4.1. Scalar centreline trajectories

Scalar centreline trajectories are important to quantify for the range of flow and injection systems explored here, given that, for a transverse jet, $s_{c}$ effectively represents the distance along which jet fluid is exposed to the cross-flow and thus liable to mix. Moreover, early JICF studies (Holdeman Reference Holdeman1993; Margason Reference Margason1993) suggest a correspondence between jet penetration (or a higher trajectory) and improved mixing. And while many transverse jet experiments over the years have studied the effects of momentum flux ratio $J$ and jet Reynolds number on jet trajectory, as summarized in Karagozian (Reference Karagozian2010), few have separately examined the effects of density ratio or injector geometry on trajectory in a systematic way.

The scalar centreline trajectories for $Re_{j}=1900$ transverse jets injected from the flush nozzle, for several different density and momentum flux ratios, are plotted in figure 7. Note that here and in other trajectory plots, centreline concentration decay plots and centreplane-based mixing metrics, data point markers do not represent actual resolution of data, which is much finer than shown. It has been previously demonstrated for the flush nozzle (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012, Reference Getsinger, Gevorkyan, Smith and Karagozian2014) that absolutely unstable jet shear layer conditions correspond to a density ratio of $S=0.35$ for all momentum flux ratios $J$ , as well as for all values of $S$ when $J\leqslant 10$ . Otherwise, for $J>10$ and $S\geqslant 0.55$ , the conditions correspond to convectively unstable upstream shear layers. The data in figure 7 show that, at a fixed $J$ , reducing $S$ (and thus increasing the velocity ratio $R$ for a fixed jet Reynolds number) had the effect of reducing penetration, or lowering the trajectory of the jet. This reduction was particularly significant at higher fixed $J$ values. The reduction in jet penetration with a decrease in density ratio was consistent with the fact that, in keeping the momentum flux ratio constant at a fixed $Re_{j}$ , decreasing the density ratio required both jet and cross-flow velocities to be increased, in some cases, significantly. These changes simultaneously reduced the cross-flow boundary layer thickness, thus enabling a more immediate influence of the cross-flow in turning the jet, hence reducing penetration. Cross-flow entrainment variation could also affect jet trajectory, but entrainment differences would have to be quantified to verify this contention. The effect of an increased jet velocity at lower jet densities could also affect the potential core, and hence mixing; these issues will be analysed in § 4.2.

Figure 8 highlights the effect of injector configuration on the trajectory of equidensity transverse jets for several different momentum flux ratios. Per Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014), only the $J=5$ condition shown for the flush nozzle and flush pipe represented a JICF with an absolutely unstable upstream shear layer; the effect of the external co-flow significantly weakens the elevated jet’s shear layer instabilities at $J=5$ . The flush pipe-injected jets were observed overall to penetrate farther into the cross-flow than either the flush nozzle- or elevated nozzle-injected jets for the higher range of momentum flux ratios shown ( $J=41$ and 12), consistent with a DNS study by Muppidi & Mahesh (Reference Muppidi and Mahesh2005) on the effects of jet velocity profile on the trajectories of the equidensity JICF. This observation suggested that jets with a parabolic velocity profile and thicker jet boundary layer at its exit could penetrate more deeply than for other jets at the same $J$ and $Re_{j}$ values, possibly in part due to the higher centreline velocity for the parabolic profile if $U_{j}$ is matched. Interestingly, the only condition in the experiments for which the flush pipe approached the trajectories of the nozzle-generated jets was for the absolutely unstable conditions (for the flush jets) at $J=5$ , where experiments have shown that the upstream jet momentum thickness for the flush nozzle is relatively large (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007) and where recent simulations (Mahesh & Iyer Reference Mahesh and Iyer2013) suggest that cross-flow fluid may travel into the flush jet orifice itself at such strong cross-flow conditions. It should be noted that the momentum flux was matched here using mass-averaged (and not momentum-averaged) velocity, and this also could have contributed to differences observed here. Figure 8(a,b) also indicates that, for equidensity higher momentum flux ratios ( $J=41$ and 12), the trajectory of the flush nozzle was similar to that of the elevated nozzle, suggesting a minimal influence of the cross-flow boundary layer on trajectory for flush injection in this regime. At the lower value of $J=5$ , the increased cross-flow velocity has been documented (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007) to create a greater degree of vertical co-flow upstream of the elevated nozzle, and this probably caused the slight difference between the flush and elevated nozzle trajectories. The implications of these observations for jet mixing and entrainment will be explored in § 4.2.

While the jet trajectories shown in figures 7 and 8 employed non-dimensionalization by diameter $D$ , alternative length scales can account for flow conditions, especially given that the high- $J$ transverse jet interacts with the cross-flow over a longer spatial distance ( $s$ or $s_{c}$ ) than does a low- $J$ transverse jet, certainly affecting jet–cross-flow interactions and the evolution of vortical structures (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014). Early experimental studies on the JICF (Keffer & Baines Reference Keffer and Baines1963) suggest that using the product of momentum flux ratio and jet diameter, $JD$ , to scale jet trajectory leads to successful data collapse in the range $36\leqslant J\leqslant 100$ , although later experiments and simulations (Broadwell & Breidenthal Reference Broadwell and Breidenthal1984; Smith & Mungal Reference Smith and Mungal1998; Hasselbrink Reference Hasselbrink1999; Muppidi & Mahesh Reference Muppidi and Mahesh2005) indicate improved data collapse with $RD$ or $\sqrt{J}D$ scaling. Yet $RD$ scaling appears to be relevant only for relatively high $R$ values; per Smith & Mungal (Reference Smith and Mungal1998), scaled trajectories for $R=5$ tend to differ from those for $R\geqslant 10$ . The computational studies of Muppidi & Mahesh (Reference Muppidi and Mahesh2005) show that transverse jets of equivalent $R$ but different cross-flow and jet velocity profiles have different trajectories; as noted earlier, jets with thinner boundary layers tend to penetrate less, and conditions where the cross-flow boundary layer thickness is smaller, due to increased cross-flow velocity, also have lower penetration. Interestingly, even when an additional length scale is introduced to represent these profile differences, the $RD$ -based trajectory collapse occurs for a higher velocity ratio, $R=5.7$ , but is much less successful for $R=1.52$ . Recent JICF scaling analysis by Forliti, Salazar & Bishop (Reference Forliti, Salazar and Bishop2015) suggests use of a scale factor $B$ which is a function of $J$ as well as empirical drag and entrainment coefficients, but this too focuses on somewhat higher values of $R$ ( ${\geqslant}4$ ).

