Dynamic wall shear stress measurement using event-based 3d particle tracking

Event-based vision (EBV), also termed dynamic vision sensing (DVS), is a new upcoming field within the field of computer vision and is inspired by the spiking mode of operation of the eye’s retina. Contrary to conventional frame-based imaging, EBV only records changes of image intensity (i.e., contrast changes) on the pixel level, triggering a positive event (\(+1\)) for increasing intensity and a negative event (\(-1\)) for a decreasing intensity change. The typical threshold of the intensity change trigger is on the order of 10–20% but can be fine-tuned. As the pixels on the detector respond individually, the events appear asynchronously throughout the detector area resulting in a continuous stream of data, with each event datum \(E_i = E_i(\textbf{x},t,p)\) consisting of pixel coordinates \(\mathbf {x_i} = (x_i,y_i)\), a time stamp \(t_i\), and a polarity \(p_i \in \{+1,-1\}\) indicating the direction of the intensity change. Unlike to conventional imaging, intensity is not directly available and the random nature of the asynchronous stream of events necessitates completely different data processing algorithms that are subject of current research. For a recent review of the technology and underlying concepts, the reader is referred to the topical review by Gallego et al. (2022).

After original prototype and conceptual development of the technology in 1990s, affordable and ready-to-use hardware based on EBV only recently has become available with current sensor resolutions of 1 MPixel. This has broadened the range of applications as testified in a steadily increasing number of publications (see, e.g., Robotics and Perception Group 2023; Gehrig and Scaramuzza 2024).

The application of EBV for the visualization and measurement of fluid flows is by no means new. Initial work was performed by Drazen et al. (2011) on particle tracking velocimetry (PTV) of dense particles in a solid–liquid two-phase pipe flow using an EBV sensor of \(256 \times 256\) pixels and continuous laser (5W) illumination. Ni et al. (2012) used an EBV array of 128\(\times\)128 elements to demonstrate microparticle tracking (\(\upmu\)PTV) with \(12\,\upmu \text{m}\) microspheres and were able to detect Brownian motion. Using a stereoscopic EBV system, Wang et al. (2020b) implemented a 3d PTV system allowing them to reconstruct three-dimensional tracks combining 2d tracking results from two cameras. Their flow experiment consisted of a small hexagonal cell with a stirrer inducing a swirling flow containing O(\(100\,\upmu \text{m}\)) polystyrene spheres. First PTV measurements in an air flow were performed by Borer et al. (2017) using three synchronized EBV cameras (128\(\times\)128 pixels) to track helium-filled soap bubbles (HFSB) in volumes up to about 1 m side length using white light LED arrays for illumination. The flow was only sparsely seeded allowing individual particles to be tracked with final data sets containing up to O(1 000–10 000) tracks. More recently, Rusch and Rösgen (2023) re-implemented this concept as a real-time 3d PTV system enabling live flow field reconstruction.

The work presented herein extends upon the recently introduced event-based imaging velocimetry (EBIV) concepts (Willert 2023; Willert and Klinner 2022) and introduces a 3d-3c Lagrangian particle tracking (LPT) system in a macroscopic imaging configuration with a magnification of O(\(10\,\upmu \text{m}/\text{pixel}\)), thereby capable of resolving the flow at the viscous scale. In comparison with previous event-imaging implementations, much higher seeding densities are achieved. However, due to the high data load, the captured sequences of event data currently cannot be processed in real time and have to be analyzed in an offline fashion, that is, after completion of the measurement.

To demonstrate the viability of the proposed technique, the setup is used to acquire the near-wall trajectories of tracers within the viscous sublayer of a TBL, specifically to estimate the unsteady WSS. In this sense, the work addresses the current shortcoming of measurement techniques capable of providing reliable data of the unsteady WSS vector. The acquired data can be directly compared to readily available direct numerical simulations (DNS) at matching Reynolds numbers for this canonical TBL flow.

As pointed out in the review by Örlü and Vinuesa (2020), the measurement of the unsteady WSS remains a challenge with very few approaches capable of measuring the unsteady WSS directly, with the exception of a few micro-mechanical implementations such as the shear stress imaging device by Kimura et al. (1999) which relies on the measurement of the actual shear force acting on an array of transducers. The majority of WSS measurement devices rely on an indirect measurement, typically of the near-wall velocity in the region dominated by viscous forces, namely the viscous layer at the wall, which extends out to about 5 viscous units \(l ^* = \nu \,/u_\tau\). Here, \(\nu\) is the kinematic viscosity of the fluid and \(u_\tau\) the friction velocity which itself is related to the WSS \(\tau _w\) and the fluid’s density \(\rho\) by \(u_\tau = \sqrt{\tau _w / \rho }\).

