Lattice Boltzmann model for non-ideal compressible fluid dynamics

This paper introduces a novel lattice Boltzmann model using first-neighbour lattices and quasi-equilibrium correction terms to simulate compressible, non-ideal fluid flows with thermodynamic consistency and numerical stability, successfully validating its accuracy through quantitative simulations of shock-drop interactions at Mach numbers up to 1.47.

The Gist

Imagine you are trying to simulate how a fluid behaves on a computer. For a long time, computers were great at simulating "ideal" fluids—like water flowing gently in a river or air moving slowly around a wing. These fluids follow simple, predictable rules.

But what happens when the fluid is under extreme pressure and heat, behaving like a dense gas that is almost a liquid, or a liquid that is almost a gas? This is the world of non-ideal compressible fluids. Think of it as a fluid that is "stressed out" and acting strangely, refusing to follow the simple rules of the ideal world. This happens in advanced technologies like supercritical CO2 turbines and organic energy cycles.

The problem is that existing computer tools struggle with these stressed fluids. They either crash, give wrong answers, or require such massive computing power that they become useless.

This paper introduces a new, smarter way to simulate these tricky fluids using a method called the Lattice Boltzmann Method (LBM). Here is how the authors' new approach works, explained through simple analogies:

1. The "Two-Track" System

Most old simulation methods try to track everything (mass, speed, energy) with a single, complicated set of rules. The authors realized this was like trying to drive a car while simultaneously juggling a dozen balls—it gets messy and unstable.

Instead, they built a two-track system:

• Track A (The Crowd): One set of rules tracks the fluid's density and speed (how many particles are there and where are they going).
• Track B (The Energy): A second, separate set of rules tracks the total energy.
  By separating these, the computer doesn't get confused. It's like having a dedicated traffic controller for cars and a separate one for fuel, ensuring neither system crashes the other.

2. The "Quasi-Equilibrium" Attractor

In physics, fluids naturally want to settle into a calm state called "equilibrium." However, in these extreme conditions, the fluid is constantly being pushed and pulled, so it never quite settles.

The authors invented a clever trick called a "quasi-equilibrium attractor."

• The Analogy: Imagine a dog chasing a ball. The ball represents the "perfect calm state." The dog represents the fluid. In a normal situation, the dog runs straight to the ball.
• The Problem: In this extreme fluid, the ball keeps moving away or changing shape. If the dog just chases the ball blindly, it might run off a cliff (the simulation becomes unstable).
• The Solution: The authors gave the dog a "GPS" that predicts where the ball will be a split second later, based on how the wind (pressure) and terrain (density) are changing. This "shifted" target allows the dog to run smoothly without falling off the cliff. This ensures the simulation stays stable even when the fluid is moving very fast or changing density rapidly.

3. Fixing the "Spurious" Heat

When fluids move fast, they generate heat. In standard computer models, the heat sometimes flows in the wrong direction or creates fake "ghost" heat that doesn't exist in reality.

• The Analogy: It's like a thermostat that thinks the room is freezing because it's measuring the draft from a window, not the actual room temperature.
• The Fix: The authors added a specific "correction term" to their equations. This acts like a filter that removes the fake drafts, ensuring the heat flows exactly as physics dictates (Fourier's Law), even in these extreme, non-ideal conditions.

4. The "Shock Tube" and "Liquid Column" Tests

To prove their new method works, they didn't just do math; they ran extreme tests:

• The Shock Tube: They simulated a sudden explosion of pressure (a shock wave) moving through a dense gas. In normal gases, these waves behave one way. In these "non-ideal" gases, the waves can do something weird: a "rarefaction shock" (a wave that spreads out but still acts like a sharp shock). Their model successfully predicted this strange behavior, which older models missed.
• The Liquid Column: They simulated a high-speed shock wave hitting a drop of liquid. This is a very hard test because the shock bounces, reflects, and tears the drop apart. Their model handled the collision perfectly, matching real-world experiments where the liquid drop flattens and expands exactly as it should.

Why This Matters

The authors claim that their method is fast, stable, and accurate. It uses a simple grid (like a standard chessboard) rather than needing a super-complex, stretched-out grid. This means scientists can now simulate these extreme, high-speed, non-ideal fluid flows on standard computers with high precision.

In summary: The paper presents a new "driver's manual" for computer simulations that allows them to handle fluids under extreme stress without crashing. By using a two-track system and a smart "GPS" for the fluid's energy, they can accurately predict how these complex fluids behave in high-speed scenarios, opening the door to better designs for advanced energy systems.

Technical Summary

Technical Summary: Lattice Boltzmann Model for Non-Ideal Compressible Fluid Dynamics

Problem Statement
Non-ideal compressible fluid dynamics (NICFD) addresses fluid behaviors in near-, trans-, and super-critical regimes, where fluids deviate significantly from ideal gas laws. These regimes are critical for modern energy technologies, such as organic Rankine cycles and supercritical CO₂ turbines. However, experimental data in these states are scarce and difficult to acquire. While Direct Numerical Simulation (DNS) is necessary to understand the complex physics (e.g., shock structures, turbulence in dense vapors) and generate engineering databases, existing numerical tools often struggle with thermodynamic consistency, stability, and the accurate recovery of hydrodynamic limits for non-ideal equations of state (EOS). Specifically, conventional Computational Fluid Dynamics (CFD) methods face challenges in the thermodynamically unstable spinodal region where the adiabatic speed of sound squared becomes negative, requiring special treatments like Maxwell's equal-area rule. The Lattice Boltzmann Method (LBM), while successful for incompressible and multiphase flows, has historically neglected non-ideal compressible thermodynamics in its kinetic formulations, often relying on ad hoc regularization or extended stencils that compromise efficiency or Galilean invariance.

