Improved third-order scheme in pseudopotential lattice Boltzmann model for multiphase flows

This paper proposes and validates an improved third-order scheme for pseudopotential lattice Boltzmann models that effectively suppresses spurious velocity oscillations at phase interfaces without adding computational complexity, thereby ensuring more reliable simulations of multiphase flows.

The Gist

Imagine you are trying to simulate how oil and water mix (or separate) inside a computer. Scientists use a digital tool called the Lattice Boltzmann Model (LBM) to do this. Think of this model as a giant, digital grid of tiny tiles. On each tile, little "particles" bounce around, carrying information about density and speed. By watching how these particles interact, the computer can predict how real fluids behave.

However, there's a problem. When the computer tries to draw the line between the oil (liquid) and the water (gas), it sometimes gets jittery. Instead of a smooth, calm boundary, the digital fluid starts to "shiver" or vibrate unnaturally. In the scientific world, this is called spurious velocity oscillation.

Think of it like trying to walk across a bridge that is slightly wobbly. Even if you are just standing still, the bridge shakes under your feet. In a fluid simulation, this shaking creates fake currents that mess up the results, making the computer think the fluid is moving when it should be calm, or changing how fast a drop falls.

The Problem: The "Grid" vs. The "Flow"

The researchers in this paper discovered that this shaking gets worse depending on how the fluid flows relative to the computer's grid:

1. Grid-Aligned: The fluid flows straight along the grid lines (like driving down a perfectly straight highway).
2. Grid-Oblique: The fluid flows diagonally across the grid (like cutting across a checkerboard).

They found that the standard math used to calculate the fluid's movement had a hidden flaw. It was missing a specific "correction term" that acts like a shock absorber. Without this shock absorber, the digital bridge shakes, especially when the fluid is moving diagonally or when the simulation tries to be very precise about the pressure between the two fluids.

The Solution: A Smarter "Shock Absorber"

The authors proposed an Improved Third-Order Scheme.

To use an analogy: Imagine you are driving a car with a suspension system. The old system (the original scheme) worked okay on straight roads but made the car bounce wildly when you turned a corner or hit a bump. The new system (the improved scheme) adds a smart, adaptive shock absorber.

• How it works: The new math looks at how the fluid is moving and how strong the "push" between the oil and water molecules is. It then adds a tiny, calculated correction to the movement equations.
• The Magic: This correction is designed to cancel out the "shaking" exactly where it happens. It's like adding a counter-vibration that perfectly neutralizes the wobble, leaving the fluid smooth and stable.
• No Extra Cost: The best part is that this new system doesn't make the computer work harder. It's just a slightly smarter way of doing the same math, so it runs just as fast as the old version.

Testing the New System

The team tested their new "shock absorber" in three different scenarios:

1. The Straight Highway (Plane Poiseuille Flow): They simulated fluid flowing between two flat walls.

   • Result: The old method made the fluid vibrate near the oil-water boundary. The new method made the flow perfectly smooth, matching the theoretical ideal.

2. The Curved Road (Annular Shear Flow): They simulated fluid flowing in a circle (like oil in a pipe).

   • Result: Even on this curved path, the old method caused jitter. The new method kept the flow calm and accurate, proving it works even when the interface isn't a straight line.

3. The Falling Drop (Droplet Falling): They simulated a drop of liquid falling through a tube.

   • The Big Surprise: This test showed why the old method is dangerous. Because the old method had those fake vibrations, it created extra "drag" (friction).
   • The Outcome: In the old simulation, the drop fell slower and stayed in the middle of the tube. In the new, accurate simulation, the drop fell faster and actually moved toward the wall, spinning as it went. The old method was lying to the scientists, making the drop seem heavier and slower than it really is.

The Takeaway

This paper is like a mechanic fixing a car that has a hidden vibration issue. They found out why the car was shaking (a missing correction term in the math), built a better suspension (the improved scheme), and proved that without this fix, your car (or your fluid simulation) will give you the wrong speed and handling data.

By using this new method, scientists can now trust their computer simulations to show exactly how fluids behave, without the annoying "digital shivers" getting in the way. This is crucial for designing better engines, understanding oil extraction, or even studying how blood flows in our bodies.

Technical Summary

1. Problem Statement

The pseudopotential Lattice Boltzmann (LB) model is a widely used mesoscopic approach for simulating multiphase flows due to its ability to naturally handle complex interface dynamics. However, the model suffers from spurious velocity oscillations (often called interfacial velocity slip) near phase interfaces, particularly when simulating flows with shear or curved interfaces.

• Specific Issue: Previous studies (e.g., Hou et al., 2016) identified these oscillations in two-phase Poiseuille flow but only analyzed the case where the flow and interface were aligned with the lattice grid.
• Limitations of Existing Solutions:
  • The origin of oscillations in grid-oblique cases (where the interface is not parallel to the grid) was unclear.
  • Proposed fixes in prior literature often sacrificed the clear microscale physical picture of the classical pairwise interaction force or introduced complex modifications that were not universally applicable.
  • These oscillations lead to significant errors in velocity profiles and can distort physical phenomena, such as drag force calculations and droplet dynamics.

