3.1 Shear layer instabilities for different jet mixtures

Prior experimental studies provide extensive evidence that the USL for the equidensity flush nozzle-injected JICF, consisting of pure $\text{N}_{2}$ injected into air, becomes globally unstable when $J$ is brought to or below $J_{cr}\approx 10$ (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010), or that the low-density JICF consisting of He–$\text{N}_{2}$ mixtures becomes absolutely/globally unstable when $J\lesssim 10$ and/or $S\lesssim 0.40$ (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012). Among the most compelling pieces of evidence for this transition is the significant alteration in hot-wire-based USL spectral characteristics with changes in flow conditions, although there are many additional documented features associated with this transition (Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010). The hot wire is traversed along the USL trajectory coordinate $s/D$ (see figure 1), determined based on a power-law fit to the maximum gradient in the local vertical velocity field. For the low-density JICF ($S<1$) and hence for continuously varying density fields, which present difficulties in hot-wire velocity calibration, the USL trajectory as well as spectral characteristics may be determined based on uncalibrated hot-wire voltage (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012) since the uncalibrated hot-wire signal still yields the same frequency content at various locations as a calibrated hot wire.

Table 1.

For various jet gas constituents with increasing acetone mole fraction, $\unicode[STIX]{x1D713}$, shown are mean jet velocity and absolute viscosity required for matching $Re_{j}=1900$ for two different density ratios, $S=1.0$ and 0.35. Values of the estimated critical momentum flux ratio $J_{cr}$ based on hot-wire measurements are also given.

In the present experiments, spectral characteristics of the JICF were acquired via hot-wire anemometry for the same overall flow conditions ($Re_{j}$, $J$, $S$) but with different mixtures of species in the jet fluid. These mixtures corresponded to experiments employing different optical diagnostics. Relevant parameters are quantified in table 1 for both equidensity and low-density ($S=0.35$) flush nozzle-injected jets for the case $Re_{j}=1900$. Gas constituents included pure nitrogen associated with earlier equidensity experiments (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012) and mixtures of nitrogen and helium with acetone for simultaneous PLIF/PIV ($\unicode[STIX]{x1D713}=0.112$) and for higher-resolution PLIF-only experiments ($\unicode[STIX]{x1D713}=0.218$) typically used to quantify molecular mixing (Gevorkyan et al. Reference Gevorkyan, Shoji, Getsinger, Smith and Karagozian2016). Gas constituents corresponding to the simultaneous PLIF/PIV experiments did not include DEHS oil so as to enable hot-wire usage for spectral measurements.

Figure 3 shows fine-resolution spectral magnitude contour plots for the equidensity JICF, representing the hot-wire response to vertical velocity fluctuations measured along the USL trajectory $s/D$, for a fixed jet Reynolds number ($Re_{j}=1900$) with different momentum flux ratios $J$. Data are shown for different mixtures of species in the jet fluid, corresponding to different mole fractions of acetone: $\unicode[STIX]{x1D713}=0$ for the pure $\text{N}_{2}$ jet (approximately the same as for PIV-only experiments), $\unicode[STIX]{x1D713}=0.0112$ for typical simultaneous PLIF/PIV experimental conditions and $\unicode[STIX]{x1D713}=0.218$ for the typical PLIF-only experiments. Interestingly, while the overall change in USL spectral characteristics from convectively unstable (CU) behaviour at $J=41$ to absolutely unstable (AU) or globally unstable (GU) behaviour at $J=5$ was similar, there were subtle differences as transition from CU to AU took place, based on $\unicode[STIX]{x1D713}$. Interestingly, despite the careful matching of $Re_{j}$, $S$ and $J$ values here, the mixture of constituent gases in the jet affected the evolution of USL, both qualitatively and quantitatively. This observation will be discussed and explored further below. At $J=41$, $20$ and $12$, multiple spatially evolving instabilities in the USL were observed, with frequency shifting along the trajectory $s/D$ resulting from tonal interference between the successively strengthening shear layer instability and the hot-wire probe (Hussain & Zaman Reference Hussain and Zaman1978; Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012). The average of the shifting frequencies was considered to correspond to a fundamental frequency $f_{o}$, as done in prior studies (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010; Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012). Subharmonic development, representing vortex merger, was more prominent at these higher $J$ values, while higher harmonics appeared to be strengthened with a reduction in $J$ as cross-flow velocity $U_{\infty }$ increased. These spectral characteristics corresponded to the behaviour of a convectively unstable USL, as discussed in prior JICF stability studies (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010; Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012). But with the addition of increasing concentrations of acetone, and with the presence of helium and nitrogen designed to create $S=1$, the location of initiation of the instability ($s/D$) was delayed, especially as $J$ was reduced, suggesting a slight weakening of the shear layer instability with increasing acetone concentration. There were also increases in the estimated average fundamental frequency $f_{o}$ and hence Strouhal number $St_{o}$ associated with the initiation of the instability as $\unicode[STIX]{x1D713}$ was increased.