Aside from hotwire anenometry (HWA) and the micro-pillar method (Brücker et al. 2007; Große and Schröder 2008), most indirect WSS measurement techniques are particle-based methods, that is, variants of laser Doppler anemometry (LDA) or particle imaging. Among these, the following offer the desired combination of unsteady measurement of the WSS vector on a reasonable field of view (FOV), that is, they are not limited to point-wise measurements:The following article introduces a measurement configuration that can capture fields of the unsteady WSS vectors by tracking the motion of particles within the viscous sublayer using the novel event-imaging approach. The paper is organized as follows: the specifics of the event camera-based 3d-3c system are followed by a description of the data processing including the employed particle tracking algorithm and an error assessment. The results section concentrates on the variety of data that can be derived from the particle tracking data including near-wall flow statistics and derived WSS. The discussion positions the herein introduced event-based 3d PTV among existing approaches and addresses shortcomings of particle imaging-based WSS estimation.

The 3d-3c particle tracking system comprises a triplet of event cameras (Prophesee EVK4, Sony IMX636 sensor, \(1280 \times 720\) pixels, \(4.86\,\upmu \text{m}\) pixel size) in a photogrammetric configuration, that is, arranged in a manner to capture a common, relatively thin volume of interest. Scheimpflug mounts on the two off-normal cameras allow a common plane of focus for all three cameras (Fig. 1a). The three cameras are synchronized with an external 1 MHz source to ensure a common time base. In addition, reference pulses at 100 Hz allow precise alignment of the separately recorded event sequences with a resolution of \(1\,\upmu \text{s}\) with respect to one another (cf. Figure 2). This is necessary since the cameras operate in a continuous mode and cannot be started from a common well-defined trigger and consequently require a posteriori re-alignment of the streamed data.

In the present application, the tracking system is mounted below the wind tunnel section and observes the bottom layer of the TBL through a 1-mm thin glass window with anti-reflective coating. This domain is illuminated with a \(\approx 0.5\)-mm thin light sheet introduced from the side of the wind tunnel with a slight inclination (\(\approx 0.5^\circ\), cf. Figure 1b). The light sheet is oriented such that all cameras receive the light scattered by the tracers at a common scattering angle of \(90^\circ\). This results in similar illumination intensities on all three detectors and avoids angle-dependent Mie scattering differences between the cameras.

At a working distance of about 200 mm, a common FOV of about \(12.0 \times 7.5 \,\text {mm}^2\) is captured (magnification \(m = 0.48\) with 10 \(\upmu \text {m/pixel}\)). The pulsed laser (Innolas/Iradion, Nanio-Air 532-10-V-SP) is operated at a pulsing frequency of 5 kHz with an integral power of about 1–2 W and is synchronized to the camera time base (see Fig. 2). The macro-objective lenses (Nikon Micro-Nikkor 55 mm / 2.8) are stepped down to \(f_\# = 8\).

Water-based tracer particles of about 1–2\(\,\upmu \text {m}\) and a lifetime of about 10 min are provided by a fog generator (HazeBase Classic, base*M fluid). As detailed in Appendix 1, the temporal response of these tracers is estimated to be in the range of \(3\,\upmu \text{s}< t_p < 12\,\upmu \text{s}\) with Stokes numbers well below unity, guaranteeing good flow following properties.

For the measurements, event recordings of up to 60 s duration are acquired at wind tunnel speeds of \(U_\infty = 5.2,\,7.5\, \text {and} \,10 \,\text{m}/\text{s}\). Table 1 summarizes specific aspects of the acquired raw data such as event data rate and the amount of actual data streamed to the host computer. Table 1 also provides the TBL’s characteristic parameters which were obtained by high-speed profile PIV (Klinner and Willert 2024).

Keeping the laser energy and light sheet position constant, the seeding density is varied by more than one order of magnitude resulting in a corresponding variation in the event data rates. The EBV camera settings—so-called biases—are adjusted for minimal pixel refractory time (“dead time” after event detection) and to favor positive (\(+1\) events) as these were found to be more responsive than the bright-to-dark contrast changes (\(-1\) events). The imbalance between positive and negative contrast change detection is a detector-specific behavior, a further investigation thereof being beyond the scope of this article. Figure 3 intends to provide an impression of the event data acquired by one of the cameras at two different seeding concentrations.

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Elemental for reliable 3d-PTV is an accurate camera mapping which allows a transformation from image space into object space and back. This is generally achieved using established camera calibration procedures.