2. Methodology

The authors employed a rigorous discrete-level theoretical analysis combined with numerical validation to address the problem.

A. Theoretical Analysis

• Framework: The study utilizes the Multi-Relaxation-Time (MRT) LB equation with a third-order scheme.
• Derivation: The authors derived finite-difference velocity equations for two-phase Poiseuille flow at the discrete lattice level.
  • Grid-Aligned Case: Analyzed flow where the interface is parallel to the lattice grid.
  • Grid-Oblique Case: Analyzed flow where the interface is at a 45° angle to the grid.
• Identification of Error Terms: By analyzing the derived equations, the authors identified specific terms in the discrete source term ($Q_m$) responsible for the spurious oscillations. They found that these oscillations arise from the interaction between the third-order moments and the pairwise interaction force, specifically when the parameter $\epsilon$ (related to thermodynamic consistency) is non-zero.

B. Proposed Solution: Improved Third-Order Scheme

Based on the analysis, the authors proposed a modified discrete source term ($Q_m$) for the third-order moments ($m_4$ and $m_6$).

• Key Innovation: Unlike the original scheme (where $Q_{m,4}$ and $Q_{m,6}$ were artificially set to zero), the improved scheme defines these terms as functions of the macroscopic velocity ($u$) and the square of the pairwise interaction force ($F_{int}^2$).
• Formulation:
  • For the grid-aligned case, $Q_{m,6}$ is derived to cancel the oscillation-inducing term.
  • For the grid-oblique case, a heuristic derivation based on symmetry and the grid-aligned result yields a combined term $Q_{m,4+6}$.
  • The general form ensures that under static conditions ($u=0$), the new terms vanish, preserving the original scheme's behavior for equilibrium states.
• Complexity: The scheme introduces no additional conceptual or computational complexity compared to the original model.

3. Key Contributions

1. Discrete-Level Origin Identification: The paper theoretically identifies the exact mathematical origin of spurious velocity oscillations in both grid-aligned and grid-oblique configurations, proving they stem from specific third-order source terms.
2. Universal Improvement Scheme: The proposed improved third-order scheme effectively suppresses oscillations in both aligned and oblique cases while retaining the classical pairwise interaction force physics.
3. Validation of Grid-Oblique Limitations: The study demonstrates that previous improvements (e.g., Hou et al.) may fail when the interface is not aligned with the lattice grid, highlighting the necessity of the new oblique-case derivation.
4. Physical Impact Demonstration: The paper quantifies how spurious oscillations lead to unphysical results in dynamic scenarios, specifically overestimating drag forces and altering droplet falling patterns.

4. Numerical Results

The authors validated the theory through three distinct numerical tests:

• Two-Phase Plane Poiseuille Flow:

  • Grid-Aligned: The original scheme showed significant velocity oscillations for $\epsilon > 0$. The improved scheme eliminated these, matching the theoretical Navier-Stokes profile perfectly.
  • Grid-Oblique: The original scheme exhibited large deviations and oscillations even when $\epsilon=0$. The improved scheme suppressed these, maintaining agreement with theory.
  • Error Metrics: Relative errors ($Err$) were reduced by orders of magnitude (e.g., from ~12% to <0.4% in high $\epsilon$ cases) using the improved scheme.

• Two-Phase Annular Shear Flow (Curved Interface):

  • Simulated flow between concentric circles with a curved gas-liquid interface.
  • The original scheme showed pronounced oscillations in both horizontal and 45°-inclined directions.
  • The improved scheme produced smooth velocity profiles across the interface, confirming the scheme's effectiveness for curved geometries.

• Droplet Falling in a Vertical Channel:

  • Simulated a liquid droplet falling under gravity.
  • Original Scheme: The droplet fell along the centerline with a higher terminal velocity, suggesting an underestimation of drag due to spurious oscillations.
  • Improved Scheme: The droplet migrated toward the channel wall (a physically realistic behavior due to wall effects) and exhibited a ~28% lower terminal velocity (at specific conditions), indicating a more accurate calculation of drag forces.
  • Conclusion: The spurious oscillations in the original scheme artificially reduced the effective drag, leading to incorrect flow patterns and velocities.

5. Significance

This work is significant because it resolves a long-standing numerical artifact in the pseudopotential LB model without compromising its physical foundation.

• Reliability: It ensures that simulations of multiphase flows (especially those involving shear, curved interfaces, or moving droplets) yield reliable, physically consistent results.
• Broad Applicability: The improved scheme is applicable to complex engineering problems (e.g., nuclear engineering, materials processing) where accurate prediction of interfacial dynamics and drag forces is critical.
• Theoretical Rigor: By deriving the solution from a discrete-level analysis rather than empirical tuning, the paper provides a robust theoretical basis for future developments in LB multiphase modeling.

In summary, the paper provides a mathematically grounded, computationally efficient, and physically consistent solution to suppress spurious velocity oscillations, significantly enhancing the predictive capability of the pseudopotential LB model for complex multiphase flows.