2.1. Conservation equations for a planar plume in a co-flow

In quiescent surroundings, measurements show that the bell-shaped cross-stream profiles of velocity and buoyancy are self-similar and well represented by Gaussians (Rouse, Yih & Humphreys Reference Rouse, Yih and Humphreys1952; Lee & Emmons Reference Lee and Emmons1961; Chen & Rodi Reference Chen and Rodi1980). However, Riley & Tveitereid (Reference Riley and Tveitereid1984) note that a co-flow flattens the velocity profile. A long-standing and proven approach for modelling plumes in a wide range of settings has been to make the simplifying assumption that the profiles are top hats. As such, and without clear justification to choose otherwise, we too adopt this simplification. It is readily shown (Appendix A) that the respective equations for the conservation of mass, momentum and buoyancy may then be written

(2.3a–c)\begin{equation} \frac{\textrm{d}}{\textrm{d} z}[bw] = 2u_e, \quad \frac{\textrm{d}}{\textrm{d} z}[b{w}^{2}] = bg' + 2w_au_e \quad\mbox{and} \quad \frac{\textrm{d}}{\textrm{d} z}[bwg'] = 0,\end{equation}

where it is clear that the co-flow, manifesting as the second term in (2.3b), serves to transport momentum into an entraining plume ($u_e>0$ signifies flow from the environment across the plume perimeter). To close this system of differential equations, the classic entrainment assumption as proposed by Taylor (Reference Taylor1945) is adopted. Accordingly, the horizontal velocity at the plume perimeter is assumed to be directly proportional to a characteristic vertical velocity, in this case, that of the plume relative to the co-flow. Thus,

(2.4)\begin{equation} u_e = \alpha(w - w_a), \end{equation}

where $\alpha (>0)$ is the entrainment coefficient.

In his discussion on the development of the entrainment assumption, Turner (Reference Turner1986) affirms that even the simplest assumption of a constant entrainment coefficient is surprising in its ability to accurately predict flow behaviours over a wide range of problems and scales. Indeed in quiescent surroundings, the treatment of pure (Morton et al. Reference Morton, Taylor and Turner1956), forced (Morton Reference Morton1959) and lazy plumes (Hunt & Kaye Reference Hunt and Kaye2005) by means of a constant entrainment coefficient has offered much insight. Following in this spirit, we also assume $\alpha =\textrm {const.}$ in our simplified treatment of a plume in a co-flow. While we acknowledge that measurements from one of the earliest studies of planar plumes, namely Rouse et al. (Reference Rouse, Yih and Humphreys1952), suggest a Gaussian entrainment coefficient of $\alpha _G=0.16$, a number of other measurements support a value closer to 0.10. For example, Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1989) report $\alpha _G=0.11$; both Yuan & Cox (Reference Yuan and Cox1996) and Paillat & Kaminski (Reference Paillat and Kaminski2014) report $\alpha _G = 0.126$; and Parker et al. (Reference Parker, Burridge, Partridge and Linden2020) report $\alpha _G=0.10$. Given that $\alpha = \sqrt {2}\alpha _G$, a top-hat entrainment coefficient of $\alpha =0.14$ would not appear to be unreasonable for use in a simplified model. For information on variable entrainment coefficients, the reader is referred to Kotsovinos & List (Reference Kotsovinos and List1977) and van den Bremer & Hunt (Reference van den Bremer and Hunt2014).

Substituting for $u_e = \alpha (w - w_a)$ into (2.3) gives

(2.5a–c)\begin{equation} \frac{\textrm{d}}{\textrm{d} z}[bw] = 2\alpha(w - w_a), \quad \frac{\textrm{d}}{\textrm{d} z}[b{w}^{2}] = bg' + 2\alpha(w - w_a)w_a \quad \mbox{and} \quad \frac{\textrm{d}}{\textrm{d} z}[bwg'] = 0. \end{equation}

In our framework, $\alpha w_a >0$ and so the role of a general co-flow is now clear: the momentum flux increases locally if the term $(w-w_a)$ is positive and decreases if ${(w-w_a)}$ is negative. From the statement of volume conservation in (2.5a–c), ${(w-w_a) < 0}$ corresponds to a detrainment of fluid from the plume. Detrained fluid likely remains in the proximity of the plume perimeter, thereby altering the density of the environment through which the plume propagates. As such, we restrict our attention to releases that entrain fluid, i.e. to co-flows for which $(w -w_a) > 0$. Evaluating the integrals in (2.1a–c), the respective fluxes per unit length of volume, specific momentum and buoyancy for top-hat profiles are

(2.6a–c)\begin{equation} Q = bw,\quad M = b{w}^{2} \quad \mbox{and} \quad B = bwg'. \end{equation}

Accordingly, (2.5a–c) reduces to

(2.7a–c)\begin{equation} \frac{\textrm{d} Q}{\textrm{d} z} = 2\alpha\left[\frac{M}{Q} - w_a\right], \quad \frac{\textrm{d} M}{\textrm{d} z} = \frac{BQ}{M} + 2\alpha\left[\frac{M}{Q} - w_a\right]w_a, \quad \frac{\textrm{d} B}{\textrm{d} z} = 0.\end{equation}

