2.1 Governing equations

Consider an axisymmetric turbulent buoyant jet without swirl in a field of background HIT with zero mean flow. The Reynolds-averaged governing equations in cylindrical coordinates (axial direction $z$ , radial direction $r$ and azimuthal direction $\unicode[STIX]{x1D703}$ ) of continuity, axial and radial momentum (in conservative form) and the tracer mass conservation are given by

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\overline{w}}{\unicode[STIX]{x2202}z}+\frac{1}{r}\frac{\unicode[STIX]{x2202}r\overline{u}}{\unicode[STIX]{x2202}r}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}(\overline{w}^{2}+\overline{w^{\prime 2}})}{\unicode[STIX]{x2202}z}+\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}r(\overline{u}~\overline{w}+\overline{u^{\prime }w^{\prime }})=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\overline{P}}{\unicode[STIX]{x2202}z}+g\frac{\overline{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}(\overline{u}~\overline{w}+\overline{u^{\prime }w^{\prime }})}{\unicode[STIX]{x2202}z}+\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}r(\overline{u}^{2}+\overline{u^{\prime 2}})=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\overline{P}}{\unicode[STIX]{x2202}r}+\frac{\overline{v}^{2}}{r}, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}\frac{\unicode[STIX]{x2202}\overline{c}}{\unicode[STIX]{x2202}r}+\overline{w}\frac{\unicode[STIX]{x2202}\overline{c}}{\unicode[STIX]{x2202}z}=-\frac{1}{r}\frac{\unicode[STIX]{x2202}r\overline{u^{\prime }c^{\prime }}}{\unicode[STIX]{x2202}r}-\frac{\unicode[STIX]{x2202}\overline{w^{\prime }c^{\prime }}}{\unicode[STIX]{x2202}z}. & \displaystyle\end{eqnarray}$$

In the above equations, $(u,v,w)$ corresponds to the velocity components in the ( $r$ , $\unicode[STIX]{x1D703}$ , $z$ ) directions, $g$ is the acceleration due to gravity (acting in the $-z$ direction), $\unicode[STIX]{x1D70C}_{a}$ is the ambient density and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{a}$ is the local density difference. An overline represents time averaging and a prime represents the velocity or concentration fluctuation about the mean. The viscous diffusion terms have been neglected since they are negligible away from solid boundaries for high-Reynolds-number flows (Hussein, Capp & George Reference Hussein, Capp and George1994). Following Hussein et al. (Reference Hussein, Capp and George1994), integration of the radial momentum equation with respect to $r$ from $r$ to infinity, substituting the $z$ -derivative of pressure into the $z$ -momentum equation, and integrating over the entire jet cross-section will result in the conservation equation of total momentum flux:

(2.5) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}\frac{\text{d}}{\text{d}z}\int _{0}^{\infty }[\overline{w}^{2}+\overline{w^{\prime 2}}-\overline{u^{\prime 2}}]r\,\text{d}r=2\unicode[STIX]{x03C0}\int _{0}^{\infty }g\frac{\overline{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}r\,\text{d}r+\underbrace{2\unicode[STIX]{x03C0}[r(\overline{u}\overline{w}+\overline{u^{\prime }w^{\prime }})]_{0}^{\infty }}_{=0\text{ for HIT}}. & & \displaystyle\end{eqnarray}$$

It has been assumed that $\overline{u^{\prime 2}}\approx \overline{v^{\prime 2}}$ , as observed in a buoyant jet in quiescent ambient (Wang & Law Reference Wang and Law2002); this is reasonable when the jet structure has not been significantly altered by the background turbulence, as is the region of interest in this study. Also note that the shear stress, mean flow and the $z$ -gradient of the root-mean-square (r.m.s.) velocity of the background HIT are zero when $r\rightarrow \infty$ , which results in an equation with the second-order terms (the turbulence contribution) that is exactly the same as that of a buoyant jet in quiescent ambient (Wang & Law Reference Wang and Law2002).

Equation (2.5) can also be arrived at using control volume (CV) analysis. Equation (2.2) is a result of the $z$ -momentum balance of the ring-shape CV as shown in figure 1 (see appendix A). The lateral net $z$ -momentum flux into the CV is represented by the term $(1/r)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r)r(\overline{u}\overline{w}+\overline{u^{\prime }w^{\prime }})$ , which includes the shear stress $\overline{u^{\prime }w^{\prime }}$ . Noting that the shear stress is identically zero in HIT (e.g. Pope Reference Pope2000) external to the jet and that $\overline{u^{\prime }w^{\prime }}$ is equal and opposite for positive and negative $\unicode[STIX]{x1D703}$ within the jet, the net contribution of the term to the $z$ -momentum balance of the CV is hence zero (note that this is not so for other types of turbulence). Integrating the remaining terms in the momentum balance equation along $r$ from 0 to  $\infty$ , and noting that $(1/\unicode[STIX]{x1D70C})\unicode[STIX]{x2202}_{z}\overline{P}\approx -\unicode[STIX]{x2202}_{z}\overline{u^{\prime 2}}$ (e.g. Hussein et al. Reference Hussein, Capp and George1994 and Shabbir & George Reference Shabbir and George1994), will result in (2.5).

Figure 1.

Control volume of a buoyant jet in background HIT.

In equation (2.5), the left-hand side represents the rate of change of total momentum $M$ in the jet direction, which consists of the momentum flux contributed by the mean flow,

(2.6) $$\begin{eqnarray}\displaystyle M_{M}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\overline{w}^{2}r\,\text{d}r, & & \displaystyle\end{eqnarray}$$

and that contributed by the turbulence or the second-order terms,

(2.7) $$\begin{eqnarray}\displaystyle M_{T}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }[\overline{w^{\prime 2}}-\overline{u^{\prime 2}}]r\,\text{d}r. & & \displaystyle\end{eqnarray}$$

The right-hand side represents the local buoyancy force, which increases the total momentum of the buoyant jet along $z$ . In quiescent ambient, the ratio of $M_{T}$ to $M$ is typically only about 10 % (Papanicolaou & List Reference Papanicolaou and List1988), and a first-order integral model that gives good predictions of the jet mixing characteristics can be developed by considering only the momentum flux contributed by the mean flow (Lee & Chu Reference Lee and Chu2003). However, with background turbulence, it is expected that the relative contribution of turbulent momentum flux will become progressively more important (Hunt et al. Reference Hunt, Eames and Westerweel2006) and needs to be considered.

