Lagrangian and Eulerian measurements in high-speed jets using Multi-Pulse Shake-The-Box and fine scale reconstruction (VIC#)

Abstract

Accurate measurement of high-speed flows in the presence of elevated levels of shear and turbulence is a challenging yet necessary endeavor to understand ubiquitous flows that are of great engineering importance. While Eulerian methods, such as Particle Image Velocimetry, represent the traditional approach, Lagrangian alternatives, such as Particle Tracking Velocimetry, have witnessed a resurgence recently due to improved technology and interest in Lagrangian analysis methods. In this research, a recently developed implementation of a volumetric Lagrangian technique for tracking particles in densely seeded flows, namely, Multi-Pulse Shake-The-Box (MP-STB) with the specific implementation referred to as Four-Pulse Shake-The-Box is described and its performance in high-speed jet flows is evaluated. The MP-STB technique is based on recent developments in the Shake-The-Box method by Novara et al. (Experiment in Fluids 60(3):44, 2019) and uses low-speed cameras combined with a double-exposed image acquisition strategy and multi-pulse tracking. Its use of four laser pulses in quick succession with an uneven pulse timing scheme allows for high-accuracy estimates of velocity and acceleration, and repeated ensembles of short-duration, time-resolved measurements in realistic high-speed flows. Experiments with circular jets operating at exit Mach numbers of 0.31 and 0.59 in two different configurations, namely, free jets and jets impinging on a ground plate located 4.75 jet diameters away from the nozzle, were performed to evaluate MP-STB. Scattered four-particle tracks from MP-STB were mapped onto a regular Eulerian grid through the Fine Scale Reconstruction implementation of the VIC# data assimilation method by Jeon et al. (2018). Unique information, including acceleration fields, is presented for these well-known canonical flows. Comparisons with traditional Eulerian measurements from Tomographic PIV, Stereoscopic PIV, and planar PIV are provided to validate the accuracy and comparative cost of volumetric MP-STB measurements combined with the VIC# data assimilation technique.

Graphic abstract

1 Introduction and background

Direct measurement of flow quantities in high-speed turbulent flows is crucial to fully understand the physical mechanisms involved, but it is challenging due to the short time scales inherent to such flows. Non-intrusive velocity measurements are possible in such flows using techniques such as Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV), but measurement of quantities such as acceleration has proven challenging, especially when volumetric data are desired. Calculating acceleration from velocity data sourced through PIV is subject to severe limitations in terms of its accuracy unless the velocity data are acquired with sufficiently high temporal resolution. Such time-resolved PIV (TR-PIV) can be prohibitively expensive and challenging even for relatively low-speed flows, requiring the use of expensive high-speed lasers and high-speed cameras. Even when measurements are fully time-resolved, TR-PIV is currently limited in its ability to resolve regions of high-shear due to limited spatial resolution. The spatial averaging over interrogation windows, inherent to PIV methods that are based on cross-correlation analysis, can be overcome by various PTV methods which track individual particles. But until recently, PTV methods were generally limited to low seeding densities (particle image density around 0.005 particles per pixel, ppp) until the introduction of Shake-The-Box (Schanz et al. 2016).

Shake-The-Box (STB) is a Lagrangian particle tracking method that can handle particle image densities comparable to levels normally encountered in Tomographic-PIV. Schanz et al. (2016) initially introduced the STB technique and presented an efficient and highly functional form capable of handling particle image densities of up to 0.125 ppp and successfully applied it to fully time-resolved measurements of a water jet (exit velocity = 0.43 m/s) to obtain dense Lagrangian particle tracks. STB made this possible by leveraging the Iterative Particle Reconstruction (IPR) technique presented in Wieneke (2012) for its ‘shaking’ step and introduced the concept of particle prediction based on time-resolved data to successfully reconstruct individual tracks even for high particle image density cases. This implementation of STB was also applied to study the motion of a thermal plume convecting at velocities up to 0.35 m/s and the dense particle tracks were used to compute Lagrangian Coherent Structures (Huhn et al. 2017). In both cases, the recordings were fully time-resolved.

The desire to apply STB to realistic high-speed flows with the additional requirement of high temporal resolution led to the development of a multi-pulse version of STB [Multi-Pulse STB, Novara et al. (2016b)]. This technique, using illumination systems consisting of dual cavity lasers to produce multiple pulses within an overall duration tens of microseconds, allows for intermittently time-resolved measurements of even high-speed flows. Application of this MP-STB tracking technique on multi-pulse experimental data obtained using imaging systems consisting of six or more cameras was performed first by Novara et al. (2016b) and then Geisler et al. (2016) for moderate high-speed flows with incoming velocities of 36 m/s and 80 m/s, respectively. The multi-pulse algorithm was also applied to synthetic data by Novara et al. (2016a), who studied the flow over an axisymmetric step at Mach 0.7 using synthetic particle images generated from a Zonal Detached Eddy Simulation. Godbersen et al. (2019) applied the technique to experimental data obtained at Mach 0.5 and 0.84 using multiple imaging systems consisting of a total of eight cameras and orthogonally polarized light produced by four lasers. All of these experimental studies used complicated, and relatively expensive, setups consisting of a large number of cameras divided into several imaging systems that separate the multiple laser pulses based on their polarization to capture singly exposed images.

An alternative approach (Novara et al. 2019) is to use two low-speed, dual-cavity lasers optically combined and synchronized to provide a sequence of four non-uniformly spaced pulses but does not require imaging systems to record pulses with different polarization. Instead, in the four-pulse case, for example, each image of standard dual-frame PIV cameras is exposed twice by each of the two lasers leading to a doubling of the particle image density, since particles are imaged twice within the same image frame. This double exposure necessitates a means of identifying and tracking which two particle realizations among the multitude of particle images within each frame correspond to the same physical particle. A comparison of this multi-exposed image acquisition approach with previous polarization-based methods is presented in Novara et al. (2018). Irrespective of the image acquisition strategy, the iterative Multi-Pulse STB algorithm described in Novara et al. (2016b), with some modifications discussed further below, is used to track and identify four-pulse particle tracks. Using multi-exposed images recorded using four cameras, Novara et al. (2019) made measurements in turbulent boundary layer flows up to 35.48 m/s and provided velocity and acceleration estimates.

