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# Lagrangian Measurements and Physics-Informed Neural Network for Rayleigh-Bénard Flow Reconstruction

Anne Sergent, Soufiane Mrini, Elian Bernard, Didier Lucor

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Anne Sergent, Soufiane Mrini, Elian Bernard, Didier Lucor. Lagrangian Measurements and Physics-Informed Neural Network for Rayleigh-Bénard Flow Reconstruction. 26th International Congress of Theoretical and Applied Mechanics (ICTAM2024), Aug 2024, Daegu, South Korea. hal-04924332

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Distributed under a Creative Commons Attribution 4.0 International License
# LAGRANGIAN MEASUREMENTS AND PHYSICS-INFORMED NEURAL NETWORK FOR RAYLEIGH-BÉNARD FLOW RECONSTRUCTION

Anne Sergent*<sup>a)</sup>, Soufiane Mrini**, Elian Bernard*** & Didier Lucor**

*Sorbonne Université, Paris, France; Lab. Interdisciplinaire des Sciences du Numérique, Université Paris-Saclay, Orsay, France

**CNRS, Lab. Interdisciplinaire des Sciences du Numérique, Université Paris-Saclay, Orsay, France

***Lab. Physique, Ecole Normale Supérieure de Lyon, CNRS, Lyon, France

*Summary* The aim of this work is to reconstruct the temperature field from Lagrangian measurements in a turbulent Rayleigh-Bénard flow using a Physics Informed Neural Network. First, the spatial concentration of particles in the experimental dataset is examined using a 3D Voronoï analysis with a view to merging the experimental and numerical databases.

## INTRODUCTION

Turbulent buoyant convection is a spontaneous physical process present in many natural systems as well as in engineering applications. Many efforts have focused on coherent flow structures and in particular plumes, that originates from convective transport. Lagrangian measurements are particularly well-suited to the study of coherent structures. However, simultaneously measuring the local temperature and velocity fields remains a challenge [1], and tracer particles are more commonly used to follow the fluid velocity field.

Recently, many works have been done to infer hidden quantities from partial data in space or time using deep neural networks. For example, inference of the full continuous three-dimensional (3-D) velocity has been obtained by deploying physics-informed neural networks (PINNs) to 3-D temperature snapshots measured by a Schlieren technique [2]. Following this approach, we built a reduced model to deduce 3-D flows from a sparse temperature field using a dataset retrieved from Direct Numerical Simulation (DNS) of turbulent Rayleigh-Bénard convection [3]. In a similar line, Lagrangian velocity measurements can be thought of as partial velocity data.

In the present work, we use a PINN (figure 1) to reconstruct the temperature field from Lagrangian measurements in a Rayleigh-Bénard turbulent flow. DNS data in the same range of Rayleigh ($Ra$) number (figure 2) are used for preliminary tests with a future perspective of merging experimental and numerical databases. However, using deep learning techniques with a mixture of numerical and experimental data poses some challenges as the latter tend to be noisier. Besides, as NN models are sensitive to the spatial and temporal sampling of the data, we are primarily interested in characterising the spatial aspect for the experimental data.

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Figure 1. The framework of Physics Informed Neural Network. The inputs are space coordinates $\mathbf{x} = (x, y, z)$ and time $t$. The outputs are velocity $\mathbf{U} = (u, v, w)$, pressure $p$ and temperature $T$. The Boussinesq PDEs are imposed as physical laws, and approximated using automatic differentiation operators. The multi-objective loss is made up of the PDE residuals and the label error (here the velocity). In this work, we consider data obtained numerically by DNS as well as experimental measurements.

