2.1 Idealised entrainment

Throughout the present paper, the discussion will be restricted to statistically stationary, vertically oriented axisymmetric jets in an unstratified incompressible fluid under the Boussinesq approximation. Following Morton et al. (Reference Morton, Taylor and Turner1956), the ambient fluid is assumed to be entrained at a rate proportional to some characteristic velocity within the jet, and contributions from fluctuations about the mean are neglected. By neglecting volumetric heating and assuming a ‘top-hat’ profile, conservation equations for mass, momentum and buoyancy can be written as

(2.1a-c ) $$\begin{eqnarray}\frac{\text{d}}{\text{d}z}(b^{2}u)=2\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}bu,\quad \frac{\text{d}}{\text{d}z}(b^{2}u^{2})=b^{2}g^{\prime },\quad \frac{\text{d}}{\text{d}z}(b^{2}ug^{\prime })=0,\end{eqnarray}$$

where $b(z)$ is the jet radius, $u(z)$ is the characteristic velocity scale, $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}$ is the entrainment coefficient (assumed at this stage to be a constant), $g^{\prime }(z)=g(\unicode[STIX]{x1D70C}_{\infty }-\unicode[STIX]{x1D70C})/\unicode[STIX]{x1D70C}_{\infty }$ is the modified gravity for jet and ambient densities $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}_{\infty }$ , respectively, and $g$ is the acceleration due to gravity.

2.2 Hunt’s proposal

Hunt (Reference Hunt, Davies and Valente1994) presented the following analysis in a discussion of atmospheric jets and plumes. The idea is that by accounting for the streamwise momentum flux due to turbulence, an increase in the relative turbulent transport of streamwise momentum could lead to a decrease in mass flux. Specifically, the square of the velocity in the mean specific momentum flux is modified to account for turbulent momentum flux by $\overline{u^{2}}=\bar{u}^{2}+\overline{{u^{\prime }}^{2}}$ , where the bar denotes an average that will be formalised below. The momentum equation (2.1) can be rewritten as

(2.2) $$\begin{eqnarray}\frac{\text{d}\bar{M}}{\text{d}z}=B-\frac{\text{d}M^{\prime }}{\text{d}z},\end{eqnarray}$$

where $\bar{M}=b^{2}\bar{u}^{2}$ and $M^{\prime }=b^{2}\overline{{u^{\prime }}^{2}}$ are the mean and turbulent momentum fluxes, respectively, and $B=b^{2}\overline{g^{\prime }}$ is the buoyancy force (note that the original notation in Hunt (Reference Hunt, Davies and Valente1994) used $F$ , but $B$ is used here for convention).

Integrating (2.2) between two planes, $z_{1}$ and $z_{2}$ , gives

(2.3) $$\begin{eqnarray}\bar{M}_{2}=\unicode[STIX]{x1D706}\bar{M}_{1},\quad \text{where}~\unicode[STIX]{x1D706}=1+\left(\frac{1}{\bar{M}_{1}}\int _{z_{1}}^{z_{2}}B\,\text{d}z\right)-\left(\frac{M_{2}^{\prime }-M_{1}^{\prime }}{\bar{M}_{1}}\right).\end{eqnarray}$$

The specific mass flux $Q=b^{2}\bar{u}$ is related to the mean specific momentum flux through $Q^{2}=b^{2}\bar{M}$ , and so it is then possible to relate the change in mass flux through these two planes in terms of $\unicode[STIX]{x1D706}$ ,

(2.4) $$\begin{eqnarray}\frac{Q_{2}}{Q_{1}}=\frac{b_{2}}{b_{1}}\sqrt{\frac{\bar{M}_{2}}{\bar{M}_{1}}}=\frac{b_{2}}{b_{1}}\sqrt{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

From this, Hunt deduced that any positive forcing, $B$ , will lead to an increase in entrainment, whereas ‘a tendency for the jet to break up into individual eddies’, so that the relative turbulent transport of streamwise momentum increases, will result in a decrease in entrainment. As will be seen below, the second deduction needs careful interpretation; considering changes in relative turbulent transport of streamwise momentum is only valid under the assumption that the mean momentum flux is independent of $z$ , which is not necessarily the case. Consequently, the deduction needs to be modified slightly: the mass flux can decrease if there is a relative increase in absolute turbulent momentum flux, rather than an increase in relative turbulent transport of streamwise momentum. This change of wording is better described symbolically, and involves considering the term

