Representative Phenomena of Cyclic Turbulent Combustion in High-Pressure Fuel Sprays

Cyclic variations in conventional diesel combustion engines can lead to large differences in engine out emissions even at steady operation. This study uses an optically accessible constant-pressure flow chamber to acquire fuel injections in quick succession to analyze mixing, auto-ignition, and combustion of diesel-surrogate n-heptane using multiple high-speed optical diagnostics. Prior studies have utilized fewer injections and/or they rely on analysis of ensemble average behavior. These approaches do not yield information on injection-to-injection variation or provide confidence in utilizing individual injection measurements for high-fidelity computational fluid dynamics(CFD) model validation. In this study, a large set of 500 injections is used to obtain global parameters including liquid length, vapor penetration length, ignition delay time, and lift-off length. Results for multiple injections are presented to illustrate large injection to injection variations. Potential sources for these variations are analyzed to conclude localized, small scale turbulence and rate of injection variations as the likely sources. Then, a statistical method based on z-scores is proposed and implemented to identify instantaneous injections that best represent the bulk data-set of jet boundaries measured independently by three different diagnostics. This synthesis of statistics-guided screening of data set and ensemble-average analysis offers higher confidence for CFD model validation relying upon both a representative single and average injection results.


Introduction
The margins for gains in combustion efficiency for transient combustion events found in compression ignition 'diesel' engines have become smaller. Increased understanding of the fundamental physics that governs the spray breakup, vaporization, fuel-air mixing, ignition, and combustion processes, still offers the opportunity to mitigate in-cylinder criteria emissions (e.g. CO, NOx, THC) and total greenhouse gas (GHG) production through the use of drop-in biomass derived fuels. However, relating properties of these emerging fuel candidates to expected 'engine out' emissions remains a significant challenge in the screening effort. Part of this challenge can be attributed to the lack of clarity obtained from combustion experiments, even with exhaustive optical diagnostics. From droplet combustion to spray vessels, rapid compression machines, and optical engines, the ability to study various flow and combustion processes becomes increasingly challenging. Typically this results in a limited number of fuel injection events (in the case of spray chambers) acquired at steady operating conditions. The analysis then focuses on ensemble average behavior that may not represent an individual experiment/injection. As evident in the literature and investigated in this study, ensemble average analysis does have significant limitations since the injection-to-injection variation in the combustion process can be vast even under very steady operating conditions. Further, in both spark and compression ignition engines, the cycle-to-cycle variations represent a significant challenge to researchers and engine calibrators to push efficiency and reduce emissions. Kircher et al. (2022) demonstrated through simulation calibrated to experimental results that local variations in temperature coupled with effects of negative temperature coefficient (NTC) drive a wide range of main ignition delays. In that case, as ignition occurs from near piston top center up to 10 • later resulting in the maximum cylinder pressure for a cycle observed being 36% higher than lowest cycle. Experimentally, Aleiferis et al. (2015) studied the impact of ambient flow field on the spray under non-and flash boiling sprays and noted large impact for flash boiling cases. To provide some measure of turbulence they report mean and RMS spray images results though focus on discussion of ensemble average behaviors.
The limitations of ensemble average results are not unique to combustion phenomena. For example, to simplify and streamline the costly manufacture of aircraft cockpits, the United States Army Air Force measured key dimensions of over 4,000 of its pilots in the 1950's. Ten of the most crucial design dimensions were chosen to confirm the one-sizefits-all design outlook, and the number of pilots that fell into the middle 30% of all parameters was determined to be zero. Not a single pilot was 'average' by the liberal definition of falling within the middle 30% of ten key parameters. Reducing this definition to include only three parameters excluded all but 3.5% of the pilots (Rose 2016). A popular 1945 contest centered around the 'Norma' statue (Creadick 2010) arrived at the same conclusion; even in seemingly repeatable circumstances, nature exhibits remarkable diversity, and being normal never looks 'normal'.
Beyond extracting typical behavior from experimental results, the challenge also extends to the reliability of model based predictions as they are built and calibrated against measurements. While increasingly high-fidelity computational fluid dynamics (CFD) models of spray, combustion, and flame stabilization offer insights into the transient combustion process (Pei et al. 2013), these models must be validated against experimental data. Large Eddy Simulations (LES) offer a significant advance in predicting localized turbulent combustion behaviors over Reynolds Average Navier Stokes (RANS) models (Liu and Haworth 2010;Li et al. 2017;Schmitt et al. 2015). However, it is not possible to meaningfully compare results from one experiment to a single LES realization (Sornette et al. 2007) because of the large injection to injection variations. Though both LES and RANS approaches can predict fluid flow and some global combustion parameters reasonably well (Som et al. 2012), the fidelity and accuracy required to predict the bulk behavior (e.g. cylinder pressure) versus emissions is vastly different. This is true experimentally as well, for example, Bizon et al. (2009) utilized an optically accessible engine to report large injection to injection variations of in-cylinder luminosity at conventional diesel conditions, but less significant differences in pressure-traces. Likewise, Wright observed repeatable ignition delay and pressure-traces but "substantial variations" in ignition location while testing n-heptane combustion at high pressures/temperatures in a constant volume chamber (Wright et al. 2010). Donkerbroek et al. (2011) observed significant cycle-to-cycle variations in OH* chemiluminescence (OH*CL) signal in an optical engine. Pal et al. (2018) identified a single representative pressure trace out of a sample of cycles in spark-ignition engines. Maes et al. (2020) reported results from 200 injection events quantifying variation in soot mass at a specific location in the spray. Payri et al. (2015) and Wanstall et al. (2019) both used probability maps to define spray boundaries under various conditions. It is clear that sufficiently large datasets are therefore necessary to extend the understanding of injection to injection variations and to support CFD model validation. These datasets should provide spatiotemporally resolved measurements, typically from optical diagnostics. A rigorous statistical analysis could then identify/isolate a smaller set or a single case representing the entire datasets.
Large data-sets, while important for reliable statistical analysis, are prone to include a wide range of disparate outliers. Xuan et al. describe massive injection to injection variations in soot formation across 30-40 injections, recorded simultaneously with two colorpyrometry (2CP) and diffuse back-illumination (DBI); an otherwise nominal injection had a propensity for 'non-sooting' at particular thermodynamic conditions; this behavior was attributed to strong fluctuations of in-cylinder air flow caused by the piston movement Xuan et al. (2019). Other work by Panagiotis has shown the strong impact of cyclic incylinder pressure oscillations on average values Kyrtatos et al. (2014) in engine tests, and highlights the potential of LES in explaining these wide variations, as done by Li and others Li et al. (2017); Schmitt et al. (2015). Therefore, combustion models must be capable of predicting the range of variation, but model validation requires a reference 'nominal' behavior, which is difficult to isolate in a transient, turbulent combustion process Clark and Kook (2018). This work seeks to identify the 'nominal' behavior from a large dataset of repeated injection/combustion experiments in a constant pressure flow chamber (CPFC) to provide context to LES modeling efforts, as has been done in other fields where deviation from normalcy proves to be widespread.
The stochastic nature of turbulent flows has produced a long history of statistical tools in combustion research. Turbulent combustion models often involve assumptions such as assumed probability distributions to predict key phenomena at limited spatial resolution Burluka et al. (1997). Modern combustion systems take advantage of improved abilities to collect vast amounts of data-optical diagnostics are increasingly common, as are high sampling rates-to better utilize statistical tools. For example, modern aviation gas turbines collect over 2.5TB of data per day, and terms such as Combustion Machine Learning (CombML) have been coined Ihme et al. (2022). Though this work does not leverage advanced statistical tools, the process of identifying 'nominal' or representative behavior from amongst a large data-set provides a targeted approach compared to data reduction through averaging, which can lead to misleading observations. The methodology of 1 3 identifying a representative individual injection can be extended to various diagnostics or conditions to delineate most appropriate single-event results.
This work presents a methodology to describe the key features and deviations among transient, reacting turbulent jets to support improved modeling and model validation of the same. The methodology is built on multiple injections at fixed test conditions to illustrate the limitations of ensemble average analysis and to make the case for a representative injection. Hundreds of transient spray combustion experiments are performed in quick succession in a high-pressure, high-temperature environment, and fuel injection, mixing, and combustion processes are probed at high spatial and temporal resolutions simultaneously by three optical diagnostics, each focused on different aspects of the jet behavior. The diagnostics include Rainbow Schlieren Deflectometry (RSD) to visualize the early, noncombusting jet Reggeti et al. (2021); Parker et al. (2021), OH*CL to detect ignition and reaction zones Blocquet et al. (2013), and 2CP to quantify produced soot mass Xuan et al. The remainder of the paper presents the specifics of the experimental apparatus, test conditions, diagnostics and data processing in Sect. 2, followed by quantitative and visual analysis of the injection-to-injection variations in Sects. 3.1-3.2, and demonstrates stability of test conditions in Sect. 3.3.1. Finally, a methodology to identify a representative injection is presented in Sect. 3.4 together with a set of representative injections before summary and conclusions.