Figure 3.

Spectral contour maps of the hot-wire-based velocity disturbances along the USL trajectory coordinate $s/D$ for the JICF at $Re_{j}=1900$ and $S=1.0$ with varying $J$ and acetone mole fraction $\unicode[STIX]{x1D713}$, corresponding to jets with $\text{N}_{2}$ only ($\unicode[STIX]{x1D713}=0$, reproduced from original data in Getsinger et al. (Reference Getsinger, Hendrickson and Karagozian2012)), the jet mixture for simultaneous PLIF and PIV ($\unicode[STIX]{x1D713}=0.112$) and the PLIF-only experiments ($\unicode[STIX]{x1D713}=0.218$). The colour bar represents disturbance amplitude in dB.

As the momentum flux ratio was reduced to the range close to initiation of an absolutely unstable USL for the pure nitrogen jet, near $J=10$ (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010; Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012), figure 3(d) indicates that the effect of acetone in the jet mixture was not only to delay the onset of the instability, but to cause its characteristics to be associated with convective instability rather than the pure-tone, spatially invariant absolute instability observed for $\unicode[STIX]{x1D713}=0$ at this flow condition. Even at $J=8$ (figure 3e), there was evidence of a weakening of the shear layer, with a very low level of tonal interference, especially at $\unicode[STIX]{x1D713}=0.218$, which typically would not be classified as characteristic of GU flow. At a much lower value of $J=5$, shown in figure 3(f), all jet mixtures produced the typical AU/GU behaviour, with a strong pure-tone instability initiated close to the jet exit, a clear fundamental frequency $f_{o}$ and the absence of subharmonic disturbances. For these lower $J$ values there were nevertheless small increases in $St_{o}$ with increasing $\unicode[STIX]{x1D713}$.

To verify that the phenomena observed in figure 3 reflected actual USL behaviour and that acetone itself was not affecting hot-wire performance, spectral measurements were also made outside of the USL, where there was little or no acetone present. For these spectral measurements the hot-wire probe was initially situated at a fixed USL trajectory location then traversed upstream, in the negative $x$ direction, fixing $y$ and $z$ locations. There were no substantive alterations in spectral character (other than strength of the instabilities) as the hot wire was moved outside of the USL into the air cross-flow and, in addition, $St_{o}$ did not vary. Thus it was determined that alterations in the USL transition and $J_{cr}$ for the acetone-seeded JICF reflected actual physical phenomena and were not affected by the measurement method.

Similar alterations in fundamental $St_{o}$ values and $J_{cr}$ were observed in other acetone-seeded JICF experiments for a range of operating conditions, where deviations from established values of $J_{cr}$ occurred when acetone was present in the jet fluid. Yet overall trends associated with the variation in $St_{o}$ with $J$ were consistent with prior equidensity flush nozzle-injected JICF results in the absence of acetone (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007). This is shown in figure 4 for various equidensity flow conditions, with different $Re_{j}$ values, gas constituents and two different nozzle diameters. For a fixed $Re_{j}$ and acetone concentration, as $J$ was reduced from a large value by increasing the cross-flow velocity, frequency $f_{o}$ and hence $St_{o}$ values from hot-wire-based spectra corresponding to convectively unstable flow always increased before reaching a peak at $J\approx J_{cr}$, and then decreased after the transition to global instability for $J<J_{cr}$. Figure 4 demonstrates a minimal influence of jet diameter for the same Reynolds number (e.g. comparing black and purple symbols), but a somewhat larger influence of $Re_{j}$ and $\unicode[STIX]{x1D713}$ on $J_{cr}$ (e.g. comparing green and dark blue symbols). Increasing the mole fraction of acetone appeared to reduce the value of $J_{cr}$, seen as the $J$ value at which the closed or filled symbols became open ones for a given set of conditions in figure 4. For a given $S$ and $Re_{j}$, a smaller $J_{cr}$ corresponded to an increase in the cross-flow velocity $U_{\infty }$ required to create transition to absolute instability. Increasing acetone concentrations also caused an increase in $f_{o}$ and hence in $St_{o}$, consistent with spectral contour plots (figure 3).

