2.1. Facility and diagnostics

The experiments were conducted in a low-speed wind tunnel with a cross-section of $152\,\mathrm{mm} \times 152\,\mathrm{mm}$ and a length of 1.42 m. The boundary layer grows on the smooth, bottom surface of the wind tunnel, and is tripped using a cylindrical rod. A small tube with a diameter of $d_i/\delta =0.026\ (0.625\,\rm mm)$ was used to inject an acetone–air mixture at two heights $h_i/\delta =0.135$ and $h_i/\delta =0.307$ which correspond to the boundary-layer logarithmic and wake regions, respectively. We refer to these two injection cases as log-injection and wake-injection cases in this paper. This mixture was injected isokinetically at a velocity of $u_{\textit{inj}}$ at the two injection points such that $u_{\textit{inj}}=\overline {U}(y_{\textit{inj}})$ . Measurements were made starting $1.3\delta$ downstream from the injection point, with the field of view spanning $1.4\delta \times 2.0\delta$ in the streamwise–wall-normal plane. The schematic shown in figure 2 illustrates the general experimental set-up.

Figure 2. (a) Top view of wind tunnel set-up and cameras. (b) Schematic side view of experimental set-up. The field of view captures stage 2 evolution from figure 1.

To measure the flow field, simultaneous measurements of scalar and velocity fields were performed using two-component PIV and Ac-PLIF, with synchronised imaging performed from the opposite sides of the laser sheet (figure 2 a). A single dual-cavity Quantel Q-Smart Nd:YAG laser provided both 532 nm ( $380\,{\rm mJ}$ per pulse) and 266 nm ( $90\,{\rm mJ}$ per pulse) laser pulses at 10 Hz required for the simultaneous measurements. An extensive optical set-up was necessary to separate and recombine the desired wavelengths from the Nd:YAG laser for optimal performance. Since the raw quadrupled Nd:YAG beam contains $1064\,\mathrm{nm} + 532\,\mathrm{nm}+266\,\mathrm{nm}$ , a series of optics were used to isolate the $532\,\mathrm{nm}+266\,\mathrm{nm}$ wavelengths and dump the $1064\,\mathrm{nm}$ into a beam block. The combined $532\,\mathrm{nm}+266\,\mathrm{nm}$ beams were then focused through a series of spherical lenses and a sheet-forming cylindrical lens. This focused the laser into a thin sheet in the streamwise–wall-normal plane and was projected into the wind tunnel from the optical set-up underneath. Lastly, a BNC Model 575 timing box was used to control the timing of the flash-lamp and Q-switch delay for both laser heads, and the exposure window for both cameras. The PIV camera captured images from both laser pulses while the Ac-PLIF camera captured only the first pulse.

The Mie-scattered light was used to perform planar PIV in the field of view shown in figure 2 using a 4 MP Phantom V641 camera imaging through a $532\,$ nm band-pass filter. A Laskin nozzle filled with olive oil provided the seeding particles necessary for the PIV. The PIV image processing was performed using LaVision Davis 10 with a recursive, multi-pass refinement to a final interrogation window size of $16 \times 16\,{\rm px}^2$ with 50 % overlap. Lastly, in post-processing, outlier vectors in the PIV field were filled in using interpolation from the surrounding vectors.

For the injected scalar, an acetone–air mixture was created using an acetone bubbler apparatus consisting of a vertical steel pipe submersed in an acetone bath. Pressurised air entering from the top end of the pipe is forced through small holes at the bottom end of the pipe submerged in an acetone bath. A heater is wrapped around the acetone bath to increase the temperature and the acetone saturation concentration of the injected plume. The resulting acetone–air mixture created from the bubbler was diverted into two streams. One of the streams was controlled electronically using an MKS-GM50A mass flow controller, and injected into the boundary layer through the injection tube. The excess acetone was vented away. In this way, the injected acetone–air mixture is controlled such that the injected plume is isokinetic with the mean of the incident boundary-layer flow. The Ac-PLIF was performed by exciting the injected acetone gas with the $266\,\mathrm{nm}$ laser sheet. The fluoresced light from injected acetone was then imaged with an Andor Zyla sCMOS camera with a 500 nm short-pass filter. The use of this filter greatly reduced the stray fluorescence and reflections from the 266 and 532 nm light by the surrounding materials (mainly the walls/structure of the wind tunnel). The acetone concentration was small enough that absorption of the laser energy across the scalar field is minimal, and we could use an ‘optically thin’ assumption (Crimaldi Reference Crimaldi2008). This avoids the need for corrections of Beer–Lambert attenuation. The fields of view of both the PIV and PLIF cameras overlap and both fully capture the laser sheet. To spatially calibrate both fields and overlay them on the same physical space, a symmetric aluminium calibration target visible to both cameras was imaged in the location of the laser sheet. This allowed the PIV and Ac-PLIF data to be projected onto the same plane and simultaneous fields to be calculated. The final calibration and magnification resulted in a vector field resolution of $0.53\,\rm mm$ and a scalar field resolution of $0.078\,\rm mm$ .