Experimental Setup
The Constant Pressure Flow Chamber (CPFC) used in this study is illustrated in Fig. 1 is a continuous flow system, maintaining a nearly quiescent ambient environment in which to study fuel injection processes and enable the acquisition of about ∼500 injections in two hours Parker et al. (2021); Wanstall et al. (2019Wanstall et al. ( , 2020; Reggeti et al. (2021). Pressurized, electrically preheated air is supplied to the chamber at an average speed of about 0.5 m/s (nearly three orders of magnitude less than the fuel injection velocity) in a counter flow arrangement, as shown in Fig. 1. The flow velocity is less than that in direct-injection studies Aleiferis et al. (2015), and is much lower than the ambient velocity in diesel engines Xuan et al. (2019). Air enters through a diffuser section followed by a flow conditioner consisting of six 0.5 mm thick, 100 micron mesh screens to break up eddies and uniformly distribute the flow across the CPFC. The air exits at the top of the rig through four 3 mm diameter holes, placed symmetrically around the injector tip. An upstream dome regulator controls air supply pressure, and the air flow rate is regulated by a downstream control valve. Fuel is injected by a Bosch CRIN3-18 injector modified to have a single 104 m hole at the tip to create a spray along the axis of the rig.
A pneumatic pressure multiplying pump (300:1 ratio) provides fuel pressurization up to 2000 bar and is controlled remotely via an air pressure regulator feeding the pump. This type of pump results in low frequency (0.2-1 Hz) pressure fluctuations as the piston cycles. After reaching the desired ambient pressure and temperature conditions, the injector is triggered on the rising-edge of fuel pressure passing into the narrow tolerance window of 15 bar. An injection-to-injection dwell time of at least 8 s ensures that fresh air has completely displaced any residual combustion products (longer dwell times occur as the fuel pressure cycles before the next rising-edge trigger point). The fuel temperature is maintained at 355 K by circulating coolant in a jacket around the injector. The ambient air is maintained at nominal conditions of 800 K and 3 MPa (13 kg∕m 3 ). Table 1 shows the test conditions and their variations during the experiment.