Figure 9 shows the trajectories of equidensity transverse jets injected from all three injectors in the present studies for $41\geqslant J\geqslant 5$ , using different scaling of the physical coordinates via $D$ , $JD$ and $\sqrt{J}D$ . If any of these parameters were to capture the relevant physics associated with the jet trajectory (e.g. entrainment or spreading rate), trajectory data would collapse onto a single curve and the flow would be considered to be self-similar. It is clear from figure 9 that none of the length scales used to scale physical coordinates achieved significant data collapse in this rather broad range of $J$ values: $JD$ scaling seemed to produce the highest degree of data collapse, especially for the flush and elevated nozzle-injected cases in the range $41\leqslant J\leqslant 8$ ; $J=5$ data also appeared to collapse reasonably well for the flush nozzle with $JD$ scaling, but the $J=5$ elevated nozzle-injected jet penetrated more deeply into the cross-flow in $JD$ -scaled space as compared with the other cases. The elevated JICF at this low- $J$ condition corresponds to very weak shear layer instabilities, in contrast to absolute instabilities for the flush injectors. The degree of data collapse with $JD$ scaling for the flush pipe-based trajectories was not as favourable in this flow regime as it was for other injectors, consistent with trajectory trends shown in figure 8.

Figure 9.

Scalar centreline trajectories of $S=1.00$ transverse jets for various momentum flux ratios, for various injectors: (a,d,g) flush nozzle, (b,e,h) elevated nozzle and (c,f,i) flush straight pipe. Data are shown for various physical coordinate scalings: $D$ , $JD$ and $\sqrt{J}D$ , in (a–c), (d–f) and (g–i), respectively.

Figure 10.

Maximum mean concentration ( $C_{m}/C_{o}$ ) decay trends along the scalar centreline trajectory coordinate $s_{c}$ for equidensity transverse jets at various momentum flux ratios, for various injectors: (a,d,g) flush nozzle, (b,e,h) elevated nozzle and (c,f,i) flush straight pipe. Data are shown for various physical coordinate scalings: $D$ , $JD$ and $\sqrt{J}D$ in (a–c), (d–f) and (g–i), respectively.

4.2. Maximum mean concentration decay

The decay in the maximum mean scalar concentration $C_{m}$ along the jet centreline has been commonly used to characterize mixing for transverse jets (Smith & Mungal Reference Smith and Mungal1998); the centreplane-based acetone PLIF images in the present study may be used to extract such data. Figure 10 highlights the maximum mean concentration decay trends in $C_{m}/C_{o}$ along the scaled centreplane trajectory $s_{c}/D$ for equidensity transverse jets injected from all three injectors. In figure 10, the trajectory coordinate $s_{c}$ is used rather than the downstream distance $x$ , given the ability for the former to represent local exposure of jet fluid to cross-flow, and hence to represent the evolution of molecular mixing more effectively. More direct comparisons of maximum centreline concentration decay, $C_{m}/C_{o}$ , are also more straightforward among different flow conditions with $s_{c}$ -based quantification. Mixing metric evolution with downstream distance $x$ will be examined in a later section. Figure 10 displays scaling of distance $s_{c}$ by $D$ , $JD$ and $\sqrt{J}D$ . Simple plotting of centreline concentration decay as a function of $s_{c}/D$ , in figure 10(a–c), demonstrated the effects of injector and $J$ values on the length of the potential core, defined as the locus at which the concentration $C/C_{o}$ started to drop below unity. On the other hand, scaling of downstream decay with $s_{c}/\sqrt{J}D$ , shown in figure 10(g–i), enabled effective matching of potential cores for flush nozzle and flush pipe injectors for virtually all flow conditions, and thus a more straightforward comparison of concentration decay slopes downstream of the potential core. This centreline concentration decay scaling with $\sqrt{J}D$ is also observed for the flush nozzle-injected JICF at higher momentum flux ratios, $25\leqslant J\leqslant 100$ (Smith & Mungal Reference Smith and Mungal1998), to enable potential core lengths to be matched. The present studies showed that none of the scaling shown in figure 10 enabled collapse of potential core lengths for the elevated nozzle, however, consistent with the influence of vertical co-flow at higher cross-flow velocities. Scaling of centreline concentration decay with $JD$ (figure 10 d–f) did not allow collapse of the potential cores for any injector.

