3.1 Normalisation of mean profiles

The profiles for the centreline mean velocity and mass fraction, shown in figure 3, start off very close to constant in the region of the flow nearest the pipe orifice due to the presence of an unmixed potential core and then decay asymptotically with distance as the jet spreads and mixes. Centreline measurements are normalised by the peak values measured close to the orifice, and the Favre averages are computed in a density-weighted sense. The smaller density ratio for the air jet causes this decay to happen more rapidly because mixing is more easily induced in lighter fluids with smaller momentum. This behaviour is consistent with variable-density, low-speed shear layers, where mixing layer growth is a function of the velocity and density ratios of the two layers, as observed by Brown & Roshko (Reference Brown and Roshko1974) and analytically described by Dimotakis (Reference Dimotakis1986). The faster mixing of lighter fluids versus heavy was also observed in the direct numerical simulations of Rayleigh–Taylor instability by Livescu & Ristorcelli (Reference Livescu and Ristorcelli2008), where they observed asymmetry in the mixing process.

Centreline velocity development near the jet exit is usually normalised using the effective diameter, $d_{eff}$ . Thring & Newby (Reference Thring and Newby1953) defined a far-field value for this quantity as $d_{eff\infty }=d_{0}s^{1/2}$ that gives the diameter of an equivalent jet of ambient fluid with the same momentum flux as the variable-density jet. This normalisation is implicit in the scalings outlined for the momentum-dominated region in the review of Chen & Rodi (Reference Chen and Rodi1980), and many researchers have used this form for comparing the development of jets of varying density ratios both near the orifice and in the far field (Pitts Reference Pitts1986, Reference Pitts1991a ,Reference Pitts b ; Panchapakesan & Lumley Reference Panchapakesan and Lumley1993b ; Amielh et al. Reference Amielh, Djeridane, Anselmet and Fulachier1996).

Figure 3.

Axial history of velocity and mass fraction along the centreline of the developing jets. The dotted lines are the fitted decay laws for the combined data shown in figures 4 and 5.

Variable-density jets in coflow require a definition that accounts for the development of the jet as it transitions from pure fluid exiting from the orifice with an arbitrary velocity profile to a far-field profile with density very near the ambient (Sautet & Stepowski Reference Sautet and Stepowski1995). In the case with coflow, $d_{eff}$ as a function of downstream distance from the jet exit becomes

where

Here $\unicode[STIX]{x1D70C}_{eff}$ is the mean density of the jet at a given downstream distance, weighted by the excess momentum flux, and $d_{eff}$ is the diameter of a jet of pure ambient fluid that would have the same induced momentum flux at that station as the variable-density jet being studied. $d_{eff}$ now evolves from $d_{0}$ near the origin to $d_{eff\infty }$ in the far field.

Rather than using excess velocity, Sautet & Stepowski (Reference Sautet and Stepowski1995) suggest normalising using excess momentum,

where the axial positions, $x_{1}$ , have been adjusted for a virtual origin $x_{u_{CL}}$ . In (3.3) $\tilde{u} _{1,CL}$ is the Favre-averaged centreline axial velocity. The equivalent normalisation for the mass fraction is

This scaling is applicable for $X_{I}\leqslant 0.5$ , the conditions of the current study.

3.2 Mean velocity and concentration development

Figure 4 shows the centreline velocity decaying more slowly as the jet density ratio increases. This is in contrast with the results of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ), who showed a slight increase and Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996), who showed saw no change, as shown in figure 4. On the other hand, the current result is in agreement with the simulations of Chassaing et al. (Reference Chassaing, Harran and Joly1994), which show decay rate decreasing with increasing density ratio, for $s=1.0$ and 1.43 in the intermediate region of the jet, between $X_{I}=0.5$ and 5.0. Furthermore, as reviewed in that work, previous experiments overall have also shown a decreasing trend for the decay rate with increasing density ratio. Table 3 shows the calculated centreline decay rates and virtual origins for the jets according to (3.3). The heavy $\text{SF}_{6}$ jet velocity decays less quickly than air jets from most current and previous experiments. In addition, the virtual origin seems to be decreasing with increasing density ratio, which can also be observed in the previously reported results.

Table 3 lends some insight into velocity development for the range of variable-density experiments that have been performed. The table shows the measurement regions span a range of distances across the momentum-dominated or intermediate development regions of the jets. This likely contributes to the considerable scatter between experiments by different researchers, in addition to differences in jet initial conditions (pipe flow versus top hat exit profiles) and the way the curve fits were done on the data.

Figure 4.

Development of the centreline velocity pseudo-similarity scaling for momentum-dominated regions using the initial centreline velocity. Previously published linear fits are included for comparison, with the flow conditions described in more detail in table 3. Symbols are used for reference only.

Table 3.

Fitting parameters for centreline velocity development for the experiments presented in figure 4 as well as selected numerical results.

a Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) fitted all three cases simultaneously with a single slope and different virtual origins, and report reduced centreline velocity decay rates, likely due to confinement effects at their facility.

b Chassaing et al. (Reference Chassaing, Harran and Joly1994) are fitted to simulations, but compared with experiments under the same conditions.

Figure 5.

Development of the centreline mass fraction pseudo-similarity scaling in the momentum-dominated region. Previously published linear fits are included for comparison, with the flow conditions described in more detail in table 4. Symbols are used for reference only.

Table 4.

