3.1.1. Validation

Boxx et al. (Reference Boxx, Pareja and Saini2022) used high-speed (10 kHz) stereoscopic particle image velocimetry (SPIV) for both non-reacting and reacting flows along with dual-plan laser-induced fluorescence of hydroxyl radical (OH-PLIF) to study flame attributes at 7 bar. The diagnostics details can be found from Boxx et al. (Reference Boxx, Pareja and Saini2022). Figure 4 compares the measured and computed mean value of $\widetilde {u}_x$ and $ \widetilde {u}_z$ , along with their root-mean-square (r.m.s.) values, for the non-reacting flow at 7 bar. The r.m.s. values are computed using the resolved velocity variances as $\sqrt {\langle \sigma _{u_i, \textit {res}}^2 \rangle }$ . Figures 4( $a$ ) and 4( $d$ ) present the cross-stream variation of $\langle \widetilde {u}_x\rangle$ and $\sqrt {\langle \sigma _{u_{x,\textit {res}}}^2\rangle }$ , respectively, measured at 10 mm upstream of the jet exit ( $x = -5D$ ). These are compared with LES results extracted at four axial positions: $x = -5D$ , $-25D$ , $-50D$ and $-100D$ . The computed values at all four locations overlap since the cross-flow is fully developed, and also agree well with the measurements at $x = -5D$ . Also, the presence of the jet is not felt at this $x$ location. Streamlines plotted on the mid-axial and $z = 2D$ planes (not shown) suggest that the jet’s influence is felt by the cross-flow, as indicated by the deflections of the streamlines, only after $x = -2.5D$ . This upstream location does not change with pressure since $J$ is kept constant for the four cases of table 1. The strong agreement in figures 4( $a$ ) and 4( $d$ ) demonstrates that the inflow velocities and turbulence characteristics of the cross-flow, which are obtained from a separate LES of the channel flow, are well matched with the experiment. Figures 4( $b$ ) and 4( $c$ ) compare the mean velocity components in the mixing region, while figures 4( $e$ ) and 4( $f$ ) present the corresponding r.m.s. values. The good agreement between simulation and experiment for both the mean and fluctuating components of velocity confirms the reliability of the numerical model for studying scalar mixing under the operating conditions of interest.

Figure 4. Comparisons of measured ( $\circ$ ) and computed () ( $a$ , $b$ ) mean axial and ( $c$ ) cross-stream velocities, and ( $d$ – $f$ ) their respective variances at four axial locations for the case P7-NR. The variations of $\langle \widetilde {u}_x \rangle$ and $\sqrt {\langle \sigma _{u_x, \textit {res}}^2} \rangle$ , both normalised by $U_j$ , at several locations upstream of the jet are shown in panels ( $a$ ) and ( $d$ ).

3.1.2. Jet trajectory and cross-sectional asymmetry

The spatial evolution of the jet centreline, called as jet trajectory, is of fundamental interest. This trajectory can be defined using the velocity or scalar concentration, though this choice can influence the results. The scalar-based definition is often preferred in studies focusing on mixing as it directly reflects the distribution and penetration of jet fluid into the cross-flow. For instance, Smith & Mungal (Reference Smith and Mungal1998) defined the trajectory as the locus of maximum jet fluid concentration in the mid- $y$ plane, where the local maximum is obtained from the radial distribution of the scalar as illustrated in figure 5( $b$ ). Figures 5( $a$ ) and 5( $b$ ) show this trajectory for the case P1-NR, identified using the time-averaged mixture fraction field, $\langle \widetilde {\xi } \rangle$ , where $\langle \widetilde {\xi } \rangle = 1$ corresponds to pure jet fluid and $ \langle \widetilde {\xi } \rangle = 0$ to the cross-flow fluid. However, this trajectory exhibits discontinuity around $x/D \approx 12$ , arising from the asymmetric distribution of the jet fluid about the $y=0$ plane. As shown in the top row of figure 6 for the case P1-NR, the jet fluid is concentrated more in one (left) lobe of the CVP, which leads to the discontinuity in the trajectory. Due to this, the scalar-based trajectory is not used in this study. Alternative definitions have been proposed in the past: Fearn & Weston (Reference Fearn and Weston1974) used the locus of maximum velocity magnitude, but Kamotani & Greber (Reference Kamotani and Greber1972) showed that velocity-based trajectories typically penetrate 5 %–10 % deeper than scalar-based ones, as seen in figure 5. The current study adopts the streamline-based trajectory used by Hasselbrink & Mungal (Reference Hasselbrink and Mungal2001) to avoid the limitations of the other two methods. The trajectory is defined as the streamline originating from the jet exit centre. This approach avoids discontinuities from asymmetry and yields a trajectory that aligns more closely with scalar-based definitions (see figure 5), consistent with those reported by Yuan & Street (Reference Yuan and Street1998). A curvilinear orthogonal coordinate system ( $s$ - $n$ - $y$ ) is then constructed using the streamline-based jet trajectory. As illustrated in figure 5( $a$ ), the $s$ -axis follows this trajectory, with the distance along $s$ computed as a piecewise integral along the streamline. The $n$ -axis represents the local normal direction in the $x$ – $z$ plane, perpendicular to the trajectory, while the $y$ -axis is along the lateral direction.

Figure 5. ( $a$ ) Spatial distribution of $\langle \widetilde {\xi } \rangle$ in the mid- $y$ plane for the P1-NR case. ( $b$ ) Corresponding radial variations of $\langle \widetilde {u}_{{mag}} \rangle$ and $\langle \widetilde {\xi } \rangle$ at some locations along the jet, within the same plane. Overlaid on these plots are the trajectories, defined as the locus of peak values of $\langle \widetilde {u}_{{mag}} \rangle$ () and $\langle \widetilde {\xi } \rangle$ (), and the streamline originating from the jet centre ().

Figure 6. Spatial distribution of $\langle \widetilde {\xi }\rangle$ in the jet cross-section for three $s/D$ locations.