Figure 4.

Variation in Strouhal number $St_{o}$ for the fundamental USL mode based on jet diameter and mean jet velocity $U_{j}$ as a function of $J$ for the equidensity flush nozzle- and larger flush nozzle-injected JICF. Data are shown for $Re_{j}$ and acetone mole fractions $\unicode[STIX]{x1D713}$. Filled and open symbols represent the hot-wire-determined convectively and globally unstable USL, respectively.

The reduction in hot-wire-estimated $J_{cr}$ with increasing concentrations of acetone in the jet fluid is recorded in table 1 for both equidensity and low-density flush nozzle-injected jets for experiments with a fixed $Re_{j}=1900$. Because jet density and Reynolds number were matched in these experiments for each $J$ value studied, and because the absolute viscosity of the mixture did vary among the cases for different concentrations of acetone, as determined via the Wilke formula (Bird et al. Reference Bird, Stewart and Lightfoot1960) and the Reichenberg method (Poling et al. Reference Poling, Prausnitz and O’Connell2001), the actual mean jet velocity $U_{j}$ at the exit plane necessarily differed slightly for the different acetone concentrations with otherwise constant bulk flow properties. The changes in $J_{cr}$ observed in figure 4 and in table 1 with acetone addition may be interpreted in the context of flow in the vicinity of the USL near the injection location, as will be shown.

3.2 Analogy to counter-current shear layer

Iyer & Mahesh (Reference Iyer and Mahesh2016) computationally demonstrate the phenomenological similarity in the USL instability transition for the flush nozzle-injected transverse jet to that for the CCSL based on the CCSL velocity ratio $R_{1}$ in the centre plane, defined in (1.1). The present study explored this similarity based on numerous experimental datasets, using velocities extracted from PIV-only data for density ratios $S=1.00$, $0.70$, $0.55$ and $0.35$ (without the presence of acetone in the jet), taken from Getsinger (Reference Getsinger2012), and also using the PIV portion of simultaneous PLIF and PIV experiments at $S=1.00$ and $0.35$, many of which were taken from Shoji (Reference Shoji2017) and Gevorkyan et al. (Reference Gevorkyan, Shoji, Peng and Karagozian2018). Hence the acetone mole fractions in this comparison were relatively small, $\unicode[STIX]{x1D713}=0.112$.

To extract the mixing layer (CCSL) velocity ratio $R_{1}$ from PIV-based data, the same method as that used computationally in Iyer & Mahesh (Reference Iyer and Mahesh2016) was applied. First, the USL was defined by the magnitude of mean vorticity in the out-of-plane direction, $|\unicode[STIX]{x1D714}_{y}|$, as shown, for example, in figure 5(a) for the equidensity JICF with $J=20$. The normal direction to the USL, $n/D$, may be defined from the $|\unicode[STIX]{x1D714}_{y}|$ field, displayed using multiple lines close to the jet exit in figure 5(a). As done in the numerical studies (Iyer & Mahesh Reference Iyer and Mahesh2016), mean vertical velocity distributions then can be extracted at different locations $s/D$ just downstream of the jet exit, along the layer-normal direction $n/D$, as shown in figure 5(b). These velocity distributions were observed to have regions of both negative vertical velocity upstream of the USL centre and positive vertical velocity within the jet but outside of the USL, including a fairly uniform velocity profile near the jet centre, consistent with a nozzle-generated top-hat-like exit profile (examples of which, for a range of conditions, may be found in Gevorkyan et al. (Reference Gevorkyan, Shoji, Peng and Karagozian2018)). The near-field velocity profiles in figure 5(b) appeared quite analogous to those for a CCSL, and hence $V_{1}$ and $V_{2}$ could be extracted from such PIV-based data to evaluate the CCSL velocity ratio $R_{1}$ via (1.1). More details on the method of estimation of $R_{1}$ for the JICF may be found in Iyer & Mahesh (Reference Iyer and Mahesh2016). In the present experiments, the region upstream of the layer for which negative (vertically downward) flow occurred extended to distances in the negative layer-normal direction that depended on $J$. This negative flow region extended to approximately $n/D\approxeq -0.5$ for higher $J$ values such as $J=41$, but for lower $J$ values such as $J=5$, this region extended further, to $n/D\approxeq -2.0$.

