Nakao et al. (2025) Reconstruction of thermally-driven flows using Lagrangian particle data assimilation

Identification

• Journal: Scientific Reports
• Year: 2025
• Date: 2025-10-14
• Authors: Atsushi Nakao, Daisuke Noto, Takatoshi Yanagisawa, Yuji Tasaka, Tatsu Kuwatani
• DOI: 10.1038/s41598-025-19724-x

Research Groups

• Institute of Systems and Information Engineering, University of Tsukuba, Japan
• Research Institute for Marine Geodynamics, Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Japan
• Department of Earth and Environmental Science, University of Pennsylvania, United States
• Faculty of Engineering, Hokkaido University, Japan

Short Summary

This study develops a four-dimensional variational (4DVar) Marker-in-Cell method to reconstruct time-dependent temperature fields and the Rayleigh number in thermally driven flows by assimilating sparse Lagrangian particle trajectories from laboratory experiments. The method successfully reconstructs hidden thermal and flow structures and predicts future evolution beyond the assimilation window.

Objective

• To develop and apply a four-dimensional variational (4DVar) Marker-in-Cell method to reconstruct time-dependent temperature fields, flow structures, and the unobservable Rayleigh number from sparse and noisy Lagrangian particle trajectory data in thermally driven flows, and to assess its predictive capability.

Study Configuration

• Spatial Scale: Laboratory experiment in a rectangular vessel (0.050 m height × 0.200 m width × 0.006 m thickness), modeled as a two-dimensional domain (0.200 m × 0.050 m) with a numerical grid of 150 cells (0.001 m width) × 50 cells (0.001 m height). Particle tracking spatial resolution was approximately 0.00011 m.
• Temporal Scale: Particle tracking data recorded at 10 Hz (0.1 s temporal resolution), with an assimilation time window of 2 minutes (120 s). The numerical model used a time step of 0.02 s. Observed oscillatory motion had a period of approximately 90 s.

Methodology and Data

• Models used: Four-dimensional variational (4DVar) Marker-in-Cell method (4DVarMiC) based on governing equations for two-dimensional, incompressible, highly viscous fluid, neglecting inertial effects:
  • Equation of motion: ∇⁴ψ = Ra₂D ∂T/∂x (stream function formulation)
  • Heat balance: ∂T/∂t + uᵀ ⋅ (∇T)ᵀ = ∇²T
  • Particle advection: dxᵢ/dt = uᵢ
  • Adjoint equations for stream function, temperature, and particle positions.
• Data sources: Laboratory experiment of Rayleigh–Bénard convection (RBC) in a narrow rectangular vessel filled with a dilute xanthan gum aqueous solution (Prandtl number = 70). Passive tracer particles were visualized by a 0.001 m thick laser sheet, and their trajectories were recorded as digital image sequences. Observed particle trajectories (positions and velocities) were used for assimilation.

Main Results

• The 4DVarMiC successfully reconstructed time-dependent temperature fields and flow structures from two minutes of sparse particle track data.
• The method accurately inferred the Rayleigh number (Ra₂D = 3 × 10⁵, corresponding to an experimental Raexp = 1.6 × 10⁶), a key unobservable control parameter in thermal convection.
• Reconstructed temperature fields were physically plausible (remaining within boundary temperature bounds) when the optimal Ra₂D value was used.
• The 4DVarMiC solution demonstrated predictive capability for future evolution beyond the assimilation window, with simulated root mean square velocity (uRMS) and Nusselt number (Nu0) closely matching observations.
• The study identified that sufficient data resolution and uniformity in spatial and temporal particle distribution are crucial for effective assimilation.
• Limitations included reduced estimation accuracy in regions with sparse particle observations (e.g., due to dim laser illumination) and near the thermal boundaries.

Contributions

• This study presents the first demonstration of simultaneously estimating both the temperature field and the Rayleigh number from sparse Lagrangian observations using an inverse approach (4DVarMiC), a capability generally unfeasible in forward modeling frameworks.
• Unlike previous methods, this approach does not assume a thermal steady state and directly infers both the thermal structure and the Rayleigh number from partially observed Lagrangian data.
• It highlights the utility of 4DVar for both retrospective reconstruction and forward prediction of convective behavior beyond the assimilation window, offering an advantage over methods like Physics-Informed Neural Networks (PINNs) which are typically limited to interpolation.
• The research addresses the challenge of relating the experimental Rayleigh number (Raexp) of a quasi-2D system to the 2D model Rayleigh number (Ra₂D), providing a data-informed framework for dimensional scaling.
• The developed framework offers a robust tool for analyzing unsteady thermally or compositionally driven flows in geophysical and engineering systems where temperature and Rayleigh number are not directly measurable.
• The explicit modeling and tracking of individual Lagrangian particles within the Marker-in-Cell framework enable the method's extension to component-specific transport processes in solutal or thermo-compositional systems.

Funding

• JSPS KAKENHI Grants (22K14131 and 25K00228)
• Joint Research Programs of the Earthquake Research Institute, University of Tokyo (2024-B-01 and 2025-B-01)

Citation

@article{Nakao2025Reconstruction,
  author = {Nakao, Atsushi and Noto, Daisuke and Yanagisawa, Takatoshi and Tasaka, Yuji and Kuwatani, Tatsu},
  title = {Reconstruction of thermally-driven flows using Lagrangian particle data assimilation},
  journal = {Scientific Reports},
  year = {2025},
  doi = {10.1038/s41598-025-19724-x},
  url = {https://doi.org/10.1038/s41598-025-19724-x}
}