The current study uses the same double-exposed image acquisition strategy coupled with Multi-Pulse STB particle tracking, but with a few key differences compared to Novara et al. (2019). These include a uneven pulse timing scheme that is significantly different and a modified implementation of the Multi-Pulse STB particle tracking algorithm. To differentiate this unique implementation from the other multi-pulse variants of STB that have been developed so far, the technique used in the present study is also referred to as Four-Pulse Shake-The-Box (4P-STB) and has been recently introduced under that moniker to end-users of a commercial system produced by LaVision. Since the current implementation is based on the working principle of iterative reconstruction and tracking proposed by Novara et al. (2016b, 2019), the technique will still be referred to as Multi-Pulse Shake-The-Box (MP-STB) in the present study. To reiterate, the current implementation is unique and there are key differences compared to previous implementations with the same name. Details of the multi-exposed image acquisition strategy and particle tracking scheme used as part of MP-STB are provided in Sect. 2.2. The ability of MP-STB to temporally resolve realistic high-speed flows over short intermittent time periods in the presence of steep spatial gradients is critically assessed by performing fully three-dimensional volumetric measurements of velocity and acceleration fields in two types of well-known high-speed, high-shear flows, namely, free jets and impinging jets.

The high-speed (Mach 0.31, 0.59) test flow configurations of free and impinging jets have been well studied in the literature. Information on various aspects of axisymmetric free jets is available in most standard fluid mechanics text books, including Pope (2000). While most of the information in Pope (2000) pertains to fully developed (self-similar region) jet flow, a few studies have looked at the flow field of a axisymmetric free jet in the developing region close to the nozzle exit (Lau et al. 1979; Bridges and Wernet 2011; Feng and McGuirk 2016; Salkhordeh et al. 2019). A comprehensive review of studies on axisymmetric free jets can be found in Bridges and Wernet (2011) and Feng and McGuirk (2016). The impinging jet configuration has also been well studied, especially in the context of enhancing heat transfer between the jet and impingement plate. Studies by Tummers et al. (2011), Jambunathan et al. (1992), Ho and Nosseir (1981), Hall and Ewing (2006), Gutmark et al. (1978), Donaldson and Snedeker (1971) and Donaldson et al. (1971) have all looked at various aspects of the impinging jet flow field under different conditions. There have even been volumetric data obtained through time-resolved tomographic PIV for a range of impingement distances (Violato et al. 2012). While numerous recent studies have focused on heat transfer under jet impingement, early studies such as those by Gutmark et al. (1978) and Donaldson and Snedeker (1971) provide detailed information on the iso-thermal impinging jet flow field. Both free and impinging jets present challenging high-shear, high-turbulence flow conditions, and distinct practical implementation concerns; the contribution of the current study is the rigorous characterization of Multi-Pulse Shake-The-Box to measure complex high-speed flows via Lagrangian Particle Tracking and comparison with traditional Eulerian PIV techniques. Unique information in the form of volumetric, time-resolved Lagrangian particle tracks, and acceleration fields are also presented for these well-studied high-speed canonical flows.

2 Experimental methods

Fig. 1

Schematic of experimental setup used in MP-STB and PIV. For free jets: a 3D view and b top view of $x_{F}-z_{F}$ plane (cameras 1 and 2 are located underneath cameras 4 and 3, respectively). For impinging jets: c 3D view and d top view of $x_{I}-z_{I}$ plane. In a and c, coordinate origins are located along jet centerline, in the nozzle exit plane for free jets (a) and coincident with the impingement plate for impinging jets (c). In b, dash–dot line indicates planar PIV measurement plane (camera used for planar PIV shown in silhouette). In d, cameras 2 and 3 are also used for SPIV measurements in a plane (indicated in top view by dash–dot line) parallel to impingement plate. Dotted lines indicate field-of-view of cameras for MP-STB and TPIV. Subscripts: F free jet, I impinging jet. Objects and distances are not to scale

2.1 Jet facility description

Experiments were performed in the Short Take-Off and Vertical Landing (STOVL) jet facility located at Florida Center for Advanced Aero-Propulsion (FCAAP) at the Florida State University. A single circular jet was operated during this study, with the jet stagnation chamber being supplied dry compressed air from a reservoir at 3.44 MPa (absolute) through appropriate high-pressure control valves capable of supplying a total pressure ($P_{0}$) ranging up to 827 kPa (absolute). The jet exited into a quiescent environment in the experimental facility with ambient conditions at standard atmospheric pressure ($P_{\mathrm{ambient}} = 101.325$ kPa) and temperature ($T_{\mathrm{ambient}} = 295\pm 3$ K). The jet is operated under iso-thermal conditions, with the stagnation temperature equal to $T_{\mathrm{ambient}}$. The jet is produced by a converging nozzle (exit diameter, $D = 2.54$ cm, used for non-dimensionalizing all spatial quantities) attached to the stagnation chamber through a pipe outfitted with multiple flow straighteners to condition the flow upstream of the nozzle. $P_{0}$ is adjusted appropriately to account for pressure losses within the pipe. Under these conditions, the jet is primarily characterized by its nozzle pressure ratio (NPR), which is the ratio of stagnation pressure ($P_{0}$) to ambient pressure ($P_{\mathrm{ambient}}$). NPR was maintained within $\pm 0.01$ of its nominal value during every individual test case for all experimental conditions tested in the course of this study. In both jet configurations, experiments were performed at two distinct NPR values (NPR $\approx 1.07,1.27$) corresponding to nominal Mach numbers of 0.31 and 0.59 at the nozzle exit and, based on D and the nominal jet exit velocity ($U_{\mathrm{jet}} = 106; 195$ m/s) along the centerline, the corresponding Reynolds numbers are $\mathrm{Re} = 1.83\times 10^{5}$ and $\mathrm{Re} = 3.70\times 10^{5}$, respectively.

While the experimental facility is not anechoic, efforts were taken to minimize acoustic reflections. For free jet experiments, the jet issued into the chamber, such that no reflective surfaces, except for the nozzle and support structures themselves, were present within hundreds of jet diameters. For impinging jet experiments, a rigid ground plate with a smooth impingement surface was placed at a nozzle-to-impingement plate distance of 4.75D from the nozzle exit. For all experiments, sub-micron sized tracer particles were introduced into the ambient environment using a seed particle generator (TSI Model 9307-6), while the jet itself was seeded using a custom Laskin nozzle seeder that introduced aerosolized droplets of avocado oil into the compressed air supply upstream of the stagnation chamber.