(2.5) $$\begin{eqnarray}\frac{1}{\bar{M}}\frac{\text{d}M^{\prime }}{\text{d}z},\quad \text{rather than}~\frac{\text{d}}{\text{d}z}\left(\frac{M^{\prime }}{\bar{M}}\right).\end{eqnarray}$$

The two expressions are clearly equivalent if the mean momentum flux is constant, but may differ when it is not. An alternative approach to this analysis is presented in § 2.5 following a summary of the previous work in the literature.

2.3 Results of Narasimha and co-workers

The experiment pioneered by Bhat et al. (Reference Bhat, Narasimha and Arakeri1989) involved an aqueous jet, in which an acid was added to the jet fluid, and the ambient fluid was deionised. This allowed for the use of electrodes to inject heat into the system selectively. Since the jet fluid was acidic, it conducted, and so was heated as the fluid passed between the electrodes. The ambient fluid, however, did not conduct, and so was not heated. The domain in which heat was added is referred to here as the heat injection zone (HIZ), and the regions below and above are referred to as the pre-HIZ and post-HIZ, respectively. The selective heating process was designed to be analogous to the latent heat release associated with condensation in atmospheric plumes. Moreover, it was shown that the rate of heating was approximately proportional to tracer concentration.

Four papers presenting results followed. Elavarasan et al. (Reference Elavarasan, Bhat, Narasimha and Prabhu1995) and Bhat & Narasimha (Reference Bhat and Narasimha1996) looked at off-source heating in jets. Venkatakrishnan et al. (Reference Venkatakrishnan, Bhat, Prabhu and Narasimha1998) and Venkatakrishnan, Bhat & Narasimha (Reference Venkatakrishnan, Bhat and Narasimha1999) performed experiments in plumes. The focus here is on the results of Bhat & Narasimha (Reference Bhat and Narasimha1996), henceforth referred to as BN, and the main conclusions are summarised as follows. Upon heating, a visual narrowing of the radial extent of the jet was observed, the decay of the centreline streamwise velocity was arrested, and it can even be reversed. The mean streamwise velocity and scalar profiles remained close to Gaussian throughout. The mass flux at first increased more rapidly than in the unheated jet, but was nearly constant further downstream. The shape of the turbulent velocity distribution was modified by heating and was not recovered rapidly in the post-HIZ zone. The RMS streamwise fluctuations remained almost constant throughout the HIZ, and so the normalised values decreased. It was concluded that the original proposal of Hunt (Reference Hunt, Davies and Valente1994) was not verified by these data.

A comparison was made between the measured mass flux and the mass flux derived by assuming a constant entrainment coefficient. The former was significantly lower. It was concluded that ‘the assumption of constant entrainment coefficient was not at all realistic in the HIZ, and indeed that it suffers a drastic reduction at high values of the heating parameter’. This is, of course, in contrast to the intuitive idea that the entrainment coefficient would increase towards a plume-like value.

The scalar and velocity widths were also contrasted. It was observed that although the visual thinning was reflected in the scalar width, an increase in the velocity width occurs. This behaviour was taken into account when the integral model was compared with the experimental data. It was shown that by assuming a change in the ratio of velocity and scalar widths, the decay in velocity was more accurately predicted than by assuming the ratio to be constant. However, there was still disagreement with the experimental data. The conclusion was that the width ratio has to be taken into account but that further work was required before a convincing model could be formulated. This suggests that the mass flux data should be reassessed. Since velocities at the edge of the jet were difficult to measure, the profiles are assumed to be self-similar and the mass flux is derived from the centreline velocity and the jet width. However, using the scalar width naturally resulted in an underestimation of the mass flux. This issue will be readdressed below.