Fitting parameters for centreline mass fraction decay for the experiments presented in figure 5 as well as selected numerical results.

a Djeridane et al. (Reference Djeridane, Amielh, Anselmet and Fulachier1996) fitted all three cases simultaneously with a single slope and different virtual origins.

b Chassaing et al. (Reference Chassaing, Harran and Joly1994) are fitted to simulations, but compared with experiments under the same conditions.

Unlike decay rates of the centreline velocity, the trend of decreasing mass fraction decay rate with increasing density ratio is consistent across experiments and simulations, as shown in figure 5. The mass fraction decay we observe along the centreline agrees well with the pseudo-similarity scaling proposed by Chen & Rodi (Reference Chen and Rodi1980) for the momentum-dominated region but is slower than the decay reported by many previous researchers. A complete list of the fitted parameters is given in table 4. Most noticeable in this table is the clear effect of jet density ratio on the position of the virtual origin, as calculated using equation (3.4). This effect was observed by the Pitts (Reference Pitts1986) experiments shown in table 4 and Chassaing et al. (Reference Chassaing, Harran and Joly1994).

3.3 Jet spreading rates

Jet spreading rates, defined using velocity and concentration half-widths, vary considerably among jet types and among measurement quantities chosen. When we use axial velocity, $\tilde{u}$ , mean volume fraction, $\bar{X}$ , and Favre-averaged mass fraction, ${\tilde{Y}}$ , we can determine the spreading rates using

where $q$ can be either $\tilde{u}$ , $\bar{X}$ or ${\tilde{Y}}$ . The fits to (3.5) are shown in figure 6. The air jet spreads faster than the $\text{SF}_{6}$ jet, as shown by the data points and slopes ( $K_{q}$ ) in table 5. In the air jet, mass and volume fraction show equivalent spreading rates. But for the $\text{SF}_{6}$ jet, the mass fraction spreads 5 % faster than the volume fraction. With both jets, the trend is that using velocity to predict jet spreading rate will result in smaller spreading rates than when mass and volume fraction are used. The result is that significantly different virtual origins are calculated from these three different quantities. This result is aligned with the near-nozzle ( $x_{1}/d_{0}=2$ ) observations of the jets. More mixing is occurring early in the evolution of the $\text{SF}_{6}$ jet, because it shows clear spreading in both the velocity and mass fraction profiles. The volume fraction profile is narrowing, indicating that more mixing is occurring in the shear layer near the nozzle, while the $s=1.2$ jet spreads very little in this region.

Figure 6.

Development of jet widths with distance calculated using (3.5) for velocity, mean volume fraction and mass fraction. The short, heavy lines are the experimental data and the light lines are the fitted functions.

Table 5.

Parameters for jet spreading rates (slopes), $K_{q}$ , and virtual origins, $x_{q1/2}/d_{0}$ for velocity, volume fraction and mass fraction.

a Chassaing et al. (Reference Chassaing, Harran and Joly1994) are fitted to simulations, but compared with experiments under the same conditions.

3.4 Self-similarity of velocity and mass fraction

Although our measurements are within the momentum regions of both jets, in our downstream measurements ( $15<x_{1}/d_{0}<31$ ), we see some self-similar-like behaviour. The mean radial profiles of the axial and radial velocities (figures 7 and 8), and the mass fraction (figure 9), are shown using two different types of normalisations to demonstrate the evolution of the mean jet properties from the jet exit to the self-similar-like flow conditions.

Figure 7 shows local excess axial and radial velocities at three downstream locations scaled by the centreline excess velocity, with radial coordinate by the velocity half-width, $r_{\tilde{u} _{1/2}}$ (the radius at which the axial speed is half the centreline). The jet exit conditions are closely matched to the log-law exit profile that would be expected for pipe flow, although slightly blunter near the jet centreline. The match to an empirical $1/7$ th power pipe flow profile (not shown) is also quite good, demonstrating that the mean exit flow corresponds to fully developed pipe flow. A small velocity deficit is observed outside of $x_{2}/d_{0}=0.6~(x_{2}/r_{\tilde{u} _{1/2}}=1.2)$ that is a remnant of the wake region created by the square ends of the jet pipe walls; this is recovered to the free stream value by $x_{2}/d_{0}=0.7~(x_{2}/r_{\tilde{u} _{1/2}}=1.4)$ .

Figure 7.

Radial profiles of excess axial and radial velocity normalised by local centreline excess velocities and half-width. Expected axial asymptotic behaviour shown in red.

The scaling in figure 7(a) shows excellent collapse of the air and $\text{SF}_{6}$ axial velocities at each downstream position to a single profile. By $16d_{0}$ downstream, the excess velocity in both jets has reached a profile very similar to the Gaussian curve that is typically seen in single-fluid jets discharging into quiescent conditions. In the inner region within the half-width, the fit is nearly exact.

Both jets achieve self-similar conditions in the last two measurement stations ( $x_{1}/d_{0}=16.3$ and $x_{1}/d_{0}$ = 31.4), as shown in figure 8(a). Using

the profiles for the air and $\text{SF}_{6}$ jets collapse neatly onto the Gaussian profiles

where the width coefficient $A$ is defined by the velocity half-width spreading rate and the expression (Chen & Rodi Reference Chen and Rodi1980)

The spreading rate is larger at all times for the air jet than for the $\text{SF}_{6}$ , indicating faster development of the less dense jet. As mentioned previously, this effect is also observed in shear layers (Soteriou & Ghoniem Reference Soteriou and Ghoniem1995) and Rayleigh–Taylor unstable flows (Livescu & Ristorcelli Reference Livescu and Ristorcelli2008). Figure 8(a) shows that both variable-density jets are tending toward a self-similar state in the momentum-dominated region of their development, though not the same state.