The distribution of $\langle \widetilde {\xi } \rangle$ in different $y$ – $n$ planes along the jet trajectory is shown in figure 6, revealing an asymmetric distribution of the jet fluid about the mid- $y$ plane for all the pressures listed in table 1. In the near-nozzle region, $s/D \lt 1.8$ , the distribution is nearly symmetric, but this symmetry breaks in the region $1.8 \lt s/D \lt 20$ , as observed from figure 5. Previous studies have reported similar observations. For equi-density jets with $R = 5$ , 10 and 20, Kuzo (Reference Kuzo1996) observed that asymmetry becomes more pronounced when $Re_j$ falls below a particular value, which increases with $R$ . Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014) noted that such asymmetry correlates with the shear layer instability in their experimental investigations. Specifically, jets issuing from a wall-mounted converging nozzle with $1800 \leqslant Re_j \leqslant 3000$ exhibited a transition in the shear layer behaviour – from absolutely unstable to convectively unstable – as $J^2$ was increased above 10 or $S$ above 0.4. In the present study, although $J^2 \approx 35$ places the jet in the convectively unstable regime, where asymmetries are expected, $S = 0.14$ remains well below the convective instability threshold suggested by Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014). It is also important to note that the $Re_j$ in this study is significantly higher than those of Getsinger et al. (Reference Getsinger, Gevorkyan, Smith and Karagozian2014), except for case P1. Furthermore, a past DNS study of an equi-density JICF with $R=6$ and $Re_j=5000$ (Muldoon & Acharya Reference Muldoon and Acharya2010) suggested that asymmetry may also be influenced by boundary conditions.

The unstructured tetrahedral grid used in this study could, in principle, introduce asymmetries via the boundary conditions. However, an inspection of the spatial grid distribution reveals no asymmetry within the grid tolerance of 1 $\unicode{x03BC} \textrm {m}$ . Uniform profiles are prescribed at the inlets for all scalar quantities and the velocity field shows no evidence of asymmetry. Moreover, the same grid and similar boundary conditions are employed across all simulations. Despite this consistent set-up, variations in the asymmetry of $\langle \widetilde {\xi }\rangle$ are observed across cases, as seen in figure 6. In the P1-NR case, the jet fluid is concentrated on the $-y$ side (towards the reader) while at higher pressures, the concentration shifts towards the $+y$ side. This behaviour is particularly evident in the last column of figure 6. This shift suggests that neither the grid nor the boundary conditions are the likely sources of the symmetry-breaking phenomenon. Instead, the asymmetry is attributed to the convective instability characteristics of the shear layers, which are primarily governed by $Re_j$ . As a result, the direction of asymmetry – whether towards the $+y$ or $-y$ side – is equally likely if the simulations are repeated.

Figure 7. Influence of pressure on the jet trajectory based on the central streamline. The inset shows the variation of the cross-flow boundary layer thickness with pressure.

The streamline-based trajectory is shown in figure 7 for the four cases. The jet penetration depth is higher in P1-NR and it decreases with increasing pressure. These differences arise from neither the grid nor the momentum flux ratio, since both of these are kept the same for these cases. Increasing the pressure from 1 to 15 bar increases the cross-flow Reynolds number by a factor of approximately 14.4. Consequently, the cross-flow boundary layer becomes thinner (see the inset of figure 7) and this layer is defined as the wall-normal distance where the cross-flow velocity reaches 80 % of the centreline value as suggested by Muppidi & Mahesh (Reference Muppidi and Mahesh2005). These boundary layer thicknesses are calculated for the four axial locations considered for figure 4( $a$ ), and this thickness at $x = -25\,D$ is shown in the inset of figure 7 for the four pressures. The Reynolds number for the cross-flow increases with $p_0$ , as listed in table 1, and hence the boundary layer becomes thinner. The thickness shown in the inset does not vary significantly with $x$ since the cross-flow is observed to be fully developed as demonstrated in figure 4( $a$ ). The jet in the P1-NR case is issuing into a cross-flow with a thicker boundary layer, hence, it experiences relatively lower cross-flow momentum in the near-nozzle region. Thus, it penetrates deeper into the cross-flow compared with those for the elevated pressures. This is similar to the behaviour observed by Muppidi & Mahesh (Reference Muppidi and Mahesh2005) in their DNS study of incompressible JICF at atmospheric pressure.

3.1.3. Mixing performance

The jet entrains and mixes with the surrounding fluid, leading to its spatial spread. This mixing is typically quantified using the centreline decay of $\langle \widetilde {\xi } \rangle$ and unmixedness, $g_\xi = \langle \sigma _{\xi }^2 \rangle \big /\langle \widetilde {\xi } \rangle (1- \langle \widetilde {\xi }\rangle )$ , where $\sigma _{\xi }^2$ denotes the variance of the mixture fraction. The normalisation quantity is essentially the maximum variance obtained when the fluid is stirring but neither mixing nor diffusing (Dimotakis & Miller Reference Dimotakis and Miller1990). The total variance $\sigma _{\xi }^2$ is the sum of both the resolved and subgrid variances. The variation of $\langle \widetilde {\xi } \rangle$ along $s$ is often regarded as a reliable indicator of a jet’s overall evolution due to entrainment and mixing, and thus it is commonly used to characterise mixing in the JICF configuration (Smith & Mungal Reference Smith and Mungal1998). This variation is shown in figure 8 for the four pressures along with several power-laws reported in previous studies (Broadwell & Breidenthal Reference Broadwell and Breidenthal1984; Smith & Mungal Reference Smith and Mungal1998; Hasselbrink & Mungal Reference Hasselbrink and Mungal2001; Su & Mungal Reference Su and Mungal2004). The mixture fraction decreases monotonically with increasing $s$ , following an initial region close to the jet exit. The initial portion of the jet, known as the potential core region (PCR), is marked in the figure. The PCR behaves similar to an axisymmetric mixing layer, where similarity solutions are generally not expected. Its length, $s_0$ , is the distance along $s$ where $\langle \widetilde {\xi } \rangle$ drops to 0.9. This length is approximately $1.8D$ and remains independent of pressure. Based on the entrainment vortex sheet model of Coelho & Hunt (Reference Coelho and Hunt1989), Hasselbrink & Mungal (Reference Hasselbrink and Mungal2001) derived an analytical expression for $s_0$ . A leading-order approximation of the expression suggests that $s_0 \propto S^{1/2}/J$ . Since $S$ and $J$ are held constant, $s_0$ remains unchanged for the conditions studied.