Scaling quantities on the source conditions, we introduce

(2.8a–d)\begin{equation} \hat{Q} = \frac{Q}{Q_0},\quad \hat{M} = \frac{M}{M_0},\quad \hat{B} = \frac{B}{B_0}, \quad \zeta = \frac{\alpha z}{b_0}, \quad \varOmega_0 = \frac{w_a}{w_0}, \end{equation}

and define the source Richardson number as

(2.9)\begin{equation} \varGamma_0 = \frac{B_0Q_0^{3}}{2\alpha M_0^{3}}. \end{equation}

With reference to § 1, note that $w_{p0}=w_{0}$ in our definition (2.8a–d) of the co-flow strength $\varOmega _0$. From (2.7c), the buoyancy flux is invariant and in dimensionless form $\hat {B} = 1$. As previously established, in quiescent surroundings $\varGamma _0 = 1$ corresponds to a pure plume release, $\varGamma _0 < 1$ to forced releases and $\varGamma _0 > 1$ to lazy releases (e.g. van den Bremer & Hunt Reference van den Bremer and Hunt2014). In terms of the dimensionless quantities (2.8a–d) and (2.9), the equations in (2.7) become

(2.10a–c)\begin{equation} \frac{\textrm{d} \hat{Q}}{\textrm{d} \zeta} = 2\left[\frac{\hat{M}}{\hat{Q}} - \varOmega_0\right], \quad \frac{\textrm{d} \hat{M}}{\textrm{d} \zeta} = 2\varGamma_0\frac{\hat{B}\hat{Q}}{\hat{M}} + 2\left[\frac{\hat{M}}{\hat{Q}} - \varOmega_0\right]\varOmega_0, \quad \frac{\textrm{d} \hat{B}}{\textrm{d} \zeta} = 0.\end{equation}

As required, (2.10a–c) reduces to the governing equations for a planar plume in quiescent surroundings (Lee & Emmons Reference Lee and Emmons1961) on setting $\varOmega _0 = 0$. As defined, the Richardson number and co-flow strength arise naturally through the non-dimensionalisation of the governing equations.

2.2. Change of variable: $\varGamma$-based approach

The local Richardson number

(2.11)\begin{equation} \varGamma = \frac{BQ^{3}}{2\alpha M^{3}} = \frac{bg'}{2\alpha w^{2}}, \end{equation}

may be interpreted as a ratio of the local mass, momentum and buoyancy fluxes and, thereby, offers direct insight into the local dynamical behaviour of the plume. For example, if $\varGamma$ decreases in the streamwise direction, as observed in our results in § 4, this trend signifies that the role of the momentum flux is increasing relative to the buoyancy flux. Moreover, based on the value of $\varGamma$ relative to unity, the plume may be classified locally as forced, pure or lazy. In their study of planar plumes in quiescent surroundings, van den Bremer & Hunt (Reference van den Bremer and Hunt2014) adopt the approach developed by Hunt & Kaye (Reference Hunt and Kaye2005) whereby the conservation equations are recast in terms of $\varGamma$, the objective of this change of variable being to enable the dynamical variability to be solved for directly. Following this approach, we first note that

(2.12)\begin{equation} \frac{\textrm{d} \varGamma}{\textrm{d} z} = \frac{1}{2\alpha}\left[\frac{Q^{3}}{M^{3}}\frac{\textrm{d} B}{\textrm{d} z} - \frac{3BQ^{3}}{M^{4}}\frac{\textrm{d} M}{\textrm{d} z} + \frac{3BQ^{2}}{M^{3}}\frac{\textrm{d} Q}{\textrm{d} z}\right]. \end{equation}

Substituting for $\textrm {d} B/\textrm {d} z, \textrm {d} M/\textrm {d} z \text { and } \textrm {d} Q/\textrm {d} z$ from (2.7) into (2.12) and introducing the dimensionless plume width and velocity as

(2.13a,b)\begin{equation} \beta = \frac{b}{b_0} \quad \text{and} \quad \omega = \frac{w}{w_0}, \end{equation}

respectively, yields the required differential equation governing the streamwise variation of $\varGamma$, (2.14a). Similar manipulation of the two remaining conservation equations, (2.7a) and (2.7b), yields the associated differential equations for the streamwise variation of $\beta$ and $\omega$

(2.14a)\begin{gather} \frac{\textrm{d} \varGamma}{\textrm{d} \zeta} = 6\frac{\varGamma}{\beta}\left[1 - \varGamma - 2\frac{\varOmega_0}{\omega} + \left(\frac{\varOmega_0}{\omega}\right)^{2}\right], \end{gather}

(2.14b)\begin{gather}\frac{\textrm{d} \beta}{\textrm{d} \zeta} = 2\left[2 - \varGamma - 3\frac{\varOmega_0}{\omega} + \left(\frac{\varOmega_0}{\omega}\right)^{2}\right], \end{gather}

(2.14c)\begin{gather}\frac{\textrm{d} \omega}{\textrm{d} \zeta} = 2\frac{\omega}{\beta}\left[\varGamma - 1 + 2\frac{\varOmega_0}{\omega} - \left(\frac{\varOmega_0}{\omega}\right)^{2}\right]. \end{gather}

The dimensionless source conditions become

(2.15a–c)\begin{equation} \varGamma = \varGamma_0,\quad \beta = 1,\quad \omega = 1 \mbox{ on } \zeta = 0. \end{equation}

On setting $\varOmega _0 = 0$, (2.14) reduce to the governing equations derived by van den Bremer & Hunt (Reference van den Bremer and Hunt2014) $-$ the sole differences are factors of two which result from our scaling being based on the full width of the plume, rather than the half-width. For the purpose of reference the solutions for a quiescent environment ($\varOmega _0=0$) are given in Appendix C.

