3.3.1 Shear layer determination

In order to compare the velocity and scalar gradient quantities utilizing simultaneous acetone PLIF and stereo PIV measurements, a method for determining the location of the dominant upstream and downstream shear layers between jet and cross-flow fluid was developed. Utilizing the mean jet trajectory based on concentration maxima from the PLIF data, a transformation from the $x{-}z$ coordinate system (shown in figure 1) to the $s_{c}-n$ (centreline jet normal) coordinate system was implemented, similar to the method utilized for JICF mixing studies (see Gevorkyan et al. (Reference Gevorkyan, Shoji, Getsinger, Smith and Karagozian2016) for details on the transformation method). After the transformation was implemented, the shear layer location was determined as the locus with the maximum scalar dissipation rate along a ray normal to each $s_{c}/D$ position. This analysis yielded the local shear layer coordinate $s_{l}$ and its local layer-normal direction $n_{l}$ . An average shear layer trajectory coordinate $s$ , used in prior JICF instability studies (Megerian et al. Reference Megerian, Davitian, de B. Alves and Karagozian2007; Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010a ) is shown in figure 1. A similar method was utilized to track the downstream (lee-side) shear layer location. All average scalar dissipation rates and strain rates shown in this study were calculated at the instantaneous shear/mixing layer location determined from this method, and then averaged over the set of images for the coordinate location  $s_{c}$ .

3.3.2 Maximum scalar dissipation rate and minimum principal strain rate

As noted previously, scalar mixing structures in turbulent flows tend to exhibit a layer-like topology, with the scalar gradient direction aligned with the minimum principal (compressive) strain axis. Invoking this approximation of the underlying physics (Peters Reference Peters1986) for mixing layers in the present JICF study, one can compare scalar dissipation rates calculated from the PLIF measurements to the minimum principal strain rates calculated from the simultaneous PIV data at the same location to see if there were correlations among the trends. This comparison was administered for both the upstream and downstream mixing layers of equidensity flush nozzle- and flush pipe-injected transverse jets at the same momentum flux ratios. $\unicode[STIX]{x1D712}$ was determined via (1.2) from PLIF scalar data, utilizing the binary diffusivities for the acetone–gas jet mixtures into air as relevant for $S=1.00$ and $S=0.35$ conditions, given in table 1. One could argue that the diffusion of acetone alone into air might be appropriate for incorporation in (1.2) because it is acetone fluorescence intensity that is quantified in PLIF imaging (Kothnur & Clemens Reference Kothnur and Clemens2005). But because it is the mixture of acetone and jet fluid that contains the particles incorporated in PIV-based quantification of strain rate, and because this mixture is what actually diffuses into air, especially in the relative near field of the jet, in the present studies we utilized the diffusivities for acetone mixtures shown in table 1 for evaluation of scalar dissipation rates via (1.2).

The magnitude of the minimum principal (compressive) strain rate $\unicode[STIX]{x1D716}_{min}$ was determined from PIV data by taking the absolute value of the negative strain rate extracted from the diagonalized strain rate tensor shown in (1.1), which consists of two strain rate components with opposite signs. This minimum principal compressive strain rate value was then averaged at the local shear layer location over 500 image sets.

Trends in the variation in average scalar dissipation rate $\overline{\unicode[STIX]{x1D712}}$ and minimum principal strain rate $\overline{\unicode[STIX]{x1D716}}_{min}$ along the upstream and downstream mixing layers for the equidensity flush nozzle-injected and flush pipe-injected transverse jets showed similar general behaviour as a function of $s_{c}/D$ (Gevorkyan Reference Gevorkyan2015), but with notable differences, especially in the jet’s near field. Such differences suggest that the minimum principal axis did not necessarily align with the scalar gradient vector. Figure 18 highlights the magnitude of the average angle difference, $|\unicode[STIX]{x1D6FC}|$ , between the minimum principal axis of strain determined from the PIV data and the scalar gradient vector direction from the PLIF data for the different injector flows. Angle difference data are shown for both the upstream shear or mixing layer (UML) and downstream or lee-side mixing layer (DML). If there were perfect alignment between the minimum principal axis of strain and the scalar gradient vector direction, as in a flow field of pure straining, the average angle difference would be zero along the jet trajectory (Kothnur & Clemens Reference Kothnur and Clemens2005).