2.2 Multi-pulse shake-the-box (MP-STB)

Volumetric Lagrangian particle tracking using Shake-The-Box is an iterative process and details of the basic and multi-pulse variants of the STB method are available in (Schanz et al. 2016) and (Novara et al. 2019), respectively. As described in Sect. 1, the current implementation of MP–STB is derived from the technique presented in Novara et al. (2019). Therefore, only attributes unique to the current MP-STB technique will be discussed in detail herein. For all experiments, the jet and ambient environment were seeded with tracer particles and illuminated using a ‘quad-pulse’ Nd:YAG laser. This laser combines two 200 mJ/pulse, dual-cavity Quantel Evergreen lasers to produce four spatially coincident laser beams at 532 nm wavelength, with groups of four pulses occurring at 15 Hz. The laser beams were expanded and shaped using a series of optical lenses to illuminate a volume, whose sizes are as shown in Fig.1 from different views of the laser volume for the two test configurations.

For free jet experiments, as indicated in Fig. 1a and b, the laser volume was expanded to encompass from the nozzle exit one half of the emergent jet. The illuminated volume was imaged at 15 frames per second using four LaVision sCMOS dual-frame cameras ($2560\times 2160$ pixels) fitted with 105 mm Nikon lenses and appropriately aligned Scheimpflug adapters, and oriented, such that they formed four vertices of a square that were approximately equidistant from the jet centerline. In this configuration, two cameras (3,4) were in forward scattering at $f_{\#}=11$ and two cameras (1,2) were in backward scattering at $f_{\#}=8$ with respect to the direction of laser emission. For impinging jet experiments, as indicated in Fig. 1c and d, the laser volume was expanded to illuminate the flow field close to the surface of the impingement plate. All four cameras with 105 mm lenses and appropriately aligned Scheimpflug adapters were placed in a linear fashion and oriented in backward scattering at $f_{\#}=11$ and images were acquired at 15 frames per second. The particle image density was nominally around 0.08 ppp for free jet experiments and 0.10 ppp for impinging jet experiments, with slight variations between cameras due to their relative difference in viewing angle with respect to the Mie scattering profile. Since measurement data are obtained only in the common region imaged by all cameras, efforts were made to maximize overlap between fields of view for all four cameras. For every test case, 450 snapshots were recorded and processed.

Timing strategy: Figure 2a shows the laser pulse timing strategy used for acquiring double-exposed images. Pulses 1 and 4 were produced by one of the dual-cavity lasers and pulses 2 and 3 were produced by the other laser, with pulse separation time intervals dt1 (pulses 1–2), dt2 (pulses 2–3), and dt3 (pulses 3–4). Pulses 1 and 2 occurred during the exposure time of Frame 1 and pulses 3 and 4 occurred during the exposure time of Frame 2. One of the key features of MP-STB used in the present study is the long–short–long-pulse separation strategy, with dt1>dt2<dt3 and dt1=dt3. This is contrary to the short–long–short scheme implemented in Novara et al. (2019) and predominantly used to acquire the data discussed in that study.

Particle tracking: The present study used the long–short–long-pulse separation strategy, leading to a manifestly new method of tracking individual particles to form accurate four-pulse (particle/position) tracks. It is worth noting that Novara et al. (2019) also tested the long–short–long-pulse separation scheme and mentioned the concomitant particle tracking approach, and found it to provide more accurate estimates of velocity and acceleration, but they deferred any further investigation to a future study. In this tracking method used in the present study, particles within each double-exposed frame were identified and their precise location and intensity calculated using the Iterative Particle Reconstruction (IPR) technique (Wieneke 2012), which includes the eponymous ‘shaking’ step to accurately determine the position by iteratively moving particles in small increments in space. As a next step, the particle tracking scheme used in MP-STB as shown schematically in Fig. 2b was applied. Particle positions corresponding to pulses 2 and 3 were identified first based on dt2 (which determines the search radius, $\delta _{2p}$) by searching the matching regions in Frame 1 and Frame 2, respectively, and a two-particle track was constructed. This track was then linearly extrapolated outwards, effectively both in time and space, based on the values of dt1 and dt3 to form the projected positions coincident with pulses 1 and 4, respectively. These are indicated by the open, red circles in Fig. 2b. Using these projected positions as the origin, a search was undertaken within appropriate search radii, $\delta _{4p}$, to find matching true particle images. Search radius $\delta _{4p}$ is much larger than $\delta _{2p}$, with $\delta _{4p}$ determined based on maximum expected acceleration. Once particles corresponding to pulses 1 and 4 were identified, a second-order polynomial was fit to the four particle positions (solid, black circles in Fig. 2b) to form the four-particle track, which provides velocity and acceleration. A limitation of MP-STB becomes apparent in regions of the flow field that have velocities close to zero. This leads to overlapping particle images corresponding to pulses $1+2$ and $3+4$. In this situation, a two-pulse STB analysis is performed and the acceleration is set to zero.

Fig. 2

a Timing strategy for double-exposed image acquisition with dt1 > dt2 < dt 3, and b particle tracking scheme for double-exposed recordings. In b black, solid circles indicate actual particle positions at four time instants within search radii ($\delta _{2p}, \delta _{4p}$) and red, open circles indicate estimated positions of particles 1 and 4 extrapolated (blue solid arrows indicate direction) from two-particle (particles 2 and 3) track. Naming conventions similar to Novara et al. (2019). Red dashed circles indicate search regions

Once all four-particle tracks were identified, the particles that successfully contributed to complete tracks were back-projected onto the image plane and a residual was calculated with respect to the original image. The residual images have significantly fewer particle images, and consequently lower apparent seeding density, thereby alleviating the complexity and cost of the tracking problem. The residual image was then reconstructed using Iterative Particle Reconstruction in the next iteration. Details of this iterative STB process, the particle tracking method excepted, can be found in Novara et al. (2019). An iterative median filter was also applied to remove spurious particle tracks. After a few ($\sim 5-7$) iterations, a resultant flow field with dense four-pulse particle tracks, with velocity and acceleration estimates at the mid-point of each track, was computed.

Initial experiments performed as part of the present study used primitive adaptations of the MP-STB technique (provided by B. Wieneke) applied through non-public test versions of LaVision DaVis 8.4, and the results were used to further develop MP-STB. Ultimately, the results presented in this paper were processed using the publicly available implementation of the MP-STB technique, alternatively named Four-Pulse Shake-The-Box, available in LaVision DaVis 10.0.5. For both jet configurations, calibration images of a two-level calibration plate (LaVision Type $106-10$) were acquired, and the mapping between image planes and physical space was obtained. To increase the relative positional accuracy between cameras, an a posteriori correction of the mapping function using the volume self-calibration procedure described by Wieneke (2008) and as implemented in LaVision DaVis 10.0.5 software was performed. It should be noted that the volume self-calibration procedure was performed using double-frame images of the actual jet flow fields acquired with only a single laser pulse illuminating each frame. Volume self-calibration using the singly exposed images reduced the calibration errors to less than 0.05 pixels for all cameras. Furthermore, the reconstruction of particle images through the Iterative Particle Reconstruction (IPR) method in the presence of spatially varying particle shapes due to various optical distortions was improved by calculating the optical transfer function (OTF) as described in Schanz et al. (2012). Specific test conditions and relevant parameters used for processing the data as part of the MP-STB technique are provided in Table 1, along with representative estimates of the number of particles triangulated and successful four-particle tracks obtained for the two experimental configurations.