To solve for the streamwise velocity, dividing (2.14a) by (2.14c) yields $\textrm {d} \varGamma /d\omega = -3\varGamma /\omega$, which may be integrated straightforwardly to give

(2.16)\begin{equation} \omega = \left(\frac{\varGamma_0}{\varGamma}\right)^{{1}/{3}}. \end{equation}

The corresponding solutions of (2.14a)–(2.14c) for $\varGamma$ and $\beta$ are dealt with in § 4. Whilst $\varOmega _0$ does not appear explicitly in (2.16), whose forms appears identical to the solution derived for the $\varOmega _0 = 0$ case (Appendix C), the effect of a co-flow is to modify the variation of $\varGamma (\zeta )$ relative to the $\varOmega _0=0$ case.

3.1. Asymptotic velocity $\omega _f$

In the far field, $\varGamma$ remains positive and it is expected that $\textrm {d} \varGamma /\textrm {d} \zeta \to 0$, i.e. the plume approaches a dynamically invariant behaviour. Thus, while (2.14b) and (2.14c) are unchanged, (2.14a) reduces to

(3.1)\begin{equation} 0 = 1 - \varGamma - 2\frac{\varOmega_0}{\omega} + \frac{\varOmega_0^{2}}{\omega^{2}}. \end{equation}

Substituting (3.1) into (2.14c) gives

(3.2)\begin{equation} \frac{\textrm{d} \omega}{\textrm{d} \zeta} = 0 \implies \omega = \omega_{f} =\textrm{const.}, \end{equation}

where the subscript $f$ is used to denote the far-field value of a quantity. On physical grounds $\omega _f > 0$ as the plume and the environment are co-flowing. Substituting (2.16) into (3.1) leads to the following cubic polynomial in $\omega _f$:

(3.3)\begin{equation} \omega_{f}^{3} - 2\varOmega_0\omega_{f}^{2} + \varOmega_0^{2}\omega_{f} - \varGamma_0 = 0, \end{equation}

whose discriminant can be written as

(3.4)\begin{equation} \varDelta = \varGamma_0(4\varOmega_0^{3} - 27\varGamma_0). \end{equation}

The asymptotic velocity is, in general, not equal to the co-flow velocity. Indeed, these velocities are equal only for a true jet ($\varGamma _0 = 0$), as is readily confirmed on substituting $\omega _f=\varOmega _0$ into (3.3). With $k \in \{0,1,2\}$ and defining the constant $C_k$ as

(3.5)\begin{equation} C_k = \frac{1}{\sqrt[3]{2}} \sqrt[3]{(2\varOmega_0^{3} - 27\varGamma_0 - \sqrt{\varGamma_0(729\varGamma_0 - 108\varOmega_0^{3})})}\left(\frac{-1 + \sqrt{-3}}{2}\right)^{k}, \end{equation}

where $\sqrt [3]{()}$ indicates the principal root, the roots of (3.3) are

(3.6)\begin{equation} \omega_f = \frac{1}{3}\left(2\varOmega_0 -C_k - \frac{\varOmega_0^{2}}{C_k}\right). \end{equation}

For $\varGamma _0 > 4\varOmega _0^{3}/27$, (3.3) has one real-valued root as the discriminant is strictly negative. Moreover, this root is positive as may be reasoned as follows: the curve ${y=\omega _{f}^{3} - 2\varOmega _0\omega _{f}^{2} + \varOmega _0^{2}\omega _{f} - \varGamma _0}$ intersects the ordinate at $y = -\varGamma _0$ and, as there is a single real root and $\lim _{w_f \to +\infty }(y) = +\infty$, the curve must intersect the abscissa once, at a positive value of $w_f$. The other two roots are complex conjugates and are therefore discounted on physical grounds, leaving a single admissible root for the far-field velocity. In this case, the valid root is obtained on setting $k = 1$. Noting that the local co-flow strength $\varOmega$ can be expressed as

(3.7)\begin{equation} \varOmega = \varOmega_0/\omega, \end{equation}

figure 2 plots the direction of the vector $\{\textrm {d} \varGamma /\textrm {d} \zeta, \textrm {d} \varOmega /\textrm {d} \zeta \}$ in $(\varGamma,\varOmega )$ space. This vector plot demonstrates that the stable solutions of (3.3) are those for which $\varOmega = \varOmega _0/\omega < 1$ as, in this region, the vectors point towards these solutions.

Figure 2.

Vector plot showing the direction of the vector $\{\textrm {d} \varGamma /\textrm {d} \zeta, \textrm {d} \varOmega /\textrm {d} \zeta \}$ in $(\varGamma, \varOmega )$ space. Locations where $\textrm {d} \varGamma /\textrm {d} \zeta = \textrm {d} \varOmega /\textrm {d} \zeta$ = 0 are shown in red. These locations are (0,0), (1,0), (0,1) and points satisfying the equation $\varGamma = \varOmega ^{2} - 2\varOmega + 1$ as shown by the red curve. Vectors have been normalised by their magnitude so as to aid interpretation of the plot.