Figure 8.

Profiles of normalised axial and radial velocity using radial similarity variable (3.6).

Figure 9.

Radial profiles of mass fraction (a) normalised by local centreline value and half-width and (b) with respect to self-similarity coordinates. Expected asymptotic behaviour is shown in red.

The profiles for mass fraction, shown in figure 9(a), are very nearly top hat in profile. In the downstream locations, the jet mass fraction has begun to assume a nearly Gaussian profile for both jets, just as the velocity profile does. As with the velocity profiles, the jet half-width and centreline mass fraction do an excellent job of collapsing both the near-jet and downstream profiles (figure 9 a). Using self-similarity variables based on mass fraction also yields excellent collapse of downstream concentration profiles (figure 9 b), just as it did for the velocity profiles, although the self-similar states in the momentum region are not the same for the two jets.

4.1 Radial profiles of Reynolds stresses

To explore the magnitude and distribution of these effects, radial profiles of Reynolds stress tensor components are plotted in both Reynolds- and Favre-averaged forms in figure 10. The mean magnitude of the fluctuating axial and radial components of velocity are plotted in figure 10 versus the downstream distance, normalised by the excess centreline momentum according to

for the Reynolds-averaged r.m.s. fluctuations and

for the Favre-averaged root mean square (r.m.s) fluctuations. The shear stress component is also normalised by the excess centreline momentum:

This normalisation follows the functional form for the decay of the centreline velocity (3.3) suggested by Sautet & Stepowski (Reference Sautet and Stepowski1995) and yields a better collapse of the downstream profiles. It does, however, lead to smaller normalised turbulence intensities than the normalisation by $\bar{u}_{1,CL}-\bar{u}_{\infty }$ used by Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996), among others. When discussing results reported by Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996), we adjust their values to match (4.2).

Figure 10.

Radial profiles of Reynolds stresses. Jet material is denoted as (blue) for air; and (black) for $\text{SF}_{6}$ ; downstream location is given by ——, $x_{1}/d_{0}=0.6$ ; – - – - –, $x_{1}/d_{0}=2.5$ ; $\cdots \cdots \,$ , $x_{1}/d_{0}=16.3$ ; – – –, $x_{1}/d_{0}=31.4$ ; with (heavy lines) for Favre averages; and (light lines) for Reynolds averages.

Figure 10 shows the radial profiles of the Reynolds stresses at four different axial locations. At the upstream locations, any differences between Favre and Reynolds averaging is slight, and the curves lay on top of each other; the same is true for the air jet results downstream ( $x_{1}=16.3d_{0}$ and $30.5d_{0}$ ) but not the $\text{SF}_{6}$ . Additionally, several variable-density effects are observable in comparisons of the central portion of the two jet flows, within about $x_{2}/r_{\tilde{u} _{1/2}}<2$ . For the two axial locations near the jet exit, the shear is high, and we expect the flow to behave like a variable-density shear layer. Two-dimensional shear layers mix asymmetrically towards the lighter side of the layer, and the mixing entrainment is proportional to $s^{1/2}$ (Brown & Roshko Reference Brown and Roshko1974; Dimotakis Reference Dimotakis1986) and governed by large-scale coherent structures. In the jet shear layer, the magnitude of the normal stress components has a strong peak, shifted slightly radially outward for the heavy jet. At $2.5d_{0}$ , the shear layer has spread considerably and has begun to disturb the potential core while the magnitude of the fluctuations has slightly increased, due to localised production tied the large vortices characteristic of shear layer mixing. The shear layer effects are gone by $16d_{0}$ , where the profiles for both air and $\text{SF}_{6}$ have assumed very nearly the shapes expected for single-density free jets. At the downstream locations, the magnitudes of the Reynolds stresses have diverged, with peak values of air 20–30 % larger than those of $\text{SF}_{6}$ .

This difference in peak normal stress locations is also present in the data of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ) and Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996), although neither set of authors discussed it. Panchapakesan & Lumley did not provide results near the nozzle, but showed for the axial fluctuation a peak near $0.4r_{\tilde{u} _{1/2}}$ in the self-similar region of the air jet, and a peak near the centreline for helium. For Amielh et al., in the upstream shear layer there were no clear shifts in the peak for the axial fluctuations at $0.2d_{0}$ , but a shift is present in the radial component (0.7, 1.0 and $1.1r_{\tilde{u} _{1/2}}$ for He, air and $\text{CO}_{2}$ ), and for the axial component at $20d_{0}$ ( ${\approx}0.4,0.5$ and $0.6r_{\tilde{u} _{1/2}}$ ). There may also be a similar effect present in the axial fluctuations at $2.0d_{0}$ , but it is more difficult to compare due to the very rapid development of the helium jet. Taking the results of these previous studies together with the present, the data seem to suggest a consistent skewing of the location of peak turbulent stresses outward for jets with density ratios $s>1$ , and inward for jets with $s<1$ . This is probably reflective of the difference in momentum transfer between a dense jet in lighter surroundings versus a light jet in a denser medium; the less dense material will be more easily accelerated and thus accumulate larger fluctuations more easily, skewing the profiles toward the direction of less mass.

Table 6.