Diagnostics
This experiment used three optical diagnostics to simultaneously acquire data for each injection consisting of liquid injection, vaporization, mixing, auto-ignition, flame stabilization, combustion, and soot formation, followed by venting of the combustion products. The diagnostics are RSD, OH* chemiluminescence (OH*CL), and 2CP. A schematic depiction of these diagnostics in Fig. 2 shows the arrangement of each high-speed camera. Table 2 provides additional details such as camera spatial (pixel) resolution, sampling rate, and exposure time for each diagnostic.
Rainbow Schlieren Deflectometry is used to quantify the deflection angle of light rays passing through a medium, in this case the vapor-jet region of the fuel spray. As shown schematically in Fig. 3 from left to right, broadband light (from an Energetiq EQ-99X fiber coupled broadband light source) is first refocused by two 75 mm focal length, 50 mm diameter lenses onto a 3 mm high and 100 m wide rectangular aperture located at the focal point of a 75 mm diameter, 250 mm focal length achromatic doublet lens Wanstall et al. (2017); Agrawal and Wanstall (2018); Wanstall et al. (2019). The collimated broadband light is directed through the test chamber. Density gradients deflect the light rays (some light will also be absorbed or scattered by liquid jet/spray) which exit through the parallel window on the chamber. Then, a matching achromatic doublet lens refocuses the light rays onto a rainbow-filter-a transparent rectangular strip with very fine, digitally printed color (hue) gradations-placed at the focal point as shown in Fig. 3. The undeflected light rays (dashed blue ray) pass through the cyan color at the center of the filter, which forms the background color in the schlieren image. A deflected light ray will focus at some (transverse) distance away from the center of the filter (solid blue ray). Thus, rays are color coded based on the transverse displacement on the rainbow filter. The filtered rays create the color (hue) schlieren image, while absorption and/or scattering by liquid region will reduce the ray intensity that is recorded independently (red ray). A Photoron Nova S9 color camera, in conjunction with a 50 mm Nikon Nikkor lens, records the schlieren images as described in Table 2. The RSD has been used to quantify the density field in non-reacting jets Wanstall et al. (2020Wanstall et al. ( , 2021, although in this work, intensity and hue signals are used to detect liquid and vapor boundaries, respectively. The specific methodology to identify liquid and vapor boundaries is presented in prior work Wanstall et al. (2019Wanstall et al. ( , 2017. The RSD diagnostic improves upon traditional shadowgraphy and other black-andwhite imaging techniques Parker et al. (2021); Agrawal and Wanstall (2018). Fo example, both the presence and degree of jet fluctuations can be quantified with the RSD hue signal. As in traditional schlieren and shadowgraphy, frame-to-frame deviations can be revealed by subtracting consecutive images. However, hue makes the resulting values more meaningful and, in this work, three distinct bands of frame-to-frame huevariation are utilized. Lowest band pertains to the nearly imperceptible fluctuations at each pixel location in the image background, which are often very nearly zero in the quiescent flow in the CPFC. Highest band of variations refers to the pixels within the jet core, where the dense fuel jet (large hue displacements) moves quickly (large frame-toframe shifts). Between these two hue bands is the shear layer of the jet periphery, where high intensity and background-like hues make it difficult to distinguish the jet from the background. By utilizing the hue signal of RSD and subtracting subsequent RSD image, this region becomes easily distinguishable as the true boundary of the vapor jet. This method of vapor detection is used in this study together with a typical threshold sensitivity analysis.
OH*CL at the 310 nm wavelength is an indicator of the reaction zone in the spray. As shown in Fig. 2, OH*CL signal is directed towards an Invisible Vision ultraviolet (UV) intensifier equipped with a 310 nm band pass filter and UV focusing lens. The intensifier utilizes a phosphorous screen to produce visible light from UV (with a gate time of 70 s in these experiments) which is acquired by a monochromatic Photron SA5 camera as described in Table 2. Per ECN definition ECN EN (2020)(2020), the farthest upstream location detected via chemiluminescence is the reported lift-off length. The algorithm recommended by the ECN was applied to OH*CL images to reliably differentiate OH*CL signal from the noise-floor; the first frame of detectable OH*CL signal was considered the ignition time.
The dataset includes full field 2CP measurements of the thermal radiation from soot, which dominates flame luminosity in the visible spectrum. The general layout of the 2CP optics and camera can be seen in Fig. 2. The flame luminosity passes through a 50/50 beamsplitter, which transmits half of the light and reflects the other half at 90 • angle. Band-pass filters of central wavelengths (650 nm and 550 nm) each with 10 nm full width at half maximum bandwidths are attached to the beam splitter outlets. Each spectral signal is reflected by a turning mirror towards a knife-edge prism mirror, and then the two signals travel along parallel paths to the camera. This optical design ensures that both spectral signals have equal path lengths and are imaged on the same camera sensor without inducing parallax and path length errors common in existing 2CP designs. Further details of the optical setup and calibration procedures for this 2CP design can be found in Reggeti et al. (2019Reggeti et al. ( , 2022. A Phantom v7.3, 14-bit, monochromatic camera with a Nikon Nikkor 105 mm lens is used as per Table 2. In this work, the relative difference in soot radiation intensity between injections is considered in relation to RSD and OH*CL signals.