Methodology
The authors propose a novel kinetic model and its Lattice Boltzmann realization designed to recover the compressible Navier–Stokes–Korteweg (NSK) equations for non-ideal fluids. The methodology is built on the following pillars:

1. Double Distribution Function (DDF) Approach: The model employs two distribution functions: $f$ for mass and momentum, and $g$ for bulk energy ($\rho E$). This decoupling allows for the use of classical first-neighbour lattices (specifically D3Q27) without requiring extended stencils.
2. Quasi-Equilibrium Attractors: The collision operator utilizes a double-relaxation BGK-like structure involving a local equilibrium attractor ($f^{eq}, g^{eq}$) and a shifted quasi-equilibrium attractor ($f^\star_\lambda, g^\star_\lambda$).
   • The local equilibrium is defined using a reference temperature parameter $\theta = P/\rho$ (thermodynamic flow work) rather than the thermodynamic temperature $T$, ensuring the correct width of the Maxwellian distribution independent of internal energy.
   • The shifted equilibrium introduces corrections for the Korteweg force (surface tension), bulk viscosity, and heat flux. It employs shifted values for flow velocity ($u^\star$), reference temperature ($\theta^\star$), and thermodynamic temperature ($T^\star$).
3. Thermodynamic Consistency:
   • The reference temperature $\theta$ is set to $P/\rho$, a formulation that ensures the model remains valid for non-ideal EOS (e.g., van der Waals).
   • A correction term in the heat flux ($q^c_\lambda$) is introduced to compensate for spurious non-Fourier components arising from density gradients in non-ideal fluids, ensuring the recovery of Fourier's law.
   • The shift in the reference temperature allows the bulk viscosity to be an independent, tunable, and positive-definite parameter, circumventing the issue of negative bulk viscosity that can arise in standard BGK models for non-ideal pressures.
4. Discretization: The model uses a second-order accurate explicit scheme derived from trapezoidal integration along characteristics. It employs a standard D3Q27 lattice and handles source terms (Korteweg force) via the shifted equilibrium, avoiding explicit source term discretization errors.

Key Contributions

• New Kinetic Framework: The paper introduces a kinetic model that recovers the target hydrodynamic equations (mass, momentum, and bulk energy balance) for compressible non-ideal fluids using a minimal kinetic stencil (first-neighbour).
• Positive-Definite Dissipation: By constructing the model via quasi-equilibrium attractors, the authors ensure that the Navier–Stokes dissipation rates (shear and bulk viscosity) are positive-definite and Galilean invariant, a common failure point in previous LBM attempts at compressible non-ideal flows.
• Thermodynamic Robustness: The model naturally handles the thermodynamic instability regions (spinodal) where the speed of sound becomes imaginary in the inviscid limit, as the LBM inherits propagation along fixed discrete characteristics rather than relying on hyperbolic PDE solvers that require special treatment in these regions.
• Validation of Non-Classical Phenomena: The study provides the first quantitative validation of LBM for shock-drop interactions at Mach numbers up to 1.47 in a non-ideal fluid context, demonstrating the model's ability to capture complex wave structures.

Results
The model was validated across a hierarchy of increasingly complex configurations:

1. Linear Stability and Dispersion: Simulations of sound speed, shear wave dissipation, and thermal conductivity showed excellent agreement with analytical solutions for a van der Waals fluid fitted to nitrogen properties across a range of Mach numbers and reduced temperatures.
2. Multiphase Consistency: The model accurately reproduced liquid-vapour co-existence densities and interface profiles, demonstrating second-order convergence with grid refinement and agreement with Maxwell's equal-area rule.
3. Non-Classical Shock Tubes: Three 1-D shock tube cases were simulated to probe non-classical wave dynamics characterized by the fundamental derivative of gas dynamics ($\Gamma$).
   • Cases with $\Gamma < 0$ correctly captured the formation of rarefaction shocks (sharpening expansion waves) and compression fans (spreading compression waves), phenomena distinct from classical gas dynamics.
   • The model successfully transitioned between classical ($\Gamma > 0$) and non-classical ($\Gamma < 0$) regimes within a single simulation.
4. Shock–Liquid-Column Interaction: A 2D simulation of a planar shock wave interacting with a saturated liquid column near the critical point ($M_a = 1.47$) was performed.
   • The model captured complex wave structures including transmitted shocks, reflected shocks, Mach stems, and triple points.
   • Quantitative comparison of the liquid column width evolution against experimental data and previous numerical results showed good agreement.
   • Thermodynamic analysis revealed that the shock interaction drives the surrounding vapour into a supercritical state while the liquid column remains marginally subcritical, accurately capturing the strong thermodynamic coupling.

Significance and Claims
The authors claim that this work extends the applicability of Lattice Boltzmann methods to high-speed, non-ideal compressible flows with a minimal kinetic stencil. The primary significance lies in providing a thermodynamically consistent and numerically stable framework that does not rely on extended stencils or ad hoc regularization. The model is presented as a reduced kinetic theory that targets the NSK hydrodynamic limit, capable of handling the full range of non-ideal behaviors, including the spinodal region, without the numerical instabilities often associated with conventional CFD in these regimes. The successful simulation of shock-drop interactions at high Mach numbers serves as a stringent validation, demonstrating the method's potential for investigating complex flows in supercritical technologies and phase-transition-driven processes.