2.2. Corrections to Ac-PLIF

The raw fluorescence images, $I_e$ (around 2640 uncorrelated, instantaneous snapshots), require several corrections to convert image intensities to quantitative mixture fraction, $\xi$ , measurements. In addition to the fluorescence images of the injected plume, background and laser-sheet profile, $I_{cal}$ images were also taken after each experiment. Background images were taken in the same conditions as those of the actual experiment but with no acetone being injected. This accounts for laser scatter and stray fluorescence/reflections. The laser-sheet profile images were taken by creating a stagnant chamber of acetone vapour of uniform concentration in the wind tunnel, and taking a small ensemble of fluorescence images.

It must be noted that due to the large laser energies involved and the presence of two wavelengths (380 mJ at 532 nm + 90 mJ at 266 nm), the anti-reflective coating and the optics, especially the sheet-forming cylindrical lens, were found to be susceptible to minor damage during the experiment. Due to this, the laser sheet develops striations and local changes in energy as is shown in figure 3(a) that translate into noise on turbulent statistics. It was found that this can be compensated to an extent by applying a row-wise parabolic fit to the laser-sheet profile that provides an ‘ideal sheet’ (figure 3 b) and accounts for these local imperfections. This was used in lieu of $I_{cal}$ to correct for spatial variations in laser energy.

Finally, after excluding a handful of outlier images (due to laser pulse malfunction), each of the fluorescence images (I) were corrected for background (BG) and laser energy (LS) variations to compute the acetone concentration fields (relative to acetone mole fraction in a calibration experiment), and scaled to injection concentration ( $I_0$ ) to yield the mixture fraction fields, $\xi$ :

(2.1) \begin{equation} \xi =\frac {I-{\rm BG}}{{\rm LS}-{\rm BG}}\times\frac {1}{I_0}. \end{equation}

Examples of the LS field, two illustrative initial fields, I, and the corresponding mixture fraction fields, $\xi$ , are shown in figure 3 for log- and wake-injection cases. Median and Gaussian filtering (3 pixels) were used to remove the speckle and high-frequency noise, respectively. The final fields were mapped to world coordinates using LaVision Davis 10 based on the calibration image so that they overlie onto the PIV vector field down to pixel resolution.

Figure 3. (a) Calibration laser intensity image, $I_{cal}$ . (b) Ideal laser intensity field, LS. (c,e) Example fluorescence image (I) and mixture fraction field ( $\xi$ ), respectively, for the log-injection case. (d,f) The same for the wake-injection case.

2.3. Boundary layer and injection parameters

Preliminary PIV data were gathered for three different cases to characterise the incident boundary layer, ensure isokinetic injection of the acetone and capture the effects of the physical presence of the injector on the boundary layer and the presence of acetone on the boundary-layer profile. This was to ensure that the injector pipe and acetone injection would not significantly alter the boundary-layer physics. This was conducted for both log injection and wake injection in three PIV experiments performed in the field of view of interest. (i) First, the unperturbed boundary layer without the injector was measured. This is referred to as the ‘baseline case’. Then, (ii) the injector was positioned at respective heights without injecting any acetone and the boundary-layer profiles were measured. This is referred to as the ‘no injection’ case. Finally, (iii) the injector was positioned at respective heights and acetone was injected under isokinetic conditions (relative to the mean flow). This is referred to as the ‘isokinetic injection’ case, which is the focus of the the current study.

The mean boundary-layer profiles are shown for all three cases for the log and wake injection in figure 4(a,b). For both injection locations, the ‘no injection’ case seems to shift the velocity profile downwards from the wall until around $y^+=300$ due to blockage of the injector stem. The ‘isokinetic injection’ case seems to help the boundary layer recover and more closely resemble the canonical boundary layer compared with the ‘no injection’ case. Except for minor differences, the boundary-layer profiles follow the same trend for baseline and isokinetic injection cases.

Figure 4. Boundary-layer profiles (a,b) and Reynolds stresses (c,d) for both injection locations.

We also compare the changes in in-plane Reynolds stresses, as shown in figure 4(c,d). The injection, no flow case and isokinetic injection case for the streamwise Reynolds stress deviate from the canonical case close to the wall, possibly due to the wake of the vertical injector stem. Away from the wall, the values of all three cases converge on the canonical value for the streamwise stress. Interestingly, there seems to be a jump in all three stresses at a location just before $y^+=200$ for both injection locations. This jump is more prominent and noticeable for the wake injection. Nonetheless, for both cases the stresses quickly converge back to a common value consistent with the canonical profile. Overall, the wake injection has more persistent deviation from the canonical case than the log injection, and for both cases the streamwise Reynolds stress has the greatest deviation from the canonical profile. Despite these deviations, the overall trends of the three stresses for canonical and isokinetic injection cases are very similar.

The relevant boundary-layer characteristics for the canonical case are summarised in table 1, and these values are used for all normalisations in the current work. Note that we use $\delta$ to refer to $\delta _{99}$ defined in table 1 for the reminder of the paper. The boundary-layer inner-scale parameters, namely $u_{\tau }$ , $y^*$ and $ \textit{Re}_\tau$ , were estimated by finding the best fit of the measured mean boundary-layer profile to the zero-pressure-gradient composite profile proposed by Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009).