Image Alignment and Analysis
The images of an injection using each of the three diagnostics will be presented to assess the fuel jet. Different wavelengths/optics used for each diagnostic require different arrangements and focus. Further, the regions of interest for the three optical methods are not identical, but they do have significant overlap to help with the alignment. Because images are recorded simultaneously by three high-speed cameras with different resolutions, it aids analysis to consolidate the information into a single, spatially aligned image. As shown in Fig. 2, all three cameras view the jet along the center-plane of the CPFC (OH*CL and 2CP view angles are exaggerated for illustration). The structure of the reaction front observed across the three diagnostics aligns very closely, the slight difference in view angles is insignificant. Figure 4 presents the images recorded simultaneously by each optical diagnostic. The RSD (recorded in red-green and blue colors) is presented with hues as recorded; saturation and intensity have been increased for clarity. The OH*CL intensity (recorded in monochrome) is presented in yellow, and soot luminescence at 650 nm (also in monochrome, from 2CP) is shown in magenta with varying brightness according to the raw signal intensity; both OH*CL and 2CP images have been outlined in red where the signal fades to background noise. These colors were chosen to provide maximum visual contrast among the diagnostics.
In the rightmost panel of Fig. 4, the OH*CL and 2CP intensities are overlaid on the RSD image-wherever both OH*CL and 2CP are present, the intensities are added in their respective color-spaces. Thus, in the composite image, pure yellow corresponds to OH*CL without any concurrent 2CP intensity, and vice versa for pure magenta; orange results where both colors are present. In Fig. 4, the OH*CL/2CP signals cover entirely the bright regions of hue in the RSD image representing the ignition kernel, and confirm that the spatial alignment of the three diagnostics is precise. All three diagnostics are line-of-sight measurements, and thus, they do not provide local information about the flame structure in the jet core. However, the composite image illustrates the spatial non-uniformities and non-monotonic trends in mixing (hue signal) in radial and axial Fig. 4 Combustion was simultaneously recorded by Rainbow Schlieren (in color), OH*CL (BW camera, shown in yellow), and two-color pyrometry (BW, shown in magenta) as shown in Fig. 2. Images from each diagnostic, from the same time, are shown here, along with a 'composite' image to the right, in which the OH* and 2CP intensities are overlaid on top of the RSD hues (note the alignment with extreme hues in the left RSD frame). Red outlines were added to these intensities for heightened contrast directions, auto-ignition, and soot behavior, and it is the basis for the analysis and methodology developed in this study.

Results
This study makes the case that turbulent combustion phenomena should be depicted by representative instantaneous behavior rather than ensemble average behavior. Figure 5 shows profiles of liquid length, vapor penetration length, and lift-off length for the present data set. The results based on ensemble average images (average of 500 individual images) are shown in red, and the averages injections processed individually are shown in black. The 95% confidence interval of the sample (CI) bands are included for each parameter to quantify injection to injection variations of controlled test conditions. In agreement with the literature Wright et al. (2010), the CI for lift-off length is much larger than that for liquid-and vapor-penetration lengths because dissimilarities magnify as combustion takes place. With the exception of vapor penetration length, the ensemble average lengths are shorter/closer to the injector tip than the average results.
These results show that the global parameters determined from ensemble average images are different from those obtained from individual images and then averaged. Further, the CI of each parameter cannot be determined from the ensemble average analysis. The difference between ensemble average and average results arises because the relationship between image signal and flow/combustion parameters derived from it are non-linear. Thus, ensemble averaging effectively functions as a low-pass filter. For example, with an ensemble average image, the changeover from the background to signal is smoother and less clearly defined, which makes boundary detection more arbitrary compared to individual images with steeper gradients at the boundary. By changing thresholds used for boundary detection between individual and ensemble results it could be possible to find a boundary that matches the average results. However, without knowing the average behavior this analysis could not be performed and thus, it would offer little additional insight.
In the following sections, the variation among selected injections will be explained and compared with ensemble average behavior. Next, stability and repeatability of the Liquid and vapor penetration profiles are shown, along with the developing lift-off length. Black lines give the mean and 95% CI for the entire dataset, and the red lines display the same values as determined from the ensemble-average images experimental test conditions will be evaluated to determine potential relationship with injection to injection variations. Then, correlations between test conditions and global parameters such as jet penetration, ignition delay, and lift-off length will be analyzed. Finally, the case for a representative injection will be made and a methodology for to identify it will be presented.
The composite images in Fig. 6 show time evolution for several injections near ignition to illustrate the variability in the test results. Each column refers to a single injection event, and is labelled for reference from A to F. The rightmost column contains the ensemble-average images for the complete data-set and will be discussed in Sect. 3.2. Rows of Fig. 6 correspond to different time-steps after start of injection (aSOI), with the first three images depicting consecutive times typically after ignition, and the last row after 1 ms to represent the quasi-steady behavior before the flame impinges on the screens located downstream. Fig. 6 Select times aSOI are shown for six single-shot injections, as well as ensemble-average images (right). Three frames show the times by which most injections achieve ignition (1.8, 1.9 and 2.0 ms aSOI), and an additional image at 3 ms is provided, after the beginning of the quasi-steady period. Each frame is a composite image of the videos recorded simultaneously by several diagnostics, in the same style as Fig. 4. As labelled in the top left; Rainbow Schlieren Deflectometry (RSD) is shown upstream (image backgrounds removed for clarity), OH*CL intensity is overlaid in yellow, and 2CP intensity in magenta. The OH*CL and 2CP intensities have been normalized across all frames shown