Table 1 MP-STB processing parameters and statistics

2.2.1 Scattered particle tracks to uniform Eulerian grid: fine scale reconstruction (VIC#)

While the Lagrangian particle tracks along with velocity and acceleration estimates provided by MP-STB are useful on their own, it is beneficial to map them on to a uniform grid, so that various derivative quantities such as vorticity and Q-criterion may be easily calculated. It is also easier to visualize certain coherent flow structures on a uniform grid. The mapping of scattered Lagrangian particle tracks on to a uniform Eulerian grid is accomplished through the Fine Scale Reconstruction implementation of the VIC# technique (Jeon et al. 2018) by assimilating information from measured particle tracks and the Navier–Stokes equations in the form of the vorticity transport equation and an assumption of incompressibility (divergence-free flow). The suitability of applying VIC# to flows at Mach 0.31 and 0.59 is supported by the low levels of normalized divergence of velocity, $(\nabla \cdot U)D/U_{\mathrm{jet}}$, observed in instantaneous flow fields (Supplementary Material 1) at test conditions. The VIC# technique, which is a modified version of the Vortex-in-Cell-plus (VIC+) method proposed by Schneiders and Scarano (2016), is used to convert scattered particle tracks obtained through MP-STB to a uniform Eulerian grid. It is important to note that there is inherent spatial filtering when a data assimilation technique such as VIC# is applied to MP-STB data. For every test case, 250 snapshots were reconstructed to a regular grid and statistical quantities calculated. Statistical convergence of mean flow quantities occurred within 150 images or less for all cases tested. For free jet and impinging jet experiments in the present study, vector spacing of $\sim 0.7$ mm resulting from grid spacing of 16 voxels and 24 voxels, respectively, was used in converting volumetric fields containing Lagrangian particle tracks to a uniform grid through VIC#.

2.3 Particle image velocimetry (PIV)

The experimental configurations considered in the present study were also measured using various mature PIV techniques such as planar PIV, Stereoscopic PIV (SPIV), and Tomographic PIV (TPIV). Comparisons of results from MP-STB + VIC# with those extracted from the PIV measurements are performed to assess the present MP-STB results.

2.3.1 Tomographic particle image velocimetry (TPIV)

TPIV is an Eulerian technique that measures instantaneous, three-dimensional velocity fields, but is also significantly more expensive in its computational requirements compared to MP-STB, as will be discussed later in this paper. Theoretical aspects of TPIV have been described in the existing literature (Elsinga et al. 2006; Scarano 2012) and only details specific to the current setup are discussed herein. In the present study, the same image acquisition hardware setup used for MP-STB (described in Sect. 2.2) was used for TPIV data acquisition, with one exception, namely, each image frame was singly exposed. This was accomplished by firing only one dual-cavity laser contained within the ‘quad-pulse’ laser, such that each image frame has one laser pulse illuminating it. The four sCMOS cameras and the laser volume were oriented in the same manner with the same lenses and corresponding $f_{\#}$’s as in MP-STB, as shown in Fig. 1a–d for the free and impinging jet configurations. The particle image density was slightly lower, $\sim 0.06$ ppp, than that of MP-STB, with small variations between cameras due to differences in the relative viewing angles. Two hundred and fifty double-frame images were acquired for each test case at 15 Hz. Statistical convergence of mean flow quantities occurred within 150 images for all cases tested. Initial calibration and subsequent a posteriori correction of the mapping function using volume self-calibration, as described previously for MP-STB to improve relative positional accuracy, were also performed for TPIV. The acquired image data were processed using LaVision DaVis 10.0.5. Iterative tomographic volume reconstruction of the intensity distribution, using the MLOS and SMART (Atkinson and Soria 2009) algorithms with intermediate Gaussian volume smoothing (Discetti et al. 2013) and implemented in LaVision DaVis as the fastMART algorithm, was performed. Three-dimensional velocity vector fields were calculated by direct 3D cross correlation of the reconstructed intensity distribution volumes using a multi-pass algorithm with interrogation volumes of size $96 \times 96 \times 96$ voxels for the first pass and $32 \times 32 \times 32$ voxels with 75% overlap for the final pass. Spurious vectors were removed through a universal outlier detection method (Westerweel and Scarano 2005). Details of relevant parameters used during image acquisition and TPIV processing in the present study are provided in Table 2 and additional information regarding the TPIV method is available in Sellappan et al. (2018).

Table 2 TPIV test conditions and processing parameters

2.3.2 Planar particle image velocimetry (planar PIV)

Two-dimensional, two-component (2D–2C) PIV measurements were performed in the free jet configuration using one of the dual-cavity lasers described above that was synchronized with a double-frame (LaVision sCMOS) camera. A laser sheet of thickness $\sim 1$ mm, indicated by the dash–dot line in Fig. 1b, was aligned streamwise at $z_{F}/D=-0.08$ parallel to, and offset by $\sim 0.08D$ from, the jet axis. The sCMOS camera was equipped with a 55 mm lens ($f_{\#}=5.6$) and acquired 1000 image pairs at 15 Hz for each test case to resolve the time-averaged statistics of the flow field. Statistical convergence of mean flow quantities occurred within 200 images or less for all cases tested. A time separation between images forming a pair of 4 µs and 2 µs was used for the Mach 0.31 and 0.59 free jets, respectively. The images were processed using multiple passes in LaVision DaVis 10.0.5, with two initial passes made with an interrogation window of $64\times 64$ pixels and then a refined, third, fourth, and fifth pass using a $32\times 32$ pixel interrogation window. A 50% overlap was used for the initial passes, and a 75% overlap used on the final three passes, along with removal of spurious vectors. The resultant vector field has a 0.39 mm spatial resolution and a maximum measurement uncertainty within the region of interest of $\sim 5-6\%$, calculated using correlation statistics (Wieneke 2015), in the shear layer.