Table 1. Incident boundary-layer characteristics.

2.4. Simultaneous PIV–PLIF snapshots

Four representative instantaneous fields are shown in figure 5 that display the overlapped velocity and mixture fraction data for log and wake injection. Gaussian low- and high-pass filters of lengths $1.25\delta$ and $2\times$ vector spacing, respectively, were applied to the velocity field to filter out the large-scale local advection and high-frequency noise (Adrian et al. Reference Adrian, Christensen and Liu2000a ). This demonstrates the ability to perform simultaneous field measurements of quantitative scalar mixture fraction and turbulent velocity flow fields and qualitatively highlights the turbulent dynamics responsible for scalar transport within the boundary layer. We utilise this capability to capture the spatial organisation of the scalar dispersion relative to the coherent motions of the boundary layer akin to those obtained by point measurements by Talluru et al. (Reference Talluru, Philip and Chauhan2018) and Talluru & Chauhan (Reference Talluru and Chauhan2020), and calculate scalar two-point auto- and cross-correlations using simultaneous spatial measurements.

Figure 5. Instantaneous snapshots of simultaneous mixture fraction ( $\tilde {\xi }$ ) and band-pass-filtered velocity fields for (a,b) log-injection and (c,d) wake-injection cases.

4.1. Two-point mixture fraction correlation

The spatial scalar fields enable us to compute the multi-point turbulent statistics directly, without the need for Taylor’s hypothesis. The two-point mixture fraction correlation, $R_{\xi \xi }$ , defined by (4.1), is computed to understand the spatial coherence of the scalar:

(4.1) \begin{equation} R_{\xi \xi }(\Delta x, \Delta y; x_r,y_r)=\frac {{\langle \xi^{\prime}(x_r,y_r)\xi^{\prime}(x_r+\Delta x,y_r+\Delta y) \rangle }}{{\langle \left (\xi^{\prime}(x_r,y_r)\right )^2\rangle }}. \end{equation}

These correlations represent the general shape, extent and inclination of the mixture fraction neighbourhood around $(x_r, y_r)$ (where a certain value of mixture fraction, $\xi^{\prime}$ , is detected). Previous studies (Miller Reference Miller2005; Talluru & Chauhan Reference Talluru and Chauhan2020) using point measurements (and Taylor’s hypothesis) have shown that the two-point mixture fraction correlation fields are inclined at an angle to the horizontal, where the location of maximum correlation above the reference location ( $y\gt y_r$ ) is ahead in streamwise location ( $x \gt x_r$ ), and vice versa. To investigate this phenomenon spatially using the current data, the correlation fields were calculated using (4.1) for $y_r = h_i$ at various streamwise locations, $x_r$ . Further, to achieve reasonable convergence, these fields were locally averaged over streamwise spans of $0.65 \delta$ and $0.7 \delta$ for the log and wake injection, respectively (assuming local homogeneity over those distances). To find the inclination angle, $\alpha$ , the maximum $R_{\xi \xi }$ was found for each wall-normal location and a line was fitted to these points. This process is illustrated in figure 10(a,c). The angle between this line and the horizontal is the angle of inclination, similar in approach to that of Talluru & Chauhan (Reference Talluru and Chauhan2020) (who did this using temporal, point measurements).

Figure 10. Angle of maximum mixture fraction correlation for (a,b) log injection and (c,d) wake injection. (a,c) Illustrative local correlation maps at $x/\delta =1.9$ for the two cases.

Figure 10(b,d) shows the variation in the inclination angle, $\alpha$ , with increasing downstream distance for log- and wake-injection cases. The inclination angle for the log-region injection decreases downstream showing a change of roughly $5^\circ$ in inclination angle across the measurement domain. This is likely due to the strong straining and rotation that the scalar plume experiences early on as it is transported away from the wall due to the turbulent mechanisms. Figure 10(d) shows that, for wake injection, the inclination angle is roughly constant. This can be associated with the smaller shear experienced by the plume when dispersed within the wake region.

Talluru & Chauhan (Reference Talluru and Chauhan2020) calculated and discussed this inclination angle using temporal measurements with two synchronised point sensors. From their measurements, they observed an inclination angle ranging from $29^\circ \text{ to }31^\circ$ depending on the injection height. Similar two-point scalar correlations using scalar field measurements were also performed by Miller (Reference Miller2005) for various downstream distances. They observed that the correlation contours start isotropic close to the injection location (i.e. $\alpha \approx 90^\circ$ ) and then increasingly incline with downstream distance (i.e. decreasing $\alpha$ ). These trends are similar to those observed in the current work. The exact streamwise variation of this angle close to a point source (i.e. stage 1 and stage 2 mixing) and the dependence on the boundary layer and injection/injector parameters are not studied in sufficient detail. It appears that these scalar concentrations approach the inclination angles comparable to that of $R_{uu}$ at very large distances from the injector (Tavoularis & Corrsin Reference Tavoularis and Corrsin1981; stage 3). Close to the injector, we suspect that this angle is a strong function of the advection velocity, distance and the mean vorticity experienced by the plume. From this standpoint, the relative size of the plume compared with the boundary-layer thickness ( $\sigma _y/\delta$ ) becomes relevant, which is directly proportional to the injector diameter. This is significantly different between the current work ( $d_i/\delta =0.026$ ) and Talluru & Chauhan (Reference Talluru and Chauhan2020) ( $d_i/\delta \approx 0.005$ ) which likely accounts for the differences between the two works. However, it is still unclear how the plume parameters ( $x/\delta$ , $y/\delta$ , $\sigma _y/\delta$ etc.) influence this inclination angle and further studies are necessary to understand these relationships.