Description of Individual Injections
In Fig. 6, injections A-E are outliers in various ways, while Inj. F is the best representation of the overall data set, as will be shown in Sec. 3.4. Injection A ignites very late at 2.0 ms aSOI and it's upstream vapor-jet maintains a constant cone-angle indicating consistent inert mixing. By 3.0 ms aSOI, the quasi-steady jet has produced OH*CL reaction zones with intensity and location similar to other injections, although the soot luminescence is negligible. Next, Inj.B shows a typical ignition delay time (IDT) of 1.8 ms aSOI; reactions start with a small ignition kernel that propagates upstream and spreads across the jet-width. Notice that this injection has very robust OH*CL signal, shown by bright yellow in Fig. 6, and the shortest steady lift-off length at 3.0 ms aSOI when modest soot luminescence (magenta) is also visible.
Injection C represents the sootiest of the injections observed. This injection ignited at 1.6 ms aSOI (not shown), and the ignition kernel spread quickly to produce strong OH*CL and soot luminescence signals in a large view area by 1.8 ms aSOI. The subareas of yellow/magenta at 1.8 ms aSOI show that the high-intensity OH*CL/2CP regions do not align, although they are increasingly homogeneous at later times. The jet achieves a stable lift-off between 2.0 − 3.0 ms aSOI, and the lift-off length is shorter and flame shape is different from most other injections. At 3.0 ms aSOI, soot can be detected upstream in the premixed lift-off region, as well as in the downstream region near the screens at impingement.
Injection D shows typical ignition behavior at 1.8 ms aSOI when a weak 2CP signal is present. The OH*CL is imperceptible at 1.8 ms aSOI, barely visible at 1.9 ms aSOI, and is stronger at 2.0 ms aSOI. The ignition kernel propagates upstream very slowly compared to other injections, and it does not spread quickly to cover the width of the jet. Quasi-steady behavior with a faint soot signal occurs at the typical lift-off length of 35-40 mm. Interestingly, this injection has strong vapor-jet protrusions on the left side of the reaction zone at axial locations spanning from 40 mm to the end of the RSD field of view, and a smaller protrusion on the right upstream.
Injection E shows multiple OH*CL ignition kernels at 1.8 ms aSOI, which quickly widen to cover a larger and more homogenized region than those for Inj.B &D at 1.9 ms aSOI. In this case, the lift-off length is within the normal range, but soot signal is not visible either in the developing or quasi-steady jets. Injection F has a large ignition kernel (OH*CL signal) and small soot luminescence at 1.8 ms aSOI. Both OH*CL and 2CP signals quickly widen to spread across the jet width, grow axially, and intensify homogeneously. By 3.0 ms aSOI, the jet has a typical lift-off length and relatively small amounts of soot in the combusting region.
Lastly, the composite ensemble-average images in the right most column show symmetry about the centerline as random variations in instantaneous RSD, OH*CL and 2CP images are effectively filtered out. One additional feature that is most clearly visible in the RSD portion of the ensemble images is the reversal in hue about the centerline. Near the injector, the left size of the jet is darker blue and the right side is cyan. However, the colors reverse direction near the lift-off length as heat release by combustion switches direction of density gradients in the radial direction from relatively high density/low temperature reactants to low density/high temperature combustion products.
The ensemble average images show fairly uniform OH*CL and 2CP signals throughout the core at ignition; note that the red color is a blend of yellow and magenta. The reaction zone occupies nearly the entire jet head at ignition and during next few time steps (1.8-2.0 ms aSOI), similar to the individual injections. At quasi-steady period (3.0 ms aSOI), the spray transitions to stronger OH*CL signal (more yellow) but still with detectable soot luminescence in the core region. Figure 6 demonstrates drastic and significant differences in combustion behavior from injection to injection. Besides ignition timings, differences also occur in the axial location, signal intensity, and propagation of the initial ignition kernels. These disparities are outside the error bounds of the spatial and temporal resolutions of the diagnostics. In Fig. 6, ignition kernels are seen over an axial range of approximately 20 mm, and the axial length of those kernels ranges up to 15 mm. Most ignition kernels (yellow/orange) are approximately the same size (see Injs. B/D/E), although 5-10% of the injections had multiple, distinguishable kernels as seen in Inj. E. The 2CP signal or soot cloud (magenta/orange) at ignition is much more varied; Injs. C/D have strong signal, Inj. F has weak signal, and Injs. A/B/E have almost no signal. While no injection had an ignition kernel so large (or dim) as that shown by the ensemble average image, a few early, robustly reacting injections (e.g., Inj. C) are close to it at 1.8 ms aSOI.