Figure 8. Variation of $\langle \widetilde {\xi }\rangle$ with $s/D$ for the four cases under non-reacting conditions.

The flow in the near-field region (NFR) has often been described as similar to the fully developed regime of a free turbulent jet without significant deflection in the cross-flow direction. Accordingly, Hasselbrink & Mungal (Reference Hasselbrink and Mungal2001) observed that the jets in this region follow a self-similar solution similar to that of free jets, where $\langle \widetilde {\xi } \rangle$ decays as $s^{-1}$ . However, this assumption typically holds for jets with $J \gg 10$ . The present study has $J \lt 10$ ; hence, early jet deflection and the formation of the CVP lead to a deviation from the axisymmetric jet behaviour and accelerated decay of $\langle \widetilde {\xi } \rangle$ . As a result, a steeper decay of approximately $s^{-1.3}$ is observed, which is also independent of pressure for the reasons noted above. The NFR extends up to $s \approx 30D$ in the P1-NR case and up to $20D$ in the P15-NR case. Beyond this point lies the far-field region (FFR), where the jet has fully evolved into a coherent CVP translating with the cross-flow. As the jets reach their fully developed state, the entrainment diminishes, yielding a slower decay of $\langle \widetilde {\xi } \rangle$ following $s^{-2/3}$ trend – again showing no significant dependence on pressure.

Figure 9. ( $a$ ) Typical spatial variation of $g_{\xi }$ in the $y = 0$ plane. The variation in $y$ - $n$ planes at locations marked in panel ( $a$ ) are shown in panels ( $b$ – $d$ ). The distribution is shown here for the case P1-NR.

The centreline decay of the mixture fraction does not capture the full spatial distribution of jet fluid, specifically the asymmetric features observed in figure 6. Therefore, a global quantification of the degree of mixing is imperative. Although turbulence is responsible for stirring the jet and cross-flow fluids at large scales, it indirectly enhances mixing by increasing the mixture fraction gradient, $\nabla \xi$ . It is also well known that this gradient is directly proportional to $ \sigma _\xi ^2$ and thereby proportional to $g_\xi$ . It is important to note that $g_\xi$ represents the degree of segregation between two fluids, rather than a direct measure of the mixing rate. Its spatial distribution highlights the regions of strong scalar gradients – a condition conducive to intense local mixing. In regions of spatially uniform $\langle \widetilde {\xi } \rangle$ , $\sigma _\xi ^2$ and $g_\xi$ are zero, whether the fluid is fully mixed or completely unmixed. The maximum value of $g_\xi = 1$ occurs in regions where the two fluids are in contact but remain unmixed. Intermediate values arise where the fluids are diffusing. Hence, $g_\xi$ can be used for characterising local mixing. Figure 9 shows the spatial distribution of $g_\xi$ for case P1-NR. Its distribution in the $y=0$ plane, shown in figure 9( $a$ ), indicates that the highest value of $g_\xi$ occurs in the near-nozzle region, peaking near the end of the PCR ( $s/D \approx 1.8$ ), where the windward and leeward jet shear layers merge. This region corresponds to intense shear-driven roll-up and high scalar gradients, suggesting the potential for the most intense mixing. Beyond the PCR, $g_\xi$ decreases with increasing $s$ , indicating a reduction in fluid segregation and thereby the progression in mixing. However, large values of $g_\xi$ persist in localised regions. For example, at $s/D = 3$ (see figure 9 $b$ ), high mixture fraction gradients appear on both sides of the jet, producing two local maxima in $g_\xi$ (at $n \simeq \pm D$ ). Further downstream, the influence of the leeward recirculation zone diminishes, and thus there is a single peak at $s/D = 6$ . The strength of the local maxima reduces further by $s/D = 12$ due to upstream mixing. This pattern reflects how the entrainment of cross-flow fluid and the action of CVP stretch the jet laterally in the $y$ -direction, distributing the jet fluid around the periphery of the CVP and expanding the area over which it is distributed. As a result, scalar gradients and local segregation level reduce. These spatial patterns are almost the same for all cases listed in table 1.

Figure 10. Variation of the integrated unmixedness, $\Lambda$ , with $s/D$ .

For quantifying the rate at which the mixing progresses along the jet, another measure of unmixedness, $ \Lambda$ , can be defined as

(3.1) \begin{equation} \Lambda (s) = \frac {1}{A}\int _A\;g_\xi \;\textrm {d}A = \frac {1}{A}\int _A \frac {\left\langle \sigma _\xi ^2\right\rangle } {\langle \widetilde {\xi }\rangle (1-\langle \widetilde {\xi }\rangle )}\;\textrm {d}A, \end{equation}

