Consistent multiple-relaxation-time lattice Boltzmann method for the volume averaged Navier-Stokes equations

This paper proposes a consistent multiple-relaxation-time lattice Boltzmann method that decouples void fraction from density and employs a penalty source term to eliminate spurious velocities, thereby accurately recovering the volume-averaged Navier-Stokes equations with second-order accuracy for complex fluid-solid multiphase flows.

The Gist

Imagine you are trying to simulate how water flows through a sponge, a forest of trees, or a crowd of people moving through a busy hallway. In physics, this is called a fluid-solid multiphase flow. The challenge is that the "holes" (where the fluid moves) and the "solid parts" (where it can't) change size and shape constantly.

Scientists use a set of rules called the Volume-Averaged Navier-Stokes equations (VANSE) to describe this. Think of these rules as the "traffic laws" for fluids moving through obstacles.

However, the computer program used to solve these laws, called the Lattice Boltzmann Method (LBM), has been struggling with a specific problem: Spurious Velocities.

The Problem: The "Ghost Wind"

Imagine you are standing perfectly still in a room full of furniture. Suddenly, a computer simulation says, "Hey, there's a wind blowing!" even though no one turned on a fan. This is what happens in older simulation methods when the density of obstacles changes sharply (like going from a dense forest to an open field).

The old math gets confused at the edges where the "holes" in the material change size. It creates fake forces that push the fluid around, creating ghost winds (spurious velocities) that ruin the accuracy of the simulation. It's like trying to measure the temperature of a room with a thermometer that starts shaking violently whenever you move it near a wall.

The Solution: A New "Traffic Cop"

The authors of this paper, researchers from Tsinghua University and Beijing Institute of Technology, have invented a new, smarter way to run these simulations. They call it the MRTLB-VANSE method.

Here is how they fixed the problem, using simple analogies:

1. Decoupling the "Density" from the "Holes"

In the old method, the computer treated the amount of "stuff" (density) and the amount of "empty space" (void fraction) as a single, tangled knot. When the empty space changed, the density calculation got messy, causing the ghost winds.

The Fix: The new method introduces a provisional rule (a temporary equation of state). Imagine you are organizing a party. Instead of counting every single guest and every empty chair separately and getting confused, you create a temporary "guest list" that ignores the chairs for a moment. This allows the computer to calculate the flow without getting tripped up by the changing number of obstacles. It untangles the knot.

2. The "Penalty Source Term" (The Traffic Cop)

Even with the new rule, the computer still makes tiny, subtle math errors when calculating how the fluid twists and turns (viscous stress). In the old days, these small errors added up and broke the simulation, especially when the fluid was moving fast or the obstacles were very dense.

The Fix: The authors added a Penalty Source Term. Think of this as a highly skilled Traffic Cop standing in the middle of the intersection.

• The fluid particles are cars.
• The math errors are cars drifting slightly out of their lane.
• The Traffic Cop (the penalty term) sees a car drifting and gently nudges it back into the correct lane before the drift causes a crash.

This "nudge" ensures that the simulation remains perfectly balanced, even when the fluid is moving through complex, changing environments.

3. The "Multiple-Relaxation-Time" (MRT) Engine

Older methods used a "Single-Relaxation-Time" (SRT) engine, which is like a car with only one gear. It works fine on a flat road (simple flows) but struggles on steep hills (complex, high-viscosity flows).

The Fix: The new method uses Multiple-Relaxation-Time (MRT). This is like a car with a multi-gear transmission. It can shift gears automatically to handle different speeds and terrains. This makes the simulation much more stable and accurate, allowing it to handle everything from slow-moving honey-like fluids to fast-moving water, even when the "holes" in the material change abruptly.

Why Does This Matter?

Before this paper, if you wanted to simulate:

• Blood flowing through a complex network of capillaries.
• Water filtering through soil that is dissolving or changing shape.
• Particles moving in a fluidized bed (like sand in a factory).

...you often had to accept that your simulation would have "ghost winds" or would crash when the obstacles got too dense.

This new method removes those ghost winds. It allows scientists to simulate these complex, real-world scenarios with high precision and stability. It's like upgrading from a shaky, hand-drawn map to a high-definition GPS that never loses signal, no matter how twisty the road gets.

In a Nutshell

The researchers built a smarter, more stable computer engine for simulating fluids moving through obstacles. By untangling the math regarding empty space and adding a corrective "traffic cop" to fix small errors, they eliminated the fake winds that used to ruin these simulations. This means we can now model complex natural and industrial processes (like pollution filtering, blood flow, or oil extraction) with much greater confidence.

