Lagrangian particle tracking

Lagrangian particle tracking (LPT) is a method used in fluid mechanics to analyze the motion of particles when subjected to a flow field. It provides a Lagrangian perspective, in which the flow is described by tracking fluid parcels or tracers over time, rather than observing changes at fixed locations as in the Eulerian frame. In experimental studies, LPT is typically performed using three-dimensional particle tracking velocimetry (3D-PTV). Neutrally buoyant tracer particles are introduced into the flow, and their positions are recorded using high-speed cameras and stereo reconstruction techniques. The resulting particle paths allow for the study of turbulent structures, transport phenomena, and time-resolved Lagrangian statistics. In computational fluid dynamics, LPT refers to the numerical simulation of discrete particles embedded in a continuous flow field. The fluid phase is typically solved in an Eulerian framework, while the particle phase is resolved using Lagrangian mechanics. This approach, also termed discrete particle simulation (DPS), is particularly suited to situations where particle–fluid coupling is weak, such as dilute multiphase flows (such as aerosols), particle deposition in the human airways, and environmental particle transport. Applications of LPT also include cases where coupling is not negligible, which require more advanced numerical methods such as the discrete element method (DEM). Examples of these cases include industrial mixing, combustion modelling, sprays, and fluidized beds. Beyond engineering and turbulence research, LPT has been widely adopted in environmental modelling. Its capacity to resolve particle motion over complex terrain and large scales makes it suitable for studying the dispersion of atmospheric pollutants. In regional air quality assessments, LPT methods have been used for both forward simulations (predicting particle transport from known sources) and inverse modelling (inferring sources from observed concentrations). These techniques have proven effective in identifying transboundary pollution pathways and assessing exposure risks.

Experimental field The main goal of Lagrangian particle tracking in the experimental field is to extract data about the flow field, such as the flow velocity, acceleration (the material derivative), and pressure fields in the Lagrangian frame.

Extracting data about the flow field is done by using particle tracking velocimetry (PTV) methods. PTV methods use a lower particle concentration compared to particle image velocimetry (PIV) methods, allowing individual particles in the flow field to be followed. Typical particle image densities for PTV methods are between ∼ 0.005 {\displaystyle \sim 0.005} and ∼ 0.02 {\displaystyle \sim 0.02} ppp (particles per pixel) while PIV experiments can be performed with higher particle densities. In both PTV and PIV methods, the flow is seeded with neutrally buoyant particles (typically fluid ones in gaseous media and solid ones in fluid media) that are sufficiently small to follow the flow streamlines. This is verified when the Stokes number is sufficiently small, the typical condition being S t < 0.1 {\displaystyle St<0.1} . The particles are illuminated twice at given intervals, and a high definition camera is used to capture an image each time particles are illuminated. Each pair of images can be elaborated to extract the velocity field. Images are then post-processed to obtain the particles' velocity and positions. Several PTV methods have been proposed, and their particular setup changes are based on the specific needs of the experiment. The number of cameras used varies between one for the standard 2D-PTV method and up to six in state-of-the-art setups. Recent advancement in cameras (high frequency CMOS and CCD sensors), illumination technology (high frequency lasers and scalable LED illumination), calibration (volume self-calibration method), and post-processing algorithms (Shake-the-Box and iterative particle reconstruction techniques) make it possible to develop complex setups, such as time-resolved 3D-PTV with particle image densities that reach PIV-level density ( ∼ 0.2 {\displaystyle \sim 0.2} ppp).

When performing PTV experiments, typical difficulties include:

Irregular scattering behavior caused by the particles' different sizes, shapes, rotation velocities, and positions with respect to the illumination source Background intensity and light reflection caused by the model, which can diminish the signal-to-noise ratio (SNR) Image distortions due to improper calibration or camera astigmatism, which can lead to non-Gaussian particle imaging, deteriorating the quality of post-processing results

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