Lagrangian Flow Field Reconstruction Based on Constrained Stable Radial Basis Function

Abstract

Recent advances in three-dimensional (3D) high seeding density time-resolved Lagrangian particle tracking (LPT) techniques have made diagnosing fluid flows at high resolution in space and time under a Lagrangian framework feasible and practical. But challenges persist in developing LPT data processing methods. A promising processing method should accurately and robustly reconstruct, for example, particle trajectories, velocities, and differential quantities from noisy experimental data. Despite numerous algorithms available in the LPT community, they may suffer from some issues, such as unfavorable accuracy and robustness, lack of physical constraints, and unnecessary projection from Lagrangian data onto Eulerian meshes. These challenges may limit the application of the 3D high seeding density time-resolved LPT techniques.

In this thesis, a novel 3D Lagrangian flow field reconstruction method is proposed to address these challenges. The proposed method is based on a stable radial basis function (RBF) and constrained least squares (CLS). The stable RBF serves as a model function to approximate trajectories and velocity fields. The CLS provides a framework to facilitate regression and enforce physical constraints, further enhancing the reconstruction performance. The stable RBF and CLS work together to reconstruct 3D Lagrangian flow fields with high accuracy and robustness.

The proposed method offers several advantages over the algorithms currently used in the LPT community. First, it accurately reconstructs particle trajectories, velocities, and differential quantities in 3D, even from noisy experimental data, while satisfying physical constraints such as divergence-free for incompressible flows. Second, it does not require projecting Lagrangian data onto Eulerian meshes, allowing for direct flow field reconstruction at scattered data locations. Third, it effectively mitigates experimental noise in particle locations. Last, the proposed method enables smooth spatial and temporal super-resolution with ease. These advantages exhibit that the proposed method is promising for LPT data processing and further applications in data assimilation and machine learning.

Systematic tests were conducted to validate and verify the proposed method. Two-dimensional and 3D validations were performed using synthetic data based on exact solutions of the Taylor-Green vortex and Arnold-Beltrami-Childress flow with added artificial noise. The validations show that the proposed method outperforms baseline algorithms (e.g., finite difference methods and polynomial fittings) under different flow conditions. The method was then verified using experimental data from a 3D low-speed pulsing jet, showing its reliable performance. Based on these validations and verification, it is demonstrated that the proposed method can process experimental LPT data and reconstruct Lagrangian flow fields with accuracy and robustness.