2.3.3 Stereoscopic particle image velocimetry (SPIV)

Two-dimensional, three-component (2D–3C) stereoscopic PIV measurements were performed in the impinging jet configuration using one of the dual-cavity lasers described above and two LaVision sCMOS cameras equipped with Scheimpflug adapters and 105 mm lenses ($f_{\#}=11$). Using appropriate sheet forming optics, a $\sim 2$ mm-thick laser sheet was created at $z_{I}/D = 0.425$ parallel to the impingement plate (shown by dash–dot line in Fig. 1d). Two cameras (2, 3) were used to acquire 500 double-frame images at 15 Hz for two impinging jet cases at exit Mach numbers of 0.31 and 0.59 with corresponding pulse separation times of 5 µs and 2.5 µs, respectively. Images were acquired, de-warped, and processed in LaVision DaVis 10.0.5 using a multi-pass, stereo-cross-correlation algorithm, with two initial passes made with an interrogation window of $64\times 64$ pixels and 50% overlap, and then a refined, third and fourth pass using a $32\times 32$ pixel interrogation window with 75% overlap, along with removal of spurious vectors. Statistical convergence of mean flow quantities occurred within 150 images or less for all cases tested. Three-component velocity fields obtained had an in-plane spatial resolution of $\sim 0.27$ mm with typical measurement uncertainties of $6-8\%$ within a distinct donut-shaped region containing the highest shear stresses, and relatively lower measurement uncertainty in the jet core. Additional information and comparison of various uncertainty quantification methods can be found in Sciacchitano et al. (2015).

3 Free jets—Mach 0.31 and 0.59

High-speed jets issuing freely into the ambient environment were measured through MP-STB, TPIV, and planar PIV. Fully three-dimensional Lagrangian particle tracks were obtained using MP-STB within a region close to the nozzle exit, as indicated in Fig. 1a and b. TPIV and planar PIV measurements of similar flow fields were also obtained to allow the comparisons of the MP-STB+VIC# results with established techniques.

Fig. 3

Instantaneous snapshot of Lagrangian particle tracks colored by normalized axial velocity, $u/U_{\mathrm{jet}}$, for free jets at a Mach 0.31, and b Mach 0.59. Only one particle shown at mid-location of each four-particle track along with attendant velocity vector, with a 26750 and b 15904 particle tracks displayed in each snapshot. Flow is from left to right

Fig. 4

Instantaneous snapshot of Lagrangian particle tracks colored by normalized axial acceleration for free jets at a, b Mach 0.31, and c, and d Mach 0.59. Only one particle shown at mid-location of each four-particle track along with attendant velocity vector. a, c Three-dimensional view with flow going from left to right, and b, d plan view in flow-normal direction of particle tracks within $0.75 \le x_{F}/D \le 1.25$

Fig. 5

Instantaneous Eulerian flow field reconstructed through VIC#. Multiple planes (flow normal: $x_{F}/D=0.25,1.00,1.75,2.50,3.25$; streamwise: $z_{F}/D=0$) show normalized axial velocity, $u/U_{\mathrm{jet}}$, for free jets at a Mach 0.31 and b Mach 0.59. Flow is from right to left

Fig. 6

Instantaneous Eulerian flow field reconstructed through VIC#. Contours of normalized axial acceleration, $A_{x}D/U_{\mathrm{jet}}^2$, along streamwise plane at $z_{F}/D=-0.25$ for free jets at a Mach 0.31, and b Mach 0.59. Iso-surfaces of normalized Q-criterion (iso-value = 0.005) superimposed and colored by normalized axial velocity. Flow is from left to right

Fig. 7

Radial profiles of a normalized axial velocity and b normalized turbulent kinetic energy extracted from time-averaged VIC# flow fields for free jets at $x_{F}/D=1 (\mathrm{blue}-\mathrm{triangle}), 2 (\mathrm{red}-\mathrm{circle}), 3 (\mathrm{green}-\mathrm{square})$. In a, $\Diamond (\mathrm{black}-\mathrm{diamond})$ indicates velocity data extracted from Lau et al. (1979) for free jet at Mach 0.28 and $x_{F}/D=2$. Radial coordinate ($\eta ^{*}$) specified along arbitrary azimuthal angle as $\eta ^{*} = (R-R_{0.5})/x_{F}$, similar to Lau et al. (1979), with $R_{0.5}$ being radial location (R) where $u=0.5U_{\mathrm{jet}}$

3.1 Lagrangian four-pulse particle tracks

Four-particle tracks were obtained for free jets at Mach 0.31 and 0.59. Representative examples of the instantaneous flow fields at Mach 0.31 and 0.59 are presented in Fig. 3a and b, respectively. The instantaneous snapshots show particles colored by normalized axial velocity, $u/U_{\mathrm{jet}}$, with each displayed particle being located in the middle of the four-particle tracks to take advantage of the fact that accuracy of the estimates in velocity and acceleration is highest at the mid-point (Novara et al. 2019). Although complete track information and all four particle positions are available for every track, given the high density of particle tracks obtained through MP-STB, only one particle obtained from the polynomial fit evaluated at the time mid-point of the four-pulse recording sequence is displayed per track in all relevant figures in the current paper to improve visual clarity. Certain features common to high-speed jets are readily apparent in Fig. 3, such as the high-velocity jet core extending streamwise across the domain ($0\le x_{F}/D \le 4$) and the appearance of large-scale structures within the shear layers as $x_{F}/D$ increases. In both cases, the shear layers are extremely thin close ($x_{F}/D < 1$) to the nozzle and only a few sporadic tracks serendipitously located at that instant within the shear layer, as indicated by particles with intermediate velocities ($\approx 0.25-0.95\times u/U_{\mathrm{jet}}$), are present. While traditional PIV techniques that use interrogation windows/volumes are likely to smear (smooth out) such thin shear layers due to the inherent spatial averaging, MP-STB does not suffer from this issue and careful implementation is likely to enable the capture of such thin shear layers. As the jets grow downstream, coherent structures created due to the Kelvin–Helmholtz instability within the shear layers are clearly visible even in these instantaneous snapshots as indicated by clusters of particles colored accordingly with intermediate velocities. These shear layer structures grow spatially as they convect downstream and eventually lead to evolution of the jet into a fully developed, self-similar state far downstream.