5.1. Scaled probability of concentration at different injection points

Gifford (Reference Gifford1959) presented the meandering plume model (also discussed in Marro et al. (Reference Marro, Nironi, Salizzoni and Soulhac2015)) in which the plume mixing occurs through two simultaneous phenomena captured via their PDFs. The first component of this model is the meandering of the centre of mass ( $p_m$ in (5.1)) of the plume where large-scale structures transport the entire plume along the wall-normal and spanwise directions. The second component of the theory encompasses the local dissipation of the plume through small-scale turbulence and random molecular motion ( $p_{cr}$ in (5.1)):

(5.1) \begin{equation} p(\xi ;x,y,z)=\int _{0}^{\infty }\int _{-\infty }^{\infty }p_{cr}(\xi ,x,y,z;y_m,z_m)p_m(y_m,z_m; x)\, {\rm d}y_m\, {\rm d}z_m, \end{equation}

with $\xi$ denoting the mixture fraction, $x,y,z$ denoting the streamwise, wall-normal and spanwise positions, respectively, and $y_m$ and $z_m$ denoting the $y$ and $z$ locations of the centre of mass. Critically, the model assumes that the two processes are independent of each other, an approximation that holds in high- $ \textit{Re}$ boundary layers when the plume diameter $D_p \ll \delta$ (i.e. $p_{cr} = fn(\xi , x, y-y_m,z-z_m\ \textrm {only})$ ). This implies that, in the frame of reference of the plume’s centre of mass, its statistical characteristics (such as distributions of concentration, inclination angle, aspect ratio etc.) for a given $x$ will be independent of $y$ and $z$ . In other words, if the plume meanders to a given location, then its statistical characteristics should be identical to those of other wall-normal locations (though the probability of this meandering, $p_m$ , will be different).

Figure 12. Normalised PDF of $\tilde {\xi }$ at different wall-normal distances from the injection location for (a) log-injection and (b) wake-injection cases.

We evaluate this condition via the PDF of the average mixture fraction, $\tilde {\xi }$ , for both the log- and wake-injection cases. Figure 12 shows the scaled (by maximum value) blob concentration probabilities at different wall-normal distances from the injection point. The data suggest that while the average total concentration decreases further from the injection point, the distribution of blob concentration at an arbitrary distance from the injection point seems to be identical. Phenomenologically this suggests that blobs of different concentrations indiscriminately meander from the injection line such that the statistical distributions of the concentration of the blobs at any wall-normal distance form the injection line are identical. This supports the statistical independence of the meandering and dispersion mechanisms described in the meandering plume model. However, further studies, ideally with a well-converged dataset and a range of injection locations, are necessary to confidently establish the statistical independence between wall-normal location and distribution of blob mixture fraction.

5.2. Relationship between blob density and area

Using the blob database, we try to understand the mechanisms by which the discrete blobs ‘break up’ and ‘disperse’ into a continuous field, as was shown in figure 1. Both these processes change the area of the scalar blobs observed. An ideal ‘breakup’ occurs due to locally diverging vector topology (saddle-like topologies in plane), and preserves the average concentration of scalar in the blob (i.e. $\tilde {\xi }$ ). The ‘dispersion’ occurs due to micromixing by small-scale local turbulence ( $l\ll D_p$ ), and preserves the amount of substance in the blob (i.e. $\tilde {\xi }\tilde {A}$ ). Even though the transport is a three-dimensional phenomenon, these two processes can be statistically visualised using the distribution of blob areas and concentrations in the symmetry plane.

Figure 13. Joint PDF of $\tilde {\rho }$ and area $\tilde {A}$ for (a) log-injection and (b) wake-injection cases.