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Figure 2. Plume visualisation at a particular instant obtained by Direct Numerical Simulation ($Ra = 10^{10}$ and $Pr = 4.4$)

<footnote><sup>a)</sup>Corresponding author. E-mail: anne.sergent@lisn.upsaclay.fr</footnote>
## METHOD

We consider the fluid flow within a water-filled (Pr = 4 – 5) parallelepiped Rayleigh-Bénard cavity at Ra ≃ 10¹⁰. Same geometrical and physical parameters are applied to experimental and numerical set-ups. Two datasets are available. i) The DNS dataset is obtained by solving the Boussinesq equations and using the in-house parallel solver SUNFLUIDH. ii) The experimental measurement section covers a half-cavity (top or bottom). The experimental data are collected for a duration of 200 seconds by sequence with an acquisition frequency of 80Hz. Around 5000 tracer particles are detected on each frame (see figure 3-a for illustration). These datasets are used to train a PINN architecture similar to [3].

## RESULTS

Although the methodology initially consists of testing the PINN architecture on a DNS database, we propose to characterize the experimental spatial distribution of the tracer particles beforehand so that the neural network, previously trained on the DNS data, can then be adjusted to suit the Lagrangian nature of the experimental data. To describe the spatial concentration of the particles, we use a 3-D Voronoï analysis to determine and quantify the possible presence of clusters and voids. It is well-known that a random but uniform particles distribution yields a p.d.f. of the normalized Voronoï volumes ($V/\tilde{V}$ with $\tilde{V}$ being the mean Voronoï volume) which is well described by a $\Gamma$-distribution [4] (here $\Gamma(5, 5)$ in 3-D). Deviations from $\Gamma(5, 5)$ indicates either the presence of clusters for the smallest $V$ or of voids for the largest $V$. The figure 3-b displays the $V/\tilde{V}$ p.d.f. of the experimental dataset. It shows that the quantity of clusters does not depend on the size of the viewing box, while the bigger the box, the more voids there are. However, the relative pdf (figure 3-c) enables to count the total volume filled either by clusters or by voids. In our case, less than 5% of the study volume is covered by clusters, while 8 to 9% is associated with voids. We conclude that a DNS training dataset randomly selected in the 3-D space but with a uniform distribution is sufficient to reproduce the position of more than 85% of the particles in the Lagrangian measurement campaigns at our disposal. Nevertheless, the random selection of the particle positions in the DNS dataset can be refined to get closer to the experimental p.d.f.s.

<figure><img src="image_5.png" /><figcaption>Figure 3. Example of Lagrangian measurements. (a) A few particle trajectories in the measurement volume. (b) Normalized Voronoï volume p.d.f for a typical experiment (Ra = 2 · 10¹⁰, around 5000 particles per instant, 320 instants): Voronoï diagrams for two study volumes (Box 3 ∈ Box 1); dotted line associated with a random Gamma Γ(5, 5) distribution. (c) Ratio of the measured and theoretical p.d.fs: The two lines ($V_c^*$ and $V_v^*$) delineate the volumes corresponding to clusters (on the left) and the volumes associated with voids (on the right). In the middle, particles are shown to be randomly dispersed.</figcaption></figure>

This work benefits from the French National Research Agency funding (ANR-22-CE30-0018-02). The DNS database has been built using granted access to the HPC resources of IDRIS under allocation 2a0326 made by GENCI.

## References

[1] Gasteuil Y., Shew W.L., Gibert M., Chillà F., Castaing B. and Pinton J.-F.: Lagrangian Temperature, Velocity and Local Heat Flux Measurement in Rayleigh-Bénard Convection. PRL 99:234302, 2007.

[2] Cai A., Wang Z., Fuest F., Jeon Y.-J., Gray C. and Karniadakis G. E.: Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. J. Fluid Mech. 915: A102, 2021.

[3] Lucor D., Agrawal A. and Sergent A.: Simple computational strategies for more effective physics-informed neural networks modeling of turbulent natural convection. J. Comp. Phys. 456: 111022, 2022.

[4] Ferenc J.S. and Néda Z.: On the size distribution of Poisson Voronoï cells. Physica A 385: 518-526, 2007.