where $A$ is the jet cross-sectional area in the $y$ – $n$ plane. The iso-contours of $\langle \widetilde {\xi } \rangle = 0.003$ , corresponding to the global mixture fraction, bound the area used for (3.1). Hence, the values corresponding to the pure oxidiser are excluded from the calculation. The quantity $\Lambda$ represents the cross-sectionally averaged normalised mixture fraction variance. High values of $\Lambda$ correspond to predominantly segregated mixtures, whereas low values indicate a well-mixed fluid. The variation of $\Lambda$ along the jet trajectory is shown in figure 10 for the four cases. The variation of $\Lambda$ exhibits a trend similar to that of $\langle \widetilde {\xi } \rangle$ (see figure 8). The fluids are largely segregated in the potential core region (PCR), resulting in high $\Lambda$ values, which decrease along the jet due to turbulent and molecular mixing. Since $\Lambda$ incorporates variance across the entire jet cross-section, including off-centre regions, the decay in the NFR begins slowly. Beyond $s/D \approx 7$ , the decay follows an $s^{-1}$ trend, which is slightly slower than the $s^{-1.3}$ decay observed for $\langle \widetilde {\xi } \rangle$ . While the decay of $\Lambda$ in the NFR is observed to be less sensitive to $p_0$ , notable pressure dependence emerges in the FFR. As shown in figure 10, the decay exponent magnitude decreases as $p_0$ is reduced, suggesting that mixing in the FFR improves with increasing pressure. It is important to note that $\Lambda$ represents the variance normalised by $\langle \widetilde {\xi } \rangle (1-\langle \widetilde {\xi } \rangle )$ . The behaviour of the area-weighted mean mixture fraction, $\int _A \langle \widetilde {\xi } \rangle\, \textrm {dA}/A$ , exhibits a trend similar to that shown in figure 8 with a far-field decay rate of approximately $s^{-2/3}$ that is largely independent of $p_0$ . This suggests that the decay of $\langle \widetilde {\xi } \rangle (1 - \langle \widetilde {\xi } \rangle )$ is also insensitive to $p_0$ in the FFR. Consequently, the observed sensitivity of $\Lambda$ to $p_0$ emerges primarily from $\langle \sigma _\xi ^2 \rangle$ variations. As discussed previously, mixing in the FFR is dominated by turbulence. Increasing $p_0$ raises the cross-flow Reynolds number, enhancing turbulent dissipation and accelerating the decay of scalar variance. These second-order statistics are more sensitive to turbulence effects than the mean mixture fraction. Although local mixing intensity varies slightly across the cases, the overall trend of $\Lambda$ remains largely unaffected. This indicates that the peak $g_\xi$ near the end of the PCR and its gradual spreading in the downstream, as observed in figure 9, does not vary significantly with pressure.

3.1.4. Entrainment rate

Fundamentally, a jet entrains the surrounding fluid into its region of influence, causing the mass flow rate across this region to increase with distance along the jet trajectory, $s$ . This entrained mass flow rate, $\dot {m}_e$ , up to a location $s$ , can be obtained through $\dot {m}_e = \dot {m}(s) - \dot {m}_j$ , where $\dot {m}(s)$ and $\dot {m}_j$ are the mass flux, $\rho u_i n_i$ , integrated over the jet cross-section at location $s$ and the jet exit, respectively. The variations of $\textrm {d}\dot {m}_e/\textrm {d}s$ obtained from the LES results are shown in figure 11( $a$ ) for the four cases. The entrainment rate varies along the jet due to a shift in the dominant entrainment mechanism. The Kelvin–Helmholtz instabilities cause the shear layers to roll up in the near-nozzle region, initiating the formation of the CVP. As shown by Nair et al. (Reference Nair, Wilde, Emerson and Lieuwen2019), the swirling strength of these shear-layer vortices increases with $s$ , which increases the entrainment rate, reaching a peak rate between $s/D = 6$ and $8$ . Further downstream, after the jet had bent and almost levelled off, the dominant mechanism transitions to turbulence generated by the jet itself since the fluid that has emerged from the jet travels in the same direction as the cross-flow. However, this turbulence is typically less energetic than the shear-layer-driven turbulence in the near-field, resulting in a lower entrainment rate (Kankanwadi & Buxton Reference Kankanwadi and Buxton2023). Additionally, the local rate of jet-to-cross-flow entrainment increases proportionally with pressure.

Figure 11. Variations of ( $a$ ) $\textrm {d}\dot {m}_e/\textrm {d}s$ and ( $b$ ) $K_e$ with $s/D$ .

The entrainment rate of the jet in a co-flowing stream was studied extensively in the past, and the resulting scaling can be expressed as (Hill Reference Hill1972)

(3.2) \begin{equation} \hat {m}_e = \frac {\dot {m}_e}{\dot {m}_j} = K_e\left (\frac {s}{D}\right ) S^{-\frac {1}{2}}, \end{equation}

where $s$ and $S$ are respectively the axial distance from the jet exit and the density ratio defined earlier. The dimensionless number $K_e$ is called the entrainment rate coefficient, which is the normalised rate of entrainment and is calculated as

(3.3) \begin{equation} K_e(s) = \frac {\textrm {d}\dot {m}_e}{\textrm {d} s} \left (\frac {D}{\dot {m}_j}\right ) S^{\frac {1}{2}}. \end{equation}

The past measurements of jet in co-flow showed (Hill Reference Hill1972) that $K_e$ varied with $s$ , as depicted in figure 11( $b$ ), and it plateaued to a constant value of 0.32 in the far-field region. In the JICF configuration studied here, the relatively large values of $K_e$ , in the near-field region, $s/D \lt 25$ , are attributed to the increased jet surface area resulting from circumferential stretching and wrinkling during the CVP formation, as discussed in § 3.1.3. In the far-field region, $K_e$ values for both free jets and JICF converge, reflecting the similarity in the dominant entrainment mechanism. Furthermore, the collapse of $K_e$ across different pressures suggests that while the absolute entrainment rate increases with pressure, the overall mixing pattern remains unaltered when $R$ , $S$ and $J$ are kept constant. It is worth noting that among these parameters, only $\dot {m}_j$ varies across the four cases, further reinforcing this conclusion.

3.2.1. Validation

Following ignition, jet and cross-flow fluids react in regions having flammable mixtures. These flame (or reacting) regions are identified in experiments using planar laser-induced fluorescence (PLIF) of hydroxyl (OH) radicals (Hartung et al. Reference Hartung, Hult, Kaminski, Rogerson and Swaminathan2008; Boyette et al. Reference Boyette, Guiberti, Magnotti and Roberts2019; Angelilli et al. Reference Angelilli, Ciottoli, Guiberti, Galassi, Pérez, Boyette, Magnotti, Roberts, Valorani and Im2021). Hydroxyl radicals are present in both the flame and burnt products, but at differing levels. Hence, the flame regions can be qualitatively demarcated through the spatial gradient of OH fluorescence (Hartung et al. Reference Hartung, Hult, Kaminski, Rogerson and Swaminathan2008). The magnitude of the spatial gradients, calculated from the time-averaged OH-PLIF and OH mole fractions ( $X_{{OH}}$ ) from the LES, are compared in figures 12( $a$ ) and 12( $b$ ). The general features, spatial extent and the flame locations of both the windward and leeward flame branches from the LES agree well with experimental observations of Boxx et al. (Reference Boxx, Pareja and Saini2022). In addition, the contours of the time-averaged reaction rate shown in figure 12( $c$ ) also compare well with the spatial gradient of OH, suggesting that the flame regions are captured well in the simulations. Notably, some discrepancies are observed on the jet’s leeward side, where OH radicals persist in the burnt products despite negligible local reaction rates. This is also verified using instantaneous images from diagnostics and LES, but not shown here. These comparisons remain qualitative, as direct quantitative evaluation of flame-related properties such as temperature or scalar concentration is not possible due to the lack of corresponding experimental data. However, PIV measurement for the reacting flows at the 7 bar case, P7, allows for a quantitative evaluation as it reflects the effect of heat release on the local flow dynamics.