Figure 5.

(a,c) Normalized magnitude of vorticity in the out-of-plane direction, $|\unicode[STIX]{x1D714}_{y}|$, for $J=20$ and $J=5$, respectively. (b,d) Vertical mean jet velocity distributions in the USL trajectory-normal direction, $n/D$, at variable trajectory locations, $s/D$, corresponding to $J=20$ and $J=5$, respectively, with the same line colours as in parts (a,c). Flow conditions correspond to the equidensity, nozzle-injected JICF with $Re_{j}=1900$ and $\unicode[STIX]{x1D713}=0.112$.

Figure 6.

Data points show $R_{1}$ versus $J$ for the JICF at different density ratios $S$ for a fixed $Re_{j}=1900$. Dashed lines corresponding to the colour of data points represent the theoretical velocity ratio, $R_{1,cr(th)}$, for the 2-D CCSL, while the grey region represents the range of experimentally observed values, $R_{1,cr(ex)}$. Circles represent $R_{1}$ values extracted from PIV-only experiments, while triangle symbols are based on data from simultaneous PLIF/PIV experiments with acetone at $\unicode[STIX]{x1D713}\approx 0.112$. Filled and open symbols represent convectively and absolutely/globally unstable conditions, respectively, determined via hot-wire anemometry. In (a), the purple squares represent $R_{1}$ determined from equidensity DNS by Iyer & Mahesh (Reference Iyer and Mahesh2016) at $Re_{j}=2000$.

Resultant $R_{1}$ values for a range of experimental conditions with the flush nozzle are shown in figure 6(a) for the equidensity JICF and in figure 6(b–d) for the low-density JICF at density ratios $S$ equal to 0.70, 0.55 and 0.35, respectively. Filled and open symbols represent convectively and absolutely/globally unstable transverse jet USL conditions, respectively, determined experimentally via hot-wire-measured spectral characteristics such as those shown in figure 3. In addition, for most of the cases in figure 6(a–d), separate PIV-based proper orthogonal decomposition (POD) mode coefficient plots were examined to verify consistency with AU shear layer identification. As described in detail in Meyer, Pedersen & Özcan (Reference Meyer, Pedersen and Özcan2007) and Gevorkyan et al. (Reference Gevorkyan, Shoji, Peng and Karagozian2018), using snapshot POD analysis of the PIV-based jet centre plane data, plots of the POD coefficients of the first and second modes become circular when there is a periodic travelling wave flow structure consistent with an absolutely unstable USL. Such behaviour in the PIV-based experiments was used to verify consistency with the stability condition for the USL identified via hot-wire anemometry and indicated by the open circles in figure 6. The dashed lines in each panel indicate the established (theoretical) 2-D CCSL velocity ratio, $R_{1,cr(th)}$, for transition from a CU to AU shear layer for the given density ratio. These values are predicted by inviscid linear stability theory for the planar CCSL with $S=1$ from Huerre & Monkewitz (Reference Huerre and Monkewitz1985). For $S<1$, theoretical values of $R_{1,cr(th)}$ were extracted from the study by Pavithran & Redekopp (Reference Pavithran and Redekopp1989) for the CCSL, although it is noted that use of the Crocco–Busemann relation to compute density profiles can be inaccurate in representing the near field and thus in prediction of stability characteristics for light and hot jets (Coenen & Sevilla Reference Coenen and Sevilla2012). For all cases in figure 6, dashed lines theoretically distinguish convectively unstable conditions ($R_{1}<R_{1,cr}$) from absolutely unstable conditions ($R_{1}>R_{1,cr}$) for the CCSL. The transition from filled to open symbols in the panels enables estimation of the experimentally determined value or range of values for the critical velocity ratio, $R_{1,cr(ex)}$, indicated by the shaded regions in figure 6.

