similarity solution in the near-source region is

(Hunt & Kaye Reference Hunt and Kaye2005; Devenish, Rooney & Thomson Reference Devenish, Rooney and Thomson2010), where $M=Vw$ is the half-plume momentum flux. Thus, ${\it\Gamma}=1$ for a plume in which the fluxes are in balance with the similarity solution, while ${\it\Gamma}\rightarrow 1$ as $z\rightarrow \infty$ for plumes with arbitrary source conditions. Initial conditions for the integration of (5.4) can be generated by first choosing a small value of $p$ (here 0.05) and hence the corresponding value of $\bar{A}$ . Choosing a value of ${\it\Gamma}_{0}={\it\Gamma}(0)$ then sets the initial value of $\bar{w}$ using (5.5), and hence also $\bar{V}$ . Solutions can then be compared for initial conditions of pure, forced or lazy plumes (respectively ${\it\Gamma}_{0}=1$ , ${\it\Gamma}_{0}=0.1$ , ${\it\Gamma}_{0}=10.0$ ).

Equations (5.4) were integrated upward in the range $0.00036\leqslant \bar{z}\leqslant 0.72$ using the fourth-order Runge–Kutta method with step size ${\rm\Delta}\bar{z}=0.00036$ . The value of $p$ is obtained from the half-plume area at each height step, for input to the next increment calculation.

Results are shown in figure 3. From the height at which the value of $p$ is closest to unity, the pure, forced and lazy plumes begin to merge at $\bar{z}=$ 0.088, 0.084 and 0.090 respectively. These results do not include an adjustment for the virtual source, which is discussed further in § 7.

Figure 3.

Results of integrating the unstratified plume equations (5.4). Pure ( ${\it\Gamma}_{0}=1$ ), forced ( ${\it\Gamma}_{0}=0.1$ ) and lazy ( ${\it\Gamma}_{0}=10$ ) initial conditions are represented by solid, dashed and dot-dashed lines respectively. The solid straight lines show the similarity scalings for axisymmetric ( $p<1$ ) and line ( $p>1$ ) plumes. The dotted vertical lines mark $\bar{z}=0.088$ , where $p$ is closest to 1 for the pure plume.

Taking ${\it\alpha}=0.1$ indicates that the pure-plume row begins to merge at $z/a\approx 2.76$ . This is approximately 30 % lower than the height for full merging observed for two equal plumes by Kaye & Linden (Reference Kaye and Linden2004). However, it appears to be comparable with, or perhaps higher than, the height at which merging was observed to begin for two plumes, from the example in figure 13 of Kaye & Linden (Reference Kaye and Linden2004).

Figure 3 shows that the solutions approach the similarity predictions for axisymmetric plumes and line plumes below and above the merging height respectively. The lazy-plume solution shows that $p$ may vary non-monotonically with $\bar{z}$ , in this case because of flow contraction above the lazy-plume source. Fitting a line to the $\bar{V}$ data for $0.5\leqslant \bar{z}\leqslant 0.72$ and extrapolating backward indicates that the line-plume virtual origin is above the source in all cases, with values of $\bar{z}=$ 0.038, 0.033 and 0.040 for the pure, forced and lazy cases respectively.

6. Merging jets in an unstratified environment

Where the source and the environment are both of the same density, then $B=0$ and the equations represent the merging of jets. Equations (5.1) combine to show that $M$ is now constant and (5.1b ) becomes

or

upon non-dimensionalising using

If $w=M^{1/2}a^{-1}{\hat{w}}$ and $\hat{V}/{\hat{w}}=\hat{A}=A/a^{2}$ , then conservation of $M$ implies that $\hat{A}{\hat{w}}^{2}=1$ , and hence a choice of $p=0.05$ is sufficient to set the initial conditions in this case. It should be noted that $\hat{A}=\bar{A}$ , and $\hat{z}=\bar{z}$ apart from differences in the value of ${\it\alpha}$ between the plume and jet cases.

The jet solution is shown in figure 4, and predicts a merging height of $\hat{z}=0.052$ . Fitting a line to the $\hat{V}^{2}$ data for $0.5\leqslant \hat{z}\leqslant 0.72$ and extrapolating backward indicates that the line-jet virtual origin is also above the source, at $\hat{z}=0.035$ .

Figure 4.

Results of integrating the unstratified jet equation (6.2). The solid straight lines show the similarity scalings for axisymmetric ( $p<1$ ) and line ( $p>1$ ) jets. The dotted vertical lines mark $\hat{z}=0.052$ , where $p$ is closest to 1 for the jet.

7. Plume rise in a channel

Bush & Woods (Reference Bush and Woods1998, figure 2) present experimental results on the depth of a two-dimensional outflow, resulting from plume rise from a source at the centre of the floor of a non-rotating channel. The experimental configuration is shown in figure 5. It is shown that as the channel becomes relatively narrower, the scaled depth of the outflow decreases. This is attributed to the reduction in plume entrainment associated with the influence of the channel walls on the plume flow.