The steep gradients present in the shear layer are even more apparent when considering the acceleration fields, as shown in Fig. 4. The same instantaneous data shown in Fig. 3 are presented in Fig. 4, but with particles colored by normalized axial acceleration ($A_{x}D/U_{\mathrm{jet}}^{2}$). The highest absolute values of acceleration are found to occur in the shear layers as expected. A conspicuous banding consisting of alternating values of positive and negative acceleration (positive-acceleration, negative-deceleration) is also visible within the jet core, for both Mach 0.31 and 0.59 in the 3-D views, as shown in Fig. 4a and c, respectively, and are related to flow instabilities that produce a pulsating jet. It should be noted that this banded acceleration pattern is seen only in instantaneous fields and varies with time, and the corresponding mean acceleration field averaged over a sufficiently large number of samples will not exhibit such a banded appearance. The shear layers are even more distinctly seen in the plan view of a thin volume slice ($0.75 \le x_{F}/D \le 1.25$) shown in Fig. 4b and d, as evidenced by the thin, ring-like cluster of particles with high values of acceleration and deceleration. Overall, the qualitative similarity between free jets at Mach 0.31 and 0.59 is quite evident in both velocity and acceleration fields, as shown in Figs. 3 and 4.

3.2 Eulerian framework (MP-STB + VIC#)

The scattered four-particle tracks are mapped onto a regular Eulerian grid through VIC# as described in Sect. 2.2.1. Instantaneous Eulerian flow fields displaying normalized axial velocity at Mach 0.31 and 0.59 are shown in Fig. 5a and b, respectively, and were reconstructed using VIC# from the MP-STB data presented in Figs. 3 and 4. The streamwise and flow-normal planes clearly show the jet evolution as the compact axisymmetric jet structure close to the nozzle exit rapidly develops downstream while entraining ambient fluid with attendant shear layer growth. The instantaneous flow fields contain packets of negative axial velocity ($u/U_{\mathrm{jet}}<0$) in the shear layer, indicative of coherent vortex structures.

One method of visualizing vortex structures is through iso-surfaces of Q-criterion (Q), which is the second invariant of the velocity gradient tensor. Figure 6 shows the instantaneous Eulerian flow field reconstructed through VIC# and the normalized acceleration field, $A_{x}D/U_{\mathrm{jet}}^{2}$, along a streamwise plane ($z_{F}/D=-0.25$) plotted along with superimposed iso-surfaces of normalized Q-criterion, $Q(D/U_{\mathrm{jet}})^2$ (iso-value=0.005). The iso-surfaces are colored using normalized axial velocity. Two things are immediately apparent, namely, alternating bands of acceleration within the jet core and steep radial gradient in axial velocity over the iso-surfaces of Q-criterion. The bands of acceleration are the same feature as those seen in the MP-STB results before. They are visually more apparent in the Eulerian reconstruction and are present at both Mach 0.31 and 0.59. The iso-surfaces of Q-criterion, which indicate large-scale coherent vortex structures with some of the highest levels of vorticity in the entire flow field, bound the banded acceleration fields present within the jet core and are in the shear layer. This is as expected, since the shear layer is the only region with significant vorticity production within a free jet flow and the steep gradient in axial velocity seen radially between the inner and outer edges of these vortex structures supports this hypothesis. As such, the iso-surfaces also indicate the growth of the shear layer downstream, with the vortex structures rapidly increasing in size.

While the instantaneous flow fields obtained through MP-STB and VIC# are typical of free jet flows and appear qualitatively similar to prior observations in literature, additional quantitative measures are obtained from mean flow fields. Time-averaged Eulerian flow fields for Mach 0.31 and Mach 0.59 obtained through averaging of instantaneous VIC# snapshots are presented in Supplementary Material 2. Based on the mean flow fields, radial profiles of normalized axial velocity and turbulent kinetic energy extracted at three streamwise locations ($x_{F}/D = 1,2,3$) are plotted in Fig. 7. Since the jet is axisymmetric, the radial profiles were extracted along an arbitrary azimuthal angle. When plotted as a function of radial coordinate $\eta ^{*}$, with $\eta ^{*} = (R-R_{0.5})/x_{F}$ similar to Lau et al. (1979) and $R_{0.5}$ being the radial location where $u=0.5U_{\mathrm{jet}}$, the profiles of axial velocity collapse together independent of streamwise distance and Mach number. This matches with the observations of Lau et al. (1979) who found similar trends for various subsonic free jet flows in data measured using a laser velocimeter. Such quantitative similarity with independent measurements performed using a different technique engenders confidence in the accuracy of MP-STB and reconstruction through VIC#. The profiles of normalized turbulent kinetic energy ($T.K.E./U_{\mathrm{jet}}^{2}$) plotted as a function of $\eta ^{*}$ indicate that the turbulence production is predominantly in the shear layer region and the overall levels of T.K.E. increase with streamwise distance. This is again expected behavior in free jets. Profiles of normalized axial velocity and turbulent kinetic energy plotted as functions of radial distance from the jet centerline, $R_{F}/D$, are provided in Supplementary Material 3 to further elucidate the relationship between the shear layer region and turbulent kinetic energy.

Fig. 8

Instantaneous snapshot of Lagrangian particle tracks colored by normalized axial velocity, $w/U_{\mathrm{jet}}$, for impinging jets at a, c Mach 0.31, and b, and d Mach 0.59, with total number of particle tracks reconstructed being 17118 and 20384, respectively. Only one particle shown at mid-location of each four-particle track along with attendant velocity vector. a, b Top-view with flow directed inwards and only particles within $0.2 \le z_{I}/D \le 0.4$ shown, and b, d side-view with flow directed from top to bottom and only particles within $-0.1 \le y_{I}/D \le 0.1$ shown

Fig. 9

Instantaneous Eulerian flow field reconstructed through VIC#. Multiple planes (flow-normal, $x_{I}-y_{I}$, plane: $z_{I}/D=0.25$; orthogonal planes: $x_{I}/D=0; y_{I}/D=0$) show normalized axial velocity, $w/U_{\mathrm{jet}}$, for impinging jets at a Mach 0.31, and b Mach 0.59. Flow is directed into $x_{I}-y_{I}$ plane in $-z_{I}$ direction

4 Impinging jets—Mach 0.31 and 0.59

High-speed jets with two different exit Mach numbers (0.31, 0.59) impinging on a ground plane at a distance of 4.75D from the nozzle were measured through MP-STB, TPIV, and SPIV. Fully three-dimensional Lagrangian particle tracks were obtained using MP-STB in the impingement region, as indicated in Fig. 1c and d. TPIV and SPIV measurements of similar flow fields were also obtained to allow comparisons of MP–STB+VIC# results with established techniques.