Figure 13 shows the joint PDF of scalar density (defined as $\tilde {\rho }=\tilde {\xi }/\tilde {A}$ ) and blob areas ( $\tilde {A}$ ) for the log- and wake-injection cases. Both cases show a distribution of $\tilde {\rho }$ and $\tilde {A}$ . For a pure breakup event, where a blob breaks up into smaller blobs, the area would decrease but the average mixture fraction $\tilde {\xi }$ would be conserved. A successive progression of this process will result in a distribution of blobs cascading up a curve of $\tilde {\rho }=k_1/\tilde {A}$ ( $k_1$ is an arbitrary constant depending on the initial concentration). The solid red line represents an example of this with $k_1=0.3$ ( $k_1=0.25$ provides the best fit assuming only blob breakup). On the other hand, should the blob change in physical size due to diffusion, the area changes to conserve the material, $\tilde {m}=\tilde {\rho }\tilde {A}^2$ . This results in a cascade along the line $\tilde {\rho } = k_2/\tilde {A}^2$ ( $k_2$ is a constant depending on initial blob mass). The dashed lines in figure 13 represent this diffusion cascade for several $k_2$ values illustrating how dispersion causes the blob evolution to stray from the purely breakup case (solid line). Given an initial distribution of blobs with large area and low $\tilde {\rho }$ (bottom right of figure 13), the evolution of these blobs along a purely breakup trajectory would account for the general shape of the joint PDF including some of the spread perpendicular to the $\tilde {\rho }=k/\tilde {A}$ line. The spread of the joint PDF perpendicular to this line of $\tilde {\rho }=0.25/\tilde {A}$ arises from both the variation in initial blob size and the dispersion along the dashed lines (diffusion). The strong trends of the joint PDF along the breakup line, and weak spread perpendicular to it, indicate that blob breakup is the primary mechanism of plume evolution in the streamwise extent considered. However, as the plume disperses within the boundary layer further downstream and the scalar spectrum develops length scales $\mathcal{O}({\delta })$ , dispersion is expected to play a more significant role, limiting the utility of this discrete blob approach.

The evolution of the blob area and mean concentration as a function of streamwise location can also be investigated to determine the dominant mechanism of plume evolution. If the plume evolution at this stage is breakup-dominated, then the average area should decrease as the streamwise position increases. Since breakup does not change the mixture fraction $\tilde {\xi }$ , it should stay relatively constant. Alternatively, if the plume evolution is dispersion-dominated, the opposite trends are expected. The average area of the blobs should increase with streamwise position, with an associated decrease in the mean mixture fraction. The trends of blob mixture fraction and area as a function of streamwise location can be seen in figure 14. The evolution of the average blob area ( $\overline {\tilde {A}}$ ) shows a marked decrease across the field of view. The intermittent peaks in the average blob area (figure 14 b,d) are likely caused by the lack of convergence. The average mixture fraction of the blobs ( $\overline {\tilde {\xi }}$ ) stays relatively constant ( ${\approx}\pm 3\,\%$ ) across the field of view. The marked decrease in area with downstream location seen in figure 14(b,d) and the constant value of mixture fraction across the streamwise locations (figure 14 a,c) support the breakup-dominated plume evolution previously discussed.

Figure 14. Mixture fraction of the blobs (a,c) and area of the blobs (b,d) at different streamwise locations for log (a,b) and wake (c,d) injection.

5.3. Aspect ratio and inclination angle of blobs

Besides the size and mixture fraction, the scalar blobs are also elongated and inclined to the horizontal at a certain angle. These characteristics were also investigated using the ensemble. For each blob, an ellipse was fitted to the outline (figure 11 b) with an equivalent area moment of inertia about the major and minor axes (MATLAB 2023). The $\widetilde {\boldsymbol{AR}}$ can then be calculated as the ratio of the major to minor axes ( $a/b$ ), and $\phi$ as the counterclockwise angle between the horizontal and the major axis of the fitted ellipse. Its important to note that the blobs rarely formed perfect ellipses; however, using this approach provides a consistent method to calculate these two quantities which provide insight into the statistical nature of the blobs’ evolution. Inclination angle provides insight into how the blobs tend to rotate and how they are formed, while $\widetilde{\textit{AR}}$ gives insight into how the blobs tend to be stretched and elongated. Clear trends can be seen in figure 15 for $\widetilde {\boldsymbol{AR}}$ and inclination angle $\tilde {\rho }$ at both injection locations. A normal distribution was fitted to the inclination angle, which provides a good fit to the data. A skewed normal distribution was used to fit the aspect ratio providing a good estimate of the observed measurements.

The constants in table 2 summarise the various constants for the distribution (mean $\mu$ and variance $\sigma$ ). It can be seen that there is a clear positive preference for the inclination angle of the blobs in both injection regions. The log injection has a slightly larger inclination angle at $ 12.5^\circ$ compared with $10^\circ$ in the wake injection. This is due to larger shear that the blobs experience as they evolve from the injection height. The standard deviations for both regions are statistically identical. For the aspect ratio, both regions have a peak around 1.6–1.7 with a steep decline in PDF afterwards. Both regions show a strong preference for an elongated $(AR\gt 1)$ shape, with the majority of the blobs having an aspect ratio $1\lt AR\lt 3$ . The only notable difference between the aspect ratio distribution for the log and wake injection is the wake distribution in aspect ratio is flatter than in the log region, likely due to higher mean shear experienced by the log-injected plume and leading to a larger variation in aspect ratios.

Figure 15. Joint PDFs and PDFs of aspect ratio ( $\widetilde {\boldsymbol{AR}}$ ) and inclination angle ( $\tilde {\phi }$ ) for log- and wake-injection cases.

Table 2. Best-fit distribution constants for the log-injection (left) and wake-injection (right) cases.