Figure 5.

The experimental configuration of Bush & Woods (Reference Bush and Woods1998). The diagram shows a section along one half of the saltwater channel, of depth $H$ and width (into the page) $a$ . The position $y=0$ marks the channel halfway point. The freshwater plume source is at the origin in the centre of the channel floor, and the plume produces currents flowing in both the positive and negative $y$ directions. Measurements of current depth $h$ were made at positions $y>H$ , but always at distances greater than $a$ from the end of the channel.

In the present notation, the predicted outflow depth scale for a wide channel is derived by Bush & Woods (Reference Bush and Woods1998) by assuming that continuity implies $V_{H}=uah$ and taking $u\sim (g_{H}^{\prime }h)^{1/2}$ , where $V_{H}$ is the half-plume volume flux at the top of the channel of height $H$ , $a$ is the channel width, and $u$ , $g_{H}^{\prime }$ and $h$ are the outflow speed, reduced gravity and depth respectively. Combining these with the axisymmetric-plume similarity solution prediction of $V_{H}\sim B^{1/3}H^{5/3}$ and $g_{H}^{\prime }\sim B^{2/3}H^{-5/3}$ yields the wide-channel depth prediction of

where $C_{1}$ is a dimensionless constant determined experimentally. As the channel narrows, the plume flow departs from the similarity solution and entrains less. The experimental data show the depth to take the general form

where $r$ is an unknown decreasing function of the aspect ratio $H/a$ .

The model of the previous sections may be applied to this case, since the ‘plume in an infinite row of plumes’ is also a representation of plume rise in a channel using the method of images. This again assumes an absence of symmetry breaking by small perturbations, and neglects any boundary-layer effects from the channel walls which may act as a drag on the entrainment flow at the edges.

Thus, the equivalent model here takes $V_{H}=uah$ and $u\sim (g_{H}^{\prime }h)^{1/2}$ as above, and substitutes $g_{H}^{\prime }=B/V_{H}$ to obtain $(h/a)\propto V_{H}/(B^{1/3}a^{5/3})$ or

where, following (5.3c ),

and $C_{2}$ is a dimensionless constant to be determined. Hence, plotting $ha^{2/3}/H^{5/3}$ against $H/a$ for the data of Bush & Woods (Reference Bush and Woods1998) is equivalent to plotting $C_{2}({\it\alpha}/{\rm\pi})^{4/3}\bar{V}/\bar{z}^{5/3}$ against $\bar{z}({\it\alpha}/{\rm\pi})^{-1}$ from the (pure) plume integration. The comparison is shown in figure 6, where a typical value of ${\it\alpha}=0.1$ has been assumed. It may be seen that taking $C_{2}=1.4$ produces a satisfactory fit to the data (solid line).

Figure 6.

Comparison of the model output with the data of Bush & Woods (Reference Bush and Woods1998). The solid line shows the pure-plume model with a proportionality constant of $C_{2}=1.4$ . The dashed line shows the same model with an optimal virtual-origin correction incorporated. The symbols *, ♢, $+$ , ▵, ▫ and $\times$ correspond to scaled outflow-depth measurements in channels of width 4, 6, 9, 10, 12 and 15 cm respectively. Bush & Woods (Reference Bush and Woods1998) indicate an experimental error of approximately $\pm 0.013$ in the scaled depths.

Figure 6 also shows that the current-depth equivalent from the model data diverges at very small values of $z/a$ rather than tending to the wide-channel limit. This divergence may be attributed to the departure of the pure-plume numerical solution from the similarity scaling at small values of $\bar{z}$ , which is mainly due to the lack of virtual-origin correction. The numerical model has an initial ‘radius’ $b_{0}$ given by $b_{0}/a\approx 0.0159$ (see table 1) at the first model level $\bar{z}_{1}=0.00036$ . From the axisymmetric pure-plume similarity solution $b=6{\it\alpha}z/5$ (Morton et al. Reference Morton, Taylor and Turner1956), the depth of the virtual origin $\bar{z}_{v}$ would be given by

hence $\bar{z}_{v}\approx 0.0039$ . The optimal value for convergence at small $\bar{z}$ is found to be $\bar{z}_{v}\approx 0.0034$ , approximately 87 % of the theoretical axisymmetric-plume value (and 4 % of the pure-plume merging height). The model data with this correction (i.e. $\bar{z}$ replaced by $\bar{z}+\bar{z}_{v}$ ) are plotted as the dashed line in figure 6. The model now tends to a wide-channel value of approximately 0.13, which is also the best estimate of $C_{1}$ by Bush & Woods (Reference Bush and Woods1998). Virtual-origin correction is not discussed by Bush & Woods (Reference Bush and Woods1998), so it is possible that similar issues are responsible for some of the spread in the experimental data at low values of $H/a$ .