Source-linked AI summary
Flow over an espresso cup: Inferring 3D velocity and pressure fields from tomographic background oriented schlieren videos via physics-informed neural networks
Shengze Cai, Zhicheng Wang, Frederik Fuest, Young-Jin Jeon, Callum Gray, George Em Karniadakis
TL;DR
The paper addresses how to recover continuous 3D velocity and pressure fields when Tomo-BOS provides primarily 3D temperature observations. It uses physics-informed neural networks that combine temperature-data mismatch with Navier–Stokes and heat-equation residuals, and applies them to espresso-cup flow, with PIV comparison and parameter studies. The authors report successful field inference and systematic dependence of velocity and pressure on Reynolds and Richardson numbers.
Problem
Conventional BOS velocimetry methods are mainly optimized for tracer-particle PIV images, whereas BOS generally visualizes scalar fields; the paper therefore targets velocity and pressure inference from Tomo-BOS temperature data.
Method
A PINN represents temperature, velocity, and pressure while enforcing the Boussinesq incompressible Navier–Stokes and heat equations through residuals combined with temperature-data mismatch.
Results
The method successfully infers 3D velocity and pressure over an espresso cup from reconstructed 3D temperature data, with qualitative validation against PIV measurements.
Takeaways & Limitations
PINNs provide a route for using planar or tomographic BOS data for velocity and pressure quantification in complex fluid flows.
Takeaways & Limitations
The formulation assumes the Boussinesq approximation and heat equation, and the experiment specifies physical properties and characteristic scales for the modeled flow.
Abstract
from arXiv · showhide
Tomographic background oriented schlieren (Tomo-BOS) imaging measures density or temperature fields in 3D using multiple camera BOS projections, and is particularly useful for instantaneous flow visualizations of complex fluid dynamics problems. We propose a new method based on physics-informed neural networks (PINNs) to infer the full continuous 3D velocity and pressure fields from snapshots of 3D temperature fields obtained by Tomo-BOS imaging. PINNs seamlessly integrate the underlying physics of the observed fluid flow and the visualization data, hence enabling the inference of latent quantities using limited experimental data. In this hidden fluid mechanics paradigm, we train the neural network by minimizing a loss function composed of a data mismatch term and residual terms associated with the coupled Navier-Stokes and heat transfer equations. We first quantify the accuracy of the proposed method based on a 2D synthetic data set for buoyancy-driven flow, and subsequently apply it to the Tomo-BOS data set, where we are able to infer the instantaneous velocity and pressure fields of the flow over an espresso cup based only on the temperature field provided by the Tomo-BOS imaging. Moreover, we conduct an independent PIV experiment to validate the PINN inference for the unsteady velocity field at a center plane. To explain the observed flow physics, we also perform systematic PINN simulations at different Reynolds and Richardson numbers and quantify the variations in velocity and pressure fields. The results in this paper indicate that the proposed deep learning technique can become a promising direction in experimental fluid mechanics.
1. Introduction
BOS offers flexible, relatively inexpensive flow visualization, but conventional velocimetry methods are not well matched to its scalar-field images. This paper applies PINN-based hidden fluid mechanics to infer continuous velocity and pressure from Tomo-BOS temperature data, including an espresso-cup experiment.
- Background: BOS is less expensive and more flexible to set up than PIV and LIF.It visualizes density gradients through refraction-induced image distortion.
- Existing velocimetry: Correlation-based BOS velocimetry is mainly optimized for tracer-particle PIV images rather than scalar-field BOS images.Density tagging velocimetry was proposed to address this mismatch by treating local density variation as a transported tracer.
- Proposed approach: The paper develops a method to estimate continuous velocity and pressure fields simultaneously from 3D temperature fields measured by Tomo-BOS.The approach uses deep neural networks to represent continuous flow fields.
- Contribution: The PINN integrates natural-convection governing equations with reconstructed 3D temperature data from an espresso-cup experiment.The authors state that this is the first application of the HFM paradigm to real experimental imaging data.
- Study design: The study includes an independent center-plane PIV experiment for validation and systematic simulations across parameter settings.The paper also reports a sequence of schlieren images and supplementary evaluations of PINN performance.
2. Physics-informed neural networks (PINNs)
The PINN represents temperature, velocity, and pressure as neural-network outputs while enforcing incompressible Navier–Stokes and heat-transfer residuals. Training combines temperature-data mismatch with physics residuals to infer latent flow fields from temperature observations.
- Method overview: PINNs extend hidden fluid mechanics by inferring velocity and pressure from visualized temperature data in buoyancy-driven flow.The formulation uses the Boussinesq approximation of incompressible Navier–Stokes equations and the corresponding heat equation.