The tracer mass conservation equation (2.4) can also be integrated to give the mass conservation equation in integral form as

(2.8) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}\frac{\text{d}}{\text{d}z}\int _{0}^{\infty }(\overline{wc}+\overline{w^{\prime }c^{\prime }})r\,\text{d}r=0, & & \displaystyle\end{eqnarray}$$

and the total tracer mass flux $H$ can similarly be expressed as the sum of the contribution from the mean flow

(2.9) $$\begin{eqnarray}\displaystyle H_{M}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\overline{wc}r\,\text{d}r & & \displaystyle\end{eqnarray}$$

and that from the turbulence

(2.10) $$\begin{eqnarray}\displaystyle H_{T}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\overline{w^{\prime }c^{\prime }}r\,\text{d}r. & & \displaystyle\end{eqnarray}$$

It is useful to define two variables to represent the ratios of the total to the mean fluxes of momentum $(k_{M})$ and tracer mass $(k_{H})$ as in Wang & Law (Reference Wang and Law2002), where

(2.11) $$\begin{eqnarray}\displaystyle k_{M}=1+\frac{M_{T}}{M_{M}} & & \displaystyle\end{eqnarray}$$

and

(2.12) $$\begin{eqnarray}\displaystyle k_{H}=1+\frac{H_{T}}{H_{M}}. & & \displaystyle\end{eqnarray}$$

The radial velocity and concentration profiles of a buoyant jet in background turbulence are in general not self-similar, but are expected to be Gaussian in shape initially (after the zone of flow establishment ${\approx}6D$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Lee & Chu Reference Lee and Chu2003)) until the jet structure becomes severely affected by the background HIT (not considered in this study). If we consider the region in which the velocity and concentration profiles are Gaussian, i.e.

(2.13a,b ) $$\begin{eqnarray}\displaystyle \overline{w}=\overline{w_{c}}\exp [-(r/b_{g})^{2}]\quad \text{and}\quad \overline{c}=\overline{c_{c}}\exp [-(r/\unicode[STIX]{x1D706}b_{g})^{2}], & & \displaystyle\end{eqnarray}$$

where $b_{g}$ is the Gaussian jet half-width, defined as the radial distance from the jet centreline in which the velocity has decayed to $\exp (-1)$ of $\overline{u_{c}}$ , and $\unicode[STIX]{x1D706}\approx 1.0{-}1.2$ (Wood, Bell & Wilkinson Reference Wood, Bell and Wilkinson1993) is the ratio of the jet concentration Gaussian half-width to velocity Gaussian half-width, then a set of integral equations can be developed by integrating (2.1), (2.5) and (2.8) (replacing $c$ by $g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}$ ), and using the entrainment hypothesis (Morton, Taylor & Turner Reference Morton, Taylor and Turner1956):

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}Q}{\text{d}z}=2\unicode[STIX]{x1D6FC}\sqrt{\frac{2\unicode[STIX]{x03C0}M}{k_{M}}}, & \displaystyle\end{eqnarray}$$

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}M}{\text{d}z}=\frac{k_{M}}{k_{H}}(1+\unicode[STIX]{x1D706}^{2})\frac{FQ}{2M}, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}F}{\text{d}z}=0, & \displaystyle\end{eqnarray}$$

where $Q$ , $M$ and $F$ are the total volume, momentum and buoyancy fluxes, respectively; $\unicode[STIX]{x1D6FC}$ is the entrainment coefficient which is $\unicode[STIX]{x1D6FC}_{j}=0.057$ for a jet to $\unicode[STIX]{x1D6FC}_{p}=0.086$ for a plume (e.g. Wang & Law Reference Wang and Law2002), and varied for a buoyant jet as, for example, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{j}+(\unicode[STIX]{x1D6FC}_{p}-\unicode[STIX]{x1D6FC}_{j})(Fr_{p}/Fr_{L})^{2}$ , where $Fr_{p}=(5/4)^{1/2}(\unicode[STIX]{x1D706}/\sqrt{2})$ and $Fr_{L}=w_{c}/\sqrt{g^{\prime }b_{g}}$ are the plume and local densimetric Froude numbers, respectively (Lee & Chu Reference Lee and Chu2003). Note that the second-order statistics of the jet are not self-similar even for the range of $z/l_{c}$ considered in this study, but the effects due to the change in their cross-sectional profiles on the jet mixing characteristics have been accounted for via $k_{M}$ and $k_{H}$ . Also, the background turbulence is expected to change the entrainment coefficient derived from a quiescent background since the turbulence kinetic energy (TKE) production is enhanced and the turbulence statistics profiles are also changing downstream (see van Reeuwijk & Craske Reference van Reeuwijkl and Craske2015 for a detailed derivation and discussion of the entrainment coefficient). Our preliminary analysis showed that the change in $\unicode[STIX]{x1D6FC}$ has a maximum of about 20 % in our region of interest and the model predictions are relatively insensitive to this change (see appendix B); hence the entrainment coefficient for a quiescent background will be adopted in this study until a devoted study on the issue is available. The centreline variables of the buoyant jet can be calculated by relating them to the fluxes:

(2.17a-c ) $$\begin{eqnarray}\displaystyle \overline{w_{c}}=\frac{2M}{k_{M}Q},\quad b_{g}=\sqrt{\frac{k_{M}Q^{2}}{2\unicode[STIX]{x03C0}M}}\quad \text{and}\quad \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}}g=\frac{(1+\unicode[STIX]{x1D706}^{2})}{k_{H}\unicode[STIX]{x1D706}^{2}}\frac{F}{Q}. & & \displaystyle\end{eqnarray}$$

The effect of background HIT can be introduced into the integral equations through $k_{M}$ and $k_{H}$ . The two variables generally depend on the flow characteristics of both the jet and the background turbulence, as demonstrated in the following dimensional analysis.