For $\varGamma _0 < 4\varOmega _0^{3}/27$, there are three real-valued roots which, by Descartes sign theorem, may all be positive. The valid root in this case is identified by insisting ${\varOmega _0/w_f \leq 1}$ (equivalently $(w_f-w_a) \geq 0$), i.e. that there is no detrainment, which figure 2 demonstrates is a necessary condition for a stable solution in the far field. In this case, the valid root is obtained on setting $k = 2$.

Point A at $(\varGamma _0=0,\varOmega =\varOmega _0=0)$ in figure 2 marks the classic jet solution for which any incremental increase in the buoyancy of the release (corresponding to an increase in $\varGamma$) or co-flow strength (corresponding to an increase in $\varOmega$) drives the flow away from this unstable solution. The stable classic pure plume solution at Point B ${(\varGamma _0=1,\varOmega =\varOmega _0=0)}$ and all points along the red curve, which plots $\varGamma = \varOmega ^{2} - 2\varOmega + 1$ from (3.1), are the coordinates in ($\varGamma,\varOmega$) space where $\textrm {d} \varGamma /\textrm {d} \zeta = \textrm {d} \varOmega /\textrm {d} \zeta = 0$. Note that for $\varOmega = \varOmega _0/\omega \leq 1$ the arrows point towards the curve, indicating that those solutions are stable.

3.2. Asymptotic growth rate

Subtracting (3.1) from (2.14b) gives

(3.8)\begin{equation} \frac{\textrm{d} \beta}{\textrm{d} \zeta} = 2\left(1 - \frac{\varOmega_0}{\omega_{f}}\right) = \textrm{const.}, \end{equation}

and, hence, the relative asymptotic growth rates in co-flowing and quiescent environments are

(3.9)\begin{equation} \left.\frac{\textrm{d} \beta}{\textrm{d} \zeta}\right/{\frac{\textrm{d} \beta}{\textrm{d} \zeta}}\biggr\rvert_{\varOmega_0=0} = 1 - \frac{\varOmega_0}{\omega_{f}}. \end{equation}

For the reference case of a zero co-flow, (3.8) reduces to the positive growth rate ${\textrm {d} \beta /\textrm {d} \zeta = 2}$ in accordance with the predictions of Lee & Emmons (Reference Lee and Emmons1961) and van den Bremer & Hunt (Reference van den Bremer and Hunt2014). Integrating (3.8) gives the far-field width as

(3.10)\begin{equation} \beta = \frac{b}{b_0} = f(\zeta) + 2\zeta - 2\frac{\varOmega_0}{\omega_f}\zeta,\end{equation}

with $f(0) = 1$ and $f(\zeta ) \to C = \textrm {const.}$ for $\zeta \to \infty$. While presented here as an asymptotic solution, (3.10) is valid for all $\zeta$ for the special case of a linearly expanding plume (analogous to a pure release in zero co-flow). Further consideration of this special case (§ 4.2) offers additional insight into the role of the co-flow. Given $\varOmega _0$ and $\omega _f$ are positive constants, $\varOmega _0/\omega _f > 0$, and it is clear that the third term in (3.10) acts to reduce the plumes width, cf. the expression for $\beta$ in (C 1).

3.3. Asymptotic Richardson number

The far-field Richardson number $\varGamma = \varGamma _{f}$ is obtained on substitution of $\omega _{f}$ from (3.6) into (2.16)

(3.11)\begin{equation} \varGamma_{f} = \frac{\varGamma_0}{\omega_{f}^{3}} = 27 \varGamma_0 \left(2\varOmega_0 -C_k - \frac{\varOmega_0^{2}}{C_k}\right)^{{-}3} \quad \mbox{where } k= \begin{cases} 1, & \text{if } \varGamma_0 > 4\varOmega_0^{3}/27\\ 2, & \text{if } \varGamma_0 < 4\varOmega_0^{3}/27. \end{cases} \end{equation}

The variation of the far-field velocity (3.6) and Richardson number (3.11) with $\varGamma _0$ and $\varOmega _0$ is shown on the contour plots in figure 3. It is apparent that the velocity in the far field $\omega _f$ increases as either the strength of the co-flow or source Richardson number increases (figure 3a).

Figure 3.

Variation of asymptotic vertical velocity and Richardson number with $\varGamma _0$ and $\varOmega _0$. (a) Contours of constant $\omega _f$ with colour bar indicating the magnitude of $\omega _f$. (b) Contours of constant $\varGamma _f$ with colour bar indicating the magnitude of $\varGamma _f$.

On the abscissa $\varOmega _0 = 0$ of figure 3(b), $\varGamma _f=1$ irrespective of the source Richardson number as was to be expected based on existing results for plumes in quiescent surroundings. Clearly, the effect of the co-flow is to reduce the far-field value of $\varGamma _f$ relative to the reference case of a zero co-flow. In other words, the co-flow increases the ‘forced-ness’ of the far-field plume.