4.1 Lagrangian four-pulse particle tracks

Four-particle tracks were obtained in the measurement volume adjacent to the impingement plate for jets with exit Mach numbers of 0.31 and 0.59. Representative examples of the instantaneous flow fields at Mach 0.31 and 0.59 are presented in Fig. 8a–d. The instantaneous snapshots show particles and their attendant velocity vector colored by normalized axial velocity, $w/U_{\mathrm{jet}}$. The displayed particles are located in the middle of four-particle tracks as noted before to maximize the accuracy of velocity and acceleration estimates, and only one particle is displayed per track to improve visual clarity.

Figure 8a and b shows the top view of a thin volume slice ($0.2 \le z_{I}/D \le 0.4$) with flow directed inwards into the impingement plate located at $z_{I}/D = 0$. The jet core is apparent in both jets, as indicated by $w/U_{\mathrm{jet}}<<0$ in the central region and $w/U_{\mathrm{jet}} \approx 0$ elsewhere in the field. It is worth noting that regions farther away from the jet centerline can occasionally have $w/U_{\mathrm{jet}}>0$ due to the jet spreading outwards and away from the ground plane after impingement. The distinct lack of particles at the edges of the domain is due to non-overlap of camera views in those regions. The radial trajectory and the primary jet core are easy to discern in the side view of a thin volume slice ($-0.1 \le y_{I}/D \le 0.1$), as presented in Fig. 8c and d. The relative scarcity of particle tracks close to the surface is due to the illumination volume being slightly offset from it to reduce the intensity of light reflected from the surface. Overall, the instantaneous snapshots captured through MP-STB indicate a complex flow field containing a wide range of velocities and steep gradients for both Mach 0.31 and 0.59 jets. Again, such qualitative similarity is as expected in subsonic impinging jet flow fields.

4.2 Eulerian framework (MP-STB + VIC#)

Instantaneous Eulerian flow fields reconstructed by mapping scattered four-particle tracks onto a regular Eulerian grid using VIC# are shown in Fig. 9. Multiple orthogonal planes colored by normalized axial velocity, $w/U_{\mathrm{jet}}$, at Mach 0.31 and 0.59 are shown in Fig. 9a and b, respectively, and were reconstructed using VIC# from the MP-STB data presented in Fig. 8. The reconstructed flow fields are qualitatively similar, with the VIC# reconstruction faithfully capturing the complex jet core and the surrounding wall jet flow for both Mach 0.31 and 0.59 impinging jets. The complexity of the flow field is readily apparent, especially outside of the jet core, with patches of widely varying velocities occurring in close proximity. This highly turbulent, chaotic flow field contains patches of positive axial velocity indicating reversed flow after impingement and is a manifestation of the wall jet flow directed radially outwards from the impingement zone. The jet core itself appears distorted from its expected circular shape in the flow-normal ($x_{I}-y_{I}$) plane due to the highly unsteady flow field of an impinging jet and the magnitude of axial velocity is reduced closer to the impingement plate, as seen in the orthogonal planes. The acceleration fields (not shown) also present a similarly diverse appearance, with a mix of positive (deceleration) and negative (acceleration) values present. While the impinging jets are highly unsteady and instantaneous snapshots capture time-varying flow fields in which the shape and position of the jet core rapidly change, the mean flow fields (Supplementary Material 4) at both Mach 0.31 and 0.59 indicate axisymmetric features that are concomitant of the circular primary jets issuing from the nozzle.

5 Comparative performance of Eulerian techniques

The two jet flow-field configurations characterized above were captured through both Lagrangian particle tracking (MP-STB) and Eulerian TPIV. Both of these are volumetric techniques, albeit with inherent advantages and disadvantages due to their different measurement approaches, and the application of VIC# to the MP-STB data allows for direct comparison of the two methods. Measurements were also performed along a single 2-D plane using planar PIV (free jets) and SPIV (impinging jets) to compare and contrast results and engender confidence in the MP-STB+VIC# technique. It should be noted that assessments of MP-STB+VIC# performance derived based on Eulerian comparisons are necessarily indicative only of the combined Lagrangian particle tracking and data assimilation.

Fig. 10

Time-averaged, normalized axial velocity, $u/U_{\mathrm{jet}}$, along $x_{F}-y_{F}$ plane at $z_{F}/D=-0.08$ for Mach 0.59 free jet measured using a planar PIV, b VIC#, and c TPIV. Flow is from left to right

Fig. 11

Mean profiles of normalized axial velocity ($u/U_{\mathrm{jet}}$) and normalized vorticity ($\omega _{z}D/U_{\mathrm{jet}}$) at $x_{F}/D=0.5, z_{F}/D=-0.08$ for free jets at a Mach 0.31 and b Mach 0.59, and mean profiles of normalized axial velocity ($w/U_{\mathrm{jet}}$) at $x_{I}/D=0, z_{I}/D=0.425$ for impinging jets at c Mach 0.31 and 0.59 obtained using different measurement techniques

Fig. 12

Time-averaged, normalized axial velocity, $w/U_{\mathrm{jet}}$, along $x_{I}-y_{I}$ plane at $z_{I}/D=0.425$ for Mach 0.31 impinging jet measured using a SPIV, b VIC#, and c TPIV. Flow is directed into $x_{I}-y_{I}$ plane in $-z_{I}$ direction

Figure 10 shows the mean normalized axial velocity ($u/U_{\mathrm{jet}}$) for a Mach 0.59 free jet extracted along the $x_{F}-z_{F}$ plane at $z/D=-0.08$ from data obtained using planar PIV, MP-STB+VIC#, and TPIV. It is evident that the global flow fields are qualitatively similar, as measured by different techniques, with distinct potential core and shear layer growth. Planes extracted from VIC# and TPIV volumetric data both indicate a minor artifact close to the nozzle lip ($x_{F}/D< 0.2, y_{F}/D \approx 0.5$). Since the orientation of cameras for MP-STB and TPIV in the free jet configuration leads to oblique camera views of the nozzle lip and the direction of the laser illumination with respect to the nozzle leads to spurious background reflections from the nozzle lip, both volumetric techniques tend to produce artifacts in that region. Planar PIV does not suffer from this artifact due to its orthogonal view of the laser sheet in this particular experimental configuration.