6.1. Instantaneous observations

With the previous observations indicating that the scalar plume is highly intermittent and dispersed into discrete blobs, we can focus on the role of specific turbulence structures in developing the spatial trends. In other words, a continuous injection of the passive scalar is reorganised into intermittent and concentrated pockets by the action of turbulent structures of similar scale. Figure 16 shows instantaneous snapshots of this organisation, in an arbitrary frame of reference moving with the fluid for the two injection configurations. Also shown are the contours of swirling strength, $\lambda _{\textit{ci}}$ , that represent the local rotational motion of the fluid elements due to strong vortices (the swirling motion is not immediately evident in the vector field due to advection). The swirling strength $\lambda _{\textit{ci}}$ is defined as the imaginary part of the complex eigenvalue of the velocity gradient tensor, and local maxima in its value indicate the location of a vortex element (Chakraborty, Balachandar & Adrian Reference Chakraborty, Balachandar and Adrian2005). The current experiments are only capable of capturing the vortex elements sliced perpendicularly by the measurement plane, and the sign of the vorticity is assigned to identify the ‘prograde’ ( $\lambda _{\textit{ci}}\leqslant 0$ , clockwise) and ‘retrograde’ ( $\lambda _{\textit{ci}}\gt 0$ , anticlockwise) vortices (Adrian et al. Reference Adrian, Christensen and Liu2000a ).

Figure 16. (a–d) Instantaneous snapshots showing scalar plume together with contours of swirling strength. Vectors are shown in a frame moving at ${\approx}0.88 U_\infty$ .

A qualitative organisation of the scalar plume relative to the vortex cores is observed in figure 16(a,b) – a trend that occurs around 25 % of the vector fields. This trend is particularly dominant if the fields are conditioned on a scalar blob occurring far away from the injection location ( $y-y_{i}\gt 2\sigma _y$ ). It can be seen that a discrete scalar blob occurs preferentially between the vortex cores, and at an angle inclined ${\approx}\,15^\circ {-}18^\circ$ . Additionally, multiple vortex cores are observed to be aligned along this line with the distinct signature of the coherent vortex packet (Adrian et al. Reference Adrian, Meinhart and Tomkins2000b ; Christensen & Adrian Reference Christensen and Adrian2001; Adrian Reference Adrian2007). This alternating inclined scalar-blob–vortex-core organisation was less prevalent when the injection was in the wake region, as illustrated in figure 16(c,d). The scalar plume was still dispersed and intermittent, and the discrete scalar blobs were mostly attached to a dominant vortex core. However, coherent interlacing of scalar blobs with an inclined set of coherent vortices extending all the way to the wall was not observed for wake injection. This indicates that the action of the vortices (possibly from the detached packets) on the scalar plume is still the dominant cause of the scalar plume intermittency, however without the spatial coherence. Finally, it must be noted while interpreting these snapshots that we are only measuring a single plane, and that we are limited to observing these coherent trends only when the coherent vortex packet is aligned with the measurement/injection plane.

6.2. Statistical footprint with coherent vortex packets

6.2.1. Conditional structure for log injection

Based on these qualitative observations, we can devise a statistical approach to investigate the dominance of these trends via conditional fields. However, since the conditional averages require a large quantum of data, we estimate the same using the linear stochastic estimation approach previously employed to observe similar spatial trends (Adrian et al. Reference Adrian, Jones, Chung, Hassan, Nithianandan and Tung1989; Adrian Reference Adrian1994). The conditional field of a quantity $Q (\overline {x})$ conditioned on the occurrence of a second quantity (‘event’) $Q_E(\bar {x}_{r})$ at reference location $\bar {x}_{r}$ is given as

(6.1) \begin{align} \langle Q(\bar {x})|Q_{E}(\bar {x}_{r})\rangle \approx \frac {\langle Q(\bar {x})Q_{E}(\bar {x}_{r}) \rangle }{\langle Q_{E} Q_{E}\rangle } Q_{E}(\bar {x}_{r}). \end{align}