Reference lines shown in figure 10 represent concentration decay as various power laws: $s_{c}^{-1.3}$ , observed by Smith & Mungal (Reference Smith and Mungal1998) for transverse jets at high $J$ in the near field, prior to the aforementioned ‘branch’ point; $s_{c}^{-2/3}$ , observed by these researchers for decay after the branch point; and $s_{c}^{-1}$ , representing concentration decay for the free jet as well as the straight pipe-generated JICF at high $J$ (Su & Mungal Reference Su and Mungal2004). The present results suggested differing decay scaling for the various injectors and flow conditions. For the flush nozzle at $J=41$ and especially at $J=30$ , there was a greater near-field rate of concentration decay than even $s_{c}^{-1.3}$ scaling, after which there appeared to be a branch point, and then concentration decay at a lower rate, closer to $s_{c}^{-2/3}$ scaling. It is noted that both $J=41$ and $J=30$ convectively unstable transverse jets created by the flush nozzle had clearly asymmetric cross-sections (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014; Gevorkyan Reference Gevorkyan2015). Smith & Mungal (Reference Smith and Mungal1998) observe similar branch points for $J>25$ , attributing them to the transverse jet’s shifting from jet-like behaviour in the near field to wake-like behaviour in the far field (Hasselbrink & Mungal Reference Hasselbrink and Mungal2001). Smith & Mungal (Reference Smith and Mungal1998) find the ‘branch point’ to occur at $s_{c}/JD=0.3$ , very close to the branch point observed in the present studies for $J=41$ and $J=30$ ( $s_{c}/JD\approx 0.33$ ) as shown in the $JD$ -scaled flush nozzle-injected concentration decay trends in figure 10. There were no other injectors or equidensity jet conditions for which a branch point was observed here; studies by Su & Mungal (Reference Su and Mungal2004) for flush pipe- and elevated pipe-injected transverse jets similarly show a lack of branch point behaviour at high momentum flux ratios.

There was hence a ‘crossing over’ of decay lines for varying $J$ in the flush nozzle-injected transverse jet $C_{m}/C_{o}$ decay trends in the present experiments, more apparent in the plot of $C_{m}/C_{o}$ as a function of $s_{c}/D$ in figure 10(a). This ‘crossing over’ is also observed by Smith & Mungal (Reference Smith and Mungal1998) at higher $J$ values, and is postulated to be the physical mechanism behind a minimum in flame length observed to occur at a specific value of $R$ in reactive JICF experiments (with minima produced for flows in the range $10<R<20$ , depending on the chemical reaction). One could argue from the present results that, for a (relatively low) critical value of $C_{m}/C_{o}$ associated with a well-mixed jet, $J=30$ might provide a minimum in flame length as compared with other flow conditions, although to our knowledge there are no reactive experiments showing critical minimum flame lengths in this regime for  $J$ .

At lower $J$ values for the flush nozzle, the JICF concentration decay rate scaled between $s_{c}^{-1.3}$ and $s_{c}^{-1}$ , with absolutely unstable shear layer conditions ( $J=8$ and 5) being closer to the $s_{c}^{-1}$ scaling. While this could suggest that absolutely unstable transverse jets with lower $J$ mixed at a slower rate after potential core breakdown than for the higher- $J$ (convectively unstable) jets, one must recognize that centreplane slices of the JICF do not usually capture jet fluid that is concentrated in the cores of the CVP, and thus neglect important information concerning stirring and mixing. The elevated nozzle data in figure 10 also showed decay rates lying between $s_{c}^{-1.3}$ and $s_{c}^{-1}$ , although the majority of cases were closer to the latter, consistent with weakly convective shear layers similar to those for equidensity free jets. The flush pipe data, in contrast, showed higher overall scalar decay rates than did the flush nozzle, with decay scaling lying between approximately $s_{c}^{-1.6}$ and $s_{c}^{-1.3}$ , and with the strongest decay associated with higher $J$ values and relatively weak shear layer instability. As noted in Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014), for the flush pipe there are relatively symmetric CVPs observed at lower $J$ values, but at high $J$ values the jet cross-sections were generally asymmetric without the distinctive CVP structure. Cross-section-based mixing statistics should reveal the significance of the CVP’s absence, presence or asymmetries to the centreline decay data shown in figure 10; these will be explored in § 4.3.

Figure 11.

Maximum mean concentration decay as a function of centreline trajectory coordinate $s_{c}/D$ for flush nozzle-injected transverse jets at various density ratios $S$ , for (a)  $J=41$ , (b)  $J=12$ and (c)  $J=5$ .

Figure 11 shows the maximum mean concentration decay trends for flush nozzle-injected transverse jets for fixed values of density ratio $S=1.0$ , 0.55 and 0.35, and for fixed $J=41$ , $J=12$ and $J=5$ . Since $J$ is a constant in each of the plots, simple $s_{c}/D$ scaling is employed. As noted previously, all $J=5$ data and all $S=0.35$ data correspond to absolutely unstable conditions; the other conditions shown correspond either to relatively weak convectively unstable shear layers or, for $J=12$ , to situations where the shear layer instability is in the process of strengthening and approaching absolute instability (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012). Interestingly, for both $J=41$ and $J=12$ , potential core lengths differed for the various density ratios, with $S=0.35$ producing the shortest core lengths and with a more gradual transition to concentration decay, consistent with immediate vorticity rollup resulting from an absolutely unstable upstream shear layer (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014). The $J=5$ jets appeared to have potential core lengths that were independent of density ratio, all with a gradual rather than an abrupt concentration decay, again probably related to the fact that all cases in figure 11(c) represented absolutely unstable shear layers with rapid vorticity rollup. The equidensity jets for $J=12$ and $J=5$ actually appeared to create a higher rate of concentration decay beyond the potential core than for the lower density ratios, but the opposite trend occurred for $J=41$ , where lowering the density ratio to $S=0.35$ increased the rate of decay in the near field, and with the presence of a branch point at $s_{c}/JD\approx 0.36$ . It should also be noted that no observable branch point was measured in the rate of decay for the $S=0.55$ , $J=41$ jet. These and additional results (Gevorkyan Reference Gevorkyan2015) suggested that, when the transverse jet’s momentum flux ratio was relatively high, e.g.  $J=41$ , reducing the density ratio and creating an absolutely unstable shear layer appeared to improve mixing, that is, it increased the rate of centreline concentration decay. If one considers that a high momentum flux ratio jet behaves similarly to a free jet in the potential core region, and the fact that lowering the density ratio is observed in recent JICF experiments (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014) to create stronger and more coherent periodic shear layer vorticity rollup, as with free jets (Jendoubi & Strykowski Reference Jendoubi and Strykowski1994; Srinivasan, Hallberg & Strykowski Reference Srinivasan, Hallberg and Strykowski2010), an improvement in mixing with lowered density ratio, as seen in figure 11(a), may not be unexpected in the near field. On the other hand, when the momentum flux ratio of the transverse jet was relatively low (e.g.  $J=5$ ), with absolutely unstable shear layers irrespective of density ratio, reducing $S$ actually was observed to reduce the rate of concentration decay and possibly worsen mixing. This issue will be explored in the context of other mixing metrics as well as the nature of shear layer vorticity in subsequent sections.