3.4. Power-law solutions for a line source

On dimensional grounds, power-law solutions for the plume that develops from a line source take the form

(3.12a–c)\begin{equation} w-w_a=c_1, \quad b=c_2z, \quad g'=c_3z^{{-}1}, \end{equation}

for the unknown coefficients $c_1$, $c_2$ and $c_3$. Rajaratnam & Lal (Reference Rajaratnam and Lal1983) provide implicit relationships for the coefficients and comment that if the value of one is known, say from experiment, then the other two may be determined. We show below how each coefficient can be determined explicitly with reference to the results developed above.

Substituting for $w=w_f$ from (3.6) into (3.12a–c) leads to

(3.13)\begin{equation} c_1 = w_0(\omega_f - \varOmega_0) ={-}\frac{w_0}{3}\left(\varOmega_0 + C_k +\frac{\varOmega_0^{2}}{C_k}\right). \end{equation}

Substituting for $\zeta = \alpha z/b_0$ in (3.10) gives

(3.14)\begin{equation} b = b_0 f(\zeta) + 2\alpha\left(1 - \frac{\varOmega_0}{\omega_f}\right)z. \end{equation}

Asymptotically far from the source, the first term in (3.14) is diminishingly small relative to the second, and, hence, equating (3.14) with $b=c_2z$ from (3.12a–c) gives

(3.15)\begin{equation} c_2 = 2\alpha\left(1 - \frac{\varOmega_0}{\omega_f}\right). \end{equation}

In the far field, $\varGamma =\varGamma _f=\varGamma _0/\omega _{f}^{3}$ from (2.16) and $b=2\alpha (1-\varOmega _0/\omega _f)z$ from (3.14). Thus, given $g'=2\alpha w^{2} \varGamma /b$ from (2.11), the buoyancy in the far field may be written as $g'=c_3z^{-1}$ where

(3.16)\begin{equation} c_3=\frac{\varGamma_0 w_0^{2}}{ \left(1-\dfrac{\varOmega_0}{\omega_f}\right)\omega_f}. \end{equation}

As an aside, we note from (3.13) that $c_1/w_0=(1-\varOmega _0/\omega _f)\omega _f$ and, hence, $c_3=\varGamma _0 w_0^{3}/c_1$. The variation of the non-dimensional coefficients $k_1 = c_1/w_0, k_2 = c_2/\alpha$ and $k_3 = c_3/w_0^{2}$ with the co-flow strength and source Richardson number are plotted in figure 4. As expected, for $\varGamma _0 = 1$ and $\varOmega _0 = 0$, $k_1 = 1$, i.e. the vertical velocity above a line source is invariant.

Figure 4.

The variation of the non-dimensional coefficients $k_1 = c_1/w_0$, $k_2 = c_2/\alpha$ and $k_3 = c_3/w_0^{2}$ with (a) $\varOmega _0$ and (b) $\varGamma _0$ for the example values of $\varGamma _0 = 1$ and $\varOmega _0 = 0.5$, respectively.

4.1. Plume contraction

In the above, plume contraction ($\textrm {d} \beta /\textrm {d} \zeta < 0$) has been considered only for the co-flow $\varOmega _0=1$. Contraction in the absence of a co-flow occurs on account of the streamwise acceleration of the plume when the local Richardson number $\varGamma > 2$ (van den Bremer & Hunt Reference van den Bremer and Hunt2014). It is clear from figure 5 that a contraction and the associated necking can also occur for both forced and pure releases for a sufficiently strong co-flow. This behaviour is of particular note as only lazy plumes can exhibit a contraction in the absence of a co-flow. From (2.14b), a contraction requires that locally

(4.3)\begin{equation} \varGamma > 2-3\frac{\varOmega_0}{\omega} + \left(\frac{\varOmega_0}{\omega}\right)^{2}. \end{equation}

As $\zeta \to 0$, $\omega \to 1$ and so a contraction immediately above the source requires

(4.4)\begin{equation} \varGamma_0 > 2-(3\varOmega_0 - \varOmega_0^{2}). \end{equation}

For the co-flows of interest, i.e. $0 < \varOmega _0 \leq 1$, $(3\varOmega _0 - \varOmega _0^{2})>0$ and, hence, a co-flow lowers the threshold source Richardson number above which contraction occurs. For the strongest co-flow considered ($\varOmega _0=1$) this threshold is $\varGamma _0>0$, indicating that contraction would occur irrespective of the class of release. Setting $\varGamma _0 = 1$ reveals that ${\varOmega _0 = 3/2 - \sqrt {5/4}}$ is the minimum co-flow strength which causes a pure release to contract at source.

That the co-flow causes plume contraction at a lower source Richardson number than in quiescent surroundings can be explained by considering the acceleration of the released fluid adjacent to the source. A defining characteristic of a co-flow is the production of a strong positive streamwise velocity gradient in the near-source region for pure and lazy releases (figures 6c and 6e) a gradient that increases with $\varOmega _0$ and decreases downstream. As $\zeta \to 0, \beta \to 1, \omega \to 1 \mbox { and } \varGamma \to \varGamma _0$, and so (2.14c) reduces to

(4.5)\begin{equation} \left.\frac{\textrm{d} \omega}{\textrm{d} \zeta}\right|_{\zeta=0} = 2(\varGamma_0-1)+2(2\varOmega_0-\varOmega_0^{2}). \end{equation}

