Particle tracking velocimetry in liquid gallium flow around a cylindrical obstacle

Abstract

This paper demonstrates particle tracking velocimetry performed for a model system wherein particle-laden liquid metal flow around a cylindrical obstacle was studied. We present the image processing methodology developed for particle detection in images with disparate and often low signal- and contrast-to-noise ratios, and the application of our MHT-X tracing algorithm for particle trajectory reconstruction for the wake flow around the obstacle. Preliminary results indicate that the utilized methods enable consistent particle detection and recovery of long, representative particle trajectories with high confidence. However, we also underline the necessity of implementing a more advanced particle position extrapolation approach for increased tracking accuracy. Satisfactory tracking accuracy can be inferred from the fact that the fluctuations in the measured particle velocity are dominated by frequencies that agree sufficiently well with the expected frequencies of the cylinder wake.

Graphical abstract

Appendix

1.1 Appendix A: Parallelization and performance

The image processing pipeline as outlined above was implemented in Wolfram Mathematica using its parallel computing functionality. Due to a large number of IWs per image (~ 10²) and the small relative area of an IW (~ 3.8%), it was decided to parallelize IW processing for individual images, thus performing sets of parallel computations, as many as there are images in a sequence. IW generation is performed in serial mode, while local filtering and particle segmentation (Algorithm 1, Steps 4 and 5) are parallelized. To speed up false positive filtering (Algorithm 4), it is split into a sequence of steps, each individually parallelized, as outlined in Algorithm 5. Afterward, the assembly of the full FOV masks is performed in parallel (image composition and addition), while area thresholding and morphological opening are performed in serial mode.

The implementation as presented in this paper was tested on two machines (Windows 10):

• Intel Core i9-10980XE (18 cores/36 threads) with 256 GB 2933 MHz DDR4 RAM

• Intel Core i7-10700K (8 cores/16 threads) with 32 GB 3200 MHz DDR4 RAM

The code was first tested using a sequence of 1500 FOV images. Three test cases were run multiple times each: using all available parallel threads (hyperthreading was used since all the underlying operations for IWs are independent) on the Core i9 and Core i7 systems, and running the code on the Core i9 CPU using half of the available parallel threads. The results are summarized in Table 1.

Table 1 The results of the image processing code benchmarks. $\tau$ denotes wall time. RAM utilization accounts for the system processes

In both cases with all available parallel threads used, the CPU utilization for Algorithms 2 and 3 was consistently at 100%. For the Core i9 system with all threads utilized, all stages of Algorithm 5 combined exhibit mean CPU utilization of ~ 39% on average, with ~ 28% at minimum and ~ 83% at maximum. The global mask assembly runs with ~ 81% CPU utilization on average. CPU utilization for the Core i7 system was greater for both Algorithm 5 and global mask assembly: ~ 59% and ~ 90% on average, respectively. The discrepancies in processing time proportions and CPU utilization between the Core i9 and Core i7 systems are largely due to a considerable difference in the number of cores in favor of the Core i9 machine and the better single-core performance of the Core i7 machine. The speedup factor between the two systems systems running all available threads is ~ 1.43.

1.2 Appendix B: Particle detection density

Figures 13 and 14 show the particle detection density over frames (equivalently, in time with 100 FPS) and space for a 1500-frame image sequence. Particle count per frame (Fig. 13) is on average ~ 105 with a ~ 10% deviation, indicating consistency in particle detection.

Fig. 13

Particle count per frame (100 FPS, black) over a 1500-frame image sequence. The red curve is the averaged trend obtained via Gaussian total variation (TV) filtering (regularization parameter equal to 2) Rudin et al. (1992) and the statistically significant ($q=0.9$ quantile) value ranges about the averaged curve are indicated with the light-gray envelope. The envelope is derived by filtering the quantile spline envelopes Antonov (2014) for data with the same TV filter as the data

Fig. 14

Normalized area density of particle detection events within the FOV (Fig. 2) over a 1500-frame image sequence. Density isolines are shown in white. The density map was computed using a Gaussian kernel over the count area density with Silverman’s bandwidth estimation

Figure 14 indicates that particle detection density is considerably greater within the wake of the cylindrical obstacle-this makes sense intuitively, since particles entrapped in or travelling through the wake flow zone are slower and have longer residence times within the FOV than the particles travelling with the mean flow around the obstacle, to the top and bottom of the FOV. Hence, more detection events per unit area are generated in the wake flow region of the FOV. This means that generated detection events are physically consistent with what one would expect from the studied system.