Figure 12. Comparison of normalised spatial gradient magnitude derived from ( $a$ ) time-averaged OH–PLIF measurements and ( $b$ ) computed OH mole fraction. ( $c$ ) Reaction rate, $\overline{\dot{\omega}^*}$ . These quantities are normalised using their respective global maximum.

Figure 13. Comparisons of measured ( $\circ$ ) and computed () ( $a$ ) mean axial and ( $b$ ) $z$ cross-stream velocities and ( $c$ and $d$ ) their respective r.m.s values. The cross-stream variations are shown for four axial locations of the case P7. The values are normalised using $U_j = 135$ m s⁻¹.

The computed axial and cross-stream velocity statistics are compared with their respective measurements in figure 13 for the case P7. A very good agreement is observed for both mean and r.m.s. values. However, some disagreements are observed for the r.m.s. in the near-wall region, $z/D \lt 2$ , which is because the laser reflection from the bottom wall can yield spurious behaviour for second-order statistics. Also, the wall boundary layers are not resolved but modelled in the LES, as noted in § 2.3. For these reasons, one must be careful while comparing the measured and computed near-wall statistics. Furthermore, the average numerical grid size in the mixing region is approximately 0.15–0.4 mm. This implies turbulent structures of size larger than 9.76 $\delta _{th}$ and their influences on the flame are resolved in the LES of P7 case. The multi-scale analysis of Doan, Swaminathan & Chakraborty (Reference Doan, Swaminathan and Chakraborty2017) suggested that eddies of size $2\leqslant \delta _{th}\leqslant 5$ contribute predominantly to the flame straining. These eddies are subgrid scales for the conditions of P7, and hence, one would expect these eddies to influence the reaction rate through flame straining. However, these subgrid eddies have a small amount of kinetic energy associated with them since more than 90 % of the kinetic energy is resolved by the grid (see figure 3). Hence, the flame straining caused by these subgrid scale structures is insignificant compared with that of the resolved scales. Thus, the combustion closure based on the unstrained flamelets model, described in § 2.2, is good for the flow, thermochemical and elevated pressure conditions considered for this study. Furthermore, the good agreement observed for the measured and computed r.m.s. values suggest that the inlet turbulence conditions and the method used to generate it (see § 2.3) mimic the experimental situations well.

3.2.2. Spatial distribution of heat release rate

Typical spatial variation of $\widehat {\dot {q}} =\overline {\dot {q}}/\dot {q}_{{P1, max}}$ , where $ \overline {\dot {q}}$ is the filtered volumetric heat release rate and $\dot {q}_{{P1, max}}$ is the maximum volumetric heat release rate in a freely propagating stoichiometric (H₂+He)–air laminar flame at atmospheric pressure, are shown in figure 14( $a$ ) for the cases P1, P4 and P15. These variations are shown in the $y=0$ plane at $t = 6\tau _f$ from ignition. Also, the jet trajectories of corresponding non-reacting and reacting flows are shown for comparison along with a few $s/D$ locations used for further analyses. The contours of $\widehat {\dot {q}}$ show that the predominant heat release is in the shear layers for all the cases. Although the flames are lifted on both the windward and leeward sides on an instantaneous basis they have merged together at $s/D \approx 3.5$ in an averaged sense as suggested by the time-averaged heat release rate distribution shown in figures 14( $b$ ) and 14( $c$ ) for the atmospheric case. This merging does not occur for the elevated pressure cases, and the spatial distributions of $\widehat {\dot {q}}$ and $\langle \widehat {\dot {q}}\rangle$ are similar for the P4 and P15 cases. The density of the reactants increases with pressure as $\rho _u \propto p_0$ , resulting in an increased burning flux, $f^0 = \rho _u s_L \propto p_0^{n/2}$ , where $n$ is the order of the overall chemical reaction, which is approximately 2 for an H $_2$ –air flame (Law Reference Law2006). Since $f^0$ represents the eigenvalue of flame propagation, the windward and leeward flame branches move upstream and attach to the jet exit as suggested by the contours of $\widehat {\dot {q}}$ and $\langle \widehat {\dot {q}} \rangle$ . The leeward branch is stabilised by the recirculation zone and hence it is quite steady, robust and includes lean mixtures with relatively lower amount of heat release. The distribution of this averaged heat release rate in the jet cross-section is similar to the CVP shape as shown in figure 14(c).

Figure 14. ( $a$ ) Spatial distribution of $\widehat {\dot {q}}$ in the $y=0$ plane for the cases P1, P4 and P15. (b,c) Corresponding time-averaged variations in the $x$ – $z$ and $y$ – $n$ planes. The locations of the $y$ – $n$ planes are marked in panel ( $b$ ). The iso-lines of $\langle \widetilde {\xi }\rangle = \xi _l$ , $\langle \widetilde {\xi } \rangle = \xi _r$ and $\langle \widetilde {\xi } \rangle =\xi _{st}$ are also shown in panels ( $b$ ) and ( $c$ ). The jet trajectories for reacting and non-reacting flows are shown in panel ( $a$ ) along with various $s/D$ locations used for later analysis.