Figure 18. Magnitude of average angle difference, $|\unicode[STIX]{x1D6FC}|$ , between the minimum principal compressive strain axis and the maximum scalar gradient vector. Data shown for both the upstream mixing layer (UML) and downstream mixing layer (DML) of flush nozzle-injected and flush pipe-injected, $S=1.00$ transverse jets with (a)  $J=41$ , (b)  $J=12$ , and (c)  $J=5$ .

The results in figure 18 are similar to what one would expect for this relatively low Reynolds number, transitional transverse jet. Near the jet exit, before shear layer-generated vortex formation for the $J=41$ transverse jets occurred (for either injector), the magnitude of the angle difference was close to $45^{\circ }$ , as expected for a shear-dominant environment. For the upstream shear layer, initial vortex formation caused a drop in the angle difference, for example, as in the flush nozzle, $J=41$ UML angle difference data in figure 18(a) at $s_{c}/D\approx 2.5$ , or in the flush nozzle, $J=12$ UML data in figure 18(b) at $s_{c}/D\approx 1.5$ . This correspondence to initial vortex roll-up location may be understood via comparison with $J=41$ and $12$ images in figure 6. Figure 18 highlights the fact that the angle difference between the maximum scalar gradient vector and the minimum principal strain axis was rarely close to zero, except during the initial vortex roll up associated with the absolutely unstable $J=5$ upstream shear layer for the flush nozzle (figure 18 c), consistent with strong local straining. In general, then, a quantitative comparison of the local scalar dissipation rate to the component of strain along the scalar gradient vector direction would be a more appropriate comparison to make here, especially given the relevance of the former to reactive interfaces (Bish & Dahm Reference Bish and Dahm1995). Hence, for quantitative comparisons with both PLIF-extracted scalar dissipation rate and strain rates, the PIV-extracted strain rate was determined in the direction of the local scalar gradient, as described below.

3.3.3 Maximum scalar dissipation rate and layer-normal strain rate

One can make a direct comparison of the scalar dissipation rate to the strain rate in the scalar gradient direction, neglecting three-dimensional effects, as noted previously. This evaluation in the $y=0$ plane assumes that compressive strain normal to a scalar dissipation layer dominates the evolution of the scalar dissipation rate. Such comparison could yield insights into JICF evolution, mixing, and, indirectly, reaction characteristics. Note that in the absence of time-resolved data, the comparison between the scalar dissipation rate and strain rate is only valid when the strain rate normal to the layer is compressive, since a quasi-steady state solution to the advection–diffusion equation of the scalar is only possible when the molecular diffusion in the scalar gradient direction is balanced by compressive strain in the scalar gradient direction and extensive strain along the layer normal to the scalar gradient direction (Bish & Dahm Reference Bish and Dahm1995). If there is extensive strain in the scalar gradient direction, both strain and diffusion act to decrease the scalar dissipation rate and a quasi-steady solution to the advection–diffusion equation is not possible. Hence, in order to compare the trends among scalar dissipation rates and strain rates, any PLIF- or PIV-extracted strain rates that were negative, thus with extensive strain in the scalar gradient direction, were removed from the averaging process via an automated check. As found in the study of Kothnur & Clemens (Reference Kothnur and Clemens2005) for transitional and turbulent planar jets, extensive strain normal to the scalar gradient direction was found for the JICF to affect the evolution of scalar structures quite often, especially in regions immediately preceding vortex roll up (e.g. $z/D\approx 2.1$ for the equidensity flush nozzle-injected $J=41$ jet, as shown in figure 6 a1,b1). In order to ensure statistical significance, a qualifier was applied to the averaging process that required at least 200 data points to be contained within the averaging of layer-normal strain rate $\overline{\unicode[STIX]{x1D716}}$ and scalar dissipation rate $\overline{\unicode[STIX]{x1D712}}$ at each $s_{c}/D$ location.