Inspection of the second term of (4.5) reveals $2(2\varOmega _0-\varOmega _0^{2})>0$ for $0<\varOmega _0 \leq 1$ and that its magnitude increases with $\varOmega _0$. Thus, the presence of a co-flow enhances the acceleration of the release and a maximum acceleration occurs for $\varOmega _0 = 1$, i.e. for a co-flow that produces no shear at the source. Whether or not the release contracts is determined by the interplay between entrainment (which acts to increase plume width) and acceleration (which, by conservation of mass, acts to decrease plume width). At the source, (2.14b) reduces to

(4.6)\begin{equation} \left.\frac{\textrm{d} \beta}{\textrm{d} \zeta}\right|_{\zeta=0} = 2(2 - \varGamma_0 - 3\varOmega_0 + \varOmega_0^{2}), \end{equation}

which, combined with (4.5) yields

(4.7)\begin{equation} \left.\frac{\textrm{d} \beta}{\textrm{d} \zeta}\right|_{\zeta=0} = 2(1 - \varOmega_0) - \left.\frac{\textrm{d} \omega}{\textrm{d} \zeta}\right|_{\zeta=0}. \end{equation}

Given that $2(1-\varOmega _0)\rightarrow 0$ and $\textrm {d} \omega /\textrm {d} \zeta \rvert _{\zeta =0}$ increases as $\varOmega _0 \rightarrow 1$, (4.7) demonstrates that the acceleration dominates the interplay as the co-flow strengthens, leading to $\textrm {d} \beta /\textrm {d} \zeta \rvert _{\zeta =0}<0$, i.e. to plume contraction.

As a final remark on plume contraction, it is readily shown (Appendix E) that the Richardson number at the neck $\varGamma _n$ reduces with increasing $\varOmega _0$, the magnitude of the reduction decreasing as $\varGamma _0$ increases.

4.2. The case $\varGamma _0 < 1$

The solutions plotted in figure 5(a) show a pronounced reduction in the width of the forced release as the co-flow increases in strength, and the appearance of a local minimum, i.e. of a neck. The accompanying plot, figure 5(b) showing the streamwise variation of $\varGamma$, reveals the profound influence a co-flow has on the dynamical behaviour. In the absence of a co-flow, $\varGamma$ increases monotonically towards unity as is well documented. However, for a sufficiently weak co-flow, $\varGamma (\zeta )$ is an increasing function of $\zeta$ and leads to $\varGamma _0<\varGamma _f<1$. In other words, the relative excess of momentum flux introduced at the source is reduced downstream, where the plume is locally less forced than at the source. By contrast, for both pure and lazy releases into non-zero co-flows, $\varGamma (\zeta )$ decreases towards the far-field value $\varGamma _f<1$ following similar trends (figures 5d and 5f). Thus, their far-field behaviour is ‘forced’, characterised by a relative excess of momentum flux.

Returning to forced releases, given $\textrm {d} \varGamma /\textrm {d} \zeta |_{\zeta =0} > 0$ for $\varOmega _0 = 0$ and $\textrm {d} \varGamma /\textrm {d} \zeta |_{\zeta =0} < 0$ for $\varOmega _0 = 1$, see (2.14a) and figure 5(b) it is natural to enquire whether a particular co-flow strength, $\varOmega _0=\varOmega _0^{*}$, leads to a dynamically invariant plume. From (2.14a)

(4.8)\begin{equation} \frac{\textrm{d} \varGamma}{\textrm{d} \zeta} = 0 \!\implies\! 6\frac{\varGamma}{\beta}\left[1 - \varGamma - 2\frac{\varOmega_0^{*}}{\omega} + \left(\frac{\varOmega_0^{*}}{\omega}\right)^{2}\right]\!=0 \!\implies\! 1 - \varGamma_0 - 2\frac{\varOmega_0^{*}}{\omega} + \left(\frac{\varOmega_0^{*}}{\omega}\right)^{2}\!=0, \end{equation}

provided $\varGamma /\beta \neq 0$. Insisting $\textrm {d} \varGamma /\textrm {d} \zeta = 0$, (2.16) reduces to $\omega = 1$ which on substitution into (4.8) yields the required co-flow, namely

(4.9)\begin{equation} \varOmega_0^{*} = 1 \pm \sqrt{\varGamma_0}. \end{equation}

In conjunction with a constant plume velocity $\omega$, this co-flow necessarily results in a linearly varying plume width, as is confirmed from (2.14b)

(4.10)\begin{equation} \frac{\textrm{d} \beta}{\textrm{d} \zeta} ={\mp} 2 \sqrt{\varGamma_0} \implies \beta ={\mp} 2\sqrt{\varGamma_0}\zeta + 1 \quad \mbox{for } \varOmega_0=\varOmega_0^{*}. \end{equation}

Evidently, the smaller root $\varOmega _0^{*} = 1 - \sqrt {\varGamma _0}$ results in a linearly increasing plume width ($\textrm {d} \beta /\textrm {d} \zeta >0$) and the larger root, a linearly decreasing plume width ($\textrm {d} \beta /\textrm {d} \zeta <0$). The larger root $\varOmega _0^{*} = 1 + \sqrt {\varGamma _0}$ necessarily results in detrainment (see (4.1)) and so is not considered further.