Figure 15. Conditional probability density function (c.p.d.f.s) of $\Phi$ , $\psi$ and $\kappa$ , conditioned on $\widehat {\dot {q}}$ . The respective conditional means are shown by the solid lines ().

3.2.3. Heat release effects on mixing

It is well established that heat release suppresses entrainment (Hermanson & Dimotakis Reference Hermanson and Dimotakis1989) by influencing the shear layer roll-up. There are two possible mechanisms behind this influence: the first one being the volumetric expansion or dilatation, which alters the spatial velocity gradients, and the second one being the increase in the total kinematic viscosity of the fluid. The volumetric expansion arises from density changes caused by mixing and chemical reactions. The expansion resulting solely from chemical heat release can be quantified using the pointwise difference in dilatation (also the strain rates and kinematic viscosity discussed later) between the time-averaged reacting and non-reacting flows. This difference is expressed as $\Phi = ( \partial \langle \widetilde {u}_i \rangle /\partial x_i )_{{r}} - (\partial \langle \widetilde {u}_i \rangle /\partial x_i )_{ {nr}}/[|E|_{{nr}}]$ , where $|E| = \sqrt {2E_{ij}E_{ij}}$ is the Frobenius norm of the symmetric strain rate tensor $E_{ij}$ . The subscripts ‘r’ and ‘nr’ denote the reacting and non-reacting conditions, respectively, and the operator $[\;]$ represents the volume-weighted averaging. The top row of figure 15 shows the conditional probability density function (c.p.d.f.) of $\Phi$ , conditioned on the normalised heat release rate $ \widehat {\dot {q}}$ . The c.p.d.f.s are computed using 60 bins in each sample space, with data sampled from the reacting regions of the flow, which is defined by $\widehat {\dot {q}} \gt 0.001 \times [\widehat {\dot {q}}]$ . Due to its wide spatial extent, low heat release rates exhibit relatively higher probability. The difference in the dilatation increases with $\widehat {\dot {q}}$ , and since the maximum value of $ \widehat {\dot {q}}$ rises with operating pressure $p_0$ , the maximum $\Phi$ also increases accordingly as reflected by the conditional mean profiles. Furthermore, the density difference between the jet and cross-flow fluids yields non-zero dilatation in the mixing layers of non-reacting flow, and this mixing layer shifts spatially because of expansion effects arising from heat release in reacting flows. Hence, $ \Phi$ can take some small negative values as observed in the top row of figure 15.

The strain rate magnitude $|E|_{{nr}}$ is used for normalisation because it reflects the local kinematic deformation of the flow, encompassing both volumetric expansion and shear deformations. While the dilatation term $tr(E_{ij})$ accounts for the volumetric expansion, it does not capture changes in shear deformation due to heat release. To assess this, the change in strain rate magnitude, expressed as $\psi = |E|_{{r}} - |E|_{ {nr}}/[|E|_{ {nr}}]$ , is calculated. The middle row of figure 15 shows the c.p.d.f.s of $\psi$ with respect to $\widehat {\dot {q}}$ . These contours closely resemble those of $ \Phi$ : probability is concentrated at low $\widehat {\dot {q}}$ , and the conditional mean increases with $ \widehat {\dot {q}}$ and $p_0$ . This trend reflects the intensification of local velocity gradients due to flow acceleration in the reacting shear layer. At very high values of $\widehat {\dot {q}}$ , the conditional mean of $\psi$ decreases due to sparse sampling. It is important to note that combustion slightly increases the jet penetration depth, subtly shifting the shear layer location radially in downstream regions of $s/D \approx 5$ . Consequently, in some regions where the shear layer exists under non-reacting conditions, $\psi$ can become negative. This is particularly evident in case P1, where the flame root (and hence high $\widehat {\dot {q}}$ ) lies downstream of this region, resulting in negative conditional means of $\psi$ . However, such behaviour is not observed in the higher-pressure cases, where the flame is anchored closer to the jet exit, coinciding with the peak strain rate location in both reacting and non-reacting conditions.

The effect through the change in the local kinematic viscosity $\nu$ can be quantified as $\kappa = \nu _{{r}} - \nu _{{nr}}/[\nu _{{nr}}]$ . The bottom row of figure 15 presents the conditional p.d.f. and the corresponding mean of $\kappa$ . Notably, there could be two distinct values of $ \nu$ for a given value of $\widehat {\dot {q}}$ , leading to a reverse-C-shaped distribution in the c.p.d.f.s. The p.d.f.s (as it is the sample fraction) for high $\kappa$ are small because they arise from locally reacting (in a mean sense) mixture, which typically occupy a small region. While the time scale of chemical kinetics decreases with increasing $\widehat {\dot {q}}$ , the burnt product temperature remains unaltered. Consequently, following Sutherland’s law, the normalised change in $\nu$ remains constant with respect to $\widehat {\dot {q}}$ and $p_0$ . Comparing the conditional mean of $\Phi$ , $\psi$ and $\kappa$ reveals that the heat release affects both the local flow dynamics (via strain rate and dilatation) and the thermochemical properties (via viscosity) equally at atmospheric pressure conditions. However, with increasing pressure, the influences on flow dynamics become significantly more pronounced, whereas the effect on viscosity remains essentially constant. At 15 bar, the change in strain rate and dilatation are nearly an order of magnitude larger than the changes in viscosity. These combustion-induced changes in flow dynamics and properties alter the behaviour of the shear layer by locally laminarising the flow, thereby reducing mixing and increasing jet penetration depth (Hasselbrink & Mungal Reference Hasselbrink and Mungal2001). Furthermore, the periodic roll-up of shear layer vortices is suppressed (Nair et al. Reference Nair, Wilde, Emerson and Lieuwen2019; Saini et al. Reference Saini, Chterev, Pareja, Aigner and Boxx2021; Sayadi & Schmid Reference Sayadi and Schmid2021). Since the viscosity-related changes remain insensitive to pressure, it becomes essential to investigate how the substantially large changes in flow dynamics at high pressures govern mixing in the reacting JICF configuration.

