Estimating density, velocity, and pressure fields in supersonic flow using physics-informed BOS

This paper introduces a novel physics-informed background-oriented schlieren (BOS) workflow that utilizes physics-informed neural networks to simultaneously reconstruct accurate density, velocity, and pressure fields in supersonic flows by integrating measurement data with governing Euler and irrotationality equations, thereby overcoming the limitations of conventional methods and achieving the first PINN-based reconstruction of supersonic flow from experimental data.

The Gist

Imagine you are trying to figure out what's happening inside a swirling, invisible storm of air moving at twice the speed of sound. You can't see the air, but you know it's bending light like a funhouse mirror. This is the challenge scientists face when studying supersonic flight, rocket re-entry, and high-speed engines.

This paper introduces a clever new way to "see" these invisible storms using a technique called Background-Oriented Schlieren (BOS) and a smart computer brain called a Physics-Informed Neural Network (PINN).

Here is the story of how they did it, explained with some everyday analogies.

The Problem: The "Invisible" Storm

Imagine you are looking at a patterned wallpaper through a window. If the air outside is perfectly still, the wallpaper looks normal. But if a hot, fast jet of air passes in front of the window, the air gets denser in some spots and less dense in others. This changes how light travels, making the wallpaper look wavy or distorted.

BOS is a camera trick that measures these waves. It takes a picture of the wallpaper before the air moves (the reference) and after the air moves (the distorted image). By comparing the two, it can calculate how much the light bent.

The Catch: The camera only sees the result of the bending (the distortion), not the cause (the actual air density, speed, and pressure). It's like seeing a shadow and trying to guess the exact shape of the object casting it. There are infinite ways to create that same shadow. This is called an "ill-posed problem."

The Old Way: Guessing with Smoothness

Traditionally, scientists tried to solve this by assuming the air flow is "smooth." They used mathematical rules to say, "Okay, the air density probably doesn't jump around wildly; it changes gradually."

Think of this like trying to draw a mountain range by connecting dots with a smooth, rubbery line. If the mountain has a sharp, jagged peak (like a shockwave in supersonic flow), a smooth rubber line will just round off the peak. You lose the sharp details. The old methods often smoothed out the most important parts of the flow, leading to blurry, inaccurate results.

The New Way: The "Physics Detective" (PINN)

The authors of this paper decided to stop guessing and start using the laws of physics as a guide. They used a special type of Artificial Intelligence called a Physics-Informed Neural Network (PINN).

Here is how the PINN works, using a detective analogy:

1. The Suspect (The Flow): The PINN is a digital detective trying to reconstruct the entire story of the air flow (density, speed, and pressure).
2. The Clue (The Photo): The detective looks at the distorted photo (the BOS image) to see what the flow did to the light.
3. The Rulebook (The Physics): This is the magic part. The detective carries a rulebook containing the Euler Equations (the laws of how fast-moving gas behaves). The detective is told: "You can propose any solution you want, BUT it must obey these laws of physics."

If the detective suggests a solution where the air suddenly speeds up without a reason, the rulebook says, "No, that violates physics!" The detective has to try again.

The Workflow: Solving the Puzzle

Instead of breaking the problem into three messy steps (measuring the bend, guessing the shape, then smoothing it out), the PINN does it all at once.

• The Input: It takes the raw, distorted photos.
• The Process: It runs a simulation in its "brain" that tries to match the photo while strictly obeying the laws of aerodynamics.
• The Output: It spits out a complete 3D map of the air, showing exactly where the density is high, where the speed is fast, and where the pressure is building up.

Why This is a Big Deal

The researchers tested this on two things:

1. A Computer Simulation: They created a fake supersonic flow (a cone shape in a wind tunnel) and fed the distorted images to the PINN. The PINN reconstructed the flow almost perfectly, capturing the sharp, jagged shockwaves that the old "smooth" methods missed.
2. Real Experiments: They took actual photos from a wind tunnel in India. Even with "noise" (like grainy camera static or imperfect lighting), the PINN figured out the flow field better than any previous method.

The Superpower:
Before this, BOS could only tell you about density (how thick the air is). Because the PINN is forced to obey the laws of physics, it can also calculate velocity (how fast the air is moving) and pressure (how hard the air is pushing) without needing extra sensors. It's like looking at a shadow and being able to tell you not just the shape of the object, but how fast it was moving and how heavy it is.

The Bottom Line

Think of the old method as trying to fix a blurry photo by just smoothing it out. The new method is like using a smart AI that knows exactly how light and air interact. It looks at the blurry photo and says, "Based on the laws of physics, the only way this blur could happen is if the air was moving this fast and had this much pressure."

This is the first time this specific type of AI has been used to reconstruct real, high-speed supersonic flows from experimental data. It turns a blurry, confusing picture into a crystal-clear, multi-dimensional map of the invisible storm, helping engineers design better, faster, and safer aircraft.

Technical Summary

1. Problem Statement

Background-Oriented Schlieren (BOS) is a quantitative flow visualization technique used to characterize compressible, high-speed flows (e.g., supersonic and hypersonic) by measuring density gradients. However, reconstructing flow fields from BOS data involves a series of ill-posed inverse problems.