Figure 19. Mixing layer average maximum scalar dissipation rate (——, black), $\overline{\unicode[STIX]{x1D712}}$ and average strain rate normal to the scalar gradient direction (——, red), $\overline{\unicode[STIX]{x1D716}}$ , of $S=1.00$ , flush nozzle-injected transverse jets with $J=41$ (a1,b1), $J=12$ (a2,b2) and $J=5$ (a3,b3). Data shown for (a) upstream mixing layer and (b) downstream mixing layer. Data points with extensive strain in the scalar gradient direction are removed from averaging process.

Figure 19 compares the trends for upstream and downstream mixing layer average scalar dissipation rates $\overline{\unicode[STIX]{x1D712}}$ and average layer-normal strain rates $\overline{\unicode[STIX]{x1D716}}$ for the flush nozzle-injected, equidensity transverse jets at $J=41$ , 12, and 5. The spatial evolution of the upstream mixing layer PLIF-extracted scalar dissipation rates and PIV-extracted strain rates (figure 19 a) generally corresponded well to one another for all $J$ values when each parameter was determined along the scalar gradient vector direction. It is important to note that increases in strain rate and scalar dissipation rate were highly correlated in regions of initial upstream shear layer vortex roll up (e.g. $s_{c}/D\approx 2.7$ for $J=41$ ). The downstream mixing layer comparison for the evolution of scalar dissipation rate and layer-normal strain rate (figure 19 b) only showed correspondence for $J=12$ , and little similarity for other conditions, likely arising from three-dimensional and/or highly transient effects in the wake region of the JICF. In figure 20, the flush pipe’s scalar dissipation rate and layer-normal strain rate trends for the same values of $J$ similarly showed good correspondence in the upstream mixing layer (figure 20 a). For the downstream mixing layer region of the flush pipe-injected jets (figure 20 b) there was somewhat better correspondence between $\overline{\unicode[STIX]{x1D712}}$ and $\overline{\unicode[STIX]{x1D716}}$ than for flush nozzle conditions, especially for general trends at $J=41$ and $J=5$ . This same approach may be used to compare $\overline{\unicode[STIX]{x1D712}}$ and $\overline{\unicode[STIX]{x1D716}}$ for the lower density ( $S=0.35$ ) JICF conditions. As for equidensity conditions, results in figure 21 similarly showed good correspondence between $\overline{\unicode[STIX]{x1D712}}$ and $\overline{\unicode[STIX]{x1D716}}$ trends in the upstream layer, but poorer correspondence in the downstream shear/mixing layer, in fact, with opposing trends in spatial variation for $J=5$ . Again, it is likely that these discrepancies for the DML were due to three-dimensional/transient effects, where interactions with wake vortices drawing fluid from the wall boundary layer into the jet had an increasingly important influence (Fric & Roshko Reference Fric and Roshko1994).

Figure 20. Mixing layer average maximum scalar dissipation rate (——, black), $\overline{\unicode[STIX]{x1D712}}$ and average strain rate normal to the scalar gradient direction (——, red), $\overline{\unicode[STIX]{x1D716}}$ , of $S=1.00$ , flush pipe-injected transverse jets with $J=41$ (a1,b1), $J=12$ (a2,b2) and $J=5$ (a3,b3). Data shown for (a) upstream mixing layer and (b) downstream mixing layer. Data points with extensive strain in the scalar gradient direction are removed from averaging process.