For the equidensity JICF without acetone in figure 6(a) (circles), corresponding to PIV-only experiments from which $V_{1}$ and $V_{2}$ were extracted, AU conditions occurred when $J\lesssim 10$ according to hot-wire measurements. The corresponding value of $R_{1}$ at this AU transition was approximately 1.24, below the theoretical critical value, $R_{1,cr(th)}=1.32$. While there was a quantitative difference in the critical value of $R_{1}$, as one might expect for comparisons of a round, deflected shear layer with an idealized planar shear layer separated by infinite flow regions, the trends in figure 6(a) were consistent with the concept of the CCSL. That is, lower $J$ values producing absolute instability in the transverse jet corresponded to larger $R_{1}$ values, exceeding a critical value and thus being consistent with a self-excited CCSL. Hot-wire-determined convectively unstable conditions, with filled symbols in figure 6(a), similarly corresponded to $R_{1}<R_{1,cr(ex)}\approx 1.24$. These trends were also consistent with the DNS results for two different velocity ratios computed by Iyer & Mahesh (Reference Iyer and Mahesh2016), shown for reference in figure 6(a). These two computed data points for $R_{1}$ not only straddled the experimentally estimated value $R_{1,cr(ex)}\approx 1.24$, but also were consistent with the trends in the range of values shown by the experimental data points (circles, in the absence of acetone).

For the equidensity experiments with acetone present, $V_{1}$ and $V_{2}$ extracted from the PIV portion of simultaneous PLIF and PIV measurements showed small yet systematic reductions in $V_{1}$ and less-negative values of $V_{2}$ as compared with extracted values in PIV-only experiments at the same $J$ and $Re_{j}$. These changes are reflected in the altered values of $R_{1}$ shown by the triangle symbols in figure 6(a), where the critical condition changed to $J_{cr}\lesssim 8$, with $R_{1,cr(ex)}\approx 1.23$. The differing $J_{cr}$ values associated with USL transition with and without acetone in figure 6(a) appeared to be associated with the slightly altered USL CCSL-like velocity distributions and the resulting small differences in the values of $V_{1}$ and $V_{2}$ close to the jet’s exit, thus producing a slightly different $R_{1}$. These differences in the velocity distributions resulted from small differences in the mean exit velocity $U_{j}$ that were required to match both $Re_{j}$ and $S$ for a given $J$ when the jet mixture contained acetone, with a different overall absolute viscosity $\unicode[STIX]{x1D707}_{j}$, as noted in § 3.1. Hence, while in both equidensity cases in figure 6(a) there was consistency in trends in the USL instability transition (from CU to AU, corresponding to increases in $R_{1}$), there was a small quantitative difference in $R_{1,cr}$ as well as in $J_{cr}$.

For the low-density JICF, trends in increasing $R_{1}$ with a reduction in $J$ at matched $Re_{j}$ and $S$ were also consistent with the CCSL model, shown in figures 6(b), 6(c), and 6(d). The theoretical value of $R_{1,cr(th)}$ determined from Pavithran & Redekopp (Reference Pavithran and Redekopp1989) was somewhat larger than the PIV-based value of $R_{1,cr(ex)}$ for density ratio $S=0.70$ (figure 6b), though the band in which the transition from CU to AU took place (at $J_{cr}\lesssim 10$) was relatively wide in terms of estimating the critical $R_{1}$ range. For $S=0.55$ (figure 6c), the theoretical value of $R_{1,cr(th)}$ was slightly higher than the band at which PIV-only experiments indicated a transition from convective to absolute instability. Consistent with findings in Getsinger et al. (Reference Getsinger, Hendrickson and Karagozian2012), the very-low-density conditions for $S=0.35$ in figure 6(d) generally showed absolutely unstable USL conditions for virtually all $J$ values explored. An exception for the case with acetone present occurred at $J=41$, where the spectra were not quite the same as those associated with absolute instability, with a small degree of tonal interference. This condition for $J=41$ and $S=0.35$, which may have been marginally convectively unstable, produced CCSL velocity profiles corresponding to $R_{1}\approx 1.065$, well below that of the theoretical critical value of $R_{1,cr(th)}=1.12$ for this density ratio. We note that in Getsinger et al. (Reference Getsinger, Hendrickson and Karagozian2012) hot-wire measurements indicated that either $J<10$ or $S<0.40$ produced an absolutely/globally unstable USL, but data in figure 6(d) suggested that a $J$ value greater than $41$ at $S=0.35$ in the presence of acetone, or potentially greater than $J=52$ without acetone, could produce a convectively unstable USL, although such conditions were not attainable in the present experiments. Overall, then, there was general consistency between JICF shear layer transition and transition in a CCSL model from convective to absolute instability, even though the critical values of $R_{1}$ were typically smaller in the experiments than from the inviscid 2-D CCSL analysis. These findings suggest that the mechanism by which the JICF USL transition takes place, at least for the flush nozzle, could relate to the strengthening local counter-flow occurring at the jet exit.