4.3. The case $\varGamma _0 = 1$

As the sole restriction placed on the derivation of (4.9) is $\varGamma /\beta \neq 0$, the expression for $\varOmega _0^{*}$ holds across all three classes of release with the exclusion of the jet ($\varGamma _0=0$). For a pure plume release, (4.9) reproduces the classic result that $\varGamma$ is invariant in a co-flow of zero velocity and provides the new result that the co-flow $\varOmega _0 = 2$ causes a pure release to decrease linearly in width until $\beta = 0$, at which point the model breaks down.

As the co-flow strengthens, the plume width reduces markedly (figure 5c). The plume outline transitions from straight sided (with $\varOmega _0=0$) to resembling a lazy plume in quiescent surroundings owing to the appearance of a neck (e.g. with $\varOmega _0=1$). Moreover, the relative velocity of the plume $(w-w_a)$ decreases. Thus in the reference frame of the plume, the inertial force decreases and, consequently, the buoyancy force associated with the plume plays an increasingly dominant role – this trend ultimately leading to plume contraction (see § 4.1).

The gradient $\textrm {d} \varGamma /\textrm {d} \zeta$ reveals a strong dynamical variation in the near field of the plume, a variation that increases with $\varOmega _0$. Further downstream, the gradient weakens and the subsequent vertical variation is slow as the plume asymptotes to its far-field behaviour (§ 3.3). Notably, the release becomes increasingly forced downstream as the co-flow strengthens.

4.4. The case $\varGamma _0 > 1$

In the absence of a co-flow, $\varGamma$ rapidly converges to unity due to the excess of buoyancy relative to a pure plume (van den Bremer & Hunt Reference van den Bremer and Hunt2014). This rapid variation is also evident in the presence of a co-flow (figure 5f). Increasing the strength of the co-flow also reduces the plume width, though to a considerably lesser extent than for pure and forced releases. Moreover, the co-flow accentuates the plume neck, which narrows, forms further downstream and at a lower local Richardson number. These trends are shown in figure 7 for the example release $\varGamma _0=20$. The location and width of the neck was found numerically by locating the sign change in $\textrm {d} \beta /\textrm {d} \zeta$. The closed-form solution for $\varGamma _n$ (Appendix E) was derived by substituting (2.16) into (2.14b) and setting $\textrm {d} \beta /\textrm {d} \zeta = 0$.

Figure 7.

Variation of the plume neck with co-flow strength for the lazy release $\varGamma _0$=20; (a) $\beta _n$ vs $\varOmega _0$, (b) $\zeta _n$ vs $\varOmega _0$ (solid line), and $\varGamma _n$ vs $\varOmega _0$ (dashed line). Here, $\bullet$ (blue) show the solutions of van den Bremer & Hunt (Reference van den Bremer and Hunt2014) for a quiescent environment.

4.5. Adjustment to locally pure plume behaviour

To acquire further insight into the role of a co-flow it is informative to examine the streamwise distance over which a release adjusts to locally pure plume conditions. The ‘adjustment’ length $\zeta _{\varGamma =1}$ over which this dynamical variation occurs is defined such that $\varGamma (\zeta _{\varGamma =1})=1$ and, as a consequence, is meaningful only for releases for which $\varGamma _0>1$. In quiescent surroundings $\varGamma (\zeta ) \rightarrow \varGamma _f=1$ as $\zeta \rightarrow \infty$ for $\varGamma _0 \neq 1$, i.e. the adjustment length is infinite for $\varGamma _0>1$. It is clear, however, that a co-flow significantly alters the behaviour of $\varGamma (\zeta )$, reducing the far-field Richardson number to a value below unity (figure 5). With reference to (2.14a)

(4.11)\begin{equation} \zeta_{\varGamma=1}= \int_{0}^{\zeta_{\varGamma = 1}} \textrm{d} \zeta=\int_{\varGamma_0}^{1}\, \frac{\beta}{6\varGamma\left(1 - \varGamma - 2\dfrac{\varOmega_0}{\omega} + \left(\dfrac{\varOmega_0}{\omega}\right)^{2}\right)} \textrm{d} \varGamma . \end{equation}

Comparing the finite distance (4.11) over which the Richardson number reduces from $\varGamma =\varGamma _0\, (>1)$ to $\varGamma =1$ for different strengths of co-flow allows us to establish whether the inertia of the co-flow or the buoyancy force of the plume is dominant in controlling the dynamical behaviour. Figure 8 plots $\zeta _{\varGamma =1}$ vs $\varGamma _0$ for non-zero co-flows that vary by two orders of magnitude. Below each curve the plume is locally lazy (i.e. for $\zeta <\zeta _{\varGamma =1}$), above each curve the plume is locally forced (i.e. for $\zeta >\zeta _{\varGamma =1}$) and on the curves the plume is pure. Evidently, the adjustment to a pure plume state is now attained over a finite distance. Moreover, for $\zeta > \zeta _{\varGamma =1}$ the release continues to adjust such that asymptotically $\varGamma \rightarrow \varGamma _f <1$. As the co-flow increases in strength, $\zeta _{\varGamma =1}$ decreases and the variation of $\zeta _{\varGamma =1}$ with $\varGamma _0$ weakens $-$ note the flatness of the curve when $\varOmega _0=1$. The decrease in $\zeta _{\varGamma =1}$ can be attributed to the increase in momentum transported by the plume as the co-flow strengthens, an increase that acts to reduce $\varGamma$, see (2.11).