Figure 21. Mixing layer average maximum scalar dissipation rate (——, black), $\overline{\unicode[STIX]{x1D712}}$ and average strain rate normal to the scalar gradient direction (——, red), $\overline{\unicode[STIX]{x1D716}}$ , of $S=0.35$ , flush nozzle-injected transverse jets with $J=41$ (a1,b1) and $J=5$ (a2,b2). Data shown for (a) upstream mixing layer and (b) downstream mixing layer. Data points with extensive strain in the scalar gradient direction are removed from the averaging process.

3.3.4 Strained dissipation and reaction layer model applied to the JICF

In order to directly compare the strain rates extracted from the scalar field and from the velocity field in the JICF, the strained dissipation and reaction layer (SDRL) model (1.3) was employed (Bish & Dahm Reference Bish and Dahm1995); as noted earlier, the SDRL is based on classical flamelet models. This enabled the PLIF-based scalar data (and thus strain rate $\unicode[STIX]{x1D716}$ using (1.3)) to be compared directly to the strain rates extracted from the PIV-based velocity field data. The PLIF-based strain rate was calculated from scalar dissipation rates $\overline{\unicode[STIX]{x1D712}}$ and scalar concentration measurements utilizing an automated method of error function fitting, formulated and applied to each instantaneous scalar mixing layer location; such fitting was required in order to determine the boundary scalar values $\unicode[STIX]{x1D701}^{+}$ and $\unicode[STIX]{x1D701}^{-}$ in (1.3), which can have a significant influence on calculated strain rates. If one solves the one-dimensional scalar advection–diffusion equation, assuming quasi-steady behaviour and a locally uniform strain rate, the scalar distribution (normalized concentration $C/C_{o}$ or  $\unicode[STIX]{x1D701}$ ) takes the form shown in (3.1):

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D701}=0.5(\unicode[STIX]{x1D701}^{+}+\unicode[STIX]{x1D701}-)+0.5(\unicode[STIX]{x1D701}^{+}-\unicode[STIX]{x1D701}-)\text{erf}(n_{l}/\unicode[STIX]{x1D706}_{D}).\end{eqnarray}$$

In this equation, $n_{l}$ is the layer-normal coordinate direction, which was assumed to be the two-dimensional scalar gradient direction determined from the PLIF measurements for each mixing layer location, and $\unicode[STIX]{x1D706}_{D}$ is the length scale that results from the competition between strain and diffusion ( $\unicode[STIX]{x1D706}_{D}=\sqrt{\hat{D}_{j\rightarrow \infty }/\unicode[STIX]{x1D716}}$ in the quasi-steady state limit, where $\hat{D}$ is the binary diffusivity). In practice, an error function fit of the form shown in (3.2) was applied to the $\unicode[STIX]{x1D701}=C/C_{o}$ data in the layer-normal direction  $n_{l}$ :

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D701}=a+b\times \text{erf}\left(\frac{n_{l}-c}{d}\right).\end{eqnarray}$$

The coefficient $c$ in (3.2) represents the offset of the fit from the centre location. The boundary conditions, $\unicode[STIX]{x1D701}^{+}=C/C_{o}^{+}$ and $\unicode[STIX]{x1D701}^{-}=C/C_{o}^{-}$ , were determined by comparing (3.1) to experimental concentration data. To ensure accuracy and applicability of the fit, besides requiring that the strain rate normal to the layer determined from comparative PIV data was compressive, another qualifier in the averaging process was applied based on the Pearson correlation coefficient of the fit ( $r>0.99$ ). Additionally, error function fits assumed the magnitude of the coefficient $|c|$ to be larger than the two-pixel width in PLIF images in order to have a more precise representation of the layer centre location; points outside of this criterion were removed from the strain rate evaluation. As with the data discussed in § 3.3.3, at least 200 data points for each $s_{c}/D$ location were required in the averaging process in order to consider the average to be statistically significant (Gevorkyan Reference Gevorkyan2015). An example fit for the flush nozzle-injected, equidensity $J=5$ jet is shown in figure 22(a), utilizing the spatial resolution associated with PLIF data during simultaneous PLIF/PIV experiments. This fit to the error function is reasonably good.