Overall, there is remarkable agreement between flow fields obtained through the three different techniques. This qualitative assessment is buttressed by quantitative comparisons of velocity and vorticity profiles extracted from the volumetric data with those measured through planar PIV. Figure 11a and b shows mean profiles of normalized axial velocity ($u/U_{\mathrm{jet}}$) and z-vorticity ($\omega _{z}D/U_{\mathrm{jet}}$) for free jets at Mach 0.31 and 0.59, respectively. As expected in a free jet, close to the nozzle exit the axial velocity presents a top–hat profile and vorticity production is confined to the shear layer region. Both VIC# and TPIV exhibit a slight ($\sim 2\%$) over-shoot in velocity estimates compared to planar PIV in the inner region of the shear layer, possibly due to the previously mentioned artifact. Overall, it is evident that MP-STB+VIC# performs comparably to TPIV and planar PIV in reliably measuring high-speed free jet flows.

The performance of MP-STB+VIC# in measuring high-speed impinging jet flow fields close to the impingement zone is compared against TPIV and SPIV in Fig. 12. Mean, normalized axial velocity ($w/U_{\mathrm{jet}}$) is shown along a plane located at $z_{I}/D=0.425$ parallel to the impingement surface for a Mach 0.31 impinging jet. The data shown in Fig. 12 were measured using (a) SPIV, (b) MP-STB+VIC#, and (c) TPIV. All three techniques capture the jet core with its subtle annular feature reliably, resulting in qualitatively similar flow fields. Once again, quantitative line profiles extracted from the three data sets, as shown in Fig. 11c, also indicate the ability of MP-STB+VIC# to measure a challenging flow field with an accuracy comparable to TPIV and SPIV. Figure 11c shows the normalized axial velocity ($w/U_{\mathrm{jet}}$) extracted at $x_{I}/D=0, z_{I}/D=0.425$ for both Mach 0.31 and 0.59 impinging jets. While the three techniques provide similar measures outside of the jet core, there is a slight difference in core velocity within, especially in the TPIV data at Mach 0.31. Since the data for the three measurement techniques were obtained during chronologically separated tests, minor variability in the nozzle stagnation conditions was present and accounts for the observed variations in the measured jet core velocity.

For the impinging jet configuration, the cameras were arranged linearly, as indicated in Fig. 1d, such that all cameras have oblique views of the impingement plate. While the laser volume was slightly offset from the surface, some amount of diffuse reflection from the surface was present manifesting as background noise in the camera images. SPIV does not suffer from this as the laser sheet is thin and located relatively far from the surface. In MP-STB and TPIV, due to the direction of propagation and expansion of the laser volume, the intensity of reflected light and the concomitant image background noise increase vertically as $y_{I}/D$ increases. This leads to minor deviations in the volumetric data outside of the jet core as measured by the various volumetric techniques for $y_{I}/D > 0$. Overall, MP-STB with VIC# provides reliable measures of the impinging jet flow field at both Mach 0.31 and Mach 0.59 that match closely with the benchmark SPIV data.

Compared to TPIV, MP-STB had a significantly lower computational cost. While the specific computational cost of MP-STB depends on multiple factors including number of IPR tracking and shaking iterations, with more STB iterations needed for higher seeding density, and size of measurement domain, MP-STB generally costs an order of magnitude less computational time compared to TPIV. The cost of VIC# reconstruction is considerable and dependent on the desired vector spatial resolution, but the overall cost of the combined MP-STB and VIC# was still found to be less than an equivalent TPIV measurement. It should be noted that the hardware requirements are of course higher for MP-STB since it requires a ‘quad-pulse’ laser. In terms of data storage requirements, which is a crucial concern during prolonged measurement campaigns, MP-STB is especially advantageous, since it only stores discrete particle positions leading to an extremely small data storage footprint.

The computational cost of obtaining Eulerian information from scattered Lagrangian particle tracks can be further reduced by implementing an ensemble averaging approach (Agüera et al. 2016; Kasagi and Nishino 1991; Schanz et al. 2016; Novara et al. 2019) instead of the VIC# data assimilation technique. While it is less computationally expensive and is capable of higher spatial resolution than what is possible with cross-correlation or VIC#, the ensemble averaging approach is limited to providing only mean, and not instantaneous, estimates of velocity, acceleration, and Reynolds stress components. The ensemble averaging approach also necessitates a significantly longer data acquisition than was currently obtained in this study, since it requires large number of samples for statistical convergence within each bin, especially when the bin size is reduced to resolve thin shear layers, and is therefore not evaluated in the present study. There are also alternative data assimilation methods available, such as Flow Fit (Gesemann et al. 2016) which was implemented in Schanz et al. (2016), Godbersen et al. (2019), and Novara et al. (2019), whose computational cost was not evaluated in the present study.

6 Conclusions

Multi-Pulse Shake-The-Box (MP-STB), with the specific implementation used in this study also alternatively referred to as Four-Pulse Shake-The-Box, provides a volumetric Lagrangian method for tracking particles in densely seeded flows. The present study describes the unique features of MP-STB that allow for its robust tracking ability and enable the direct measurement of velocity and acceleration. A noteworthy feature is the potential of MP-STB to provide quasi-time-resolved measurements over the period of each four-pulse sequence, which is facilitated by the short pulse separations realizable using the ‘quad-pulse’ laser. Experiments performed in high-speed jet flows, both free jets and jets impinging on a ground plane, elucidate the capability of MP-STB to provide Lagrangian particle tracks in the presence of high levels of shear and turbulence. Notably, MP-STB successfully captured the thin shear layer close to the nozzle exit in free jets at both Mach 0.31 and 0.59. Furthermore, the ability of MP-STB to identify particle tracks in the presence of noisy background illumination was validated through measurements of the impinging jet flow field.

The scattered Lagrangian particle tracks produced by MP-STB were also mapped on to a uniform grid through VIC#. This data assimilation method facilitated the reconstruction of flow fields on a regular Eulerian grid, which allows for direct comparison of the combined performance of MP-STB+VIC# against more traditional measurement methods including TPIV, SPIV, and planar PIV. Based on these comparisons, MP-STB was found to accurately capture volumetric information in realistic high-speed flows, albeit with certain inherent advantages and disadvantages. Given these considerations, MP-STB coupled with VIC# would be an appropriate technique for future studies on high-speed subsonic flows, especially if acceleration or quasi-time-resolved volumetric information is desired. Ensemble averaging of MP-STB data at relatively low computational cost and high spatial resolution is also possible if sufficiently long data recordings are available and only mean quantities are required. The availability of acceleration measures also provide for the ability to calculate pressure fields, but pressure computation (Van Gent et al. 2017; Van Oudheusden 2013) is a topic in and of itself and, while of interest, is beyond the scope of the present paper and will be the subject of a future publication. Direct calculation of Lagrangian Coherent Structures from scattered four-particle tracks would also be advantageous and is a rich topic for future research.