Figure 12.

Maximum mean concentration decay as a function of scalar centreline trajectory coordinate $s_{c}/D$ for equidensity transverse jets injected from the flush nozzle, elevated nozzle and flush straight pipe injector configurations, for (a)  $J=41$ , (b)  $J=12$ and (c)  $J=5$ .

Figure 12 highlights the explicit effect of different injector configurations on concentration decay for equidensity jets with fixed values of $J=41$ , $J=12$ and $J=5$ . For the high momentum flux ratio, $J=41$ case (figure 12 a), the elevated nozzle- and flush nozzle-injected jets had similar concentration decay trends up to $s_{c}/D\approx 7$ , with similar potential core lengths and near-field concentration decay rates between these two cases. This would be expected based on the similar trajectories (figure 8) and shear layer stability characteristics (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007; Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014) for these nozzles at $J=41$ . Interestingly, neither the elevated nozzle nor the flush pipe exhibited branch point behaviour and the subsequent reduction in decay rate found in the flush nozzle data, consistent with the pipe injection studies by Su & Mungal (Reference Su and Mungal2004). The pipe flow-injected $J=41$ JICF in fact had a somewhat longer potential core than the nozzle flows, consistent with much weaker upstream shear layer instabilities in this regime (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014), but ultimately the jet had a much higher concentration decay rate downstream of the potential core. This trend in the pipe flow data appeared to be present for the lower momentum flux ratio cases as well (figure 12 b,c). For all momentum flux ratios, particularly at $J=12$ and 5, the elevated nozzle had the weakest rate of concentration decay of all the injectors, with scaling similar to that of the free jet, proportional to $s_{c}^{-1}$ . This scaling was consistent with the elevated jet’s significantly weakened shear layer instabilities due to upstream co-flow at higher cross-flow magnitudes (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007; Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014).

The present observations on mean centreline concentration decay, based on centreplane PLIF imaging, did not provide a clear correlation with previously determined transverse jet shear layer instabilities or structural characteristics (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014). Figure 10 indicates that the strongest rate of maximum concentration decay tended to occur for high $J$ values and convectively unstable shear layers, especially for flush injection where the jet cross-section has been shown to be asymmetric. Yet in figure 11, data for $J=41$ suggested improved concentration decay rates could be attained by lowering the density ratio and hence approaching absolutely unstable conditions which create a more rapid shear layer vortex rollup and, correspondingly, a symmetric CVP in the cross-section. While the nature of entrainment of cross-flow by a jet is clearly dependent on density differences and the formation and evolution of shear layer vortical structures, understanding the precise correlations with mixing requires exploration of additional metrics.

4.3. Cross-section-based mixing metrics

Both cross-sectional ( $y{-}z$ plane) and centreplane ( $x{-}z$ plane) based unmixedness $U$ may be evaluated from PLIF data scalable to local jet fluid concentration; the same may be said for the averaged spatial p.d.f. metric. For cross-sectional statistics, matching of the mean scaled concentration $\overline{C}/C_{o}$ over the $y{-}z$ imaging plane at a given $x$ location is required to enable comparisons among different operating conditions, as described in § 3. The question of which reference mean value to use for these cross-sectional imaging statistics was explored in detail in the present studies. Two extremes in the selection of the reference $\overline{C}/C_{o}$ , for example, could involve the highest ( $J=41$ ) and lowest ( $J=2$ ) momentum flux ratio conditions at the farthest downstream cross-sectional location analysed here, $x/D=10.5$ . Figure 13 shows the cross-section-based unmixedness evaluated over the $y{-}z$ cross-sectional plane, $U_{yz}$ , for the equidensity, flush nozzle-injected transverse jet, where cross-sectional images acquired at three distinct $x$ locations were analysed and then plotted as a function of the spatial coordinate $s_{c}/D$ , per trajectory data such as those in figure 9. The cross-section-based $U_{yz}$ was quantified in two different ways in figure 13. One (figure 13 a) involved matching each instantaneous image’s spatial mean at the given flow condition to a reference mean value corresponding to $J=2$ at $x/D=10.5$ . Figure 13(b) shows the same information for $U_{yz}$ , but with the reference mean value selected to correspond to $J=41$ at $x/D=10.5$ . Such $\overline{C}/C_{o}$ matching was achieved by adding or removing zero-valued pixels to the chosen interrogation area $A_{i}$ containing mixed jet fluid. Very little sensitivity of $U_{yz}$ values was observed for these two extremes in the mean reference value, despite the fact that the high momentum flux ratio jet had still not turned completely to the horizontal by $x/D=10.5$ . Similar observations were made for cross-section-based p.d.f. data (Gevorkyan Reference Gevorkyan2015). All subsequent cross-section-based unmixedness and p.d.f. calculations were thus chosen to employ the mean value of the equidensity, flush nozzle-injected $J=2$ JICF at $x/D=10.5$ as the reference value for $\overline{C}/C_{o}$ ; the interrogation areas were adjusted accordingly to match that reference mean.