Figure 22. Example error function fits for flush-nozzle injected, $S=1.00$ , $J=5$ transverse jets. Data shown for (a) PLIF portion of simultaneous PLIF/PIV experiments and (b) high-resolution PLIF measurements.

Figure 23. Average strain rate for the mixing layer calculated from PLIF-based scalar measurements using (1.3) (○) and average strain rate extracted from PIV in the direction normal to the scalar gradient direction (——, red), for the $S=1.00$ , flush nozzle-injected transverse jets with $J=41$ (a1,b1), $J=12$ (a2,b2) and $J=5$ (a3,b3). Data shown for (a) upstream mixing layer and (b) downstream mixing layer.

Figure 23 compares the computed mixing layer strain rate calculated from the PLIF-based scalar measurements (3.1) to the average layer-normal strain rate extracted from the PIV measurements, for the equidensity flush nozzle-injected transverse jets at momentum flux ratios $J=41$ , 12, and 5. This comparison is administered for both upstream (figure 23 a) and downstream (figure 23 b) shear layers. For $J=41$ , there was remarkable qualitative and quantitative correspondence between the PIV- and PLIF-based strain rates on both upstream and downstream mixing layers (figure 23 a1,b1), lending credence to the use of flamelet-like models for strained dissipation layers in non-reactive flows (Bish & Dahm Reference Bish and Dahm1995; Kothnur & Clemens Reference Kothnur and Clemens2005). There was similar correspondence between quantified increases in strain rates and general locations of initial shear layer vortex roll up for both the upstream and downstream layers observed in figures 6(a1) and 6(b1), respectively. As the momentum flux ratio was lowered to $J=12$ , the correlation between the PLIF- and PIV-based strain rate trends was also quite good in the UML but less so in the DML. In contrast, for $J=5$ in the near field of the UML, there were large PIV-based strain rates, consistent with strong vortex roll up taking place almost immediately, as indicated in figures 6(a3) and 6(b3). The corresponding spatial trends in PLIF-based strain rates were relatively poorly correlated with PIV-based data for $J=5$ , although correlation with vortex roll up improved in the DML region, especially further downstream of injection. As expected, the upstream strain rates for the $J=5$ absolutely unstable jet were considerably higher than those for convectively unstable $J=41$ and $J=12$ jets. The separate determination that molecular mixing for the absolutely unstable JICF tends to be greater than that for the convectively unstable condition (described in detail in Gevorkyan et al. (Reference Gevorkyan, Shoji, Getsinger, Smith and Karagozian2016)) is consistent with the present strain rate estimates and spatial trends.

It is difficult to definitively determine the exact cause of any qualitative and quantitative discrepancies for $J=12$ and 5 without time-resolved, fully three-dimensional measurements of the scalar and velocity fields. One explanation for the lack of correspondence on the downstream side of the jet for $J=12$ , for example, could be associated with the superior ability of PIV to capture the evolving transient effects in the wake, on the lee side of the jet, as momentum flux ratio was lowered, as evidenced by the transient wake vortical structures in mode 3 and mode 4 observed to be present along with UML structures in the PIV POD for $J=12$ (third and fourth rows in figure 9 b). Another possible contributor to a lack of correspondence in trends for several cases studied here is the finite response time of the scalar dissipation layer to changes in actual strain rate. As summarized in the work of Kothnur & Clemens (Reference Kothnur and Clemens2005), the scalar dissipation layer response time is dependent on both amplitude and frequency of strain rate fluctuations, and it is also dependent on whether the strain rate is temporally increasing or decreasing. This could contribute to inaccuracies in PLIF-based strain rates. Spatial resolution may play an additional role here, and this will be explored below.

Figure 24. Average strain rate for the mixing layer calculated from PLIF-based scalar measurements using (1.3) (○) and average strain rate extracted from PIV in the direction normal to the scalar gradient direction (——, red), for the $S=1.00$ , flush pipe-injected transverse jets with $J=41$ (a1,b1), $J=12$ (a2,b2) and $J=5$ (a3,b3). Data shown for (a) upstream mixing layer and (b) downstream mixing layer.