The variations of $\langle \widetilde {\xi } \rangle$ with $s/D$ for the four reacting conditions are shown in figure 16( $a$ ) along with the non-reacting case of P1 for comparison. The flame is lifted in the P1 case and hence the near-field ( $s/D \leqslant 3.8$ ) mixing is identical for the non-reacting and reacting conditions as seen in the figure. Hence, the jet trajectory is also not disturbed by the heat release as seen in figure 14. The heat release in the subsequent downstream region influences the entrainment, leading to a drop in the decay rate as seen in figure 16( $a$ ) and this continues until $s/D \approx 10$ . The $\langle \widetilde {\xi }\rangle$ varies as $s^{-0.8}$ for $3.5\lt s/D \lt 10$ and it changes to $s^{-1.13}$ for $s/D \gt 10$ . Since the flame is attached to the jet exit, the exponent of $-1.13$ is also observed for the elevated pressures from $s/D \approx 2.5$ . Except for this change, there is no influence of pressure on the mixing as observed for the non-reacting flows in § 3.1.3. This implies that the substantial changes observed in the flow dynamics does not alter the mixing characteristics unduly.

Figure 16. Variation of ( $a$ ) $\langle \widetilde {\xi }\rangle$ and ( $b$ ) $\Lambda$ with $s/D$ under reacting conditions.

The variation of $\Lambda$ along the jet centreline is shown in figure 16( $b$ ) for all cases. This variation for the non-reacting case of P1 is also shown for comparison. Under the reacting conditions, the local mixture is distributed towards the leeward side due to stirring and mixing supported by the recirculation zone. This results in a relatively lower segregation level in the near-nozzle region ( $s/D \lt 3$ ) compared with the non-reacting counterpart. As the mixing progresses, the mixture fraction variance and the segregation level decrease with increasing $s$ . As shown in figure 16( $a$ ), the mean mixture fraction is higher than its non-reacting counterpart at a given $s/D$ location within the reacting region. This is because the local heat release rate suppresses the entrainment, which preserves scalar segregation. While the variance, $\sigma _\xi ^2$ , and $\langle \widetilde {\xi } \rangle$ decrease due to molecular diffusion, $\Lambda$ increases within the flame region and attains a peak at $s/D \approx 7$ , coinciding with the local maximum heat release rate. This suggests that combustion locally suppresses turbulent stirring, lowering $\widetilde {\xi } (1- \widetilde {\xi })$ . Beyond this point, the heat release rate decreases, allowing further mixing to occur, which lowers the segregation level. For the P4, P7 and P15 cases, the predominant reaction occurs in the windward near-nozzle region and gradually decreases downstream (see figure 14). The suppression of air entrainment by the reaction leads to a relatively higher $\langle \widetilde {\xi } \rangle$ in this region, as seen in figure 16( $a$ ). Consequently, $g_\xi$ and $\Lambda$ are large for $3\lt s/D \lt 7$ . The mixing progresses further as the heat release rate decreases in the downstream region leading to a drop in $\Lambda$ . Notably, in the low heat release rate region ( $s/D \gtrsim 20$ ), the decay of $\Lambda$ follows the same scaling as in the corresponding non-reacting case. This indicates that combustion primarily influences the near-field by suppressing shear-driven turbulence, but has a diminishing impact on mixing further downstream.

3.2.4. Combustion modes

Generally, the mixing and combustion have mutual influence in partially premixed combustion, which is the case for the JICF investigated here. Hence, the combustion occurs over a wide range of mixture fractions, covering from lean to rich flammability limits. Also, the range of mixture fraction values over which combustion occurs varies along the jet centreline as discussed in § 3.2.3. To illustrate this further, the temperature variation with the mixture fraction is shown in figure 17 for the case P1. The data collected from a small $y$ - $n$ slices with approximately 1 mm thickness centred around the $s/D$ locations noted in the figure (also marked in figure 14 $a$ ) are used for the scatter plot. The fully burnt and unburnt limits are also marked using $\zeta = 1$ and 0, respectively. Also, the temporal change of the variations shown in figure 17 is negligible for the data collection period.

Figure 17. Typical variation of temperature in the mixture fraction space at $t = 6\tau _f$ in the case P1. The conditionally averaged temperature is shown using the red line. The mixing and fully reacting limits are shown using the dash-dotted line. The colour denotes the progress variable.

The temperature follows the mixing line, $\zeta = 0$ , in figure 17 for $s/D = 0.5$ since no combustion occurs in the near-nozzle region. There are some preheated mixtures having $\widetilde {\xi } \lt \xi _l$ for $s/D = 2$ location, which results from the mixing of burnt gases from the leeward side behind the PCR with unburnt gases from the corresponding windward side (see figure 14 $b$ ). The combustion in $2 \leqslant s/D \leqslant 5$ occurs in mixtures with $\xi _{{st}} \lt \xi \lt \xi _r$ , as seen in figure 14. The windward and leeward shear layers merge at $s/D = 5$ accelerating the break-up of the jet and its mixing with the cross-flow fluid as observed in figure 16( $a$ ). As a result, the $\langle \widetilde {\xi } \rangle = \xi _r$ iso-contour does not extend beyond $s/D = 5$ , see figure 14. Hence, the combustion here occurs over the entire flammable range as shown in figure 17 with some non-reacting mixtures, signified by the data points on the $\zeta = 0$ line. This preheated mixture and those created through the mixing of burnt and unburnt mixtures react at further downstream locations which is signified by the shift of the data from the $\zeta = 0$ line at the $s/D = 10$ location. A substantial amount of fuel is consumed before $s/D$ of 25 and hence the samples are within $\widetilde {\xi } = 0$ to $0.14$ .

Figure 18. Temperature variation with the mixture fraction is shown for the case P15 at $t = 6\tau _f$ and the colour denotes the reaction progress variable. The conditionally averaged temperature is shown using a red line. The mixing and fully burnt reacting limits are shown using the dash-dotted lines.

