3D Lagrangian Particle Tracking in Fluid Mechanics

In the past few decades various particle image based volumetric flow measurement techniques have been developed which showed their potential in accessing unsteady flow properties quantitatively in various experimental applications in fluid mechanics. In this article we would like to focus on physical properties and circumstances of 3D particle-based measurements and what knowledge can be used for gaining advancements in the reconstruction accuracy, spatial and temporal resolution and completeness. The natural candidate for our focus is 3D Lagrangian Particle Tracking (LPT), which allows determining position, velocity and acceleration along a large number of individual particle tracks in the investigated volume. With the advent of the dense 3D LPT technique Shake-The-Box in the past decade further possibilities for characterizing unsteady flows have been opened by delivering input data for powerful data assimilation techniques which use Navier-Stokes constraints. As a result, high-resolution Lagrangian and Eulerian data can be gained including long particle trajectories embedded in time-resolved 3D velocity- and pressure fields

Pressure reconstruction from Lagrangian particle tracking with FFT integration

Volumetric time-resolved pressure gradient fields in unsteady flows can be estimated through flow measurements of the material acceleration in the fluid and the assumption of the governing momentum equation. In order to derive pressure, almost exclusively two numerical methods have been used to spatially integrate the pressure gradient until now: first, direct path integration in the spatial domain, and second, the solution of the Poisson equation with numerical methods. We propose an alternative method by integrating the pressure gradient field directly in Fourier space with a standard FFT function. The method is fast and easy to implement. We demonstrate the accuracy of the integration scheme on a synthetic pressure field and apply it to an experimental example based on acceleration data from Lagrangian particle tracking with high seeding density (Shake-The-Box method)

Two-Pulse 3D particle tracking with Shake-The-Box

The problem of reconstructing the three-dimensional position of particle flow tracers from their projection on multiple cameras lies at the heart of several 3D particle-image-based velocimetry and Lagrangian particle tracking (LPT) measurement techniques. While cross-correlation-based techniques such as tomographic-PIV (Tomo-PIV, Elsinga et al. 2006) make use of algebraic methods (e.g. MART, Herman and Lent 1976) to reconstruct particles as intensity peaks in a discretized voxel space, triangulation-based methods (3D-PTV, Nishino et al. 1989, Maas et al. 1993 and Iterative Particle Reconstruction, IPR, Wieneke 2013, Jahn et al. 2021) leverage epipolar geometry to reconstruct individual particles as positions and peak-intensity values in the 3D domain. Despite the inherent differences concerning accuracy, robustness and computational cost between the two approaches, in both cases the 3D reconstruction represents a bottleneck when the spatial resolution (i.e. particle image density, indicated in particles per pixel, ppp) of the measurement is considered. In fact, as the number of particles to be reconstructed increases (assuming constant properties of the imaging system), the reconstruction process become increasingly difficult due to the underdetermined nature of the problem. This typically results in a lower number and positional accuracy of the reconstructed particles, as well as an increasing number of spurious particles (ghost particles, Elsinga et al. 2011) which affect the accuracy of the measurement. As a consequence, during the last decade, several methods have been developed to increase the performances of the reconstruction technique; an overview of these developments can be found in Scarano 2012 and Jahn et al. 2021

3D particle position determination and correction at high particle densities

The method of Iterative Particle Reconstruction (IPR), introduced by Wieneke in 2013, constitutes a major step towards Lagrangian Particle Tracking of densely seeded flows (Schanz et al. 2016). This work presents the advanced Iterative Particle Reconstruction method that was recently published in Experiments in Fluids (Jahn et al. 2021) and additionally aims to quantify the abilities of the method to correct for misplaced particle positions e.g. by biased triangulation or the predictive step of the Shake-The-Box scheme. The former part details numerous approaches to enhance the original IPR working principle. Most of them are easy to implement, come at low computational cost and, in combination, nearly triple the processable particle image densities. The updated method is proven to be fast, accurate and robust against image noise and other imaging artifacts. The latter will allow to derive guidelines for the layout of experiments in terms of the required acquisition frequency as a function of the expected Lagrangian accelerations in order to guarantee reliable particle tracking. Furthermore, an a-posteriori approach for quantifying the impact of particle image overlap situations on the achievable position accuracy and corresponding correction methods will be discusse

