Development and validation of a sharp interface immersed boundary method for high-speed flows

This paper presents and validates a novel sharp-interface immersed boundary method integrated with the blastFOAM library on OpenFOAM, which effectively simulates high-speed compressible flows with complex and dynamic geometries by combining slip boundary conditions, polynomial reconstruction, and multiple flux schemes to achieve accurate shock resolution without requiring body-fitted meshes.

The Gist

Imagine you are trying to film a high-speed race car zooming through a city. In the old days, to get a perfect shot, you would have to build a custom camera rig that fits perfectly around every curve of the car and every twist of the road. If the car swerved or the road changed, you'd have to stop, rebuild the rig, and start over. This is like the traditional way engineers simulate high-speed air (like jets or rockets) using "body-fitted" grids. It's accurate, but it's slow, expensive, and a nightmare to manage when things are moving.

This paper introduces a new, smarter way to do this simulation. Think of it as switching from building a custom rig to using a high-speed drone with a smart camera that flies over the city. The drone doesn't care about the shape of the car or the road; it just flies over a standard grid (like a checkerboard) and uses software to figure out where the car is and how the air is moving around it.

Here is the breakdown of what the researchers did, using everyday analogies:

1. The Problem: The "Rigid Grid" vs. The "Smart Overlay"

• The Old Way (Body-Fitted): Imagine trying to pour water into a complex sculpture made of clay. You have to mold the water container to match the clay perfectly. If the clay moves, you have to melt and reshape the container. This is what happens with traditional computer models for high-speed air. It works, but it's tedious.
• The New Way (Immersed Boundary Method - IBM): Now, imagine pouring that water over the clay while it sits on a flat table. You don't change the table; you just tell the water, "Stop here, because there's clay underneath." The water flows around the clay naturally, even if the clay moves. This is the Immersed Boundary Method (IBM). It lets the computer use a simple, square grid (the table) to simulate complex, moving shapes (the clay).

2. The Challenge: High-Speed Air is "Snappy"

Simulating slow air is easy. But simulating high-speed air (like a jet breaking the sound barrier) is like trying to catch a bullet.

• Shock Waves: When air moves faster than sound, it creates invisible "walls" of pressure called shock waves. These are like the sonic booms you hear.
• The Difficulty: In a computer, these shock waves can get blurry or cause the simulation to crash (like a glitchy video game). The researchers needed a way to make these "walls" sharp and clear without the computer getting confused.

3. The Solution: A "Ghost" and a "Slippery Surface"

The team created a new tool called blastIBFOAM. Here is how it works in simple terms:

• The "Ghost" Trick: Imagine standing next to a wall. To know what the air feels like on the wall, the computer creates a "ghost" version of the air on the other side of the wall (inside the solid object). It uses math to guess what the ghost air should be doing so that the real air behaves correctly right at the surface. This is called the Ghost-Cell Method.
• The "Slippery" Rule: For high-speed, inviscid (frictionless) flows, the air doesn't stick to the object like honey; it slides right off. The researchers added a special rule called a "Slip Boundary Condition." It tells the computer: "The air can't go through the wall, but it can slide along it perfectly." This prevents the simulation from getting stuck or creating fake friction.
• The "Sharp" Focus: They used a technique called Second-Order Polynomial Reconstruction. Think of this as a high-definition camera lens. Instead of just guessing the air pressure at a point, they use a curved mathematical line to connect the dots, making the picture of the shock wave incredibly sharp and accurate.

4. The "Traffic Cop" (Flux Schemes)

When air rushes through the computer's grid, the software needs to decide how to calculate the flow at every intersection. The researchers tested different "Traffic Cops" (mathematical schemes like HLL, AUSM+up, and Kurganov).

• Some cops are strict and slow (good for smooth traffic).
• Some are fast and aggressive (good for chaotic traffic).
• The paper found that no single cop is perfect for every situation. Sometimes one works best for a wedge shape, and another works best for a sphere. The beauty of their new tool is that it lets engineers choose the right cop for the job.

5. The Proof: Testing the New Tool

To prove their new "drone camera" works, they ran it through several extreme tests:

• The Wedge: A sharp triangle in the wind. The tool drew the shock wave perfectly sharp, just like the math predicted.
• The Moving Piston: A piston shooting forward at Mach 2 (twice the speed of sound). The tool tracked the shock wave perfectly, even though the object was moving.
• The Cylinder & Sphere: They simulated air hitting round objects. The tool handled the complex 3D shapes and the swirling air behind them (the wake) with great accuracy.
• The Airplane Wing: They simulated a supersonic jet wing. The results matched the old, slow "custom rig" methods but were much faster to compute.

Why Does This Matter?

This new method is like upgrading from a hand-drawn map to a real-time GPS.

• Speed: It's faster because you don't have to rebuild the grid every time an object moves.
• Flexibility: You can simulate complex shapes (like a rocket with fins, or a car with spinning wheels) without needing a super-computer to handle the mesh.
• Accuracy: It captures the "shock waves" (the sonic booms) with high precision, which is critical for designing safe and efficient aircraft.

In a nutshell: The researchers built a flexible, high-speed simulation tool that treats complex shapes as if they are just "ghosts" floating in a simple grid. It handles the chaos of supersonic flight with the precision of a scalpel, making it easier and cheaper to design the next generation of high-speed aircraft and rockets.

