Reconstruction of thermally-driven flows using Lagrangian particle data assimilation.

Reconstructing hidden thermal and flow structures from limited observations is a fundamental challenge in many scientific disciplines. Particles passively advected by the surrounding fluid often encode valuable information along their trajectories, but such data are typically sparse and noisy. To infer the comprehensive dynamics of thermally driven flows from such limited information, we develop a four-dimensional variational (4DVar) Marker-in-Cell method. Its application to laboratory data demonstrates successful reconstruction of the time-dependent temperature field and the Rayleigh number-unobservable yet essential for understanding thermal forcing and heat transport-by assimilating particle trajectories with the governing equations. Furthermore, our method enables prediction of future evolution beyond the assimilation window, yielding results that are consistent with actual observations. We critically assess the method's performance in light of convective dynamics, identifying the conditions under which it is effective and outlining directions for future refinement. These findings highlight the utility of 4DVar not only for retrospective reconstruction but also for forward prediction of convective behavior, offering a robust framework for analyzing thermally or compositionally driven flows in geophysical and engineering systems.