It should be noted that the CCSL model here also provided some evidence for a mechanism for the development of the self-sustained global instability in the jet’s USL. Huerre & Monkewitz (Reference Huerre and Monkewitz1990) note that, as a flow transitions from convective to absolute instability, a ‘zone of local absolute instability’ forms. As the zone grows beyond a critical size, the system can become globally unstable, with a Hopf bifurcation to a self-sustained global mode. Huerre and Monkewitz note that the existence of this region of local absolute instability is a necessary though not sufficient condition for the existence of an amplified global mode. Evidence of global instability in the flush nozzle-injected transverse jet for a small enough $J$ value, including evidence of a Hopf bifurcation (Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010; Regan & Mahesh Reference Regan and Mahesh2017), has already been documented both experimentally and computationally. The region of negative flow upstream of injection for the transverse jet, found here to grow in the negative $n/D$ direction as cross-flow velocity was increased (e.g. as in figure 5), could indeed represent this growing ‘zone of absolute instability’, providing additional evidence for transition to a global mode. This phenomenon appeared to occur even for the somewhat thick CCSLs created by this nozzle, as will be discussed below.

Additional features of the CCSL for which experimental data from the round JICF may be compared with 2-D theory include the Strouhal number associated with the fundamental shear layer instability but where the length scale of interest is the momentum thickness of the local near-field CCSL shown in figure 5. The linear stability theory of Huerre & Monkewitz (Reference Huerre and Monkewitz1985) for the equidensity 2-D CCSL suggests that the critical velocity ratio $R_{1,cr}=1.315$ corresponds to a non-dimensional frequency of 0.192, which may be written in terms of a momentum-thickness-based Strouhal number $St_{\unicode[STIX]{x1D703}}\equiv f_{o}\unicode[STIX]{x1D703}_{o}/2\bar{V}=0.0153$, where the average velocity in the CCSL takes the form $\bar{V}\equiv \frac{1}{2}(V_{1}+V_{2})$. For the 2-D CCSL in figure 5(b), the momentum thickness in the near-field region of the layer may be approximated as

(3.1)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{CCSL}=\int _{n_{V_{2}}}^{n_{V_{1}}}\frac{u(n)-V_{2}}{V_{1}-V_{2}}\left(1-\frac{u(n)-V_{2}}{V_{1}-V_{2}}\right)\,\text{d}n, & & \displaystyle\end{eqnarray}$$

where the limits of integration represent locations in the layer-normal direction at which the velocities reach the values of $V_{1}$ and $V_{2}$. Strykowski & Niccum (Reference Strykowski and Niccum1991) compare an experimentally determined Strouhal number with the theoretical value of Huerre & Monkewitz (Reference Huerre and Monkewitz1985), but alter the limits of integration for (3.1) so that the lower limit is chosen to minimize uncertainties caused by very low velocities in the vicinity of the negative counter-flow. That lower limit $n_{L}$ corresponds to the location at which $(u-\bar{V})/(V_{1}-V_{2})=-0.4$. This produces a momentum thickness designated here as $\unicode[STIX]{x1D703}_{n_{L}}$. On this basis, the Strouhal number extracted from the Strykowski & Niccum (Reference Strykowski and Niccum1991) experiments at $R_{1}=1.4$ (above the critical limit) is $St_{\unicode[STIX]{x1D703}_{n_{L}}}=0.0185$. At a velocity ratio below the critical limit for the transition to absolute instability for the CCSL, at $R_{1}=1$, Strykowski & Niccum (Reference Strykowski and Niccum1991) observe that $St_{\unicode[STIX]{x1D703}_{n_{L}}}=0.012$, in agreement with the Strouhal number associated with the free jet shear layer by Hussain & Zaman (Reference Hussain and Zaman1978). Hussain & Zaman (Reference Hussain and Zaman1978) note the prevalence of a ‘shear layer mode’ at which vortex pairing has occurred, hence is associated with the subharmonic mode for the free jet, downstream of the spatial locations at which the authors observe tonal interference by a hot-wire probe producing frequency shifting of the fundamental mode; such subharmonic modes were seen for the transverse jet at higher momentum flux ratios, e.g. in figures 3(a–c).