Figure 24 shows the trends in upstream and downstream mixing layer PIV- and PLIF-based strain rates for the equidensity, flush pipe-injected transverse jets at $J=41$ , 12, and 5. As with the flush nozzle data in figure 23, the best correspondence in PLIF- and PIV-based strain rates for both UML and DML evolution in figure 24 was observed for convectively unstable conditions at $J=41$ , with correspondence to delayed shear layer vorticity roll up as shown in figure 8, although correspondence for $J=12$ was also reasonably good. Quantitative correspondence between the PLIF-calculated strain rate and the strain rate extracted from PIV was poorest for the absolutely unstable $J=5$ case in figure 24, despite similar qualitative trends to one another. Below we briefly explore the effects of PLIF spatial resolution as a possible culprit in the quantitative differences here, which appeared to be greatest for conditions of strong vorticity roll up and high strain rates, especially in the near field.

In determining the effect of spatial resolution on the quantitative comparisons of strain rates in figure 23, higher-resolution PLIF measurements previously used to study mixing processes in detail for the JICF (Gevorkyan et al. Reference Gevorkyan, Shoji, Getsinger, Smith and Karagozian2016) were analysed using the methods outlined above. It should be noted that since no simultaneous PIV data could be taken with the higher-resolution PLIF data, mixing layer locations with extensional strain rate in the scalar gradient direction could not be removed from the averaging process, yielding uncertainty in the process of comparison. As noted earlier, the higher-resolution PLIF images had an in-plane resolution of $34~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ , whereas the lower-resolution, PLIF portion of the simultaneous PLIF/PIV measurements, had an in-plane resolution of $65~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ .

Scalar dissipation rates were determined per (1.2) for the data sets with two different resolutions. The effect of PLIF resolution on the strain rate is shown in figure 22(b) with an error function fit again used to evaluate the scalar boundary values $\unicode[STIX]{x1D701}^{+}$ and $\unicode[STIX]{x1D701}^{-}$ but now for PLIF data taken at higher resolution. There was a small yet noticeable improvement in the fit over the lower-resolution PLIF-PIV experiments, and this, in addition to the computation of $\overline{\unicode[STIX]{x1D712}}$ , altered the computed strain rates. The strain rates calculated from the error function fits of both high and lower-resolution PLIF data sets are shown in figure 25 for the UML of the equidensity flush nozzle-injected $J=41$ and $J=5$ transverse jets. Although general trends were not affected significantly by the differences in resolution for $J=41$ , the values of average local strain rate were significantly altered, with an increase in peak values of $\overline{\unicode[STIX]{x1D716}}$ for $J=41$ that in fact exceeded the magnitude of the PIV-derived strain rates for the UML in figure 23. For $J=5$ , the higher resolution data also increased peak values of $\overline{\unicode[STIX]{x1D716}}$ , in this case bringing the PLIF-based strain rates into closer alignment, qualitatively and quantitatively, to the corresponding PIV-based data in figure 23.

Figure 25. Average strain rate comparison between high-resolution PLIF imaging data (*, red) and PLIF portion of simultaneous PLIF/PIV measurements (○) for the upstream shear layer. Data shown for $S=1.00$ , flush nozzle-injected transverse jets with (a)  $J=41$ and (b)  $J=5$ .