2.2 Dimensional analysis

To simplify the problem, dimensional analysis is carried out to identify the key dimensionless parameters. The buoyant jet under consideration can be characterized by its initial volume $(Q_{o})$ , momentum $(M_{o})$ and buoyancy $(F_{o})$ fluxes. The background turbulence can be characterized by its r.m.s. velocity $u_{a}^{\prime }$ and dissipation rate $\unicode[STIX]{x1D716}_{a}$ . Then any dependent variables $\unicode[STIX]{x1D719}$ of the jet can be expressed as a function of

(2.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}\sim f(Q_{o},M_{o},F_{o},z,u_{a}^{\prime },\unicode[STIX]{x1D716}_{a}). & & \displaystyle\end{eqnarray}$$

The choice of variables to characterize can be further explained. Following Hunt (Reference Hunt, Davies and Valente Neves1994), only the smaller scales of the background turbulence can affect the jet; larger eddies can only meander the jet without affecting its structure. Thus, while the r.m.s. velocity $u_{a}^{\prime }$ and integral length scale $L_{I}$ are intuitively more relevant quantities to characterize the background turbulence, the velocity scale $u^{\prime }<u_{a}^{\prime }$ of the background turbulence with length scale/eddy size $L$ close to that of the jet $L\sim L_{J}<L_{I}$ is more the key in affecting the jet/plume structure, where $L_{J}$ is the jet/plume characteristic length scale. When $u^{\prime }\sim u_{J}$ , the jet structure will be severely affected, where $u_{J}$ is the buoyant jet characteristic velocity. This critical velocity scale $u^{\prime }$ can be estimated as $u^{\prime }\sim (L\unicode[STIX]{x1D716}_{a})^{1/3}\sim (L_{J}\unicode[STIX]{x1D716}_{a})^{1/3}\sim (\unicode[STIX]{x1D6FD}z\unicode[STIX]{x1D716}_{a})^{1/3}$ , where $\unicode[STIX]{x1D6FD}$ is the jet spread rate. This occurs when $z\sim u^{\prime 3}/\unicode[STIX]{x1D716}_{a}$ and thus it is sensible to use $\unicode[STIX]{x1D716}_{a}$ in characterizing the background turbulence. The second variable to characterize the background turbulence can be $u_{a}^{\prime }$ or $L_{a}$ . The significance of $u_{a}^{\prime }/w_{o}$ has been described by Ching et al. (Reference Ching, Fernando and Robles1995) and will be adopted in this study. If $u_{a}^{\prime }/w_{o}\gg 1$ , it is possible for the jet/plume to break up at the onset. On the other hand, if $u_{a}^{\prime }/w_{o}\ll 1$ as in this study, the jet/plume will travel to a certain distance before breaking up. It is hence another parameter of physical significance and should be included.

One possible set of dimensionless groups from (2.18) is

(2.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left[\frac{Q_{o}\unicode[STIX]{x1D716}_{a}^{5/6}}{F_{o}^{7/6}},\frac{M_{o}\unicode[STIX]{x1D716}_{a}^{2/3}}{F_{o}^{4/3}},\frac{z}{(M_{o}^{3}/\unicode[STIX]{x1D716}_{a}^{2})^{1/8}},\frac{u_{a}^{\prime }}{(F_{o}\unicode[STIX]{x1D716}_{a})^{1/6}}\right]. & & \displaystyle\end{eqnarray}$$

But a more useful set is found by combining the dimensionless groups

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left(Fr,\frac{z}{l_{M}},\frac{z}{l_{c}},\frac{u_{a}^{\prime }}{w_{o}}\right). & & \displaystyle\end{eqnarray}$$

In equation (2.20), $Fr\sim M_{o}^{5/4}/F_{o}^{1/2}Q_{o}$ is the jet densimetric Froude number and $l_{M}\sim M_{o}^{3/4}/F_{o}^{1/2}$ is the momentum length scale; when $z/l_{M}<\sim 1$ , the flow is momentum-dominated (a jet), and when $z/l_{M}>\sim 6$ it becomes buoyancy-dominated (a plume). Parameter $l_{c}$ is the critical length scale in which the jet will start to be severely affected by the background turbulence. The physical significance of $l_{c}$ was first noticed by Hunt (Reference Hunt, Davies and Valente Neves1994), who argued that when the velocity scale of the background eddies with size comparable to the jet width is close to that of the jet entrainment velocity, the jet structure will begin to be severely affected by the background turbulence. A theoretically derived relationship showed that $l_{c}=l_{cp}=(F_{o}/\unicode[STIX]{x1D716}_{a})^{1/2}$ for a plume (Hunt Reference Hunt, Davies and Valente Neves1994), while for jets, $l_{c}=l_{cj}=(M_{o}^{3}/\unicode[STIX]{x1D716}_{a}^{2})^{1/8}$ (see appendix C). It is expected that when $z\ll l_{c}$ , the jet mixing will not be significantly affected, while when $z\gg l_{c}$ , the jet mixing will be noticeably affected, or even be broken up by the turbulence. In the transition regime, the change in jet mixing characteristics has to be measured. Finally, the term $u_{a}^{\prime }/w_{o}$ indicates that, in general, with the same $z/l_{c}$ , the jet may still be affected by the ratio of the jet initial velocity and the background turbulence intensity. The importance of $u_{a}^{\prime }/w_{o}$ (if any) is to be determined experimentally. It can be seen there for pure jets, equation (2.20) can be simplified to

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left(\frac{z}{l_{cj}},\frac{u_{a}^{\prime }}{w_{o}}\right); & & \displaystyle\end{eqnarray}$$

and for pure plumes, equation (2.20) can be simplified to

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left(\frac{z}{l_{cp}}\right). & & \displaystyle\end{eqnarray}$$

In this study, for buoyant jets in which $1<z/l_{M}<5$ , whether $l_{cj}$ or $l_{cp}$ is more appropriate in the interpretation of experimental data will be discussed. Also, the importance of $u_{a}^{\prime }/w_{o}$ will be determined using the data of our experiments as well as those in the literature; the variation of the buoyant jet mixing characteristics with $z/l_{c}$ will be measured. In particular, the determination of $k_{M}$ and $k_{H}$ along $z/l_{c}$ will enable a predictive integral model to be formulated.