Calibration data in the form of image–object correspondence points is collected from recordings of a calibration target. Here, a checker-board target with \(1\,\text{mm} \times 1\,\text{mm}\) squares printed on glass is mounted parallel to the observation window and traversed in wall-normal (y) direction at increments of \(\Delta y = 250\,\upmu \text{m}\) (Fig. 4a). Due to insensitivity of the EBV to static imagery, the glass target is back-illuminated by a pulsed LED at \(100\,\text{Hz}\). Summing events over a period of \(0.5\,\text{s}\) provides high-contrast calibration images suitable for grid marker detection (Fig. 4b). The common FOV shared by the cameras, as depicted in Fig. 5, extends about \(12\,\text{mm}\) by \(7.5\,\text{mm}\) in streamwise and spanwise direction, respectively.

The accuracy of particle reconstruction in relation to the glass surface requires the knowledge of the plane spanned by the target in relation to the plane of glass surface. This is achieved by triangulation of stationary particles and dust attached to the surface of the window, which are readily detected in the raw event data as continuous triggered pixel clusters. A 2d plane fit provides the reference plane to which the reconstructed track data will be aligned (Fig. 5b). The slope amounts to about \(50\,\upmu \text{m}\) across the \(10\,\text{mm}\) FOV (inclination \(\approx 0.3\,^{\circ }\)).

A dual plane method is used to map between object and image space and to compute epipolar lines to match the particle images between the views. A particle-based residual alignment such as typically performed in 3d STB LPT is currently not applied.

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Prior to particle tracking, the acquired event recordings are temporally aligned using the external 100-Hz reference markers and then converted to pseudo-image sequences by re-sampling the event data at a frequency corresponding to the laser pulsing rate. During a sample interval, e.g., \(200\,\upmu \text{s}\) at \(5\,\text{kHz}\) laser pulsing, any given pixel is only allowed to produce at most one event. Hence, the resulting pseudo-image is binary in nature. The automated event sampling is performed on the basis of searching for the minimum in the ensemble averaged event histogram, a representative example of which is given in Fig. 6 for a laser pulsing frequency of \(5\,\text{kHz}\). In this case, the sampling period, as indicated by the red dashed lines, would begin with an offset of \(\approx\) \(175\,\upmu \text{s}\) and end \(200\,\upmu \text{s}\) later. As described in more detail in Willert (2023), the use of pulsed light intends to mitigate issues related to the delayed response, e.g., latency, of the event detector. This latency is apparent in Fig. 6b with events being registered by the detector up to \(100\,\upmu \text{s}\) after being exposed to the short pulse of light (about \(20\,\text{ns}\)). Another advantage of the pulsed light approach is the capture of stationary particles which would be “invisible" when illuminated by a continuous light source. Finally, the use of continuous illumination was found to “favor" slower particles since they have a higher likelihood of triggering events while crossing a given pixel. This would bias the measurement toward lower velocities already at the raw data stage (Willert 2023).

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Particle tracking is performed for each camera view individually by first extracting contiguous binary pixel blobs from the pseudo-images and computing their centroids (center of mass). A k-d tree-based nearest neighbor search scheme then detects tracklets across three adjacent pseudo-images and extends these via a predictor scheme to the following image frames. The tracker accepts gaps of up to one pseudo-frame to prevent a premature truncation of tracks. Given the high variability of particle velocity and paths within a very thin volume, a multiple pass scheme is implemented: first, slow moving particles are tracked and removed from the pool of particle positions. By gradually increasing the initial search radius, the next tracking passes draw candidates from a gradually reduced pool of particle positions. This process is repeated 3 to 4 times.

Using the 2d tracks for the three cameras views, reconstruction of the 3d tracks is performed using the epipolar lines of a given particle on the other two views. Using a cubic B-spline fit on the raw 3d position data yields a set of spline coefficients and provides a continuous description of the particle’s position, velocity, and acceleration along the track, that is, in time and space. The spline fit is weighted proportionally to the inverse of the residuals of the 3d particle position. Track validation is based on the residuals of the cubic spline fit (\(r_\mathrm{fit} \le 10\,\upmu \text{m}\)), a minimal track length (\(N_\mathrm{track} \ge 7)\), and maximum allowed wall-normal fluctuation (\(v_\mathrm{rms} < 0.02 U_\infty\) for \(y^+ < 2\)).