Figure 13.

Cross-sectional (constant $x/D$ ) based unmixedness, $U_{yz}$ , for equidensity flush nozzle-injected transverse jets, plotted against the scalar centreline coordinate $s_{c}/D$ . The value of $U_{yz}$ was calculated by matching each instantaneous image’s spatial mean to the spatial mean value of the temporal mean image at $x/D=10.5$ for two different flow conditions: (a)  $J=2$ and (b)  $J=41$ . Results are shown for various momentum flux ratios.

As noted in § 3, the averaged spatial p.d.f.s of jet cross-sectional scalar concentration data can determine local mixing efficiency (Shan & Dimotakis Reference Shan and Dimotakis2006). Cross-section-based spatial p.d.f.s at three different downstream locations ( $x/D=2.5$ , 5.5 and 10.5) for the equidensity flush nozzle-injected JICF are shown in figure 14(a–c). Cross-sectional p.d.f.s (and unmixedness at the same three values of $x/D$ , shown in figure 14 d) were calculated by matching each instantaneous image spatial mean to the spatial mean value of $\overline{C}/C_{o}$ as described above. Increased levels of mixing typically result in the appearance of a preferred (most probable) value of the scalar concentration, and an evolution of that preferred value towards the mean. Note that here and in subsequent p.d.f. datasets, p.d.f. data point markers do not represent the actual resolution of data, which is much finer than shown. Unsurprisingly, the p.d.f. data for each $J$ value in figure 14(a–c) suggested improved mixing with downstream distance $x$ , where, especially for lower $J$ values (5 and 2), there was an increase in the magnitude of the p.d.f. and a reduction in the concentration $C/C_{o}$ associated with that peak as downstream distance $x$ increased. These trends were similarly observed at all $J$ values, but the spatial variation in distributions at higher $J$ were not as significant as for lower $J$ . These overall trends for spatial variation in mixing with $J$ variation were also observed in the constant $x/D$ based cross-sectional unmixedness, $U_{yz}$ , plotted as a function of $x/D$ for all three downstream locations in figure 14(d). As with trends in the p.d.f., the degree of reduction in $U_{yz}$ with increasing $x/D$ was more significant for the lower $J$ values than for the larger $J$ values. Yet for both p.d.f. and $U_{yz}$ cross-section-based metrics plotted as a function of $x$ (and not distance along the jet trajectory $s_{c}$ ), the relatively high $J=20$ equidensity flush nozzle JICF provided the ‘best’ local degree of mixing (i.e. the largest peak in the p.d.f. or the lowest value of $U_{yz}$ at a given $x/D$ location) as compared with other $J$ conditions. This $J=20$ condition is documented in Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014) to have a somewhat asymmetric cross-sectional CVP structure, with unequal distributions of jet fluid between the lobes of the CVP, although the degree of asymmetry is not as severe as for $J=41$ for this injector, which appeared in figure 14 not to mix as well. The $J=20$ condition was not observed to produce the most rapid centreline concentration decay when plotted against $s_{c}/D$ , however, per figure 10. Differences in interpretation of mixing data based on parametric variation will be expanded upon below.

Figure 14.

Cross-sectional (constant $x/D$ ) based spatial p.d.f. and unmixedness $U_{yz}$ for $S=1.00$ transverse jets injected from the flush nozzle injector: (a) p.d.f., $x/D=2.5$ ; (b) p.d.f., $x/D=5.5$ ; (c) p.d.f., $x/D=10.5$ ; and (d) unmixedness $U_{yz}$ . Data are shown for various momentum flux ratios.

Figure 15(a–c) provides additional p.d.f. data from cross-sectional slices at the fixed downstream location $x/D=10.5$ for $J=41$ , 12 and 5, respectively, and for different density ratio values ( $1.0\geqslant S\geqslant 0.35$ ). Figure 15(d) provides comparative data, also extracted from cross-sectional imaging, for the constant $x$ based $U_{yz}$ as a function of density ratio $S$ . There was again considerable consistency between the p.d.f. and $U_{yz}$ metrics computed on the basis of a fixed downstream distance $x/D$ . Both metrics suggested that, irrespective of momentum flux ratio, at constant downstream position $x/D=10.5$ , reducing the jet density relative to the cross-flow at constant $Re_{j}$ actually worsened mixing at a given downstream location $x$ , especially at the lower $J$ values. Mixing (quantified via p.d.f. or $U_{yz}$ ) for the $J=41$ JICF did not change as significantly with a variation in $S$ , although, when compared with other density ratios, the equidensity JICF overall seemed to produce slightly better mixing (i.e. a sharper and lower peak in the p.d.f. and a lower overall $U_{yz}$ ) than for low-density jets.

Figure 15.

Cross-sectional (constant $x/D=10.5$ ) based metrics: (a–c) spatial p.d.f. and (d) unmixedness $U_{yz}$ for flush nozzle-injected transverse jets with density ratios $S=1.00$ , $S=0.55$ and $S=0.35$ . Data are shown for various momentum flux ratios: (a) p.d.f., $J=41$ ; (b) p.d.f., $J=12$ ; (c) p.d.f., $J=5$ ; and (d)  $U_{yz}$ versus $S$ for various $J$ values.