That is, the two-point correlation function embodies the information of the conditional field to first order, when conditioned on a single scalar quantity. Since the aforementioned coherent reorganisation was found consistently upstream when the scalar plume is transported far away from the injection location, we first choose this as the reference event, i.e. $\bar {x}_{r}=(2.75\delta , 0.3\delta \equiv h_i + 1\sigma _y)$ to estimate the concentration ( $\langle \xi^{\prime}(\bar {x})|\xi^{\prime}(\bar {x}_{r})\rangle$ ) and velocity fields $\langle \bar {u}(\bar {x})|\xi^{\prime}(\bar {x}_{r})\rangle$ . This is shown in figure 17(a), that depicts the conditional concentration fluctuation and velocity fields in the field of view. A few aspects are immediately obvious from the conditional field. The correlation near the wall at $x\approx x_{r}$ reaches negative values, which implies that a high-concentration event away from the wall is accompanied by a low concentration close to the wall at the same streamwise location. This is expected for a conserved, meandering scalar. The conditional velocity fields around the reference location, $\bar {x}_{r}$ , also exhibit a positive $v^{\prime}$ and negative $u^{\prime}$ (an ‘ejection/Q2’ event). This is entirely consistent with the turbulent flux at $\bar {x}_{r}$ , discussed earlier in § 4. The most striking observation in the conditional scalar field is that a high-concentration event at $\bar {x}_{r}$ (we refer to this region of coherence as ‘primary blob’, P) is accompanied by two ‘secondary blobs’ of positive concentration fluctuation, upstream and closer to the wall (marked by Sin figure 17 a). The line joining these discrete, high-concentration regions is inclined at ${\approx}10^\circ$ , consistent with the inclination angle of coherent vortex packets (Adrian et al. Reference Adrian, Meinhart and Tomkins2000b ; Christensen & Adrian Reference Christensen and Adrian2001; Adrian Reference Adrian2007). The robustness of these discrete secondary blobs is also observed in a statistically independent, but complementary measure in figure 17(b). Here the correlations for a positive concentration fluctuation in a region close to the wall, i.e. $\langle \xi^{\prime}(\bar {x})|\xi^{\prime}(\bar {x}_{r})\rangle$ and $\langle \bar {u}(\bar {x})|\xi^{\prime}(\bar {x}_{r})\rangle$ with $\bar {x}_{r}=(2.75\delta , 0.04\delta )$ , are shown. Not only does the anti-correlation at the outer location corresponding the primary concentration blob at $\bar {x}=(2.75\delta , 0.3\delta )$ appear (as expected), but we also see the anti-correlation blobs at the secondary locations ( $\bar {x}\approx (1.95\delta , 0.18\delta )$ and $(1.6\delta , 0.1\delta )$ ) at exactly the same locations. This relative correlation structure by two independent measures demonstrates the robustness of this spatial organisation. Further, these observations are insensitive to the exact location of the $\bar {x}_r$ chosen, as long as it is in the vicinity of the primary blob seen here. Finally, the conditional structure in figure 17(b) indicates a consistent Q3 event (i.e. $u^{\prime}\lt 0, v^{\prime}\lt 0$ ) that is unusual to observe in a boundary layer. This structure appears in the relative frame as a pair of counter-rotating vortices, and demonstrates that the scalar below the injection location occurs predominantly in the low-momentum regions, indicating a strong relation with a UMZ. This observation was insensitive to the choice of $x_r$ for all choices of $y_r\lt y_{\textit{inj}}$ .

Figure 17. Conditional fields of scalar concentration perturbation and relative fluid velocity. (a,b) Conditioned on a positive scalar fluctuation and vectors show the velocity field. (c) Conditioned on a clockwise swirl event and vectors show velocity direction. Blue circles indicate the reference locations for each event.

If we were to hypothesise that this discrete spatial coherence in scalar concentration occurs due to the action of a coherent vortex packet, as was seen in figure 16, the same should be observable in the conditional estimates based on the occurrence of a swirl event, i.e. $\langle \xi^{\prime}(\bar {x})|\lambda _{\textit{ci}}(\bar {x}_{r})\rangle$ . A swirl event for $y_{r}\gt y_{\textit{inj}}$ was found to generally have a positive mixture fraction, $\xi^{\prime}$ , below–upstream and a negative concentration above–downstream of the reference location. This velocity–concentration field is consistent with that observed for a primary blob in figure 17(a). Thus to induce the primary scalar blob at the same location, i.e. $\bar {x}=(2.7\delta , 0.3\delta )$ , a swirl event at $\bar {x}_{r}\approx (2.77\delta , 0.37\delta )$ is considered, as shown in figure 17(c). The typical signatures of a coherent vortex packet (Adrian et al. Reference Adrian, Meinhart and Tomkins2000b ; Christensen & Adrian Reference Christensen and Adrian2001; Adrian Reference Adrian2007) in the conditional velocity field ( $\langle \hat {u}(\bar {x})|\lambda _{\textit{ci}}(\bar {x}_{r}=(2.77\delta , 0.37\delta ))\rangle$ ) are reproduced accurately (for clarity, velocity direction $\langle \hat {u}|\lambda \rangle = \langle \bar {u}|\lambda _{\textit{ci}}\rangle /|\langle \bar {u}|\lambda \rangle |$ , with unit magnitude, is shown in figure 17 c). Specifically, the primary hairpin head, and two ‘younger’, secondary hairpins appear with an inclined shear flow between them. Remarkably, the conditional scalar field shows the three positive peaks associated with the primary and secondary blobs at their respective locations identified in figure 17(a). This shows direct correlation between the coherent vortex packets and the scalar concentration blobs, as was hypothesised from instantaneous observations in figure 16(a,b). Additionally, the scalar blobs seem to be isolated from each other with influx events from the upstream vortex and a VITA event (saddle-point-like vector topology, marked V; Adrian Reference Adrian2007) that serve as topological barriers for the passive scalar. This provides a dynamical mechanism based on the coherent vortex packets for origination and development of discrete coherent scalar packets.