3.1 Experimental set-up

In order to investigate how buoyant jet mixing characteristics vary with the different dimensionless parameters in a background HIT, laboratory experiments were carried out using a tank which was 2.4 m (long) $\times$ 1.2 m (wide) $\times$ 2.0 m (deep) (figure 2). The ambient turbulence was generated by a ‘randomly actuated synthetic jet array’ (RASJA; Variano & Cowen Reference Variano and Cowen2008; Khorsandi et al. Reference Khorsandi, Gaskin and Mydlarski2013) mounted at one end of the tank. This turbulence generator can be moved along the length of the tank so that the distance between the measurement point and the generator can be adjusted (usually for the purpose of adjusting the turbulence intensity at the measurement point). The generator consisted of 12 rows, each with 8 bilge pumps, and a PVC tube of diameter 2.1 cm was mounted on each pump as a jet nozzle. The centre-to-centre distance between nozzles was $S=15~\text{cm}$ . The pumps were controlled by a computer and could be switched on and off using two algorithms: (i) the ‘coin-toss’ algorithm – in which a jet is assigned a probability of being turned on at each predetermined time step and (ii) the ‘sunbathing’ algorithm – in which each jet is controlled independently with a random on time and off time with normal distribution $(\unicode[STIX]{x1D707}_{on},\unicode[STIX]{x1D70E}_{on})$ and $(\unicode[STIX]{x1D707}_{off},\unicode[STIX]{x1D70E}_{off})$ , respectively. Algorithm (ii) was found to perform better by Variano & Cowen (Reference Variano and Cowen2008) and was hence adopted in this study. The distributions were chosen to be $(\unicode[STIX]{x1D707}_{on},\unicode[STIX]{x1D70E}_{on})=(12,4)$  s and $(\unicode[STIX]{x1D707}_{off},\unicode[STIX]{x1D70E}_{off})=(108,36)$  s, similar to that used in Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013). The details of the overall turbulence statistics generated by our turbulence generator will be reported separately. The integral length scale $L_{I}$ near the jet discharge was calculated by integrating the area under the spatial autocorrelation function–distance curve of the r.m.s. velocity measured, and was found to be $L_{11}=L_{L}=8.8~\text{cm}$ and $L_{33}=L_{T}=6.8~\text{cm}$ , with an average of $L_{I}=7.8~\text{cm}$ . The dissipation rate $\unicode[STIX]{x1D716}_{a}$ was determined by the scaling law (Variano & Cowen Reference Variano and Cowen2008):

(3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{a}=\frac{A(2/3q^{2})^{3/2}}{L_{I}}, & & \displaystyle\end{eqnarray}$$

where $q$ is the TKE and $A$ is an experimental constant. The constant $A=0.49$ was obtained in our background turbulence experiments by comparing the dissipation rate obtained by structure function fit and the scaling law shown in figure 3 (following the method described by Variano & Cowen Reference Variano and Cowen2008), which will be reported in detail in a separate publication.

Figure 2.

Experimental set-up of a negatively buoyant jet in quiescent ambient.

Figure 3.

The dissipation rates obtained by the structure function against that obtained by the scaling law.

For this study, a test region in the tank with a nearly isotropic and homogeneous turbulent background with negligible mean flow was desired, which can be achieved if the region of interest was located far enough from the turbulence generator where the rate of change of turbulence intensity is smaller along $x$ . We chose $x=11.5S$ in this study, where $x$ is the distance from the RASJA. Supplementary jet background turbulence experiments were carried out at the jet discharge location for each series of buoyant jet experiments; the r.m.s. velocity and the shear stress of these experiments were ensemble averaged, and the resulting contour maps of these quantities are shown in figure 4. The spatially averaged background r.m.s. velocity components were determined as $[(u_{a})_{x},(u_{a})_{z}]=(0.0142,0.0132)$ . The r.m.s. velocity was $u_{a}^{\prime }=\sqrt{2/3q^{2}}=0.0135~\text{m}~\text{s}^{-1}$ , where $q^{2}=1/2[(u_{a}^{\prime })_{x}^{2}+2(u_{a}^{\prime })_{z}^{2}]$ is the TKE. The mean flows $\overline{(u_{a})_{x}}$ and $\overline{(u_{a})_{z}}$ were small compared to the r.m.s. velocity (table 1).

The turbulence decay length scale (Maxey Reference Maxey1987) $L_{decay}=q^{2}/(\text{d}q^{2}/\text{d}x)\approx 70~\text{cm}$ at $x=11.5S$ , and $L_{I}/L_{decay}\approx 0.1$ which was effectively homogeneous (Variano & Cowen Reference Variano and Cowen2008) and sufficient for this study. As a measure of the isotropy of the background turbulence considered, we calculated the ratio $(u_{a}^{\prime })_{x}/(u_{a}^{\prime })_{z}=1.075$ . The spatially averaged normalized shear stress $\overline{(u_{a}^{\prime })_{x}(u_{a}^{\prime })_{z}}/q=0.13$ was close to the ideal value of 0. However, from the one-dimensional spectra of a point near to the jet measurements (figure 5), the energy-containing motions were not perfectly isotropic, the same as other studies of RASJA (Variano & Cowen Reference Variano and Cowen2008). The integral length scale ratio $L_{11}/L_{33}=1.29$ was different from 2, which can be calculated from the theory of three-dimensional HIT (Pope Reference Pope2000). This reflects that inhomogeneity and anisotropy existed in the flow. A summary of the turbulence characteristics of this study is given in table 1; the overall characteristics were measurements that scattered over a region of 0.7 m (long) $\times$ 0.4 m (wide) $\times$ 1.0 m (tall), while those at the jet background were regions of 14 cm (long) $\times$ 10 cm (tall).

Figure 4.

Contour maps of various ensemble-averaged turbulence statistics of the jet background turbulence experiments.

Figure 5.

One-dimensional spectra of a point close to the jet measurement.

A turbulent vertical (negatively) buoyant saline water jet was discharged into the field of turbulence from a nozzle with internal diameter $D=1.0~\text{cm}$ and initial velocity $w_{o}=0.646~\text{m}~\text{s}^{-1}$ (flow driven by gravity using a constant-head tank and controlled by a calibrated rotameter), giving a Reynolds number of 6460 which should be sufficiently large for a turbulent jet at the onset. The measured density of the ambient fresh water $\unicode[STIX]{x1D70C}_{a}$ was about $0.997~\text{g}~\text{cm}^{-3}$ , while the saline jet had densities $\unicode[STIX]{x1D70C}_{j}=1.0165~\text{g}~\text{cm}^{-3}$ , $1.0389~\text{g}~\text{cm}^{-3}$ and $1.1655~\text{g}~\text{cm}^{-3}$ in three different series of experiments, giving $Fr=5$ , 10 and 15. Experiments of pure momentum jets with the same density of the ambient were also carried out, and enabled us to compare our measurements with those of past similar studies (Khorsandi et al. Reference Khorsandi, Gaskin and Mydlarski2013). A summary of the experimental parameters used in this study is given in table 2.

Table 1.

Summary of the background turbulence characteristics.

Table 2.

Buoyant jet discharge parameters.