Variable-Timestep Shake-The-Box (VT-STB) for flows with high dynamic range

In recent years, Lagrangian Particle Tracking (LPT) has become more and more important in the field of 3D flow measurements, largely due to the introduction of the methods of Iterative Particle Reconstruction (IPR, Wieneke 2013) and Shake-The-Box (STB, Schanz et al. 2016). Applying the STB algorithm, particles can be tracked in large numbers, with a nearly complete suppression of ghost particles. The introduced extension to the STB method extends its working range to flows with high vleocity range, i.e. large areas within the measurement region that exhibit very slow flow velocities compared to the fastest ones, which determine the applied frame-rat

Large-scale volumetric flow measurement in a pure thermal plume by dense tracking of helium-filled soap bubbles

We present a spatially and temporally highly resolved flow measurement covering a !arge volume (~o.6 m3) in a pure thermal plume in air. The thermal plume develops above an extended heat source and is characterized by moderate velocities (U~0.35 m/s) with a Reynolds number of Re~500 and a Rayleigh number of Ra~100000. We demonstrate the requirements and capa bilities of the measurement equipment and the particle tracking approach to be able to probe measurement volumes up to and beyond one cubic meter. The use of !arge tracer particles (300 µm), helium-filled soap bubbles (HFSBs), is crucial and yields high particle image quality over large-volume depths when illuminated with arrays of pulsed high-power LEDs. The experimental limitations of the HFSBs-their limited lifetime and their intensity loss over time-are quantified. The HFSBs' uniform particle images allows an accurate reconstruction of the flow using Shake-The-Box particle tracking with high partlcle concentrations up to 0.1 particles per pixel. This enables tracking of up to 275,000 HFSBs simultaneously. After interpolating the scattered data onto a regular grid with a Navier-Stokes regularization, the velocity field of the thermal plume reveals a multitude of vortices with a smooth temporal evolution and a remarkable coherence in time (see animation, supplementary data). Acceleration fields are also derived from interpolated particle tracks and complement the flow measurement. Additionally, the flow map, the basis of a !arge dass of Lagrangian coherent structures, is computed directly from observed particle tracks. We show entrainment regions and coherent vortices of the thermal plume in the flow map and compute fields of the finite-time Lyapunov exponent

Shake-The-Box particle tracking with variable time-steps in flows with high velocity range (VT-STB)

In recent years, Lagrangian Particle Tracking (LPT) has become more and more important in the field of 3D flow measurements, largely due to the introduction of the methods of Iterative Particle Reconstruction (IPR, Wieneke 2013) and Shake-The-Box (STB, Schanz et al. 2016). Applying the STB algorithm, particles can be tracked in large numbers, with a nearly complete suppression of ghost particles. The ability of the STB algorithm to discern real particles from ghost particles is largely stemming from the observation that for real particles the change of acceleration is small between time-steps, while ghost particles are generated in quasi-random locations all over the measurement volume. However, this assumption only holds if the particles exhibit a certain amount of movement relative to each other, such that the ambiguities ‘de-correlate’ within few time-steps. However, flows with a high velocity range can exhibit regions in space with only very little movement of the particles (e.g. jet flows). As the time separation has to be chosen such that the fastest particles are reliably tracked, particles in the surrounding flow might move only a fraction of a pixel in the same time. Another potentially problematic effect of slowly moving particle clouds is that situations of overlapping particle images are retained over several time-steps. In this work we propose an extension to the standard STB evaluation, which applies multiple iterations of particle tracking with variable time-separation Δ. The idea is to start the evaluation with a Δ at iteration 1 such that the slowest particles of the flow can be optimally tracked. From there, the time-separation is iteratively reduced, tracking faster particles with every iteration. Finally, the original time-separation of the recording is reached, where only the fastest particles - remain to be tracked

Dynamics of Coherent Structures in Turbulent Rayleigh-Bénard Convection by Lagrangian Particle Tracking of Long-Lived Helium Filled Soap Bubbles