• The Challenge: Standard BOS workflows require solving sequential steps: deflection sensing (converting image distortions to displacement vectors), tomographic reconstruction (recovering refractive index gradients), and integration (recovering the density field).
• Limitations of Current Methods: These steps rely heavily on regularization (e.g., Tikhonov or Total Variation penalties) to stabilize the solution. These penalties typically enforce global or piecewise smoothness, which is physically incompatible with the sharp gradients (shocks) and complex structures found in high-speed flows. This leads to "reconstruction artifacts," over-smoothing of shocks, and significant errors in the estimated density field. Furthermore, conventional methods generally only yield density, failing to provide velocity or pressure fields without additional assumptions or measurements.

2. Methodology: Physics-Informed BOS

The authors propose a new workflow termed "Physics-Informed BOS" that utilizes Physics-Informed Neural Networks (PINNs) to solve the inverse problem. Instead of sequential processing with ad-hoc regularization, this method simultaneously reconstructs the flow fields by minimizing a composite loss function that satisfies both experimental data and governing physical laws.

Key Components:

• Neural Network Architecture: A deep, feedforward neural network maps spatial coordinates $(x, r)$ or $(x, y)$ to flow variables: density ($\rho$), velocity components ($u, v$), and total energy ($E$).
• Unified BOS Measurement Model: The network outputs a density field which is fed into a Unified BOS operator. This operator directly simulates the image formation process (calculating expected image differences based on ray tracing through the density field) and compares them to the raw experimental image differences. This avoids the intermediate, error-prone step of calculating deflection vectors first.
• Physics Loss: The network is constrained by the compressible, steady 2D Euler equations and an irrotationality condition ($\nabla \times \mathbf{u} = 0$). The residuals of these equations are calculated using Automatic Differentiation (AD) and integrated over the domain to form the physics loss ($\mathcal{L}_{phys}$).
• Data Loss: The difference between the synthetic image differences (generated by the PINN's density field) and the actual experimental image differences forms the data loss ($\mathcal{L}_{meas}$).
• Boundary Conditions: Inlet conditions (known from wind tunnel settings) are enforced as an additional loss term ($\mathcal{L}_{in}$).
• Optimization: The total loss ($\mathcal{L}_{total} = \omega_{meas}\mathcal{L}_{meas} + \omega_{phys}\mathcal{L}_{phys} + \omega_{in}\mathcal{L}_{in}$) is minimized using the Adam optimizer to train the network.

3. Key Contributions

1. First Experimental PINN for Supersonic Flow: To the authors' knowledge, this is the first application of a PINN to reconstruct a supersonic flow field from experimental BOS data. Previous works were limited to synthetic data or post-processing of conventional reconstructions.
2. Direct Reconstruction of Multi-Parameter Fields: The method simultaneously recovers density, velocity, and pressure fields from a single pair of images, whereas conventional BOS typically only provides density.
3. Elimination of Non-Physical Regularization: By embedding the governing Euler equations directly into the loss function, the method avoids the "smoothing" artifacts associated with Tikhonov or Total Variation regularization, allowing for the accurate resolution of shock waves.
4. Unified Operator Integration: The approach embeds the full BOS measurement model (including ray tracing and image formation) directly into the data loss, bypassing the need for intermediate deflection sensing steps that introduce noise and errors.

4. Results

The method was validated using three scenarios:

• Synthetic Cone-Cylinder Shock (Axisymmetric):
  • Compared against analytical solutions and CFD ground truth.
  • Accuracy: The Physics-Informed BOS (PI-BOS) achieved a Normalized Root-Mean-Square Error (NRMSE) of 3.75% for density, significantly outperforming conventional methods (Simpson's 1/3 Abel: 8.34%; Tikhonov-regularized Unified BOS: 6.12%).
  • Velocity/Pressure: Successfully reconstructed velocity and pressure fields with errors under 1.5% for clean data.
  • Noise Robustness: The inclusion of the irrotationality equation ($\epsilon_5$) significantly improved stability and accuracy in noisy (single-shot) experimental data, reducing artifacts.
• Experimental Cone-Cylinder (Mach 2):
  • Applied to real wind tunnel data from the National Aerospace Laboratories (NAL).
  • The reconstructed density field showed excellent agreement with analytical cone shock tables and previous experimental studies, capturing the shock structure and expansion fan with high fidelity.
• Analytical Prandtl-Meyer Expansion Fan (Planar):
  • Demonstrated the ability to handle continuous isentropic expansions.
  • The method accurately recovered the flow field, though singularities at the corner required excluding a small region from the data loss to prevent instability.
  • Showed that adding a single "pressure tap" measurement or the irrotationality constraint significantly reduced errors in the velocity field.

5. Significance and Outlook

• Paradigm Shift: This work shifts BOS from a purely image-processing technique to a data assimilation framework. It leverages the flexibility of deep learning to enforce physical consistency without relying on heuristic smoothing.
• High-Fidelity Diagnostics: The ability to extract velocity and pressure fields from standard BOS images (without needing pressure taps or PIV) offers a powerful, non-intrusive diagnostic tool for aerospace engineering and combustion research.
• Handling Shocks: The study addresses the challenge of applying PINNs to hyperbolic equations (shocks) without artificial viscosity, demonstrating that the combination of a unified measurement model and physical constraints is sufficient to capture discontinuities accurately.
• Future Work: The authors suggest that adaptive weighting schemes for loss terms and the inclusion of entropy pair constraints could further enhance stability for more complex, viscous, or 3D flows.

In conclusion, Physics-Informed BOS represents a significant advancement in high-speed flow diagnostics, offering superior accuracy and richer physical insight compared to traditional reconstruction algorithms.