In the present experiments for the JICF, one can make quantitative comparisons with the theoretical 2-D CCSL-based Strouhal numbers by evaluating relevant CCSL momentum thicknesses, either based on the definition in (3.1) for $\unicode[STIX]{x1D703}_{CCSL}$ or based on $\unicode[STIX]{x1D703}_{n_{L}}$. Figure 7 shows the variations in momentum thickness $\unicode[STIX]{x1D703}_{CCSL}$ with $J$ for the equidensity JICF at $Re_{j}=1900$. The aforementioned differences in velocity magnitudes $V_{1}$ and $V_{2}$ with and without acetone, resulting from the effects of acetone on absolute viscosity and thus on the mean jet velocity $U_{j}$ and cross-flow velocity $U_{\infty }$ required to match $Re_{j}$ and $S$ for a given $J$, affected the CCSL velocity profiles and hence the momentum thicknesses. With the addition of acetone at the mole fraction shown for simultaneous PLIF and PIV, $\unicode[STIX]{x1D713}\approx 0.112$, figure 7 showed a reduction in non-dimensional momentum thickness $\unicode[STIX]{x1D703}_{CCSL}/D$ of approximately 17 % for the range of $J$ values shown. Corresponding values of $\unicode[STIX]{x1D703}_{nL}/D$ showed a similar reduction.

Figure 7.

Variation in momentum thickness $\unicode[STIX]{x1D703}_{CCSL}$ (scaled by $D$) for the local CCSL as a function of $J$ for the equidensity JICF at $Re_{j}=1900$. Results for data extracted from PIV-only experiments, without acetone, and simultaneous PLIF/PIV experiments, with acetone present at $\unicode[STIX]{x1D713}\approx 0.112$, are shown.

These differences in non-dimensional momentum thickness with and without acetone provided additional insights into the possible reasons for observed differences in the values of the critical CCSL velocity ratio $R_{1}$ for the equidensity cases with and without acetone shown in figure 6(a). One would expect that the quantitative nature of the instabilities should be the same, based on the flow field’s dependence on non-dimensional parameters $J$, $Re$ and $S$ for a given transverse jet condition. The differences in the USL and the instabilities with and without acetone for the same values of these non-dimensional parameters suggested that an additional non-dimensional parameter could be relevant to the nature of the instabilities, per non-dimensional analysis. This non-dimensional parameter appeared to be the ratio of the jet-to-cross-flow absolute viscosities, $\unicode[STIX]{x1D707}_{j}/\unicode[STIX]{x1D707}_{\infty }$, causing the scaled CCSL momentum thickness $\unicode[STIX]{x1D703}/D$ to be dependent on the presence of acetone, as shown in figure 7. This issue will be discussed further below.

It is noted that the relatively large momentum thicknesses in figure 7, for example, producing $D/\unicode[STIX]{x1D703}_{CCSL}$ values as large as 32 for $J=41$ with acetone and as small as 23 for $J=5$ without acetone, were thicker than those typically considered to be thin enough so that shear layer curvature effects may be considered negligible ($D/\unicode[STIX]{x1D703}$ of the order of 50, per Huerre & Monkewitz (Reference Huerre and Monkewitz1985), and larger, of the order of 230, by Strykowski & Niccum (Reference Strykowski and Niccum1991)). The momentum thickness can impact application of planar CCSL inviscid instability theory in predicting fundamental instability frequencies, as noted by Strykowski & Niccum (Reference Strykowski and Niccum1991). For example, LSA for the planar CCSL (Jendoubi & Strykowski Reference Jendoubi and Strykowski1994) suggests that the critical value of $R_{1,cr}$ can change with more significant reductions in $D/\unicode[STIX]{x1D703}$, which in the case of the JICF would correspond to lower $J$ conditions, e.g. $J=5$. Yet the actual experimental values of $R_{1}$ for this range of momentum thicknesses and $J$ values shown in figure 6(a) were very close to DNS-based results (Iyer & Mahesh Reference Iyer and Mahesh2016) for similar flow conditions. For the strongly deflected jet in cross-flow, one would expect curvature effects in the round jet as well as the increasingly non-parallel flow along the curved USL to be inconsistent with an idealized 2-D CCSL, as also noted by Iyer & Mahesh (Reference Iyer and Mahesh2016). Nevertheless, it is remarkable that the quantitative prediction of transition to JICF global instability was generally close to values of $R_{1,cr(th)}$ predicted by inviscid 2-D LSA, both with and without the presence of acetone, indicated in figure 6. Hence it was of interest to utilize momentum thickness in comparing the present instability frequencies with established ranges of Strouhal number for the planar CCSL.

Figure 8.