One may also correct in an approximate way for the effect of the PLIF measurement resolution on the strain rate calculation. An effective method for determining imaging system blur effects on the scalar gradients is outlined in the work of Wang & Clemens (Reference Wang and Clemens2004), where a line spread function (LSF) of the imaging system is determined using experimental data from a backlit razor blade traversed through the PLIF camera field of view. Using the LSF, one can determine the degradation of the scalar dissipation rate caused by the finite resolution of the entire imaging system (camera lens, intensifier and CCD array), and these were found to be very close to the resolution quantified by a target in the field of view, so the latter was most often used in the present analysis for simplicity. Calculation of the LSF was applied to the high-resolution PLIF images. The strain rates calculated from the images shown in figure 25 were corrected for the effect of imaging blur (Gevorkyan Reference Gevorkyan2015), and results for this correction are shown in figure 26, showing further increases in the peak values for $\overline{\unicode[STIX]{x1D716}}$ . After this strain rate correction was applied, the difference between the maximum strain rate calculated from the PLIF measurement for the flush nozzle-injected, $J=5$ case, and the strain rate extracted from the PIV measurements for the same case (figure 23 a3) was reduced to 18 %, a remarkable correspondence considering all the factors that could affect this comparison beyond three-dimensional and transient effects, e.g. measurement noise, day-to-day-variations and PIV resolution and/or bias effects. Interestingly, the resolution-corrected PLIF-generated strain rates for $J=41$ in figure 26(a) were qualitatively similar to but significantly higher in magnitude than the PIV-extracted strain rate data for the same case shown in figure 23(a1). Given that extensional strain effects as well as deviation in angle between minimal principal compressive strain axis and maximum scalar gradient vector (figure 18 a) were rather significant for the $J=41$ UML, one might expect somewhat poorer correspondence for these higher-resolution PLIF data, although clearly PIV resolution and other factors could be relevant to the differences observed here.

Figure 26. Resolution-corrected upstream shear layer strain rates calculated from high-resolution PLIF measurements. Data shown for $S=1.00$ , flush nozzle-injected transverse jets with (a)  $J=41$ and (b)  $J=5$ . Raw strain rates (○) and resolution-corrected strain rates (*, red).

3.3.5 Strain rates in lower density transverse jets

In this section, strain rates associated with upstream and downstream mixing layers at a lowered density ratio, $S=0.35$ for the flush nozzle-generated JICF, are discussed. For a non-unity density ratio in the flow field, a coordinate transformation is required to account for the density variation across the jet and cross-flow. The variation in density in the classical reaction–diffusion equation can be dealt with via the Howarth transformation (Howarth Reference Howarth1948), which transforms a density-variable reference frame ( $y$ ) to an effectively incompressible frame (normal coordinate  $n_{l}$ ) via:

(3.3) $$\begin{eqnarray}n_{l}=\int _{0}^{y}\left(\frac{\unicode[STIX]{x1D70C}({\hat{y}},t)}{\unicode[STIX]{x1D70C}_{\infty }}\right)\text{d}{\hat{y}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{\infty }$ is the density of the cross-flow far from the jet and $\unicode[STIX]{x1D70C}$ is the local density. Applying the Howarth transformation makes the density-variable flow field effectively incompressible, and hence strain rates for the JICF shear layers with $S=0.35$ can be calculated in the same manner as for those at $S=1.00$ , using the Howarth-transformed layer-normal coordinate,  $n_{l}$ .

To apply the Howarth transformation, the density or normalized density field is required, according to (3.3). In this study, a density field was approximated based on acetone concentration fields acquired from PLIF imaging under the assumptions of: (i) the validity of the ideal gas law throughout the entire flow field and (ii) the assumption that the flow field in this study was isothermal and isobaric, i.e. that the pressure and temperature did not dramatically change over the course of the experiments throughout the entire flow field. To obtain jet temperature, it was noted that nitrogen and helium used to create the jet were stored at laboratory conditions ( $21\,^{\circ }\text{C}$ ) and cross-flow was generated by a blower drawing the air from inside the laboratory. While the acetone temperature in the cooling chamber was controlled to be at a lower temperature than room temperature (approximately $12\,^{\circ }\text{C}$ ), the isothermal assumption was nevertheless valid because of the relatively small molecular fraction of acetone vapour within the jet fluid, maintained at $\unicode[STIX]{x1D713}\approx 0.1$ (depending on minor day-to-day variations in room temperature) and the length of time the acetone was heated up to room temperature after seeding but before injection into the test section. The pressure monitored inside the acetone cooling chamber for the entire experiments was only 4–5 kPa higher than the room pressure, suggesting the validity of the isobaric assumption. As a consequence of isobaric, isothermal flow, the normalized acetone concentration, $\unicode[STIX]{x1D701}$ , in PLIF images could be used to determine the local density ratio. The local molecular mass in the flow field can be estimated from concentration ratio $\unicode[STIX]{x1D701}\equiv C/C_{o}$ via the relation