Peak Reynolds stresses for various jet experiments.

a Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) are rescaled using (4.2) and (4.4), and were reported as $\bar{R}_{ij}$ .

Table 6 shows peak Reynolds stresses from jet experiments ranging from $0.14\leqslant s\leqslant 4.2$ . Just as in our experiments, the trend for lower turbulence fluctuations in higher density ratio jets is seen in both the results of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ) for free helium and air jets and in Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) work with helium, air and CO $_{2}$ . Our absolute values of peak fluctuations agree well with the air jet of Panchapakesan and Lumley. The overall trend of decreasing fluctuations with increasing jet density agrees with the simulations of Chassaing et al. (Reference Chassaing, Harran and Joly1994) and Harran et al. (Reference Harran, Chassaing, Joly and Chibat1996).

Shear stress profiles have odd symmetry, as expected with values very near zero on the centreline. In the upstream region, they show a nearly linear profile within the potential core, increasing sharply to a narrow peak in the shear region that spreads with distance to overtake the whole jet downstream. Due to the anti-symmetry of the profiles, the peak is clearly off centre in the developed region ( $30d_{0}$ ); near $0.7r_{\tilde{u} _{1/2}}$ for the air jet and 0.8 for the $\text{SF}_{6}$ .

Also included in figure 10 are the Favre- and Reynolds-averaged fluctuations at the downstream locations (thick versus thin lines). For the air jet, the values are coincident, but for the $\text{SF}_{6}$ jet, we see the radial extent of the reduction in $R_{ij}$ due to the variable-density conditions. This effect is caused by the variable-density terms in (4.1). We quantify the effect of the turbulent mass fluxes in figure 11, where the axial and radial turbulent mass fluxes, $a_{1}$ and $a_{2}$ are shown at the jet exit and downstream locations. The magnitude of the mass fluxes in air are always low, increasing from the jet exit and then decreasing to the furthest measurement station. These small, non-zero values are due to the air jet density ratio of $s=1.2$ . For the heavy jet, the mass fluxes are very high and localised at the nozzle exit in the shear layer region, spreading and increasing at the $x_{1}/d_{0}=16.3$ location, then decreasing but still remaining many times larger than the air values.

Figure 11.

Turbulent mass fluxes, defined in (1.4).

The relationship between turbulent mass flux and density gradients is illustrated in Balakumar et al. (Reference Balakumar, Orlicz, Ristorcelli, Balasubramanian, Prestridge and Tomkins2012) for shock-driven Richtmyer–Meshkov mixing, where they found similar effects of turbulent mass flux on the Reynolds stresses in a reshocked heavy gas curtain ( $s=4$ ). The difference in magnitude of the turbulent mass fluxes is also observed in the calculations of Harran et al. (Reference Harran, Chassaing, Joly and Chibat1996). Our experiments and their simulations show that the peaks in $a_{1}$ for $s=4.2$ are approximately $7\times$ larger than those for $s=1.2$ , as shown in figure 11(a). We observe this same ratio between peaks for $a_{2}$ , as seen in figure 11(b). The shapes of the radial profiles for $a_{2}$ are similar to Harran et al. (Reference Harran, Chassaing, Joly and Chibat1996) figure 16, showing the peak moving inward with increasing downstream position. The importance of the turbulent mass flux in modelling and understanding variable-density mixing is also described in the direct numerical simulations of Rayleigh–Taylor instabilities by Livescu & Ristorcelli (Reference Livescu and Ristorcelli2007).

In figure 11(b), the radial turbulent mass flux, $a_{2}$ , goes to zero on the centreline, while the difference between $\bar{R}_{22}$ and $\tilde{R}_{22}$ does not vanish. Therefore, the triple correlation term $\overline{\unicode[STIX]{x1D70C}^{\prime }u_{2}^{\prime }u_{2}^{\prime }}/\bar{\unicode[STIX]{x1D70C}}$ in (4.1) is not negligible there, and must be negative and of similar magnitude to the off axis $a_{2}a_{2}$ . We can use $\overline{\unicode[STIX]{x1D70C}^{\prime }u_{2}^{\prime }u_{2}^{\prime }}$ to draw some inferences about one possible description of the instantaneous mixing conditions that could give rise to this behaviour. We expect that inward fluctuations of radial velocity ( $u_{2}^{\prime }<0$ ) should be associated with decreases of local density ( $\unicode[STIX]{x1D70C}^{\prime }<0$ ) as more dilute fluid is brought inward, and $\unicode[STIX]{x1D70C}^{\prime }u_{2}^{\prime }u_{2}^{\prime }<0$ for such events. Similarly, for positive velocity fluctuations ( $u_{2}^{\prime }>0$ ) less-mixed fluid is usually being transported outward ( $\unicode[STIX]{x1D70C}^{\prime }>0$ , $\unicode[STIX]{x1D70C}^{\prime }u_{2}^{\prime }u_{2}^{\prime }>0$ ). For the net correlation to be negative, on balance the fluid being swept inward must be more pure than the denser fluid being returned outward. The fluid being brought inward by vortical motion during an entrainment event tends to be relatively pure outer fluid that begins to mix with the surrounding jet, while the material being transported outward will tend to already be dilute. At $x_{2}=0$ , of course, any motions that penetrate the jet centreline will tend to be associated with less dense than average material, and thus $\unicode[STIX]{x1D70C}^{\prime }u_{2}^{\prime }u_{2}^{\prime }<0$ for both $u_{2}^{\prime }<0$ and $u_{2}^{\prime }>0$ . This results in the net contribution of this term being negative near the centre of the jet, reinforcing the decrease of $\tilde{R}_{ii}$ versus $\bar{R}_{ii}$ caused by the $a_{i}a_{i}$ term.