2.4 Results of Agrawal and Prasad

More recently, Agrawal and Prasad have investigated the heated jet both experimentally and numerically (Agrawal, Sreenivas & Prasad (Reference Agrawal, Sreenivas and Prasad2003), Agrawal & Prasad (Reference Agrawal and Prasad2004), Agrawal, Boersma & Prasad (Reference Agrawal, Boersma and Prasad2005); the collective work is henceforth referred to as AP). In the experiments, streamwise and radial velocities were measured in addition to temperature. The results were broadly consistent with BN, but additional observations were present; those observations and key differences are summarised below.

Experimentally, the profile of the radial velocity changed from that of an unheated jet below the HIZ to one that presented inflow at all radial locations in the lower HIZ, and in the upper half of the HIZ (induced by the acceleration). A growth in the width of the velocity profile was observed in the lower HIZ  in contrast to the decrease observed in the width of the scalar profile. The mass flux was observed to increase in the lower HIZ, above which it became nearly constant. An off-centre peak in temperature was observed, which developed towards a flat-top Gaussian. It was proposed that the double-bump profile (as it was called) arises from a competition between the acid concentration and the residence time in the HIZ; the temperature increase was reported to be proportional to the acid concentration, and so a greater temperature rise would be expected at the axis, where the concentration is higher, but the fluid is moving at greater speed in the centre, so the residence time in the HIZ is lower. Radial profiles and centreline evolution of normalised fluctuating quantities were also presented. In the lower HIZ, centreline values of the streamwise velocity fluctuations were approximately 60 % higher than in the unheated case; they decreased with height, but always remained higher than in the unheated jet. The radial profiles were similar to the unheated jet (maximum at the centre and decreasing towards the edge), but at the top of the HIZ, the profile had changed shape and it was reported to be almost constant across the range measured.

Numerically, the results were also largely in agreement with the experimental work of BN. Again, radial velocity profiles showed that the heating leads to an inflow at all radial locations. An off-centre peak in streamwise velocity was also observed, but is likely due to particularly strong heating. The scalar and velocity widths grew approximately linearly in the pre-HIZ, with a ratio of about $1.2$ . A dramatic reduction in both the scalar and velocity widths was observed at the bottom of the HIZ, with the scalar width becoming narrower than the velocity. The streamwise velocity, tracer and temperature fluctuations were observed to increase in the HIZ, and all normalised centreline fluctuations were higher in the post-HIZ than in the pre-HIZ.

2.5 Entrainment decomposition

An alternative analysis is presented here that allows the different contributions to entrainment to be measured quantitatively. The approach incorporates turbulent fluctuations, as first suggested by Hunt (Reference Hunt, Davies and Valente1994), but more carefully accounts for changes in mean momentum flux. The analysis also includes the previously neglected pressure term.

Similar to Hunt, the mass and momentum conservation equations are used to derive an expression for entrainment, where entrainment is interpreted in terms of the entrainment coefficient, which is not assumed to be constant. The present treatment drops both the assumption of the ‘top-hat’ model (used for simplicity above) and the assumption of self-similarity (note that self-similarity is not prohibited).

Define the mean of a quantity $q$ in cylindrical polar coordinates $(r,\unicode[STIX]{x1D703},z)$ as

(2.6) $$\begin{eqnarray}\bar{q}(r,z)=\frac{1}{2\unicode[STIX]{x03C0}\unicode[STIX]{x0394}t}\int _{t_{0}}^{t_{0}+\unicode[STIX]{x0394}t}\int _{0}^{2\unicode[STIX]{x03C0}}q(r,\unicode[STIX]{x1D703},z,t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}t,\end{eqnarray}$$

where $t$ denotes time and $\unicode[STIX]{x0394}t$ the averaging period, and the fluctuation about the mean as $q^{\prime }=q-\bar{q}$ . The general axisymmetric statistically stationary conservation equations for mass and momentum for an incompressible buoyant jet under the Boussinesq approximation can be written as