Yet variation in cross-section-based mixing metrics with scaled downstream distance $x/D$ showed different trends when compared with variation in these metrics along the jet centreline, $s_{c}/D$ . Because $J$ varies significantly among the different cases explored, the value of the centreline distance $s_{c}$ can be quite different for the same downstream location $x$ . For direct comparison, note that figure 13(a) shows the cross-section-based (constant $x/D$ ) unmixedness $U_{yz}$ for the equidensity, flush nozzle-injected JICF plotted against the scalar centreline trajectory coordinate $s_{c}/D$ , while figure 14(d) shows the same parameter plotted as a function of the $x/D$ position at which the cross-sectional imaging took place, all with variable $J$ values. Note that data for $J=2$ in figure 14(d) are not plotted in figure 13(a) because of the inadequacy of the previously noted trajectory fits at this very low momentum flux ratio. While the values of $U_{yz}$ in each plot otherwise correspond to the same datasets, the fact that $s_{c}$ was considerably larger than $x$ at the higher momentum flux ratio conditions, due to much greater jet penetration, completely altered the comparative spatial evolution of the $U_{yz}$ metric (and, similarly, p.d.f.s).

These differences in results based on spatial scaling raise a number of questions pertaining to which equidensity JICF flow conditions (e.g.  $J$ ) produce ‘better mixing’. Knowledge of the degree of jet cross-sectional mixing at a specified location $x$ downstream of injection could be useful in many practical applications, as is knowing the momentum flux ratio $J$ required to achieve a desired level of mixing at a specific downstream location. On the other hand, a fundamental understanding of molecular mixing efficiency by the transverse jet requires mixing quantification as the jet fluid is spatially exposed to cross-flow, particularly when one considers a comparison of molecular mixing by the jet in cross-flow with that for the free jet, where distance along the jet centreline or jet shear layer represents spatial evolution (Smith & Mungal Reference Smith and Mungal1998). This issue pertains to jet cross-section-based metrics as well as to centreplane-based metrics. Given the interest in a fundamental understanding of molecular mixing for the transverse jet as compared with other canonical jet flows, variation of mixing metrics with distance along the jet, $s_{c}/D$ , will be examined in the remainder of this paper.

4.4. Comparison of centreplane- and cross-section-based unmixedness

For jet centreplane ( $x{-}z$ ) based mixing statistics, matching of the mean scaled concentration $\overline{C}/C_{o}$ over the relevant interrogation area was similarly required to enable comparisons among different operating conditions, as discussed for cross-sectional mixing metrics in §§ 3 and 4.3. Concentration values and associated local mixing metrics could be evaluated over narrow slices of the jet oriented in the jet-normal direction $n$ , with a fixed interrogation area width in the $s_{c}$ direction, as shown in figure 6(b). For such an interrogation area, the local centreplane-based unmixedness is designated $U_{c,sn}$ . Alternatively, one could explore centreplane-based mixing metrics such as unmixedness evaluated over comparably narrow vertical slices of the jet, but with a fixed width in the $x$ direction and extending vertically in the $z$ direction; this centreplane-based unmixedness is designated $U_{c,xz}$ . For a very thin interrogation area, $U_{c,xz}$ could be compared with the cross-section-based unmixedness $U_{yz}$ obtained at the same average downstream location $x$ . Recall that cross-section-based metrics were calculated by matching each instantaneous image’s spatial mean to the spatial mean value, $\overline{C}/C_{o}$ , for the temporally averaged cross-sectional image at $x/D=10.5$ for the flush nozzle-injected, equidensity, $J=2$ case. Centreplane-based metrics were calculated by matching each interrogation area’s spatial mean to the spatial mean of the temporally averaged concentration at the location $s_{c}/D=15$ corresponding to the flush nozzle-injected, equidensity, $J=5$ case. This condition was chosen because trajectory quantification was more difficult for $J=2$ . As with cross-section-based metrics, the centreplane-based metrics did not change significantly with alterations in the selected matched mean value of $\overline{C}/C_{o}$ for different operating conditions (Gevorkyan Reference Gevorkyan2015).

Figure 16(a,b) displays centreplane-based unmixedness variation along the jet trajectory, for the equidensity transverse jet injected from the flush nozzle at a range of momentum flux ratios. Here one can compare trends in local centreplane $U_{c,xz}$ based on a thin (seven-pixel-wide) interrogation area oriented in the $x{-}z$ direction (figure 16 a) and local centreplane unmixedness $U_{c,sn}$ based on a seven-pixel-wide interrogation area oriented in the $s_{c}{-}n$ direction (figure 16 b). The former may be compared with cross-sectional unmixedness $U_{yz}$ measured at three different downstream locations, shown in figure 13(a). All data here are plotted as a function of $s_{c}/D$ . Interestingly, the two centreplane-based metrics in figure 16(a,b) showed few differences in the evolution of mixing, the exception being at $J=30$ , with a somewhat greater degree of unmixedness reduction in $U_{c,sn}$ as compared with $U_{c,xz}$ . Figure 16(a,b) also shows very similar trends to the variation in cross-section-based unmixedness, $U_{yz}$ , in figure 13(a). Note that the $J=30$ case produced such severely asymmetric mean jet cross-sections for the flush nozzle that it was not possible to accurately scale the cross-sectional concentration data with respect to the centreplane concentration data. Hence no comparable quantitative cross-section-based data for this case are shown in figure 13, for example. All three plots (figures 13 a and 16 a,b) demonstrated that, generally speaking, lower local momentum flux ratio transverse jets produced lower centreplane-based and cross-section-based values of unmixedness, at least until around 20 diameters along the jet trajectory. The rate of decay in unmixedness did appear to be greater for $J=30$ and 20 for the near-field region as compared with other conditions. Nevertheless, jets whose upstream shear layers have been documented to be absolutely unstable (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007), with $J=8$ and 5, appeared to have the best overall mixing characteristics, although this advantage diminished further along the jet, near $s_{c}/D=20$ . Data for $J=41$ appeared to have the worst overall mixing characteristics, yet similar to the $J=30$ JICF, there was significant asymmetry in the $J=41$ jet’s mean cross-sectional shape (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014), which probably affected these trends.