Figure 12.

Development with downstream distance of centreline turbulence intensities.

4.2 Evolution of turbulent kinetic energy with axial distance

The current facility is not optimal for examining the full axial evolution of the jet due to optical access limitations. The access in the tunnel is currently under improvement, but for now we focus on trends that we see in the Reynolds stresses as we move downstream. The turbulent normal stresses are shown in figure 12 versus downstream distance, with the fluctuation magnitudes normalised according to equations (4.2) and (4.3). As before, both Favre- and Reynolds-averaged fluctuations are shown, and differences are only present for the two downstream locations in the $\text{SF}_{6}$ jet.

Turbulence intensities for both jets were not isotropic exiting the pipe, with axial fluctuations near 3.5 % and radial at 2.3 %, consistent with previous experimental and numerical reports for fully developed turbulent pipe flow (see, for instance, Laufer Reference Laufer1952, Wu & Moin Reference Wu and Moin2008). For both jets, the ratio of axial to radial fluctuations downstream is approximately 1.25, consistent with ranges from 0.23–0.28 axially and 0.15–0.22 radially that are reported in the literature for air. The magnitude of the fluctuations rises more quickly for the air jet plateauing by $30d_{0}$ near values of 0.235 and 0.19 in the axial and radial directions, regardless of which averaging is used. A similar structure is seen with the $\text{SF}_{6}$ jet, though the intensities may not have yet reached an asymptotic value by the final measurement location, and the magnitudes are lower, reaching only approximately 0.19 and 0.15. Our new results in the strongly non-Boussinesq regime show that an $s=4.2$ jet is trending toward lower asymptotic turbulence fluctuation levels compared to a jet with a density ratio of $s=1.2$ .

The air jet has higher turbulence intensities in figure 12 because it develops very quickly, causing the centreline velocity to drop faster than that of the $\text{SF}_{6}$ jet. The centreline excess normalisation used in (4.2) and (4.3) causes the air jet turbulence intensities to be larger than the $\text{SF}_{6}$ jet. We will separate this effect from the density fluctuation effect in a later section. The difference between Favre and Reynolds averaging is noticeable in the $\text{SF}_{6}$ jet, with an approximate 3–5 % decrease in magnitude for mass-weighted values due to variable-density effects at both $16d_{0}$ and $30d_{0}$ locations, but not in the region close to the nozzle. While not large, the difference is consistent, and the effect of the discrepancy on the mean flow will be cumulative with distance.

These results compare well to the experiments of Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) who examined jets issuing from a fully developed pipe in momentum-matched flows (rather than Reynolds number) of He, air and $\text{CO}_{2}$ ( $s=0.14,1.0,1.4$ ), and saw that using the normalisation $\bar{u}_{1,CL}{-}\bar{u}_{\infty }$ all gasses reached a plateau for axial fluctuations of approximately 0.27 by 40 $d_{0}$ , with or without coflow. This value was reached very quickly for the lighter helium flow ( ${\sim}5d_{0}$ ), but further downstream for the air and $\text{CO}_{2}$ jets $({>}30d_{0})$ . When adjusted for the normalisations in (4.2) the trends in their data become more similar to the present investigation. The new correction causes the turbulence levels to plateau much earlier, by approximately $16d_{0}$ as is seen in the present experiments. Additionally, with this normalisation the density effect on the turbulent fluctuations appear in the data of Amielh et al. as well, with the asymptotic states for the axial fluctuations becoming approximately 0.23, 0.18 and 0.17 for helium, air and $\text{CO}_{2}$ respectively, and 0.17, 0.15 and 0.14 for the radial fluctuations. The rescaled values for air are slightly lower than those measured here, but still within the lower end of the range typically reported, and show the same trend of lower turbulent fluctuations for denser jets.

The turbulent kinetic energy, $\tilde{k}$ , has also been calculated in density-weighted form from the trace of the Favre-averaged Reynolds stress tensor according to the relation

Since $u_{3}$ (in the azimuthal direction) was not measured in this study, $\tilde{R}_{33}$ was approximated by $\tilde{R}_{22}$ , which should be true identically in the case of radial symmetry along the jet centreline. The same form was used for calculation of Reynolds-averaged t.k.e. The results are plotted in figure 13 with two different scaling methods. The first is with the scaling of $(2/3\tilde{k})^{1/2}$ or $(2/3\bar{k})^{1/2}$ to match the magnitudes of the turbulence intensities for Favre or Reynolds averaging, respectively, and normalised again by the excess momentum using the relations

The second scaling in figure 13(b) uses the initial kinetic energy of the mean flow, either $\bar{K}_{0}$ or $\tilde{K}_{0}$ .

Figure 13.

Development with downstream distance of turbulent kinetic energy along the centreline, (a) using the scaling of (4.6) and (b) scaled with the initial kinetic energy of the mean flow.