Variation in Strouhal number based on momentum thickness, $St_{\unicode[STIX]{x1D703}_{CCSL}}\equiv f_{o}\unicode[STIX]{x1D703}_{CCSL}/2\bar{V}$, as a function of $J$ for the equidensity JICF case at $Re_{j}=1900$. Results for data extracted from PIV-only experiments, without acetone, and simultaneous PLIF/PIV experiments, with acetone present at $\unicode[STIX]{x1D713}\approx 0.112$, are shown.

Figure 8 shows the variation in Strouhal number based on $\unicode[STIX]{x1D703}_{CCSL}$ derived from PIV-only and simultaneous PLIF/PIV experimental datasets for the equidensity JICF at $Re_{j}=1900$. Remarkably, for a Strouhal number based on CCSL-based momentum thickness scaling rather than diameter $D$, and based on twice the average velocity in the CCSL, $2\bar{V}$ rather than average jet velocity $U_{j}$ (which does not capture the effect of the local negative flow upstream of the jet), there was a generally good collapse of the data for cases with and without acetone in the jet fluid. An exception was seen in the case of the very low momentum flux ratio jet, with $J=5$, which as noted previously had the thickest and most deflected USL. It is remarkable that $St_{\unicode[STIX]{x1D703}}$ values with and without acetone at a given $J$ value collapsed onto one another for most cases in figure 8, in contrast to the differing Strouhal numbers based on diameter and mean jet velocity in figure 4, shown by data points indicated in green and cyan. The results in figure 8 thus provided further evidence for the importance of the non-dimensional parameter $\unicode[STIX]{x1D703}/D$, related to differences in jet fluid viscosity, and the relevance of the CCSL model in interpreting JICF shear layer instability transition. For the clearly convectively unstable flows at $J=41$, for example, $St_{\unicode[STIX]{x1D703}}$ was of the order of 0.023–0.024, corresponding to $St_{o}$ values of approximately 0.7 without acetone and approximately 0.9 with an acetone mole fraction $\unicode[STIX]{x1D713}=0.112$ (figure 4). These frequencies corresponded to the fundamental mode for the convectively unstable JICF with spectral characteristics as in figure 3(a); the observed subharmonics thus had $St_{\unicode[STIX]{x1D703}}$ values of the order of 0.011–0.012. The $J=41$ condition here produced $R_{1}\approx 1.09$, below the experimentally observed critical condition for the CCSL, $R_{1,cr}\approx 1.24$. These measured Strouhal numbers associated with $R_{1}\approx 1.09$ were approximately twice as large as those observed for the free jet by Hussain & Zaman (Reference Hussain and Zaman1978) and Strykowski & Niccum (Reference Strykowski and Niccum1991) at $R_{1}=1.0$. As $J$ was reduced and the critical condition for transition to global instability approached in figure 8, the values of $St_{\unicode[STIX]{x1D703}}$ increased to the range of 0.03 ($J=10$) to 0.037 ($J=5$ with acetone), also approximately twice as large as established theory and experiments for the planar CCSL (Huerre & Monkewitz Reference Huerre and Monkewitz1985; Strykowski & Niccum Reference Strykowski and Niccum1991). The well-established trend for the CCSL, whereby Strouhal number increases rather abruptly as one approaches the critical condition (Strykowski & Niccum Reference Strykowski and Niccum1991), is consistent with observations in figure 8.

These differences in fundamental Strouhal number $St_{\unicode[STIX]{x1D703}}$ between the present experiments and the established 2-D CCSL values were probably not so much the result of slightly different CCSL velocity ratios $R_{1}$ at transition as the relatively large CCSL momentum thickness at and beyond critical conditions, as noted previously. More importantly, Huerre & Monkewitz (Reference Huerre and Monkewitz1990) note that the frequency selection criterion for global instability depends on how the absolutely unstable frequency evolves spatially, which in turn depends on the development of the base flow for the flow field. The spatial development in a counter-current, planar mixing layer is typically much slower than the spatial development observed in a transverse jet, as indicated in earlier transverse jet stability studies (Alves et al. Reference Alves, Kelly and Karagozian2007, Reference Alves, Kelly and Karagozian2008), and the global frequency determination here is not a local feature of the flow, in contrast to the determination of the critical parameter ($R_{1,cr}$) for the onset of absolute instability. Hence in many respects one would not expect the JICF Strouhal numbers $St_{\unicode[STIX]{x1D703}}$ to correspond very closely to the established values for the planar CCSL.