Figure 16.

For the flush nozzle-injected, equidensity JICF, (a) centreplane unmixedness $U_{c,xz}$ based on a thin interrogation area oriented in the $x{-}z$ direction and (b) centreplane unmixedness $U_{c,sn}$ based on a thin interrogation area oriented in the $s_{c}{-}n$ direction. Each is plotted as a function of $s_{c}/D$ for various momentum flux ratios  $J$ .

There were similar trends in the comparison between centreplane-based unmixedness $U_{c,xz}$ and cross-section-based $U_{yz}$ for other injectors, as well as for a given injector at reduced density ratios. Such comparisons are most logical given that the laser sheet for cross-sectional mixing metrics always lies in the $y{-}z$ plane. For example, figure 17 shows constant $x/D$ based centreline unmixedness $U_{c,xz}$ (a,b) as well as cross-section-based $U_{yz}$ (c,d) as a function of distance along the jet trajectory, for the elevated nozzle (a,c) and flush pipe (b,d). Data here correspond to the equidensity JICF. As with the flush nozzle (figures 16 a and 13 a), the effects of momentum flux ratio on spatial variation in $U_{c,xz}$ were very similar to those for spatial variation in $U_{yz}$ . At a given scaled distance along the jet trajectory, $s_{c}/D$ , higher momentum flux ratio jets ( $J\geqslant 20$ ) tended to mix less well than did the JICF at lower $J$ , even when these lower- $J$ conditions did not produce an absolutely unstable shear layer, as was the case for the elevated nozzle. When the upstream shear layer did become absolutely unstable, however, as was the case for the flush nozzle and flush pipe for $J\lesssim 12$ , the actual values of unmixedness at a given location $s_{c}/D$ tended to be lower than for the more weakly unstable elevated jets at the same flow conditions. Datasets missing in these plots, e.g.  $J=30$ and $J=41$ for the flush pipe’s cross-sectional statistics, were the result of cross-sectional asymmetries that prohibited an accurate determination of local jet concentration.

Figure 17.

Centreplane unmixedness $U_{c,xz}$ based on a thin interrogation area oriented in the $x{-}z$ direction (a,b) and cross-sectional unmixedness $U_{yz}$ (c,d), both plotted as a function of $s_{c}/D$ for equidensity transverse jets of various momentum flux ratios. Shown are data for (a,c) the elevated nozzle and (b,d) the flush straight pipe injector.

Figure 18.

Centreplane unmixedness $U_{c,sn}$ plotted as a function of $s_{c}/D$ on a log–log plot. Data are shown for the equidensity JICF for various momentum flux ratios $J$ , injected from (a) the flush nozzle, (b) the elevated nozzle and (c) the flush pipe.

Trends in local centreplane-based unmixedness $U_{c,sn}$ , extracted from a thin interrogation area oriented in the $s_{c}{-}n$ direction, may be compared with the more traditional maximum centreline concentration decay metric shown in figures 10–12. When similarly plotted as a function of $s_{c}/D$ and represented on a log–log plot, centreplane-based unmixedness $U_{c,sn}$ was generally observed to decay with the same trends as for concentration decay. These unmixedness trends are shown in figures 18–20, which may be compared with figures 10(a–c), 11 and 12, respectively. The general trends in the effects of $J$ , $S$ and injector type on mixing, as represented by both relative metric values and by the spatial rate of decay of the metric, were very similar between $C_{m}/C_{o}$ and $U_{c,sn}$ , although with a few important differences. For example, for the equidensity, flush-nozzle-injected JICF with variable  $J$ , as examined in both figures 18(a) and 10(a), the lowest centreline concentration and unmixedness cases at a given location $s_{c}/D$ corresponded to those for low $J$ and, in general, an absolutely unstable upstream shear layer. When the rate of decay of the metric beyond the potential core is considered, however, both $J=30$ and 20 tended to produce the largest reduction, seen in both metrics. Similar correspondence in spatial evolution between $C_{m}/C_{o}$ and $U_{c,sn}$ was observed when density ratio $S$ was reduced for the $J=41$ flush nozzle, as shown in figures 19(a) and 11(a), and for a worsening in mixing for the $J=5$ flush nozzle, shown in figures 19(c) and 11(c). The effects of injector configuration on mixing metrics quantified in figures 12 and 20 had similar correspondence with one another. One striking difference between the $C_{m}/C_{o}$ and $U_{c,sn}$ metrics, however, was the much less sharp delineation of the potential core and subsequent drop-off in $U_{c,sn}$ as compared with $C_{m}/C_{o}$ , particularly under conditions corresponding to absolutely unstable shear layers. This difference clearly resulted from the nature of a centreline-based metric, for which only potential core fluid is quantified immediately after injection, as compared with a metric such as $U_{c,sn}$ (or other centreplane-based statistical metrics), which accounts for the distribution of the scalar field across the span of the jet, thus including the effects of near-field shear layer distortion more explicitly.

Figure 19.

Centreplane unmixedness $U_{c,sn}$ plotted as a function of $s_{c}/D$ on a log–log plot. Data are shown for flush nozzle-injected transverse jets of various density ratios $S$ , with momentum flux ratios (a)  $J=41$ , (b)  $J=12$ and (c)  $J=5$ .