Trends in t.k.e. are similar to those observed in the component fluctuations, with the turbulent kinetic energy for the air jet appearing to asymptote by $30d_{0}$ to a value just above 0.20, and the $\text{SF}_{6}$ jet beginning to approach a final value near 0.16. As expected in an $s=1.2$ jet, this is close to the asymptotic value near 0.23 seen by Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) for a slightly smaller density ratio of $s=1$ , and the direction of the deviation is consistent with the observed effect of increasing density ratio, though when renormalised using (4.2) the values drop to approximately 0.21, 0.15 and 0.14 for helium, air and $\text{SF}_{6}$ , lower than the current results. As before, values from Amielh et al. (Reference Amielh, Djeridane, Anselmet and Fulachier1996) have been renormalised to match the present scaling.

The variable-density effects that lower both the peak and asymptotic values of t.k.e. along the jet centreline were studied parametrically in the numerical simulations of Harran et al. (Reference Harran, Chassaing, Joly and Chibat1996), where they observed trends similar to figure 8. The asymptotic values of t.k.e. ( ${\approx}0.15$ for air, and slightly lower for $s=5.11$ ) were mostly reached at $x_{1}/d_{0}>300$ , which is well outside the measurements of these experiments.

Although such qualitative descriptions of the jet evolution are useful for understanding the net changes in the distribution of turbulent kinetic energy with distance and how they are affected by different jet density ratios, for a more quantitative description we need to examine the full budget equations. These are derived and discussed in the following section.

5.1 Turbulent kinetic energy transport equations

To gain some insight into the mechanisms driving the variable-density effects we have observed, we examine the budgets of t.k.e. These were derived by taking one half of the trace of Favre-averaged Reynolds stress transport equations shown in (1.3). Under the assumption of statistically stationary flow, the time derivative terms were dropped. This yielded the following budget equations:

The production term ( $P^{\tilde{k}}$ ) has two parts, the first equal to the classical effect of the Reynolds stresses acting on the straining of fluctuating part, and the second a variable-density effect where the turbulent mass flux interacts with the mean pressure strain. In the case of constant-density flow, this second term vanishes. Although the molecular shearing effects ( $M^{\tilde{k}}$ ) could be calculated from the present data, trial computations showed it was typically at least 3–4 orders of magnitude smaller than the other terms, and therefore we have omitted it from the analysis. The transport term is divided into a turbulent flux part ( $D^{\tilde{k}}$ , first part on the right-hand side) consisting of a triple correlation of velocity with the local density, $\tilde{R}_{jii}=\overline{\unicode[STIX]{x1D70C}u_{j}^{\prime \prime }u_{i}^{\prime \prime }u_{i}^{\prime \prime }}$ , and fluctuating velocity–stress and fluctuating velocity–pressure correlations (the second term is flux due to fluctuating stress; the third flux from pressure fluctuations, $\unicode[STIX]{x1D6F1}^{\tilde{k}}$ ). Many previous authors, including Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ), omitted the stress flux terms under the assumption that they are negligible. This assumption, that $T^{\tilde{k}}=D^{\tilde{k}}+\unicode[STIX]{x1D6F1}^{\tilde{k}}$ was assumed to be true here but was not verified. The pressure strain ( $\unicode[STIX]{x1D6F7}^{\tilde{k}}$ ) can be neglected, since for subsonic turbulence the divergence of the density-weighted fluctuations should be very near zero.

Recasting the remaining terms in cylindrical coordinates (to take advantage of symmetries in the flow) and assuming that there is no swirl (gradients of mean quantities in the azimuthal direction are 0), we obtain the following simplified expressions:

The Cartesian components are equivalent to the cylindrical components along the centre plane of the jet with $z=x_{1}$ , $r=x_{2}$ and $(u_{1},u_{2},u_{3})=(u_{z},u_{r},u_{\unicode[STIX]{x1D703}})$ . Also, since $R_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ was not measured directly, it was assumed to be equal to $R_{rr}$ . Along the centreline this should be true by symmetry and should be valid in the outer flow by isotropy, but may be only approximately true between those points. Fortunately, many measurements including those by Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ) have shown that this is a reasonable assumption for both single-fluid and variable-density jets throughout the domain. Similarly, the $\tilde{R}_{j\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ terms were interpreted as $\tilde{R}_{jrr}$ in the expansion of both $\tilde{R}_{rii}$ and $\tilde{R}_{zii}$ . Hotwire results by Panchapakesan & Lumley and more recently by Darisse, Lemay & Benaïssa (Reference Darisse, Lemay and Benaïssa2015) suggest that this is reasonable for $\tilde{R}_{z\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ . Measurements of $\tilde{R}_{r\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) suggest that $\tilde{R}_{r\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}\approx 1/2\tilde{R}_{rrr}$ might be a better approximation, so that was used instead.

Mean pressure gradients are needed for the variable-density terms in the second part of (5.2b ), so these were calculated using the momentum conservation equations (1.1) in cylindrical coordinates.

In the case where the pressure and stress flux terms from the $T^{\tilde{k}}$ are neglected, the computed dissipation includes the sum of both effects. Analysis by Darisse et al. (Reference Darisse, Lemay and Benaïssa2015) using a model from Lumley (Reference Lumley1979) suggests that even in the case of constant-density jets, the effects of pressure flux are important, and this would be even more true for flows with density gradients. Therefore, for the present budgets we also define the pressure flux, $\unicode[STIX]{x1D6F1}^{\tilde{k}}$ , based on the computed turbulent flux, $D^{\tilde{k}}$ using the model from Lumley. The dissipation is calculated using the balance of the remaining measured and modelled terms.