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\bar{u}_{r}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(r\bar{u}_{z}\right)=0, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\bar{u}_{z}\bar{u}_{r}+r\overline{u_{r}^{\prime }u_{z}^{\prime }}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(r\bar{u}_{z}\bar{u}_{z}+r\overline{u_{z}^{\prime }u_{z}^{\prime }}\right)=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(r\overline{u_{r}^{\prime }u_{r}^{\prime }}\right)+\unicode[STIX]{x1D6FC}_{T}\bar{T}r, & \displaystyle\end{eqnarray}$$

where $\bar{u}_{r}$ and $\bar{u}_{z}$ are the mean radial and streamwise velocities, respectively, $\bar{T}$ is the mean temperature perturbation from ambient, $\unicode[STIX]{x1D6FC}_{T}=g\unicode[STIX]{x1D6FD}_{T}$ is the product of the gravitational constant $g$ and thermal expansion coefficient $\unicode[STIX]{x1D6FD}_{T}$ , and the pressure term has been approximated following Hussein, Capp & George (Reference Hussein, Capp and George1994), specifically $\unicode[STIX]{x2202}_{z}\bar{p}\approx -\unicode[STIX]{x2202}_{z}\overline{{u_{r}^{\prime }}^{2}}$ .

Integrating (2.7) over $r$ gives an expression for entrainment,

(2.9) $$\begin{eqnarray}\frac{\text{d}Q}{\text{d}z}=2\unicode[STIX]{x03C0}b_{u}\bar{u}_{\unicode[STIX]{x1D700}},\end{eqnarray}$$

where $Q=2\unicode[STIX]{x03C0}\int _{0}^{\infty }u_{z}r\,\text{d}r$ is the specific mass flux, $\bar{u}_{\unicode[STIX]{x1D700}}=\lim _{\unicode[STIX]{x1D702}\rightarrow \infty }(-\unicode[STIX]{x1D702}\bar{u}_{r})$ is the entrainment velocity and $\unicode[STIX]{x1D702}(z)=r/b_{u}(z)$ is the radial coordinate normalised by the velocity width $b_{u}(z)$ . Here, the entrainment coefficient is taken to be the entrainment velocity normalised by the characteristic streamwise velocity scale $\bar{u}_{0}$ , specifically $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}(z)=\bar{u}_{\unicode[STIX]{x1D700}}/\bar{u}_{0}$ .

Similarly, integrating (2.8) over $r$ gives

(2.10) $$\begin{eqnarray}\frac{\text{d}\bar{M}}{\text{d}z}=B-\frac{\text{d}}{\text{d}z}(M^{\prime }-P^{\prime }),\end{eqnarray}$$

where $\bar{M}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\bar{u}_{z}^{2}\,r\,\text{d}r$ is the mean momentum flux, $M^{\prime }=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\overline{{u_{z}^{\prime }}^{2}}\,r\,\text{d}r$ is the momentum flux due to the turbulence, $B=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{T}\int _{0}^{\infty }\bar{T}\,r\,\text{d}r$ is the buoyancy force and $P^{\prime }=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\overline{{u_{r}^{\prime }}^{2}}\,r\,\text{d}r$ is the pressure term.

To follow Hunt (Reference Hunt, Davies and Valente1994), it is necessary to replace the velocity on the right-hand side of (2.9) with the mean momentum flux. This can be achieved by normalising the mean streamwise velocity according to $\bar{u}_{z}(r,z)=\bar{u}_{0}(z)f(\unicode[STIX]{x1D702},z)$ , for a dimensionless shape profile $f(\unicode[STIX]{x1D702},z)$ . Note that self-similarity is not assumed, because the shape profile can vary with $z$ . Then the mean momentum flux can be written as $\bar{M}=b_{u}^{2}\bar{u}_{0}^{2}I$ , where $I(z)=2\unicode[STIX]{x03C0}\int _{0}^{\infty }f(\unicode[STIX]{x1D702},z)^{2}\unicode[STIX]{x1D702}\,\text{d}\unicode[STIX]{x1D702}$ . An expression for the entrainment coefficient can then be derived by combining (2.9) and (2.10):

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}=\frac{\sqrt{I}}{2\unicode[STIX]{x03C0}}\left[\frac{\text{d}}{\text{d}z}\left(\frac{Q}{\sqrt{\bar{M}}}\right)+\frac{QB}{2\bar{M}^{3/2}}-\frac{Q}{2\bar{M}^{3/2}}\frac{\text{d}}{\text{d}z}\left(M^{\prime }-P^{\prime }\right)\right].\end{eqnarray}$$