The results for the P15 case are presented in figure 18 for comparison. The increased pressure shifts the flame stabilisation point closer to the jet exit. Consequently, both preheated and near fully burnt mixtures are observed for the $s/D = 0.5$ location, where fuel-rich combustion predominantly occurs on the windward side while fuel-lean combustion is on the leeward side as evident in figure 14. The dominant reactions occur in highly reactive mixture fractions with low ignition delay times. These reactions preheat the downstream mixture triggering an ignition cascade that leads to subsequent combustion of stoichiometric and richer mixtures. Therefore, the combustion at $s/D =$ 2, 5 and 10 occurs across the entire flammable range. Similar to the case P1, combustion at $s/D = 15$ and 25 occurs under fuel-lean conditions. This narrows the spatial distribution of the reacting region and increases the progress variable gradient. Hence, significant variation in $\zeta$ is not observed in P15. Moreover, the heat release rate suppresses the formation of coherent structures along the shear layer. Therefore, the reacting region present closer to the jet exit in P15 reduces the rate at which the jet breaks up, which also contributes to the lack of preheated mixtures in the P15 case.

The previous discussion highlights the pressure effect in the mixture fraction range with combustion in the JICF. However, it does not elucidate the combustion modes. Both premixed and non-premixed modes are present in partially premixed combustion. Therefore, the transport equation of the reaction progress variable has two source terms as written in (2.5). The relative contributions of these two modes can be investigated using a joint p.d.f. of $\eta$ and $\gamma$ , where $\gamma$ is the sample space variable for $\Omega = \overline {\dot {\omega }}_{np}/\overline {\dot { \omega }}_{fp}$ (Massey et al. Reference Massey, Li, Chen, Tanaka and Swaminathan2023a ). It is important to note that a flame is likely to burn in a premixed mode if the reactant mixture is within the flammability limit and hence the significant burning along the stoichiometric iso-contours, observed in figure 14, is just not a characteristic of non-premixed mode alone as it includes contributions of stoichiometric premixed combustion as well. However, $\overline {\dot {\omega }}_{np}$ denotes non-premixed contribution and it comes from the neighbourhood of $\xi _{{st}}$ since $\textrm {d} ^2Y_{{\textrm H}_{2} \textrm{O}}^{\textrm b}/\textrm {d}\eta ^2$ is non-zero only in that region. The variation of $\textrm {d}^2Y_{\textrm{H}_{2} \textrm{O}}^{\textrm b}/ \textrm {d}\eta ^2$ in the mixture fraction space has been discussed by Ruan, Swaminathan & Darbyshire (Reference Ruan, Swaminathan and Darbyshire2014) for an H $_2$ –air mixture under atmospheric pressure and it does not differ significantly for elevated pressures. Also, $ \Omega$ is 0 in the fully premixed limit and $-\infty$ in the non-premixed limit.

The fractional contribution of the non-premixed mode from the lean, rich and the stoichiometric mixtures can be obtained by taking appropriate moments of the joint p.d.f. as

(3.4) \begin{align} \langle \Omega \rangle & = \underbrace {\int _{\widetilde {\xi }_{l}}^{\widetilde {\xi } _{{st}} - \Delta \widetilde {\xi }}\int _{- \infty }^{0}\gamma \;P(\eta ,\gamma )\;\textrm { d}\gamma \;\textrm {d}\eta }_{\langle \Omega _{l}\rangle } \;\; + \;\; \underbrace {\int _{\widetilde {\xi }_{{st}}-\Delta \widetilde {\xi }}^{\widetilde {\xi }_{st} + \Delta \widetilde {\xi }}\int _{-\infty }^{0} \gamma \; P(\eta ,\gamma )\;\textrm {d} \gamma \;\textrm {d}\eta }_{\langle \Omega _{{st}}\rangle } \; \; \nonumber \\ &\quad + \underbrace {\int _{\widetilde {\xi }_{{st}} + \Delta \widetilde {\xi }}^{\widetilde {\xi }_r} \int _{-\infty }^{0} \gamma \;P(\eta ,\gamma )\; \textrm {d}\gamma \;\textrm {d}\eta }_{\langle \Omega _r\rangle }. \end{align}

Here, $\Delta \widetilde {\xi } \approx 0.0026$ , which is approximately 5 % of $\widetilde {\xi }_{{st}}$ . The data used for constructing the joint p.d.f. are obtained over the final 1 $\tau _f$ period of the simulations. The moments in (3.4) are calculated using the data from different $y$ - $n$ slices to show their variations along $s/D$ , as depicted in figure 19.

Figure 19. Variations of the time-averaged fractional contribution from the non-premixed mode in the lean, $\langle \Omega _l\rangle$ , stoichiometric, $\langle \Omega _{st}\rangle$ , and rich, $ \langle \Omega _r\rangle$ , mixture fractions. The variation at atmospheric pressure is shown in panel (a) while that for 4, 7 and 15 bars are shown in panel (b).

The fractional contribution magnitude of the non-premixed mode is below 0.2 irrespective of the mixture fraction values for the range of pressures considered here. This suggests that the combustion is predominantly in premixed mode. The mixing among the fuel and entrained air on the windward side and products from the leeward side can alter the contribution of the non-premixed mode. Since the flame is lifted in the P1 case, the reaction rates are zero in the near field and start to increase specifically on the rich side until $s/D \approx 11$ , also see figure 14( $b$ ). Since a significant amount of fuel is mixed well and also burnt in region $s/D \lesssim 11$ , the combustion shifts to fuel-lean condition in the subsequent downstream locations, leading to their dominant contributions to $\langle \Omega \rangle$ , as seen in figure 19. The flame moves upstream and is attached to the nozzle as the pressure is increased and hence non-zero contributions from the rich, lean and stoichiometric mixtures are observed from the jet exit. As seen in figure 14, the predominant heat release is along the shear layers even for the elevated pressures and hence the major contribution to $\langle \Omega \rangle$ comes from the fuel-rich side. Also, this non-premixed contribution is relatively larger than $\langle \Omega _l \rangle$ and $\langle \Omega _{{st}} \rangle$ for the elevated pressures because of the attached flames. However, the fractions of non-premixed contributions decreases substantially when the pressure is changed from 4 to 15 bar suggesting that the premixed combustion mode dominates both locally and in an averaged sense despite the fact that the fuel and air are injected through separate streams.