Particle image velocimetry (PIV) was used to measure the instantaneous velocity field of the buoyant jet. The source fluid and the ambient were first seeded with Dantec $50~\unicode[STIX]{x03BC}\text{m}$ diameter polyamid seeding particles. A Dantec DualPower 100-100 laser was used to generate laser pulses at 2 $\times$ 100 mJ with a wavelength of 532 nm. A laser guide arm was used for alignment of the pulsed laser light sheet generated from the laser head with divergence angle of $33^{\circ }$ . A complementary metal–oxide semiconductor camera (Dantec SpeedSense 1040, 2320 $\times$ 1726 pixels, 14 cm $\times$ 10 cm field of view) was placed perpendicularly to, and synchronized with, the pulsed laser sheet to capture the resulting PIV images. The PIV image pairs were sampled at 50 Hz with pulsed laser sheets separated by $5000~\unicode[STIX]{x03BC}\text{s}$ , with a maximum storage duration of 20 s.

In each series of jet experiments, measurements were made separately at distances of $z=10D{-}20D$ and $z=20D{-}30D$ due to the limitation of the field of view of the camera. The laser sheet was aligned to cut through the centreline plane of the jet. Experiments of a buoyant jet in quiescent ambient were first carried out as reference cases. The jet was then discharged into the field of background turbulence and the resulting PIV images were captured. For the distance of $z=10D{-}20D$ , 10 ensembles each of 20 s duration were captured; 20 ensembles were captured for the distance of $z=20D{-}30D$ since the jet was more affected by the HIT and less steady downstream. The jets of some runs were observed to be meandered outside of the field of view, but we did not have enough statistics to justify whether this is a part of the physics or unintended flow resulting from the less than ideal background HIT (e.g. non-zero mean flow), so we opted to include all these data in the analysis.

To determine the velocity vectors, the PIV images were subdivided into interrogation windows of size 32 $\times$ 32 pixels. An adaptive-correlation method was used to determine the velocity vectors in the windows based on 50 % overlapping using the DynamicStudio software from Dantec Dynamics. This resulted in an instantaneous velocity vector map with 106 $\times$ 144 vectors. The first- and second-order statistics of each point were then derived from the instantaneous velocity maps. Note that when the refractive index of the jet is not the same as that of the ambient, the images of individual particles will be blurred, causing possible error in the PIV measurements. Zhang, He & Jiang (Reference Zhang, He and Jiang2017) attempted to assess the impact of the refractive-index mismatch in a plume experiment, and found that this had insignificant impact on the mean and turbulence statistics of the plume. We have also compared our experimental results (first- and second-order statistics) of a buoyant jet in quiescent ambient with the well-established measurements of previous studies to ensure the measurements were reliable.

3.2 Buoyant jets in quiescent ambient

Experiments of the mixing of a buoyant jet in quiescent ambient served to ensure the reliability of the measurement techniques by comparing with data in the literature. The time-averaged centreline velocity of the buoyant jet with $Fr=5,10,15$ , as well as that of a momentum jet were extracted and are given in figure 6. It has been observed that when $z/l_{M}<\sim 1$ , the normalized centreline velocity $\sqrt{M}/(w_{m}z)$ is a constant. When $z/l_{M}>\sim 6$ , the normalized velocity $\sqrt{M}/(w_{m}z)\sim (z/l_{M})^{-2/3}$ . Our experimental data fell mainly in the transition regime, $1<z/l_{M}<5$ , and the centreline velocity can be seen to follow the best-fit curve of Papanicolaou & List’s (Reference Papanicolaou and List1988) experimental data, ranging between the jet regime and the plume regime. The jet Gaussian half-widths $b_{g}$ were measured and are plotted against the downstream distance $z$ in figure 7. The spread rate of the buoyant jet was measured to be 0.104, which is consistent with the spread rate measured by Wang & Law (Reference Wang and Law2002), and within the range of jet spread rate in the literature summarized by Fischer et al. (Reference Fischer, List, Koh, Imberger and Brooks1979).

The second-order statistics of the buoyant jet were extracted from the PIV images. The turbulence intensity of the axial and radial velocity, and also the normalized shear stress are plotted in figure 8. The statistics are in general consistent with those measured by Wang & Law (Reference Wang and Law2002). The turbulence intensity of the axial velocity has a peak value of 0.20–0.30, and that of the radial velocity is slightly lower, ranging from 0.15 to 0.20. The shear stress is also consistent with that measured by Papanicolaou & List (Reference Papanicolaou and List1988) and Wang & Law (Reference Wang and Law2002). External to the jet, the shear stress is zero; inside of the jet, the normalized shear stress peaks at about 0.02, and by symmetry it is zero at the centre of the jet. Overall, our measurements are consistent with those in the literature, showing the reliability of our measurement techniques.

Figure 6.

Normalized centreline velocity of a buoyant jet. The line represents the best fit of Papanicolaou & List’s (Reference Papanicolaou and List1988) data.

Figure 7.

The growth of Gaussian jet width of a buoyant jet with distance measured in this study.

Figure 8.

Normalized second-order statistics of a buoyant jet. The solid line represents the best fit of Wang & Law’s (Reference Wang and Law2002) data.

4.1 Observation

Figure 9 shows a typical jet in quiescent ambient and background turbulence. With background turbulence, the jet can be seen to meander due to the large background eddies. The smaller eddies of the background turbulence appeared to have been incorporated into the jet, dispersing the jet further. Occasionally, there would be a sudden strong eddy acting on the jet and advecting it out of the field of view. This tended to occur more frequently at further downstream distances, as the jet velocity decayed and was more easily affected by the turbulence. The meandering and dispersion by the background eddies affected noticeably the mixing of the buoyant jet, and the effects will be presented quantitatively in the next subsection.

Figure 9.

Visualization of a jet in (a) quiescent ambient and (b) background turbulence at $z=10D{-}20D$ .

4.2 Mixing characteristics

Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) presented the experimental results of the mixing characteristics of a jet in background turbulence. It was observed that the jet centreline velocity decayed faster when the background turbulence intensity increased, or when the jet Reynolds number decreased. It is well known that the centreline velocity of a jet in quiescent ambient, when normalized by the initial jet velocity, will decay with downstream distance as $w_{m}/w_{o}\propto (z/D)^{-1}$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979; Lee & Chu Reference Lee and Chu2003). However, with background turbulence, Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) showed that the relationship no longer holds, and the decay of the velocity will depend on both the jet discharge parameters and the background turbulence characteristics. To normalize the centreline velocity, we propose using the critical length scale $l_{c}=l_{cj}$ . Figure 10 shows the normalized centreline velocity against the downstream distance normalized by $l_{c}$ with experimental data of both this study and Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013). It can be seen that the data generally collapsed on a single curve (lines are predictions from the integral model that will be discussed later), indicating that $l_{c}$ is a suitable length scale to normalize the downstream distance. When $z/l_{c}>2$ , it can be seen that the velocity has decayed to zero, implying that the jet has been broken up by the background turbulence. The dimensional analysis indicates that the velocity decay may also be dependent on the ratio $u_{a}^{\prime }/w_{o}$ . However, centreline velocity data of different ratios collapsed onto a single curve as in figure 10 suggesting that the dependence is weak before the jet is broken up. Thus the non-dimensional relationship becomes