Injection-to-Injection Variations
Although large increases in the spatial extent and intensity of OH*CL and 2CP signals occur within a frame after ignition, neither of these trends persist thereafter. Instead, further time increments show homogenization of signal boundaries and intensities as seen in Fig. 6 for all injections at 2.0 ms aSOI (except Inj. A, which hasn't yet produced strong OH*CL and 2CP signals). Injections with notable protrusions have filled in their gaps (Inj. B most obviously, and Injs. E/F are becoming more ovular). Injections B/E also have vastly reduced OH*CL and 2CP intensity gradients, and Injs. C/F maintain larger but uniform intensities at the core than those at the jet periphery. As temperatures and reaction rates increase within the jet, flame stabilization leads to more homogeneous signals at the lift-off length and downstream regions. In contrast, ensemble average images in Fig. 6 portray homogeneous combustion at all times.
The quasi-steady state at around 3.0 ms aSOI in Fig. 6 demonstrates similarity, if not consistency, between injections. Lift-off lengths vary from 30 to 45 mm, and the OH*CL signal is strong in the 5-10 mm long premixed region near liftoff before reducing in intensity in the jet core and then rising again at the jet head. The volumetric expansion caused by premixed reactions is visible in the RSD images; the hot combustion products are likely present in the radial region between the OH*CL and RSD signal boundaries. The radial expansion of OH*CL (and often 2CP) signal at around 65-70 mm is a consequence of jet impingement on the screens located downstream. Taken together, Fig. 6 demonstrate that fuel injections undergoing autoignition at diesel-like conditions are not repeatable. This observation highlights the challenge posed to combustion researchers and raises the question: what is the typical/average/nominal injection behavior?

Potential Sources of Variation
In this section, potential sources of variations in experimental results discussed above will be explored: (1) repeatability of experimental conditions to ensure accuracy and stability over the test duration to acquire large data-set, (2) assessment of global parameters (ignition, stabilization, and quasi-steady behavior) to determine if they are stochastic or predictive, and (3) stochastic differences caused by the localized, small-scale turbulent 1 3 mixing process. Each of these potential sources of variation will be investigated next to ascertain the most significant cause(s) of injection to injection variations in transient spray combustion.

Repeatability of Experimental Conditions
The CFPC used in this study is intended to perform fuel injection experiments in quick succession at nearly identical test conditions. As shown in Table 1, chamber pressure is maintained within 0.02 MPa of target (3 MPa) for 95% of injections. The ambient air temperature has 95% CI of only ±4 • C. Finally, the fuel injection pressure is within 0.71 MPa of target (98.9 MPa) for 95% of injections. Thus, test conditions are steady and repeatable to less than one percent of the set values. Figure 7 plots the individual injection test conditions versus time to demonstrate the high degree of experimental consistency and lack of drift during the test. Next, Fig. 8 present the single injection and running average of IDT, at temporal resolution of 0.1 ms limited by the framing rate of OH*CL diagnostics. These results show that the IDT converges after about 200 injections, and that the present data set of 500 injections is more than adequate.
Next, the potential for injection-to-injection variation caused by residual product gases in the chamber is analyzed, in case the dwell time between injections was insufficient to flush out the products. Figure 9 shows three key parameters (vapor jet tip penetration at 1.0 ms aSOI, lift-off length, and IDT) for each injection plotted against the values from the previous injection. In these return plots (n vs. n-1 values) of IDT (black symbols in Fig. 9), variable size symbols are used to convey the relative number of occurrences at each point. The lack of any trend, namely clusters removed from the Fig. 7 Chamber thermodynamic conditions for each ignition are presented to demonstrate high degree of experimental repeatability x=y line, supports the conclusion that consecutive injections are independent, and thus, each injection is a distinct random sample.
Overall, the analysis shows that the test conditions are repeatable with no drift, and the sample size of the tests is more than adequate. Further, the inevitable, minor differences caused by the chamber air flow are much small compared to other test platforms, for example, optical engines

Global Parameter Correlations
In this section, the impact of ambient air temperature variation (as seen in Fig. 7) is investigated to determine systematic trends in IDTs. Thus, IDT was organized in bins of the corresponding air temperature to generate Fig. 10 showing mean IDT and 95% CI for that particular bin (black solid and dashed lines). The red solid curve represents the number of injections within each bin. Figure 10 displays a slight inverse relationship between IDT and air temperature as expected, however this trend should be considered within the context of IDT accuracy of ±0.1 ms. Figure 10 demonstrates that for the majority of the injections (within the temperature range of 804-810 K), the mean IDT and CI are less than the 0.1 ms Fig. 8 Ignition delay results from OH* for 500 consecutive injections, and cumulative average (red line) demonstrating converged stationary average. The 95% CI of the mean is provided in dashed red lines Fig. 9 A return-plot (nth vs (n − 1) th injection results) is provided to demonstrate shot-toshot independence based on three key global parameters: lift-off length, vapor penetration, and ignition delay precision; less than 10 injections (out of a total of 500 injections) outside these temperature bounds have higher CI than measurement accuracy, and can be considered outliers.
Next, results are analyzed to determine any systematic correlation among parameters. For example, is the IDT inherently related to other global parameters ? The top left frame in Fig. 11 shows a histogram of IDTs for all injections, the top right and bottom left plots show histograms of lift-off length and vapor penetration length, respectively, in bins of IDTs. A scatter plot of vapor penetration length versus lift-off length is shown on the bottom right. These three parameters were chosen for demonstration because they facilitate comparison with Fig. 9, and also because they inform other parameters discussed so far. In Fig. 11, the lift-off length has similar histogram distributions for different IDTs, indicating no direct correlation between the two parameters. Similarly, no correlation exists between vapor penetration length and IDT. No discernible trend between lift-off length and vapor penetration length can be identified from the scatter plot at 1.0 ms. Results in Figs. 8, 10, 11 and 11 show that injections are independent and can be treated as random samples in this cyclic turbulent combustion process. Multiple possibly-correlated parameters were analyzed in the same way as Figs. 9, 10, and 11, and all arrived at the same conclusion.