Figure 20.

Centreplane unmixedness $U_{c,sn}$ plotted as a function of $s_{c}/D$ on a log–log plot. Data are shown for the equidensity JICF with the flush nozzle, elevated nozzle and flush straight pipe injectors, for momentum flux ratios (a)  $J=41$ , (b)  $J=12$ and (c)  $J=5$ .

4.5. Interpretation of mixing trends

Finally, it is of interest to explore the possible physical phenomena behind some of the mixing trends observed here. For low momentum flux ratios $J$ , as noted, reducing the density ratio $S$ actually worsened molecular mixing, e.g. for $J=5$ in figures 11(c), 15(c,d) and 19(c). In contrast, at higher $J$ values, reducing the density ratio from $S=1.0$ to $S=0.35$ appeared to improve mixing, as suggested for $J=41$ in figures 11(a) and 19(a), or at least not to alter the mixing very significantly, as suggested in figure 15(a,d). Data for transitional momentum flux ratios, e.g. for $J=12$ in figures 11(b), 15(b,d) and 19(b), appeared to be more similar to $J=5$ and other low momentum flux ratio cases. The upstream shear layer for $J=5$ conditions has been documented to be absolutely unstable for all density ratios, while only the $S=0.35$ conditions clearly produce absolutely unstable shear layers at $J=41$ and 12, although the latter case involved much stronger convective instabilities than at higher $J$ (Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014). These differences in the instabilities and the implications for vorticity rollup may be observed in high-resolution instantaneous centreplane images shown in figure 21 for this range of $J$ values, for equidensity and low-density ( $S=0.35$ ) conditions. As documented in Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014), all absolutely unstable shear layer conditions produced immediate or near-immediate upstream vorticity rollup; for convectively unstable conditions, the initial rollup is weaker and delayed to a location further along the shear layer. For the equidensity JICF, however, shear layer rollup appeared to draw in considerably more cross-flow fluid (of a lower scalar concentration) into the vortical structures than did the low-density JICF, irrespective of $J$ value. The $S=0.35$ images all showed shear layer vortices comprised largely of low-density jet fluid, and this clearly influenced mixing parameters when the span of the jet (either in the jet-normal direction $n$ or in the cross-sectional plane, $y{-}z$ ) was used in mixing metric quantification. Even the centreline-based metric in figure 11 captured this trend, because of the influence of the shear layer vortices on the potential core. The reduction in entrainment of higher-density fluid into a lower-density vortical structure has been seen in a number of prior reactive shear layer studies, both experimental (Hermanson, Mungal & Dimotakis Reference Hermanson, Mungal and Dimotakis1987; Hasselbrink & Mungal Reference Hasselbrink and Mungal2001) and computational (Karagozian & Marble Reference Karagozian and Marble1986).

Figure 21.

Instantaneous high-resolution centreplane (side view) acetone PLIF imaging of the flush nozzle-generated JICF with: (a)  $J=41$ and density ratio $S=1.0$ ; (b)  $J=41$ and density ratio $S=0.35$ ; (c)  $J=12$ and density ratio $S=1.0$ ; (d)  $J=12$ and density ratio $S=0.35$ ; (e)  $J=5$ and density ratio $S=1.0$ ; and (f)  $J=5$ and density ratio $S=0.35$ .

Figure 21 further demonstrates that, for a relatively weak convectively unstable shear layer, as for the equidensity JICF with $J=41$ in figure 21(a), accelerating vorticity rollup by creating an absolutely unstable shear layer (figure 21 b) did appear to improve mixing. But when the upstream shear layer instability was already relatively strong and closer to being absolutely unstable (as for the equidensity $J=12$ condition in figure 21 c), reducing the jet density and creating the absolute instability did not have a significant benefit for mixing, probably due to the reduced local entrainment of cross-flow observed in figure 21(d). It is also noted that the sizes of the vortical structures in figure 21 were not significantly altered by the alteration in density ratio, consistent with the observations of Brown & Roshko (Reference Brown and Roshko1974) on the relatively weak effect of density ratio alone in actual shear layer growth. Yet the entrainment of higher-density fluid from the cross-flow into the lower-density vortical structures did appear to affect near-field and eventually far-field mixing quantification.

Differences in mixing metrics between the flush nozzle- and flush pipe-injected equidensity JICF (e.g. in figures 12, 16 a, 17 b,d and 20), especially in the near field of the jet, may also be explained based on instantaneous centreplane images. Figure 22(a–c) corresponds to instantaneous PLIF images for the equidensity flush pipe at $J$ values of 41, 12 and 5, respectively, and may be compared with flush nozzle-generated images in figure 21(a,c,e). Additional centreplane (and cross-sectional) images comparing equidensity flush nozzle and pipe flows may be found in Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014). While the weaker shear layer instabilities for the flush pipe at higher $J$ were manifested in delayed and larger, less coherent vortices in the upstream shear layer as compared with those for the flush nozzle, the greater distortion of jet fluid upon exit into the cross-flow when momentum flux ratio was reduced, especially at $J=5$ , did appear to affect jet mixing by the pipe injector, especially in the near field. These mixing differences between flush pipe and nozzle configurations did seem to switch beyond the first few diameters along the $J=5$ jet trajectory, however (e.g. as seen in figure 20 c), so trends in absolutely unstable JICF conditions for different injectors also did not follow more straightforward trends seen for convectively unstable jet conditions.

Figure 22.

Instantaneous high-resolution centreplane (side view) acetone PLIF imaging of the flush pipe-generated, equidensity JICF with: (a)  $J=41$ , (b)  $J=12$ and (c)  $J=5$ .