Technical Summary

1. Problem Statement

The Volume-Averaged Navier-Stokes Equations (VANSE) are fundamental for simulating fluid-solid multiphase systems (e.g., fluidized beds, porous media, blood flow) by bridging microscopic interactions with macroscale behavior. While the Lattice Boltzmann Method (LBM) is a powerful tool for such simulations, existing density-based LBM schemes for VANSE suffer from two critical issues:

• Spurious Velocities: In heterogeneous void fraction fields (especially those with large gradients or discontinuities), standard schemes generate significant non-physical velocities. This stems from numerical errors in the discretized pressure correction force, which induces non-physical acceleration.
• Inconsistency and Instability: Many existing schemes fail to consistently recover the full form of VANSE with second-order accuracy. Furthermore, schemes relying on Single-Relaxation-Time (SRT) collision operators exhibit numerical instability and loss of accuracy at high viscosities (relaxation times $\tau < 0.5$).

2. Methodology

The authors propose a novel Multiple-Relaxation-Time Lattice Boltzmann Method (MRTLB-VANSE) within a density-based framework to resolve these inconsistencies. The core innovations include:

• Provisional Equation of State (EOS) & Decoupling:

  • A provisional EOS ($p = c_s^2 \kappa \rho$) is introduced by modifying the density equilibrium distribution function (using a 3rd-order Hermite expansion).
  • This modification decouples the void fraction ($\phi$) from the density ($\rho$) in the equilibrium distribution. Consequently, the pressure correction force no longer requires the discretization of the void fraction gradient ($\nabla \phi$), effectively eliminating the source of non-physical oscillations at discontinuous interfaces.
  • The force term is adjusted to $F = \nabla(c_s^2 \kappa \rho) - \nabla(c_s^2 \phi \rho)$, where $\kappa$ is a constant close to the minimum void fraction.

• MRT Collision Operator with Penalty Source Term:

  • The authors employ a Multiple-Relaxation-Time (MRT) collision operator instead of SRT.
  • To ensure Galilean invariance and correct the viscous stress tensor (which is biased by the provisional EOS), a penalty source term ($C$) is introduced in the moment space.
  • This source term explicitly cancels out unwanted numerical errors in the recovered macroscopic momentum equations, ensuring the scheme recovers the exact VANSE form.

• Theoretical Validation:

  • The consistency of the scheme is rigorously proven via Chapman-Enskog analysis, demonstrating that the macroscopic equations recovered from the MRTLB-VANSE exactly match the target VANSE (Model A) with second-order accuracy.

3. Key Contributions

1. Elimination of Spurious Velocities: By decoupling $\phi$ from $\rho$ in the equilibrium distribution, the method prevents the generation of spurious currents in discontinuous void fraction fields, a persistent issue in previous density-based LB models.
2. Consistent Recovery of VANSE: The introduction of the penalty source term in moment space corrects the viscous stress tensor errors, ensuring the recovered equations are mathematically consistent with VANSE, unlike previous models that suffered from high-order error terms.
3. Enhanced Numerical Robustness: The use of the MRT operator allows the method to remain stable and accurate across a wide range of viscosities, overcoming the limitations of SRT-based schemes at high viscosities.
4. Second-Order Convergence: The method achieves second-order convergence for both spatially and spatiotemporally varying void fraction fields.

4. Numerical Results and Validation

The paper validates the MRTLB-VANSE through several benchmark tests:

• Uniform Porous Flow: Successfully reproduced analytical solutions for porous Poiseuille and Couette flows across various Darcy numbers, confirming the model's ability to handle REV-scale flows.
• Nonuniform Particle Flow: Validated against analytical solutions for 1-D flows with spatially and temporally fluctuating void fractions, confirming the accuracy of the pressure correction term.
• Discontinuous Field (No-Flow Test): In a domain with a sharp discontinuity in void fraction, the proposed method demonstrated significantly lower spurious velocities compared to previous density-based schemes (Zhang et al., Bukreev et al., Blais et al.). It remained stable even when $\phi < 0.3$, where other schemes diverged or produced large errors.
• Convergence Tests (MMS): Using the Method of Manufactured Solutions (MMS) for complex spatiotemporal void fraction fields, the method demonstrated second-order convergence in velocity error.
  • Crucially, while a recent pressure-based scheme (Fu et al.) showed second-order convergence at low viscosity, it lost accuracy at high viscosity ($\nu = 1.0$). The proposed MRTLB-VANSE maintained high accuracy and stability across all tested viscosity ranges.

5. Significance

This work addresses a critical gap in computational fluid dynamics for multiphase systems. By developing a consistent, stable, and accurate density-based LBM for VANSE, the authors provide a robust tool for simulating complex fluid-solid interactions involving large void fraction gradients and discontinuities.

• Practical Impact: The method is directly applicable to industrial and scientific problems such as fluidized beds, porous media flow, and particulate tracking in Euler-Euler or Euler-Lagrange frameworks.
• Future Potential: The framework is easily extensible to 3D and can be coupled with other physics (heat transfer, species transport, immiscible multiphase models), paving the way for advanced multiscale modeling of complex liquid-gas-solid processes.

In summary, the paper presents a mathematically rigorous and numerically superior LBM scheme that resolves long-standing issues of spurious velocities and inconsistency in volume-averaged flow simulations.