The wall shear stress vector \(\vec {\tau }_w = [\tau _{w,x},\tau _{w,z}]\) for each validated particle position is then obtained be dividing its estimated wall parallel velocity \(\textbf{u} = [u,0,w]\) by its distance from the wall \(\Delta y\) as an approximation to the definition of WSSwith \(\mu\) representing the dynamic viscosity and the range of \(\Delta y\) limited the viscous layer (\(y^+ < 5\)). Here it is crucial that the wall distance \(\Delta y\) is corrected for any offset and tilt of the wall surface as described in Sect. 3.

Figure 9 shows two realizations of recovered near-wall tracks at \(U_\infty = 5.2\,\text{m}/\text{s}\) (\(\mathrm{Re}_\tau = 563\)). The particle positions are color coded with the local wall stress magnitude \(\mid \vec {\tau }_w \mid\). While the tracks in Fig. 9a indicate a low-shear condition and even some flow reversal, the flow topology is completely different only 8 ms later (Fig. 9b) when it is dominated by a high shear rate aligned with the mean flow direction. The shear rate partially exceeds the mean value by a factor of two. Animations of the near-wall particle motion at different playback speeds are provided as part of the supplementary material, see section Appendix 2.

A short record of \(0.2\,\text{s}\) duration shows the evolution of WSS estimates within a small (\(1\times 1\,\text{mm}^{2}\)) area in Fig. 10. The streamwise component \(\tau _{x}\) reaches negative values (reverse flow, marked red in plot) at \(t \approx 13.93\,\text{s}\) and \(t \approx 13.98\,\text{s}\), whereas the spanwise component \(\tau _{z}\) exhibits several extreme events in excess of 3-4 times the rms. Of importance is the fact that the randomness of the LPT data does not provide continuous time records for a given point. This prevents the ad hoc calculation of space–time characteristics such as frequency spectra or temporal correlations. To allow this, the random data would first need to be subjected to data assimilation or interpolation schemes.

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In the course of the experiments, event data were collected at different seeding densities to assess its influence while keeping all other parameters constant. The track detection rate given in Table 1 indicates that there is an optimum event data rate of about 8–12\(\times 10^{6}\,\text{Ev/s}\). A further increase in the event rate (= higher seeding) actually results in a reduction in the track detection rate. With increased particle image density, the likelihood of particle ambiguity and false track initialization also increases. At the highest seeding level with more than \(20\times 10^{6}\,\text{Ev/s}\) (data set 5–2), the valid track validation rate drops to less than \(5\,\%\), which is why this data set was omitted in the data analysis. Here an approach that first reconstructs the 3d particle positions followed by 3d track building, rather than tracking in 2d space for each camera view, is likely to provide better results.

Along with the increase in event rate, the saturation of the sensor readout causes increased latency in the time stamping such that pulses are no longer clearly separated; derived pseudo-frames will contain particle images from more than one pulse which cannot be separated in time. This also impacts the particle tracking performance.

Even at optimal particle image density and event data rate, the WSS determined from the tracking results showed a consistent underestimation of the spanwise WSS fluctuation \(\tau ^+_{z, rms }\). This was also found in related measurements using highly accurate microparticle tracking techniques by Kumar et al. (2021) and Klinner and Willert (2024). Here DNS is particularly helpful in explaining the underestimation: Fig. 8d provides profiles of the velocity fluctuations for all 3 velocity components. Focusing in on the near-wall region (\(y^+ < 8\)) in Fig. 8e, they are characterized by different rates of change, with \(u^+_\mathrm{rms}\) to strongest, followed by spanwise \(w^+_\mathrm{rms}\) (\(\approx 40\%\) at \(y^+ = 5\)) and wall-normal \(v^+_\mathrm{rms}\) (\(\approx 10\%\) at \(y^+ = 5\)). However, when these quantities are normalized with the mean streamwise velocity U(y) as shown in Fig. 8f, they exhibit a completely different behavior: while the quantity \(u_\mathrm{rms}/U\) shows gradual decrease, its spanwise counterpart \(w_\mathrm{rms}/U\) rapidly decreases with increasing wall distances, whereas the wall-normal quantity gradually increases from zero at the wall. The limiting values of the former two quantities, \(u_\mathrm{rms}/U\) and \(w_\mathrm{rms}/U\), at the wall (\(y=0\)), in fact, coincide with the WSS fluctuations and represent the DNS-based estimates in Fig. 14. In the context of velocimetry-based WSS estimation, the velocity must be sampled at a finite distance \(\Delta y\) from the wall. Close to the wall, both the velocity and wall distance approach zero and relative errors rapidly increase as explained in Sect. 4.1. Since the quantity \(u_\mathrm{rms}/U\) has a weaker decay compared to \(w_\mathrm{rms}/U\), the latter will always be underestimated to a much higher degree. This is illustrated in Fig. 14 by sampling the DNS data at a finite wall distance of \(y^+ = 2\) as indicated by the gray symbols. This sampling domain is comparable to that chosen for the WSS estimation in the present work and leads to a comparable underestimation of the spanwise WSS fluctuation \(\tau ^+_{z, rms }\). In principle, the underestimation can be corrected by computing the velocity variances at different wall distance intervals and extrapolating the trend toward the wall. The velocity fluctuations plotted in Fig. 11b closely follow the DNS predictions and justify the extrapolation approach.