Representative Single Injections
Having evaluated and dismissed other causes of injection to injection variations, the localized, small scale turbulent mixing and rate-of-injection variations remain as the primary sources. The localized turbulence can be observed in single-injection images whereas important details are lost in the ensemble average images, as described in Sect . 3 and Sect. 3.1. As such, it is necessary to select a single (or a set of) injection that describes/ represents the nominal/typical cyclic injection phenomena.
A large dataset is necessary to select a representative injection because turbulent combustion phenomena are remarkably adverse to repetition as shown and discussed in Sect. 3.1. Several previous studies have utilized limited data-sets to characterize spray and combustion phenomena; < 20 injections is widespread Wright et al. (2010); Pickett et al. (2011);Manin et al. (2013Manin et al. ( , 2014; Xuan et al. (2019). However, injections that are nominal or outliers cannot be determined from the limited data-sets. Thus, our prior work has Fig. 10 Mean ignition delay (and 95% CI of the mean) is presented as a function of ambient temperature. The ambient temperatures recorded for each injection were binned to the nearest 1 K. Only five injections had temperatures outside this range, and were excluded 1 3 recommended a minimum of 100 injections Parker et al. (2021); Reggeti et al. (2021) to identify a representative injection. Li has used LES to recommend at least 40 realizations to adequately describe cycle-to-cycle variation in a gasoline direct-injection application Li et al. (2017). In the present study, 500 injections, far exceeding those standards, are used to determine the representative injection(s) as discussed next.
The method to identify the most representative injection (Inj. F in Fig. 6) utilizes the concept of a z-score rating, which rates an event's deviation from the average ( ) in terms of standard deviations ( ). This follows the formula z = x− , where z = 0 indicates perfect adherence to the mean behavior, and z = +1 represents behavior consistent within one standard deviation from the mean. This method is demonstrated by applying it to determine an injection with representative image boundaries.
At each axial position and for each side of the RSD/OH*CL/2CP image boundaries, a z-score is calculated. The z-scores are calculated as radii at left (LB) and right (RB) boundaries at a given time, r LB or RB (i, ax, t) , relative to the average and standard deviations of all injections at that time/location, r LB or RB (ax, t) and r LB or RB (ax, t) . The z-scores are averaged spatially and temporally for each injection and each diagnostic to identify superlative injections across several optical diagnostics Reggeti et al. (2021). As shown in Eq. 1, the total z-score for a specific diagnostic, z tot,i is the average over all time steps (from t o to t f ) and spatial locations (from injector tip, ax = 0 , to jet tip at that time, ax = ax max,t,i ) of the combined left and right boundaries' z-scores. Finally, an overall average is obtained by dividing the total z-score by the number of axial steps and time steps, ax steps and t steps .
Because the RSD boundaries are relatively steady compared to OH*CL and especially 2CP boundaries, the z tot,i scores for each injection and diagnostic are normalized by the maximum z tot,i over all injections for that diagnostic. Thus, each diagnostic is equally weighted to calculate the average normalized total z-score across all three diagnostics. This ranking results in Inj. F in Fig. 6 to be the most representative injection. The other five injections presented in Fig. 6 are outliers in various ways having high z-scores in one or more diagnostics. Fig. 11 The distribution of observed ignition delays among 500 injections (top left) is shown with the histograms of OH* Liftoff (top right) and vapor penetration (at 1 ms, bottom left). Both histograms are binned by ignition delay to demonstrate that these events have unrelated distributions. The relationship between lift-off and vapor-jet penetration is likewise shown (bottom right)