(4.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left(\frac{z}{l_{c}}\right). & & \displaystyle\end{eqnarray}$$

This is also evident in the Gaussian half-widths measured both in this study and that of Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) (figure 11), in which the data can be seen to collapse on a single curve when $l_{c}$ is used as the normalizing length scale. The growth of the jet is in general nonlinear, and the growth rate increases as $z/l_{c}$ increases. It is generally consistent with the observation of past studies that the spreading rate increases significantly when the background turbulence intensity reaches a threshold level (Ching et al. Reference Ching, Fernando and Robles1995). However, the increase in spreading rate appears to be more gradual than what was observed in Law et al. (Reference Law, Cheng and Davidson2001), possibly due to the fact that the background turbulence generated in past studies was inhomogeneous, with a drastic increase in strength close to the downstream oscillating grid. By contrast, in this study and in that of Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013), the turbulence was more homogeneous.

Figure 10.

The normalized centreline velocity of a jet in background turbulence. Experimental data from Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) (open symbols) and this study (filled symbols). Solid and dashed lines are the integral model predictions based on the linear and exponential best-fit lines in representing the momentum decay, respectively.

Figure 11.

The normalized Gaussian half-width of a jet in background turbulence. Experimental data from Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) (open symbols) and this study (filled symbols). Dotted line represents the spread of a jet in quiescent ambient. Solid and dashed lines are the integral model predictions based on the linear and exponential best-fit lines in representing the momentum decay, respectively.

The centreline turbulence intensity of the axial velocity of a jet in turbulence is shown in figure 12. When $z/l_{c}$ is close to zero, it is expected that $w_{rms}/w_{o}$ will be similar to that of a jet in quiescent ambient (about 0.25), since the background turbulence effects have not come into play. As $z/l_{c}$ increases, the effect of the background turbulence also increases, and it can be seen from figure 12 that the turbulence intensity increases from about 0.25 to 1.0 when $z/l_{c}$ reaches 1. Within $z/l_{c}=0.25{-}1$ , the measurements collapse relatively well onto a single curve, and the dependence of $w_{rms}/w_{c}$ on $u_{a}^{\prime }/w_{o}$ is still not strong. When $z/l_{c}>1$ , the data begin to diverge, and in all cases the $w_{rms}$ level is greater than $w_{c}$ , showing the significance of the effect of background turbulence. When this occurs, it is an indication that the jet has started to be broken up by the background turbulence. Results also showed that when $z/l_{c}>1$ , the turbulence intensity of the axial velocity will be dependent on the ratio $u_{a}^{\prime }/w_{o}$ in addition to $z/l_{c}$ . When $u_{a}^{\prime }/w_{o}$ increases, the effects of the background turbulence can also be seen to increase, with $w_{rms}/w_{c}$ going as high as 5 when $z/l_{c}=2.0$ .

We now focus on the regime $z/l_{c}<\sim 1$ for model development, because the variation of $w_{rms}/w_{c}$ is more predictable. The ratio $w_{rms}/w_{c}$ generally increased nonlinearly from 0.25 at $z/l_{c}=0$ to 1.2 when $z/l_{c}=1$ regardless of the ratio $u_{a}^{\prime }/w_{o}$ . The cross-sectional profiles of the second-order statistics at selected $z/l_{c}$ are given in figure 13. Note that the data are plotted symmetrically about $r=0$ as only one side of the data was measured to increase the field of view. The excess turbulence intensity of the axial velocity $\unicode[STIX]{x0394}w_{rms}=w_{rms}-u_{a}^{\prime }$ can be seen to increase monotonically from 0.25, like that of a jet in quiescent ambient, to about 0.7 at $z/l_{c}\approx 1$ . Interestingly, the effect of the background turbulence has caused the jet to become increasingly anisotropic; the excess turbulence intensity of the radial velocity $\unicode[STIX]{x0394}u_{rms}=u_{rms}-u_{a}^{\prime }$ has stayed in the range of 0.15–0.25, and increased less by the background turbulence. On the contrary, the shear stress inside the jet can be clearly seen to be enhanced by the background turbulence, but outside of the jet the shear stress is always zero, a property of the HIT. This would imply that momentum flux will be conserved even with the presence of background HIT as discussed in the next subsection.

Figure 12.

The normalized centreline r.m.s. axial velocity of a jet in background turbulence. Experimental data from Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) (open symbols) and this study (filled symbols). At $z/l_{c}=0$ , $w_{rms}/w_{c}\approx 0.25$ (Wang & Law Reference Wang and Law2002).

Figure 13.

Cross-sectional profiles of the second-order statistics of a jet in background turbulence (mirrored about $r=0$ ). Solid lines represent the profiles for $u_{a}^{\prime }=0~\text{m}~\text{s}^{-1}$ (Wang & Law Reference Wang and Law2002).

Figure 14.

The variation of the momentum integral $M$ along $z$ with (circles) or without (squares) accounting for the second-order statistics measured in this study ( $u_{a}^{\prime }/w_{o}=0.023$ ).

4.2.1 Conservation of momentum

The momentum conservation principle of a jet in background turbulence was examined in Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013). Different from a jet in quiescent ambient, the mean momentum integral $M_{M}=\int _{0}^{\infty }2\unicode[STIX]{x03C0}r\unicode[STIX]{x1D70C}u^{2}\,\text{d}r$ was found not to be constant but to decrease along $z$ . The finding is crucial to the development of the jet integral model when there is background turbulence. In § 2, we showed that while the mean momentum is not conserved as suggested in Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013), the conservation of total momentum flux remains valid when the second-order statistics are included. Using the experimental data of this study, we re-examined the problem by including the second-order statistics of the jet in the momentum integral. The total momentum flux equation of a jet in background turbulence is given as

(4.2) $$\begin{eqnarray}\displaystyle M=\int _{0}^{\infty }2\unicode[STIX]{x03C0}(\overline{w}^{2}+\overline{w^{\prime 2}}-\overline{u^{\prime 2}})r\,\text{d}r. & & \displaystyle\end{eqnarray}$$

The integral is evaluated for $z/l_{c}=0.3{-}1.0$ using our data, and the results are plotted in figure 14. Without accounting for the second-order statistics, the momentum can be seen to decrease along $z$ , similar to that observed by Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013). The jet has an initial momentum of $M_{o}$ , but decreases to about $0.2M_{o}$ when the jet reaches $z/l_{c}\approx 1$ . However, when the second-order statistics are included, the momentum flux is closer to $M_{o}$ for $z/l_{c}=0{-}1.0$ ; it should be noted that a portion of the momentum flux might be out of the camera field of view for $z/l_{c}>0.6$ , where the shear stress is also non-zero, and this appears to have led to a drop in the momentum downstream. Overall, except for this experimental limitation, the total momentum flux can be seen to be conserved with downstream distance in the presence of background HIT.