At the highest bulk velocity of \(U_\infty = 10\,\text{m}/\text{s}\), the particle track yield was insufficient for reliable WSS estimation, in part, due to the nearly doubled mean particle displacement (compared to \(U_\infty = 5.2\,\text{m}/\text{s}\)), but also because of the proportional reduction in the viscous scale from \(\nu /u_\tau = 69\,\upmu \text{m}\) to \(\nu /u_\tau = 37\,\upmu \text{m}\). To a certain extent, a proportionally higher laser pulsing frequency could improve the measurement. However, the bandwidth limitation of the EBV camera hardware imposes a limit of about \(10\,\text{kHz}\), in particular, at increased seeding levels. Overall it was found that the data quality improves with reduced seeding density which is related to the improved particle matching using only three cameras. Adding a fourth camera in the setup would provide additional redundancy, stabilizing the 3d particle position reconstruction.

In terms of FOV and spatial resolution, the herein introduced configuration has advantages over other WSS measurement techniques reported in the literature. Covering an area of \(12\times 7.5\,\text{mm}^{2}\) (\(170x^+ \times 110z^+\)), the FOV of the present implementation is considerably larger than that of the micro-pillar technique (\(2.1 \times 2.1\,\text{mm}^{2}\)Liu et al. 2019) or DFRH (\(1 \times 1\,\text{mm}^{2}\), \(20x^+ \times 20z^+\), Kumar et al. 2021) and \(\upmu\)DH (\(1.5 \times 1.5\,\text{mm}^{2}\), \(88x^+ \times 88z^+\), Sheng et al. 2008). Similarly, the depth-from-defocus approaches have a small FOV on the order of \(1 \times 1\,\text{mm}^{2}\) (Fuchs et al. 2023; Klinner and Willert 2024). The MEMS-based WSS “imagers" by Kimura et al. (1999) provided a FOV of \(22 \times 7.5\,\text{mm}^{2}\), however, on a relatively coarse grid of sensors consisting of 3 rows spaced at \(\Delta x = 10\,\text{mm}\) with 25 sensors each spaced at \(\Delta z = 300\,\upmu \text{m}\). In this regard, the present work offers both a high spatio-temporal resolution on a FOV covering in excess of one mean wall streak spacing.

The material presented herein demonstrates the viability of event-based imaging velocimetry for accurate measurement of TBL properties by means of Lagrangian particle tracking, providing near-wall velocity profiles and WSS distributions along with derived quantities. The reduced data stream of EBV permits continuous recording on the order of minutes (or longer) using off-the-shelf computer systems for data storage. Uncertainties arising from the limited (1-bit) signal depth of the image data are accounted for by making use of the available temporal resolution of the raw data which is on the order of 5–10 kHz. Track reconstruction can be greatly improved using Wiener or Kalman filtering such as implemented by Borer et al. (2017).

Even without processing, the raw event data are well suited for the visualization of the near-wall dynamics. While this is also possible with high-speed particle imaging approaches, the inherent binary nature of the raw imagery captured by event cameras immediately provides high-contrast visualizations without additional effort (see, e.g., event data animations provided in the supplementary material, Appendix 2). In the present application, rapid spanwise modulations imparted by the passage of flow structures in the outer layers of the TBL are clearly visualized and suggest further spatio-temporal analysis of the dynamics to retrieve, for instance, the structure convection velocity.

The time-resolved data presented herein were acquired using hardware that is considerably cheaper in comparison with conventional high-speed PIV components necessary to achieve similar results and but, at this point, are unable to stream images for extended periods. Beyond this, the higher sensitivity of the EBV detectors reduces the power requirements of the laser used to illuminate the tracer particles.

The present measurements were performed at a laser pulsing frequency of \(5\,\text{kHz}\). Although not discussed here, a small portion of data was also acquired at 10 kHz and provided acceptable results in spite of a partial leakage (overflow) of some events into the following laser pulse period. Given the same magnification and the frequency limit of about 10 kHz for the utilized event camera hardware, the proposed technique should be applicable to TBL flows with friction velocities approaching \(u_\tau = 1\,\text{m}/\text{s}\).