- Neural representation: The fully connected network maps spatial-temporal coordinates to temperature, velocity, and pressure fields with trainable parameters Θ.The outputs are represented as (T, u, p) = FNN(x, t, Θ).
- Physical constraints: Physics residuals enforce the heat equation, buoyancy-driven momentum equations, and incompressibility constraint.For three-dimensional flow, residuals e1–e5 encode these governing equations under the Boussinesq approximation.
- Physical constraints: The method is ill-posed from temperature data alone, so an additional physics-encoded network constrains the feed-forward outputs by minimizing governing-equation residuals.The paper notes that velocity boundary conditions are not supplied in this formulation.
- Optimization: After optimization, feeding coordinates into the trained network yields velocity, pressure, and temperature throughout the computational domain.The algorithm initializes the network, constructs residuals using automatic differentiation, optimizes Θ with Adam, and evaluates the fields at arbitrary points.
- Optimization: Training minimizes a loss combining temperature-data mismatch Ldata with equation residuals Lres, weighted by λ.Observed-data points determine NT, while residual points can be numerous and randomly sampled using mini-batches.
3. Inference of Tomo-BOS Experiment
The Tomo-BOS experiment reconstructs temperature fields around an espresso cup and uses PINNs to infer continuous three-dimensional velocity and pressure. The inferred flow satisfies the governing equations, agrees qualitatively with independent PIV measurements, remains consistent with sparse data, and depends on encoded physical parameters.
- Experimental setup: Six cameras recorded 400 BOS images around the espresso cup, using background dot-pattern panels and 50 Hz pulsed LED illumination.The cameras covered 150° around the cup, with a projected pixel resolution of 60 µm/pixel.
- Experimental setup: Tomographic reconstruction combined six-camera displacement fields to obtain the three-dimensional temperature data used by PINN.The displacement fields were computed with subset-based ZNSSD processing, followed by calibration and tomographic reconstruction.
- Temperature reconstruction: At t = 2.0 s, PINN regressed the Tomo-BOS temperature field with less than 1% relative L2-norm error.The absolute temperature error over the spatial domain was less than 1°C.
- Inferred fields: The inferred flow gathers at the cup surface center and rises upward with increasing speed, reaching approximately 0.4 m/s maximum v-component velocity.The mean velocity over the whole space-time domain was about 0.063 m/s, and the inferred pressure is the deviation from hydrostatic equilibrium.
- Validation: Momentum-equation residuals averaged on the order of 10^-4 m/s2, while independent PIV showed similar flow patterns, velocity magnitudes, and profiles.The PIV comparison was qualitative because the experiments were independent and their flow structures were not temporally aligned.
- Sparse-data capability: PINN produced consistent velocity and pressure fields from temporally or spatially downsampled data, including training frames containing only 1/8 of the original Tomo-BOS data.For an unseen intermediate temporal frame, the relative L2-norm temperature error was 0.362%; the spatially downsampled snapshot had a 0.430% temperature error.
- Physical-parameter sensitivity: Inferred fields were sensitive to the physical parameters encoded in the governing equations, especially the Richardson number.Increasing Richardson number generally increased pressure over the cup and velocity magnitude, with an inverse pressure pattern for Re = 100 and Ri = 1.
4. Concluding Remarks
The paper presents PINNs for inferring continuous velocity and pressure fields from Tomo-BOS temperature data, integrating governing equations with measurements. Synthetic evaluation and espresso-cup experiments indicate accurate, flexible inference, including parameter sensitivity studies.
- PINNs integrate governing equations with temperature data to estimate velocity and pressure fields simultaneously.The approach does not require CFD solvers, initial conditions, or boundary conditions, and provides continuous solutions from sparse data.
- The method is first evaluated on a 2D synthetic buoyancy-driven-flow simulation with systematic parameter studies.The study varies network size, loss weighting, spatial and temporal resolution, and data noise level.
- Velocity and pressure inference is sensitive to the physical parameters encoded in the governing equations, especially the Richardson number.The sensitivity study uses velocity profiles and pressure profiles under different parameter settings while keeping training temperature data the same.
- The espresso-cup Tomo-BOS experiment successfully infers three-dimensional velocity and pressure fields from reconstructed temperature data.The inferred velocity is qualitatively validated against an independent PIV experiment.
- The results support PINNs as an accurate and flexible approach for fluid-mechanics data with varied configurations.The paper specifically demonstrates applicability to Tomo-BOS data and discusses extension to different flow types through appropriate governing equations.