First, it must be noted that the evidence presented in figure 17(a,b) does not, by itself, substantiate the conclusion that the vortex packets are responsible for the coherent scalar arrangement. They only demonstrate that the scalar is distributed in discrete packets along an inclined line. This arrangement was found to be very robust and insensitive to the choice of $x_{r}$ (for a similar $y_{r}$ ). The relation between the coherent vortices and the relative scalar field is seen only when a conditional swirl event is observed (figure 17 c). As we present it here, the equivalence in the two observations is hypothesised only due to the compelling similarities in the two independent measures, and additional trends discussed in Appendix A. Additionally, we do not notice secondary swirl in the velocity fields conditioned on concentration event in figure 17(a,b). This could be because of a combination of the following factors:

1. (i) The advection of the high-concentration blob will dominate the velocity features observed in the conditional average, rather than the swirl and the long-range spatial coherence.

2. (ii) We expect a high-concentration blob to span a larger spanwise extent than the swirl field. This results in averaging over velocity features that are not in the symmetry plane, and in decreasing the long-range correlation of coherent vortices (dominant only in the symmetry plane). The swirl condition (figure 17 c) does not suffer from this as it decays faster in the spanwise direction (due to hairpin shape) and the misaligned cases do not contribute to the correlation. Figure 17(a) represents an average that is agnostic to this distinction. We demonstrate via analysis presented in Appendix A that about $50\,\%$ of the concentration events occur coupled with a swirl event. A direct average conditioned on both events does elucidate a weak secondary swirl (in figure 22) coupled with a secondary concentration blob.

Figure 18. (a,b) Same as figure 17(a,c), but for a different reference location.

In addition to the primary blob as reference, we investigate the conditional fields based on the occurrence of a near-wall secondary blob, i.e. a discrete concentration event at a location $\bar {x}_{r}=(1.97\delta , 0.17\delta )$ . This is shown in figure 18(a). No correlated downstream–outer peak in concentration representative of the earlier primary blob is observed, indicating that the concentration fluctuations at this location are not necessarily correlated with a coherent arrangement of concentration blobs (as was seen in figure 17 a). This indicates that there are other mechanisms at play beyond that of a large attached, coherent vortex packet that discretise the injected stream of passive scalar into pockets at this location. In other words, an occurrence of a discrete blob at an injection location is not necessarily an indication of a coherent vortex packet. This is entirely consistent with qualitative observations made from the instantaneous snapshots where, on more occasions than not, a discrete scalar blob at the injection location is not accompanied by an inclined arrangement of coherent blobs. Finally, if we consider the occurrence of a vortex event ( $\langle \xi^{\prime}(\bar {x})|\lambda _{\textit{ci}}(\bar {x}_{r})\rangle$ ) close to the wall instead of a concentration event ( $\langle \xi^{\prime}(\bar {x})|\xi^{\prime}(\bar {x}_{r})\rangle$ ), we do see a corresponding coherent arrangement of discrete blobs along the inclined shear layer, as shown in figure 18(b). This definitively demonstrates the action of the coherent vortex packets in the discretisation and transport of the passive scalar along the shear layer. Finally, it is worth noting that the observations discussed here were insensitive to the exact values of $\bar {x}_r$ shown here, as long as they were in the same approximate regimes.

Figure 19. An illustrative snapshot qualitatively showing ramp-like UMZ features and relative arrangement of vortex packets and scalars. Velocity field shown relative to a moving frame of reference, $U_{rel}=0.9U_\infty$ . Magenta contours represent contours of high $|\lambda _{\textit{ci}}|$ , and scalar contours are the same as in figure 16.

We can contextualise these observations with those observed in prior works that evaluated the large-scale structure associated with scalars (Antonia & Fulachier Reference Antonia and Fulachier1989; Metzger Reference Metzger2002; Laskari et al. Reference Laskari, Saxton-Fox and McKeon2020; Eisma et al. Reference Eisma, Westerweel and van de Water2021). Specifically, the works established the conditional large-scale structure associated with ‘coolings’ (equivalently the trailing edge of a large, low-momentum bulge inclined in the streamwise direction). The works of Antonia & Fulachier (Reference Antonia and Fulachier1989) established that these structures have a saddle-like topology in local turbulence, and Laskari et al. (Reference Laskari, Saxton-Fox and McKeon2020) established the fine-scale structure of the same, demonstrating a smaller-scale vortex distribution. The role of these structures in a point-source injected scalar is perfectly demonstrated in the conditional fields (figure 17), and the illustrative instantaneous field shown in figure 19. This integrates the advection arguments of Antonia & Fulachier (Reference Antonia and Fulachier1989) and Laskari et al. (Reference Laskari, Saxton-Fox and McKeon2020), the coherent vortex packets (Adrian Reference Adrian2007) and the observance of these as UMZs. Specifically, the trailing edge of the UMZs represent the ‘cooling’ events (Antonia & Fulachier Reference Antonia and Fulachier1989), and the vortices arranged represent the finer structure observed in Laskari et al. (Reference Laskari, Saxton-Fox and McKeon2020). Both these works use temperature as a proxy for low-momentum regions. The current work demonstrates that the injected scalar ‘hitch-hikes’ on the ‘cooling events’ resulting in a large meander event. In the process, the saddle-like topology induces the spatial discretisation observed both instantaneously (figure 19) and stochastically (figure 17 c).

Figure 20. (a,b) Same as figure 17(a,c), but for wake injection.

Figure 21. Schematic of meandering of the plume by the coherent vortex packet.