To minimise the amplification of errors through repeated differentiation of higher-order correlations, mean velocity and density fields were fitted with smoothing splines that were cubic in $x_{1}=z$ and quintic in $x_{2}=r$ . Data were mirrored from both sides of the jet and symmetry conditions at the origin were imposed (odd or even, as appropriate), to prevent artefacts at $r=0$ due to slight asymmetries in the flow. Correlation terms were first calculated from the original data, and then the same smoothing and regularisation was applied. No continuity or momentum conservation constraints were imposed on the filtered fields.

In addition to computing the magnitudes of each budget quantity, an estimate of uncertainty was also calculated. In brief, the uncertainty levels on each smoothed and fitted quantity in (5.2) ( $\bar{\unicode[STIX]{x1D70C}}$ , $\tilde{k}$ , $\tilde{u} _{i}$ , $\tilde{R}_{ii}$ , $a_{i}$ , $\unicode[STIX]{x2202}\bar{P}/\unicode[STIX]{x2202}x_{i}$ and $\tilde{R}_{jii}$ ) were estimated by finding the r.m.s. deviation between the fit and the measured quantity. These values were then propagated to the final budget components by use of a Taylor series uncertainty propagation method (International Organization for Standardization 1995; Coleman & Steele Reference Coleman and Steele2009). Using this approach, the uncertainties relative to the peak of each budget component were approximately 9 % for advection, 3 % for production, 1 % for the two flux terms and 3.5 % for dissipation at a $\pm 2\unicode[STIX]{x1D70E}$ level (corresponding to a ${\sim}95\,\%$ confidence interval). Complete details and an example calculation for the $\text{SF}_{6}$ jet at $x_{1}/d_{0}=16.3$ are given in appendix A.

5.2 A modified scaling for turbulence budgets in variable-density jets

Because of the disparate density ratios of the two jets under study, the turbulence budgets should be scaled, but the traditional kinematic $\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2}}$ scaling does not account for local density variations. The $\unicode[STIX]{x1D70C}_{eff}$ (3.2) scaling proposed for variable-density jets in coflow by Sautet & Stepowski is preferable to centreline scaling that neglects spatial variations in density and velocity. For comparison of the jet eddy lengths, we include the effect of the local effective density ratio on the jet width development by using the same scaling as $d_{eff}$ (3.1) to scale the local jet half-width:

For the characteristic eddy speed, we keep the velocity scaling as $\bar{u}_{1,CL}$ since the difference between this and the effective excess momentum that we used previously, $(\bar{u}_{1,CL}^{2}-\bar{u}_{\infty }^{2})^{1/2}$ , was negligible, although in the case of strong coflow this may not be true. The scaling in (5.3) is not used for the radial spatial coordinate, $x_{2}$ , in the turbulence budgets because the $r_{\tilde{u} _{1/2}}$ scaling already accounts for the spreading effects in the radial direction caused by density variations.

5.3 Near-exit budgets

The near-field ( $x_{1}/d_{0}=0.67$ and $x_{1}/d_{0}=3.2$ ) budgets for turbulence in the air and $\text{SF}_{6}$ jets are shown in figure 14 as a function of radial position in the jet scaled by velocity half-width. Budget components are scaled by $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2},eff}$ . The production and dissipation maxima and minima are nearly balanced, and they peak near $x_{2}=r_{\tilde{u} _{1/2}}$ due to the shear layer at the nozzle exit.

By $x_{1}/d_{0}=3.2$ , shown in figure 14(b), the shear layer has expanded slightly, and it is wider for the air compared with the $\text{SF}_{6}$ jet, mimicking two-dimensional variable-density shear layer growth. This is explained by shifts in production and dissipation peaks inward to 0.8 and 0.9 $r_{\tilde{u} _{1/2}}$ for the air and $\text{SF}_{6}$ jets. Production and dissipation are becoming unbalanced, and the magnitudes of advection and turbulent flux are increasing. This dynamic will become more exaggerated downstream, driving the rapid increase in centreline t.k.e. shown in figure 13.

While to the best of our knowledge there have been no experimental budgets reported for $s>1$ jets, the shape and magnitude of the budgets is largely similar to the results shown by Djeridane et al. (Reference Djeridane, Amielh, Anselmet and Fulachier1996) in their pipe-driven air jet at $x_{1}/d_{0}=5.00$ . Due to the additional spreading at this slightly more downstream location, their curves are broader than the present results, but show comparable structure and magnitude. Their peak values of production and dissipation near 0.9 $r_{\tilde{u} _{1/2}}$ are slightly smaller than present, while convection and turbulent flux are essentially identical. Their advection and dissipation also are qualitatively similar to ours, but scatter in their displayed data points makes closer comparison difficult.

5.4 Downstream budgets

Figure 15 shows the turbulence budgets at $x_{1}/d_{0}=16.3$ and 30.5 for both jets. The scaling matches peak production and most of the budget quantities very well in the outer part of the jet ( ${>}1r_{\tilde{u} _{1/2}}$ ); the remaining small differences between the different components in the outer jet are negligible (where we have previously established that local variable-density effects are smaller), but significant differences are still apparent in the jet core.

The budget profiles are similar to the fully developed results usually seen in single-fluid jets. The shear layer has completely closed, and as previously discussed, the profiles of velocity statistics are nearly self-similar and are only slightly changing in magnitude with distance between the two points. Thus, comparisons with previous results in fully developed free jets are not unreasonable, as well as with the results of Djeridane et al. (Reference Djeridane, Amielh, Anselmet and Fulachier1996) in coflow at $15d_{0}$ .