Figure 8.

The vertical distance $\zeta _{\varGamma =1}$, (4.11), over which the release adjusts to locally pure plume conditions plotted against $\varGamma _0$ for $\varOmega _0=\{0.01,0.1,1\}.$ The plot cannot be extended to forced releases as these never attain $\varGamma = 1$ for $\varOmega _0 \neq 0$. The figure was generated by numerically integrating (2.14) and interpolating to find the height at which $\varGamma = 1$.

The weakening variation of $\zeta _{\varGamma =1}$ with $\varGamma _0$ as $\varOmega _0$ increases is indicative of co-flows having an increasingly dominant influence on plume behaviour. That $\zeta _{\varGamma =1}$ hardly varies with $\varGamma _0$ for the co-flow $\varOmega _0=1$ (figure 8 for $\varGamma _0 \gtrsim 5$) suggests that, as defined, $\varOmega _0=1$ be regarded as a ‘strong’ co-flow; this assertion is consistent with this co-flow reducing the shear at the source to zero (§ 4). However, while nuanced, the role of buoyancy on the interaction between co-flow and plume is evident in figure 8; note the decreasing spacing between the three curves as $\varGamma _0$ increases, a trend signifying that the influence of a given co-flow on the plume lessens for increasingly buoyancy dominated releases. Indeed, this result is also borne out in figure 5.

5.1. Highly forced releases $\varGamma _0 \ll 1$

For inertially dominated releases ($\varGamma \ll 1$), the governing equations (2.14) are simplified on making the approximation ${1 - \varGamma \approx 1}$ (cf. Hunt & Kaye Reference Hunt and Kaye2005). The resulting coupled differential equations can then be straightforwardly solved to show

(5.3a)\begin{gather} \varGamma = \frac{\varGamma_0}{\omega^{3}}, \end{gather}

(5.3b)\begin{gather}\beta = \frac{1}{\omega^{3/2}}\left(\frac{1 - 2\varOmega_0}{\omega - 2\varOmega_0}\right)^{1/2}, \end{gather}

(5.3c)\begin{gather}\zeta = \frac{(1-2\varOmega_0)^{1/2}}{2\varOmega_0^{2}}\left[\frac{(1-\varOmega_0/\omega)}{(1-2\varOmega_0/\omega)^{1/2}} - \frac{1 - \varOmega_0}{(1 - 2\varOmega_0)^{1/2}}\right]. \end{gather}

Re-arranging the latter for $\omega$ gives

(5.4)\begin{equation} \omega=\frac{\varOmega_0}{1-f + \left(f(f-1)\right)^{1/2}} \quad \mbox{where } f=\left(\frac{2\varOmega_0^{2}}{(1-2\varOmega_0)^{1/2}}\zeta+ \frac{1-\varOmega_0}{(1-2\varOmega_0)^{1/2}}\right)^{2}. \end{equation}

As such, the streamwise variations $\beta (\zeta )$, $\omega (\zeta )$ and $\varGamma (\zeta )$ are known functions. Figure 9 (left column) plots the streamwise variations of $(\beta,\omega,\varGamma )$ from (5.3) for a representative forced release and weak co-flow. For the case shown, namely, ($\varGamma _0 = 0.01, \varOmega _0 = 0.1$), the analytical solutions for $\beta$, $\omega$ and $\varGamma$ remain within 10 % of the full numerical solution for $\zeta < 150$, $\zeta < 7.62$ and $\zeta < 2.65$, respectively. As we have taken $\alpha = 0.14$, $\zeta = 0.5$ is equivalent to a streamwise distance of 3.6 source widths.

5.2. Highly lazy releases $\varGamma _0 \gg 1$

For buoyancy dominated releases ($\varGamma \gg 1$, so that $\varGamma -1 \approx \varGamma$) the governing equations (2.14) can again be simplified and solved to show that

(5.5a)\begin{gather} \varGamma = \frac{\varGamma_0}{\omega^{3}}, \end{gather}

(5.5b)\begin{gather}\beta = \frac{1}{\omega}\left(\frac{\varGamma_0 + 2\varOmega_0}{\varGamma_0 + 2\varOmega_0\omega^{2}}\right)^{1/4}, \end{gather}

(5.5c)\begin{gather}\zeta = \frac{1}{2\varOmega_0}\left[1 - \left(\frac{\varGamma_0 + 2\varOmega_0}{\varGamma_0 + 2\varOmega_0\omega^{2}}\right)^{1/4}\right]. \end{gather}

The expression for $\zeta$ can be rearranged for $\omega$, yielding

(5.6)\begin{equation} \omega = \left(\frac{\varGamma_0 + 2\varOmega_0}{2\varOmega_0(1-2\varOmega_0\zeta)^{4}} - \frac{\varGamma_0}{2\varOmega_0}\right)^{1/2}. \end{equation}

Figure 9 (right column) compares the analytical expressions in (5.5) and (5.6) with the numerical solutions to (2.14) for a representative highly lazy release issuing into a weak co-flow ($\varOmega _0=0.1$, $\varGamma _0=100$). The analytical solutions for $\beta, \omega$ and $\varGamma$ are within 10 % of the full numerical solutions for $\zeta < 0.016, \zeta < 0.027$ and $\zeta < 0.012$, respectively.