Technical Summary

1. Problem Statement

Numerical simulation of high-speed compressible flows (involving shock waves, expansion fans, and large density gradients) is computationally challenging, particularly when dealing with complex or moving geometries. Traditional approaches rely on body-fitted meshes, which require frequent mesh regeneration or deformation for moving bodies, leading to high computational costs and potential numerical instabilities. While Immersed Boundary Methods (IBM) offer a solution by using non-conforming Cartesian grids, many existing IBM implementations are designed for incompressible flows or use "diffused-interface" approaches that smear boundaries, reducing accuracy in capturing sharp shock waves. There is a need for a robust, sharp-interface IBM capable of handling high-speed compressible regimes (subsonic to hypersonic) with dynamic geometries while maintaining accuracy and conservation properties.

2. Methodology

The authors developed a new solver, blastIBFOAM, by integrating a sharp-interface IBM framework with the blastFOAM library within the open-source OpenFOAM platform.

• Governing Equations: The solver solves the compressible Euler equations (inviscid flow) in conservative form using a finite-volume method on a background Cartesian mesh.
• Sharp-Interface Formulation:
  • Ghost-Cell Approach: The method utilizes a Ghost-Cell-Based Immersed Boundary Method (GCIBM). Solid cells are excluded, and "ghost cells" (fluid cells adjacent to the boundary) are reconstructed to enforce boundary conditions.
  • Boundary Conditions:
    • Inviscid Flow: A slip boundary condition is implemented for velocity (zero normal velocity, zero normal gradient for tangential velocity).
    • Thermodynamics: Neumann boundary conditions (zero normal gradient) are applied for pressure and temperature.
  • Reconstruction: Flow variables at the immersed boundary (IB) points are reconstructed using a second-order quadratic polynomial interpolation. This is achieved via a weighted least-squares method using an extended stencil of neighboring fluid cells. The weights are determined by the inverse distance from the IB point.
  • Stencil Selection: An extended stencil is selected based on an angle factor, radius factor, and maximum cell rows to ensure sufficient data points for accurate interpolation.
• Numerical Flux Schemes: The study evaluates and implements multiple flux schemes to handle different Mach number regimes:
  • HLL (Harten–Lax–van Leer)
  • AUSM+up (Advection Upstream Splitting Method Plus Upwind)
  • Kurganov and Tadmor (Central schemes)
• Time Integration: Explicit time marching (Forward Euler or Runge-Kutta) is used.

3. Key Contributions

1. Extension to Compressible Regimes: The work extends existing IBM techniques (specifically the framework by Tuković and Jasak, originally for incompressible flows) to high-speed compressible flows, addressing challenges like shock capturing and dynamic boundaries.
2. Novel Slip Condition Implementation: A specific implementation of the slip boundary condition for inviscid high-speed flows is introduced, combining Dirichlet (no penetration) and Neumann (zero shear) conditions effectively.
3. Flux Scheme Comparison: A comprehensive comparative study of four distinct flux schemes (HLL, AUSM+up, Kurganov, Tadmor) within the IBM framework across various Mach numbers and geometries.
4. Validation on Moving and 3D Bodies: The solver is validated not only on stationary 2D bodies but also on moving pistons and 3D spheres, demonstrating its capability for dynamic and complex applications without mesh deformation.

4. Results and Validation

The solver was validated against analytical solutions, body-fitted simulations, and experimental data across several benchmark cases:

• Supersonic Flow over a Wedge (Ma 3.0 & 5.0):
  • The solver achieved second-order spatial accuracy (convergence rate $\approx$ 2.2) for shock angle and pressure.
  • AUSM+up and HLL schemes showed the closest agreement with body-fitted results and analytical oblique shock relations.
  • The method captured sharp shock locations with minimal numerical oscillation.
• Moving Piston (Ma 2.0):
  • Successfully simulated a piston moving at supersonic speeds, accurately capturing both the compression shock ahead and the expansion wave behind.
  • Demonstrated grid independence and insensitivity to grid orientation (tested with a 45° rotated mesh).
  • Showed a 13% reduction in computational time compared to a body-fitted solver due to the avoidance of mesh deformation.
• Shock-Cylinder Interaction (Ma 2.81):
  • Accurately resolved complex shock interactions, including reflected shocks, Mach stems, and triple points.
  • Results matched well with published schlieren visualizations and loci of triple points.
• Flow over NACA 0012 Aerofoil (Ma 1.5):
  • Captured the leading-edge shock and surface pressure distribution ($C_p$) with high fidelity.
  • Tadmor and HLL schemes provided the best agreement with body-fitted solutions.
• 3D Sphere (Ma 5.8):
  • Validated against experimental data (Machell, 1956) for shock shape and surface pressure.
  • HLL and AUSM+up schemes provided the most robust performance for 3D bluff bodies.

5. Significance and Conclusion

This study presents a flexible, efficient, and accurate tool for high-speed compressible flow simulations.

• Efficiency: By using Cartesian meshes and avoiding body-fitted mesh regeneration, the method significantly reduces computational overhead for problems involving moving or complex geometries.
• Accuracy: The sharp-interface approach preserves the geometric fidelity of the boundary, allowing for precise shock resolution and minimal numerical smearing.
• Versatility: The solver is applicable across a wide range of Mach numbers (low-supersonic to hypersonic) and geometries (2D and 3D).
• Practical Application: The developed solver is ideal for aerospace engineering applications where dynamic geometries and high-speed flows are prevalent, offering a robust alternative to traditional mesh-based methods.

The authors conclude that while no single flux scheme is optimal for all conditions, the HLL and AUSM+up schemes generally offer the best balance of accuracy and robustness for high-speed IBM simulations. The work successfully bridges the gap between sharp-interface IBM and high-fidelity compressible flow solvers.