The peaks in production, turbulent flux and dissipation in the air mixing layer have moved inward to near 0.6 $r_{\tilde{u} _{1/2}}$ from the previous position just inside the half-width, while the advection peak has moved all the way to the centreline and merged with the opposite side of the jet, as seen in previous experiments (Panchapakesan & Lumley Reference Panchapakesan and Lumley1993a ,Reference Panchapakesan and Lumley b ; Djeridane et al. Reference Djeridane, Amielh, Anselmet and Fulachier1996; Darisse et al. Reference Darisse, Lemay and Benaïssa2015). For $\text{SF}_{6}$ , the negative central lobe of the turbulent flux component has slightly shifted outwards from 0.6 to 0.75 $r_{\tilde{u} _{1/2}}$ , and the centreline value is now strongly positive instead of the negative value of the air jet.

Finally, the value of production in the $\text{SF}_{6}$ jet has become slightly negative along the centreline. Since a positive production normally indicates that energy is being transferred from the large scales of the mean flow into turbulent fluctuations, the present results suggest that along the centreline the addition of variable-density effects are instead causing the turbulent fluctuations to reinforce the mean flow, preserving the centreline velocities further downstream.

There are significant differences in the shapes of the components near the centreline between the present results for the air jet and previous findings. An uncertainty analysis for the budgets in figures 14 and 15 is included in appendix A. The uncertainty intervals demonstrated in the appendix show that the greatest uncertainties are near the peaks, not the centrelines, so the budget differences near the centrelines are physical and due to variable-density effects.

Finally, dissipation behaves similarly to that measured in other experiments, showing a dip towards zero at the centreline and peaking near ${\sim}0.6r_{\tilde{u} _{1/2}}$ at $16d_{0}$ and ${\sim}0.5r_{\tilde{u} _{1/2}}$ at $30d_{0}$ . These shapes are consistent with the shapes from Djeridane et al. (Reference Djeridane, Amielh, Anselmet and Fulachier1996) and Darisse et al. (Reference Darisse, Lemay and Benaïssa2015). Both Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993a ) and Hussein, Capp & George (Reference Hussein, Capp and George1994) also have off-axis peaks at ${\sim}0.6r_{\tilde{u} _{1/2}}$ (Darisse et al. Reference Darisse, Lemay and Benaïssa2015).

Figure 14.

Turbulence budgets near the jet exit, in the shear-dominated region. Normalised by $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2},eff}$ .

Figure 15.

Turbulence budgets normalised by $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2},eff}$ .

5.5 Mechanisms driving variable-density mixing effects

In figure 15, marked differences between the air and $\text{SF}_{6}$ jet are visible near the jet centreline for all of the components of the turbulent kinetic energy budget. To understand what causes these differences, we must look at the terms within the production, advection and turbulent flux.

Figure 16.

Turbulence production components, equation (5.2b ), normalised by $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2,eff}}$ at $x_{1}/d_{0}=16.3$ .

Figure 16(a) shows the components of turbulent production, $P^{\tilde{k}}$ , for air and $\text{SF}_{6}$ at $16d_{0}$ . Production in air is large and shows no variable-density component. $\text{SF}_{6}$ has negative total production with significant contributions from the turbulent mass flux components (second part of (5.2b )) for $x_{2}<0.5\,r_{\tilde{u} _{1/2}}$ , and slightly positive contributions for $x_{2}>0.6\,r_{\tilde{u} _{1/2}}$ . Gradient stretching and turbulent mass flux effects are equally influencing production in the $\text{SF}_{6}$ jet at the centreline. If we look at the gradient stretching components, figure 16(b), there are only small differences in the shear, radial and circumferential gradient stretching terms. All of the differences between the gradient stretching in the two jets come from the axial component, where the air jet has significantly higher axial gradient terms near the jet centreline. The total production in the $\text{SF}_{6}$ jet is negative at the centreline because, unlike the air jet, the axial component of gradient stretching is not large enough to replace what is lost due to the expansion of the jet.

Lower production in the dense jet reduces the axial flux of turbulent kinetic energy, creating large differences in total advection between the two jets, as shown in figure 17(a). Net advection ( $C^{\tilde{k}}$ , equation (5.2a )) in the air jet is large and peaks at the centreline, while for $\text{SF}_{6}$ it is near zero at the centreline with a small peak at $x_{2}=0.6\,r_{\tilde{u} _{1/2}}$ caused by the shear layer. Radial and axial gradients in the $\text{SF}_{6}$ jet are balanced, while axial gradients dominate the advection in the air jet.

In the $\text{SF}_{6}$ jet, the total negative production and the low advection cause a deficit in t.k.e. along the centreline, causing energy from the shear layer to be transported inward towards the jet centreline by correlated turbulent fluctuations, as shown by plots of the turbulent flux components ( $D^{\tilde{k}}$ , equation (5.2c )) in figure 17(b). This high turbulent flux is not visible in the air jet because the positive production and advection are enough to maintain the outward spreading of t.k.e. through turbulent flux within one half-width of the jet.

Figure 17.

Advection and turbulent flux components normalised by $\unicode[STIX]{x1D70C}_{eff}\bar{u}_{1,CL}^{3}/r_{\tilde{u} _{1/2,eff}}$ at $x_{1}/d_{0}=16.3$ into components due to fluxes in each direction.