The cross-sectional velocity and r.m.s. axial velocity profile of Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) were extracted and used to re-examine the momentum flux conservation for a jet in background turbulence. It should be noted that the r.m.s. radial velocity profiles were not reported in their paper, but from our experimental data the contribution from radial r.m.s. velocity is typically $1/3$ of that contributed by the axial r.m.s. velocity, and this is assumed in the analysis of their data. In figure 15(a), the momentum flux integrals of a jet in background turbulence, with various $w_{o}/u_{a}^{\prime }$ ratios, were computed by only accounting for the mean flow. It can be seen that the momentum flux decayed along $z$ , and when the jet reached $z/l_{c}=1.5$ , the momentum flux dropped to only about 0.1 of that of the initial momentum flux. However, when the second-order statistics were accounted for in the integral, the total momentum flux downstream maintained relatively constant (figure 15 b). This suggests that the momentum flux can only be conserved when the second-order statistics are accounted for. In other words, the conservation of momentum should be up to second order for integral modelling in order to cater for a jet discharge in a background HIT.

Figure 15.

The variation of momentum flux along $z$ of a jet in background turbulence with and without accounting for the second-order statistics (Khorsandi et al. Reference Khorsandi, Gaskin and Mydlarski2013).

Figure 16.

The variation of the momentum ratio downstream for a momentum jet in background turbulence for $z/l_{c}<2$ . Experimental data from Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) (open symbols) and this study (filled symbols). Solid and dashed lines are the linear and best-fit curve satisfying the boundary conditions $(z/l_{c},M_{M}/M)=(0,1)$ and $(z/l_{c},M_{M}/M)=(2,0)$ , respectively.

4.2.2 Predictions from integral model

For a jet in quiescent ambient, the ratio of jet momentum carried by the mean flow and the eddies remained approximately constant independent of the downstream distance as $k_{M}\approx 1.10$ (Wang & Law Reference Wang and Law2002). This also indicated that the momentum of the jet is mainly transported by the mean flow. For a jet in background turbulence, however, the percentage of jet momentum being transported by the eddies is increasing downstream, so that the value of $k_{M}$ keeps increasing. Figure 16 shows the change in momentum ratio with $z/l_{c}$ . It can be seen that most of the momentum is transported by the mean flow initially, but then the momentum becomes increasingly transported by the eddies, and the momentum ratio decreases. When $z/l_{c}$ reaches about 1.5, the momentum ratio falls to about 0.1, which indicates that most of the momentum is now transported by the eddies. The measurements of momentum ratio can be used to obtain the $k_{M}$ variation with $z/l_{c}$ . A line represented by the equation

(4.3) $$\begin{eqnarray}\displaystyle k_{M}=\frac{1}{-0.5(z/l_{c})+1}, & & \displaystyle\end{eqnarray}$$

which guarantees $M_{M}=M_{o}$ when $z=0$ , and $M_{M}=0$ when $z/l_{c}=2$ (an assumed linear decay), represents the trend of the experimental data well for $z/l_{c}<2$ , and can be adopted. It is noted that our data show a linear decay, while those from Khorsandi et al. (Reference Khorsandi, Gaskin and Mydlarski2013) appear to be exponential. An exponential fit which guarantees the same boundary conditions gives

(4.4) $$\begin{eqnarray}\displaystyle k_{M}=\frac{1}{1.076\exp [-0.1.324(z/l_{c})]-0.076}. & & \displaystyle\end{eqnarray}$$

By substituting equation (4.3) or (4.4) into the integral model outlined in § 2, the mixing characteristics of a jet in background turbulence can be predicted. Figure 10 shows the comparison of the predicted and observed centreline velocities. Solid and dashed lines are the linear and exponential best-fit curve satisfying the boundary conditions $(z/l_{c},M_{M}/M)=(0,1)$ and $(z/l_{c},M_{M}/M)=(2,0)$ , respectively. Predictions using both curves can be seen to be in good agreement with observation. Comparing to the case of a jet in quiescent ambient, the velocity decays much more rapidly to almost zero when $z/l_{c}\approx 1.5$ , i.e. the jet is now broken up by the background turbulence. Figure 11 shows the comparison between the predicted and observed spreading of a jet in background turbulence; again the predictions can be seen to be in good agreement with the experimental data for $z/l_{c}<1.5$ .

5.1 Mixing characteristics

For a buoyant jet in background turbulence, observations are generally similar to those of a pure jet in background turbulence. The buoyant jet can be seen to meander due to the large eddies, and smaller eddies change the jet structure. The $l_{c}$ in the buoyant jet experiments was chosen to be the same as that of a pure momentum jet for $1<z/l_{M}<5$ – which will eventually be justified by experimental data. As $l_{M}\sim f(Fr)$ , the non-dimensional relationship can be written as

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}^{\prime }\sim f\left(Fr,\frac{z}{l_{c}}\right). & & \displaystyle\end{eqnarray}$$

Figure 17 shows the spreading of the buoyant jet, with the length scales normalized by $l_{c}$ . It can be seen that the spreading rate has been significantly enhanced by the background turbulence, with the initial spreading rate increasing from about 0.10 for a quiescent buoyant jet to 0.18 when there was background turbulence. The growth rate increases even more downstream, when the background eddies have affected the jet more significantly. We note that the spread rate is not strongly dependent on $Fr$ for the range of $Fr$ studied, and is close to that observed in the pure jet discharge case.

The centreline velocity of a buoyant jet in background turbulence is given in figure 18. A marked dependence on $Fr$ is observed in this case. The more buoyant the discharge is, the higher the velocity will be in general. Thus, it is difficult to use a single curve to represent the velocity decay for all $Fr$ . An integral model is needed in order to predict the mixing characteristics of the buoyant jet. The necessary quantities required for the model are $k_{M}$ and $k_{H}$ , which have to be obtained experimentally by measuring the second-order statistics of the buoyant jet in background HIT (see § 5.3).