The first term on the right-hand side of (2.11) is a measure of the growth of the width (spread rate) of the jet, the second term is the contribution from buoyancy and the final term is the rate of change of turbulent momentum fluxes. This final term is the term that Hunt predicted could explain a decrease in entrainment. Note that the derivative of the absolute turbulent momentum flux is present, not the derivative of the relative turbulent fluxes. Kaminski, Tait & Carazzo (Reference Kaminski, Tait and Carazzo2005) suggested that the entrainment behaviour observed in BN could be explained by a change in the similarity structure due to heating; changes in self-similarity are included implicitly in (2.11), and a method that unifies the Kaminski approach with that taken here is a subject of future work. The present treatment preserves the essence of Hunt’s proposal (specifically that a positive forcing can lead to an increase in entrainment and that an increase in turbulent fluctuations can lead to a decrease), but highlights the different interpretation required for the application to the heated jet.

5.1 Phenomenological summary

Off-source volumetric heating of a turbulent jet arrests the decay in velocity in the HIZ. At the heating rates considered here, the profiles of the mean streamwise velocity and tracer concentration remain essentially self-similar, with distributions that are close to Gaussian, but the acceleration of the jet leads to an increased radial inflow at all radial locations. The increased radial inflow leads to a visual thinning of the jet. There is a competition between heating rate and residence time of the fluid in the HIZ. Near the axis, heating is strong due to high tracer concentration, but the residence time is low due to high velocities. In contrast, towards the edge of the jet, heating is weak because of low concentration, but the residence time is high because of low velocities. The competition between heating and residence time results in an off-centre peak of temperature, possibly because the tracer width is initially broader than the velocity width. Contrasting behaviour is observed between the profile widths of streamwise velocity and the tracer; the increased radial inflow sweeps the tracer towards the axis, but the off-centre temperature peak produces sufficient buoyancy that the velocity profile does not narrow, and may even broaden slightly.

The decomposition of the entrainment coefficient showed that there was also a local increase due to enhanced buoyancy, and a decrease due to an increase in turbulence. The decomposition was found to be inaccurate near the boundaries of the HIZ, which may be a consequence of non-lateral entrainment.

5.2 Discrepancies with the literature?

Figure 9.

Ratio of mass fluxes based on measurements of velocity and scalar width.

All of the data presented above are consistent with the data in the literature, but there are, however, differences in the interpretation. The key difference is that BN reported a slowing in the rate of increase in mass flux. Experimentally, BN concluded that heating produced a decrease in the growth rate of mass flux. The present simulations show a strong increase in growth. Experimentally, the mass flux was inferred from centreline measurements of velocity and the width of the tracer concentration profile. An interesting observation in the present study (figure 6) is the contrasting behaviour in velocity and scalar widths; a similar observation was alluded to in BN, and was reported by AP, but not taken into account in the analysis. Importantly, this observation has a significant effect on the mass flux measurements. To demonstrate this further, figure 9 compares the ratio of mass fluxes constructed using the velocity and scalar widths; specifically, $\unicode[STIX]{x1D6FD}Q_{u}/Q_{b}$ , where $Q_{u}$ and $Q_{b}$ are the mass flux measurements based of velocity and scalar, respectively, and $\unicode[STIX]{x1D6FD}$ is a shape constant that normalises the ratio in the jet. It is clear that using the scalar width to infer the mass flux in the heated cases leads to a significant underestimation, potentially exceeding 50 %.

The decomposition of the entrainment coefficient presented in (2.11) is motivated by Hunt (Reference Hunt, Davies and Valente1994), and so differs from the recent work by van Reeuwijk & Craske (Reference van Reeuwijk and Craske2015), which uses the kinetic energy equation following the approach of Priestley & Ball (Reference Priestley and Ball1955); reconciliation of the two approaches will be considered in future work.

Figure 10.

Comparison of entrainment coefficient resulting from radial integration to (a)  $r=2.15b_{u}$ and (b) the edge of the computational domain.