(3.4) $$\begin{eqnarray}M=\unicode[STIX]{x1D701}M_{j}+(1-\unicode[STIX]{x1D701})M_{\infty },\end{eqnarray}$$

where $M_{j}$ and $M_{\infty }$ are the molecular masses of pure jet and pure cross-flow fluid, respectively, which were constant during the experiments. Because jet-to-cross-flow density ratio, $S$ , can be expressed in terms of the ratio of molecular masses, $S=M_{j}/M_{\infty }$ , one can obtain the local density ratio

(3.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{\infty }}=1-\unicode[STIX]{x1D701}(1-S). & & \displaystyle\end{eqnarray}$$

Hence, because cross-flow density, $\unicode[STIX]{x1D70C}_{\infty }$ , and jet-to-cross-flow density ratio, $S$ , were known and $\unicode[STIX]{x1D701}$ was obtained from PLIF images, the density field could be determined using (3.5). Figure 27 shows examples of instantaneous centreplane density ratio fields at $J=41$ and $J=5$ for the $S=0.35$ flush nozzle-injected JICF. Clear evidence of density increasing as the jet interacted with cross-flow, especially after vortex breakdown and in the wake region, was apparent.

Figure 27. Instantaneous (a) acetone concentration ( $C/C_{o}$ ) and (b) density ratio ( $\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{\infty }$ ) field images for flush nozzle-injected transverse jets at $S=0.35$ and $J=41$ (a1,b1) and $J=5$ (a2,b2). The density field is approximated from the instantaneous concentration images from PLIF imaging.

These instantaneous density ratio fields could be determined from each instantaneous PLIF image and incorporated into the Howarth transformation. As done in § 3.3.4, the Howarth transformation was applied in the layer-normal direction at each instantaneous scalar mixing layer location, based on the instantaneous density ratio fields, to generate a new Howarth-transformed layer-normal coordinate, $n_{l}$ . After this, as before, the error function was fit to concentration profiles with respect to the Howarth-transformed coordinate, then strain rates were determined via (1.3) using the diffusivity for the acetone–helium mixture creating this $S=0.35$ jet (see table 1). The Howarth transformation was only applied to the strain rate calculation for PLIF data because strain rates from PIV data were explicitly derived from the velocity derivative in the layer-normal coordinate.

Strain rates for the flush nozzle-injected low density ( $S=0.35$ ) transverse jets, derived from both PLIF and PIV data for the upstream and downstream mixing layers, are shown in figure 28. Here, the jet-to-cross-flow momentum flux ratios explored were $J=41$ and 5, both of which are known to have an absolutely unstable upstream shear layer (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012; Shoji Reference Shoji2017). Strain rates extracted from PLIF and PIV data at both $J=41$ and 5 showed very good qualitative and quantitative agreement with one another and with vortex roll-up locations in figure 7 in the upstream mixing layer, and moderate agreement for $J=5$ in the DML. Quantitative discrepancies between the strain rates in the DML were observed for $J=5$ , especially in the near field, but further downstream this comparison improved.

Figure 28. Mean strain rate on the mixing layer calculated from PLIF-based scalar measurements using (1.3) with the Howarth transformation (○) and mean strain rate extracted from PIV in the direction normal to the scalar gradient direction (——, red) for the $S=0.35$ , flush nozzle-injected transverse jets with $J=41$ (a1,b1) and $J=5$ (a2,b2). Data shown for (a) upstream mixing layer and (b) downstream mixing layer.