3
For visualization of the z-score metric, Fig. 12 presents RSD images for a single injection (a, left), and the ensemble average image at the same time-step (b, right). The red lines overlaid on both halves of Fig. 12a/b portray mean and 95% CI locations of the left and right jet-edges observed at each axial location at this time (2.5 ms aSOI) across all injections. The average boundary does not match with the boundary of the single injection (left image), neither does it align with the ensemble average boundary (right image). This unexpected result could stem from fuel vapor plumes randomly ejecting at all axial locations on both sides of the jet, for example, owing to cavitation within the injector although further investigation will clearly be necessary; Inj. F in Fig. 6 represents an example of this phenomena. The OH*CL and 2CP images resulted in similar differences (not shown) among individual injection, average, and ensemble average boundaries.
The z-score rankings of each diagnostic versus injections are shown in Fig. 13 plotted on a log scale. The normalized z-score values range from < 10 −3 for 2CP, and < 10 −2 for others, up to 1.0. The five injections with overall lowest average z-score across all diagnostics are identified by symbols in Fig. 13. The injection with the lowest z-score for 2CP, represented by the square symbol, has the highest z-score for RSD and OH*CL for the five injections shown. This example illustrates a single injection is unlikely to exhibit representative behavior in all regards. The effect of sample size on representative injection indicated that injections with the lowest z-score alternate within only a few candidate injections, and that a sample of 100+ injections will have a suitable and easily identifiable injection candidate based on our metric. Table 3 lists the z-scores and the key global parameters for the five injections with the lowest z-scores. Each of these injections shows IDT of either 1.7 or 1.8 ms (overall average is 1.73 ms), and vapor penetration length within 3.0 mm of the average. The main difference is observed in the lift-off length. Figure 14 shows composite images at the quasi-steady for five injections with the lowest z-scores together with the ensemble average image in the last column. The consistency of injections in this small subset is clear, except for Inj. 255 showing higher soot than other injections; any of these injections could represent the present data set. Although Inj. F (262) does not have the absolute lowest total z-score, it is chosen as the most representative injection because of its overall symmetry and consistency in all three diagnostics. (1) ax steps t steps Fig. 12 RSD-images are shown in hue space for a the Representative single injection (Inj. F in Fig. 6), and b the ensemble image; both have had the background removed. Overlaid on both images, in red, is the mean and 95% CI determined from the full data-set of 500 injections While the z-score metric presented here focuses on the boundary characterization, it is reasonable to extend the analysis to include other parameters. For example global values such as ignition delay, lift-off length, etc. for each injection could be included. A number of alternative formulations of the overall ranking method were considered resulting in only minor differences in the final set of most representative injections identified.

Summary and Conclusions
An optically accessible constant pressure flow chamber was utilized to acquire 500 injections of transient, n-heptane spray reacting in high pressure, high temperature ambient air, while maintaining near-constant control of ambient and injection conditions. Analysis of high-speed images acquired by rainbow schlieren deflectometery (RSD), OH* chemiluminescence (OH*CL), and two-color pyrometry (2CP) show remarkable injection-to-injection variations visually and in key global parameters. Analysis based on ensemble averaged images is called into question because of its inability Fig. 13 Normalized total z-score distribution for each diagnostic with symbols marking the five injections with overall lowest average z-score across all diagnostics to preserve the spatial and temporal gradients in measured and derived quantities. As a result, ensemble average analysis under-predicted global parameters compared to the results obtained from the analysis of individual injections and then averaged. A rigorous examination of test conditions revealed no correlation with global parameters such as jet penetration length, ignition delay time, and lift-off length. This leads to the conclusion that the observed variations in injection-to-injection behavior are the result of a combination spatial and temporal differences in the localized, small-scale turbulence throughout the jet and differences in rate of injection through the nozzle. Next, a statistical methodology to identify representative injection(s) from a relatively large data-set was presented and demonstrated to interpret injection-to-injection variation. The analysis is based upon a z-score of user defined parameter(s), in this case, boundaries detected by each of the three diagnostics. The z-score measures the departure of a parameter from its average values, and can be normalized and weighted to combine results from different diagnostics. Using this approach, the boundaries of the spray, reaction zone, or soot cloud of each injection are reduced to a single total z-score by averaging the local z-score at each axial position and time, and assigning equal weights to each diagnostics. This methodology requires a large data-set to obtain statistically converged results, which can be challenging or even prohibitive in some combustion applications. In view of the large injection-to-injection variation, the methodology proposed here allows one to to screen an experimental data-set and determine a representative test case for further analysis. In contrast, current approaches rely on analyzing averaged behavior of localized values. The generalized approach to identify a representative injection presented here can easily be applied to any diagnostic that provides spatiotemporal data and for data sets that are sufficiently large to show convergence of the desired parameters.
The benefits of applying the present analysis to CFD development, validation, and analysis are also significant. Representative target values can be identified for models which aim to predict the fluctuating and inherently random turbulent combustion event. Additionally, experimental representative data can be compared directly with the results of a single LES, instead of performing many LES realizations for the purpose of averaging and comparing to experimental ensemble-averages. The present approach quantifies and describes the range of variations in a phenomena to a degree impossible with Fig. 14 Composite images of the five lowest average z-score injections at 3 ms after start of injection. Contrasted to large range of variation in Fig. 6, this subset exhibit clear consistency in behavior with Inj. F (262) being selected as most representative ensemble average analysis. A hitherto unavailable dimension of validation is therefore offered to the CFD community; as few as three LES realizations may be sufficient to validate the models, if the representative individual injection behavior as well as the upper and low bounds of variation, quantified by an appropriate parameter, can be captured with acceptable accuracy. Further, while rate of injection cannot be measured insitu, additional studies characterizing the range of variation for a specific injector would be helpful. As a result of these developments, the CFD parameters can also be refined to produce more physically meaningful phenomenological models.
Author Contributions JB and AA lead development of test facility. JB, AA, and AP designed the study. AP conducted the experiments, analyzed data, prepared figures, and drafted manuscript. AA and JB drafted portions and revised the manuscript. All authors reviewed the manuscript.

Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
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