The second-order statistics of a buoyant jet in background turbulence are given in figure 19 with $Fr=5{-}15$ . For the turbulence intensity of the axial velocity, it can be seen that as the distance from the source increases, the turbulence intensity also increases. Similar to the case of a pure momentum jet, the peak turbulence intensity increases from about 0.25 to 0.70 when $z/l_{c}\approx 1$ . For the turbulence intensity of the radial velocity, the effect appears not as significant, with a modest increase of turbulence intensity from 0.18 to 0.30. For the shear stress, inside the jet the shear stress has increased noticeably from 0.015 to 0.070 at $z/l_{c}\approx 1$ , but outside of the jet it is zero and the momentum equation outlined in § 2 remains valid.

Figure 17.

Spreading of a buoyant jet in background turbulence. Dash-dotted line represents the case $u_{a}^{\prime }=0$ predicted by the integral model. Solid and dashed lines are the predictions using linear and exponential best-fit curves for the momentum decay of a pure jet, respectively.

Figure 18.

Centreline velocity of a buoyant jet in background turbulence. Solid and dashed lines are the predictions using linear and exponential best-fit curves for the momentum decay of a pure jet, respectively.

Figure 19.

Second-order statistics of a buoyant jet in background turbulence: (a) $Fr=15$ ; (b) $Fr=10$ ; (c) $Fr=5$ .

5.2 Conservation of momentum

For a buoyant jet, the momentum flux will be increased by the buoyancy of the jet. The total momentum flux equation has been given in § 2. To verify the equation, the change in total momentum with $z/l_{c}$ was obtained and is shown in figure 20. The lines represent the increase in the momentum flux of the jet due to its buoyancy, and can be obtained by solving an integral model of a buoyant jet in quiescent ambient. This is represented as

(5.2) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}\int _{0}^{\infty }g\frac{\overline{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}r\,\text{d}r, & & \displaystyle\end{eqnarray}$$

which is the right-hand side of equation (2.5). The symbols in the figure represent the total momentum integral computed using the left-hand side of equation (2.5). It can be seen from the figure that the symbols are in reasonable agreement with the lines, showing that (2.5) is valid and can be used in the integral model development.

Figure 20.

The variation of momentum of buoyant jets in background turbulence with normalized downstream distance. Dashed lines represent the estimated momentum increase due to the density difference between the jet and the ambient. From top to bottom: $Fr=5$ , $Fr=10$ , $Fr=15$ , $Fr=\infty$ .

5.3 Predictions from integral model

Similar to a jet in background turbulence, if the momentum ratio decay with $z/l_{c}$ is known, then the integral model presented in § 2 can be used to predict the mixing characteristics of a buoyant jet in background turbulence. Figure 21 shows the variation of momentum ratio with $z/l_{c}$ for a buoyant jet, from which $k_{M}$ can be determined. It can be seen that the variation of the momentum ratio of a buoyant jet can be well represented by the linear equation (4.3), in which the distance is normalized by the pure jet $l_{c}$ . Thus our experimental results suggest that $l_{c}$ defined by the jet initial momentum is also appropriate for buoyant jets with $1<z/l_{M}<5$ . Alternatively, the exponential best fit of the momentum decay which guarantees $M_{M}=M_{o}$ when $z=0$ and $M_{M}=0$ when $z/l_{c}=2$ can also be used in integral modelling:

(5.3) $$\begin{eqnarray}\displaystyle k_{M}=\frac{1}{1.828\exp [-0.396(z/l_{c})]-0.828}. & & \displaystyle\end{eqnarray}$$

An additional term required to be determined in the model is the ratio of the mean to total tracer mass or buoyancy flux along $z/l_{c}$ , equivalent to $1/k_{H}$ in equation (2.12). Although it is not measured directly in our experiments, Wang & Law (Reference Wang and Law2002) found that $k_{H}\approx k_{M}$ for both jets and plumes in quiescent ambient; the recent study of Perez-Alvarado (Reference Perez-Alvarado2016) also suggested that the evolution of mean tracer mass flux downstream is similar to that of the mean momentum flux for a jet in background HIT (figure 22), and hence it is reasonable to approximate $k_{H}=k_{M}$ for our modelling purpose.

Figure 23 shows the comparison between the integral model predicted and observed centreline velocity of a buoyant jet in background turbulence. The solid or dashed lines represent the predictions with the background turbulence accounted for, while the dash-dotted lines represent the predictions in quiescent ambient. The predictions accounting for background turbulence are in good agreement with observations, while those without background turbulence can be seen to overestimate the velocity especially when buoyancy increases.

Figure 24 shows the comparison between predicted jet Gaussian half-width and observations. Most of the data collapse onto a single curve, suggesting that the dependence of Gaussian half-width on $Fr$ is weak for the range of $Fr$ studied. The solid and dashed lines are the prediction from the integral model of a buoyant jet in background turbulence using linear and exponential curve fits for the momentum decay, respectively, which show good agreement with the experimental data. In both prediction and observation, the growth rate of the jet Gaussian half-width increases from about 0.10 to 0.18. The growth of the jet can be seen to be nonlinear, and the growth rate increases as it goes downstream until the jet is broken up by the background turbulence.

Figure 21.

The variation of the momentum ratio downstream for a buoyant jet in background turbulence. Solid and dashed lines are the linear and exponential best-fit curves for the momentum decay of buoyant jets satisfying the boundary conditions $(z/l_{c},M_{M}/M)=(0,1)$ and $(z/l_{c},M_{M}/M)=(2,0)$ , respectively.

Figure 22.

The mean momentum and buoyancy flux decay of a jet in background HIT. Data from Perez-Alvarado (Reference Perez-Alvarado2016);  $u_{a}^{\prime }/w_{o}=0.013$ .

Figure 23.

Comparisons of the predictions (solid lines) of centreline velocity of a buoyant jet in background turbulence with observations (symbols). Dash-dotted lines are the predictions without accounting for the background turbulence. Solid and dashed lines are the predictions using linear and exponential best-fit curves for the momentum decay of buoyant jets, respectively.

Figure 24.

Comparisons of model-predicted (solid line) jet width of a buoyant jet in background turbulence with observation. Dash-dotted line is the prediction without accounting for the background turbulence. Solid and dashed lines are the predictions using linear and exponential best-fit curves for the momentum decay of buoyant jets, respectively.