We present spatially and temporally resolved velocity and acceleration measurements of turbulent Rayleigh-Bénard convection covering the complete volume of a cylindrical sample with aspect ratio one. Using the "Shake-The-Box" Lagrangian particle tracking algorithm, we were able to instantaneously track more than 500,000 particles in the complete sample volume (~ 1 m³), corresponding to mean inter-particle distances down to 5-8 Kolmogorov lengths. We used the data assimilation scheme "FlowFit" with continuity and Navier-Stokes-constraints to interpolate the scattered velocity and acceleration data via continuous 3D B-Splines on a cubic grid and to recover the smallest flow scales. The measurements were enabled by a dedicated bubble fluid solution, which we developed for generation of longlived helium filled soap bubbles, allowing for long-term optical flow measurements at large scales in gaseous fluids. We show Lagrangian and Eulerian visualizations of the large-scale circulation (LSC) as well as small scale structures, such as thermal plumes and turbulent background fluctuations and unveil the dynamics of their complex interplay. By employing principal component analysis in the rotating frame of the LSC, we are able to describe the characteristic dynamics of the LSC with the first three POD modes with an accuracy of 95% by using only 50% of the turbulent kinetic energy of the flow

Investigation of universal small-scale structures in turbulence using Shake-The-Box Lagrangian Particle Tracking and FlowFit

A characteristic property of turbulent flows is the presence of a universal small-scale structure consisting of a shear layer separated by two stretched vortices. This pattern becomes visible by averaging the velocity sampled in the eigenframe of the local strain rate tensor. In this contribution, the structure is detected and tracked over time to investigate its formation and decay. For this purpose, experimental data of a von Karman flow at ReÄ = 370 is used, which is analyzed with Shake-The-Box (STB) Lagrangian Particle Tracking (LPT) and the data assimilation method FlowFit. The universal structure is characterized by the time-resolved mean distribution of dissipation and pressure in the strain rate eigenframe. Using the average power balance terms along all particle trajectories passing through the direct vicinity of conditioned high-dissipation (> 7.5 ) and enstrophy (> 7.5 ) events during the time-span from -4 < t < 4 the Lagrangian energy transport mechanism through such intermittent events can be discovered, at least in a mean sense. It is shown how the impact of the particles in a high dissipative event leads to a part of the energy being converted into heat and another part into rotation in a time period of approximately 2r,;. This finding is consistent with the existence of the universal structure. Thus, a connection of Eulerian universal structures with their underlying energy exchange processes is proposed. Furthermore, instantaneous high dissipative events and the Lagrangian tracks that constitute them are shown and related to the Statistical results

Spatially and temporally resolved measurements of turbulent Rayleigh-Bénard convection by Lagrangian particle tracking of long-lived helium-filled soap bubbles

Rayleigh-Bénard convection (RBC), where a fluid is heated from below and cooled from above, is a prevalent model system to study the fundamentals of thermal convection. Typical for the turbulent RBC system is the occurrence of a large-scale circulation (LSC), which develops by self-organization of thermal plumes, erupted from the thermal boundary layers. In cylindrical samples of aspect ratios close to unity with a high degree of symmetry, the LSC reveals complex short- and long-term dynamics, which has been studied extensively in the past. Direct volumetric measurements of the LSC, however, allowing for a direct insight into the underlying turbulent processes are still rare. To bridge this gap, we performed Lagrangian Particle Tracking (LPT) by using a multi-camera setup, long-lasting, helium-filled soap bubbles and high-power LED arrays. With the "Shake-The-Box" Lagrangian particle tracking algorithm, we were able to instantaneously track up to 560,000 particles in the complete sample volume (~ 1 m³), corresponding to mean inter-particle distances down to 6-8 Kolmogorov lengths. We used the data assimilation scheme ‘FlowFit’, which involves continuity and Navier-Stokesconstraints, to map the scattered velocity and acceleration data on cubic grids, herewith recovering the smallest flow scales. Lagrangian and Eulerian visualizations reveal the dynamics of the large-scale circulation and its interplay with small scale structures, such as thermal plumes and turbulent background fluctuations. As a result, the complex time-dependent behavior of the LSC comprising azimuthal rotations, torsional oscillation and sloshing can be extracted from the data. Further, we found more seldom dynamic events, such as spontaneous reorientations of the LSC in the data from long-term measurements