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Galilean Invariant Fluid-Solid Interfacial Dynamics in Lattice Boltzmann Simulations
Binghai Wen, Chaoying Zhang, Yusong Tu, Chunlei Wang, Haiping Fang
TL;DR
The paper addresses the limited consideration of Galilean invariance in fluid-solid interface treatments, despite its presence in the lattice Boltzmann equation. It proposes a Galilean invariant momentum exchange equation using relative velocity at the interface and evaluates it numerically. The reported simulations show improved accuracy and robustness, with sufficiently small force fluctuations that time averaging becomes unnecessary.
Problem
Galilean invariance is usually not considered in fluid-solid interface treatments even though the lattice Boltzmann equation itself has this property.
Method
The paper proposes a Galilean invariant momentum exchange equation that introduces relative velocity into interfacial momentum transfer.
Results
Numerical cases support full Galilean invariance, improved numerical accuracy and stability, and accurate depiction of suspension-particle behavior in three-dimensional Segré-Silberberg simulations.
Takeaways & Limitations
Considering Galilean invariance improves fluid-solid interfacial simulation accuracy and robustness, while making time-averaged velocity and force computations unnecessary.
Abstract
from arXiv · showhide
Galilean invariance is a fundamental property; however, although the lattice Boltzmann equation itself is Galilean invariant, this property is usually not taken into account in the treatment of the fluid-solid interface. Here, we show that consideration of Galilean invariance in fluid-solid interfacial dynamics can greatly enhance the computational accuracy and robustness in a numerical simulation. Surprisingly, simulations are so vastly improved that the force fluctuation is very small and a time average becomes unnecessary.
I. INTRODUCTION
The lattice Boltzmann equation is Galilean invariant, but fluid-solid interface treatments have largely neglected this property despite the boundary’s major influence on flow. The paper introduces a Galilean invariant momentum exchange equation and reports improved accuracy, robustness, and reduced force fluctuations.
- Galilean invariance is usually guaranteed for the governing lattice Boltzmann equations but has received little attention at fluid-solid interfaces.
- The conventional momentum exchange method does not satisfy Galilean invariance and may therefore lack very high computational accuracy.
- Previous corrections diminished numerical errors from non-Galilean effects, but full Galilean invariance with high simulation accuracy remained unsatisfactory.
- The paper introduces a Galilean invariant momentum exchange equation that uses relative velocity in interfacial momentum transfer to compute hydrodynamic force.
- The algorithm is simple, independent of boundary geometries, and reported to ensure Galilean invariance and high accuracy in dynamic fluids.
- Accounting for Galilean invariance greatly improves computational accuracy and robustness, making widely used time averaging of velocity and force unnecessary.
II. LATTICE BOLTZMANN METHOD
The lattice Boltzmann method represents fluid behavior with discrete particle distribution functions evolved through local collision and advection steps. Its formulation recovers incompressible Navier–Stokes behavior in the nearly incompressible limit.
- The lattice Boltzmann equation is rooted in kinetic theory and the cellular automaton concept, with several collision-operator variants.
- The lattice Boltzmann equation can obtain incompressible Navier–Stokes equations in the nearly incompressible limit, while its dominant collision computation remains local.
- The discretized lattice Boltzmann equation uses particle distributions, discrete speeds, and a collision operator at each lattice site and time.
- Fluid mass density and momentum density are obtained by summing distribution functions and their discrete-velocity-weighted components.
- Lattice Boltzmann evolution decomposes into collision and advection, with collision updating distributions locally before streaming.
III. GALILEAN INVARIANT MOMENTUM EXCHANGE METHOD
The conventional momentum exchange formulation computes force across fluid-solid links using discrete lattice velocities, but it is not Galilean invariant for moving boundaries. The proposed formulation accounts for boundary motion through relative velocity.
- A. Conventional momentum exchange equation: Momentum exchange methods evaluate hydrodynamic force across fluid-solid links from momentum carried by distributions entering and leaving the boundary.
- A. Conventional momentum exchange equation: A moving boundary lies between adjacent fluid and boundary nodes, with boundary velocity specified at their link intersection.
- A. Conventional momentum exchange equation: Boundary treatments may use interior-fluid evolution, half-way bounce-back, or curved boundary conditions, including boundary-velocity forcing terms where required.
- A. Conventional momentum exchange equation: The conventional force equation directly uses discrete velocities for incoming and outgoing distribution functions.
- A. Conventional momentum exchange equation: The conventional equation remains conceptually unchanged across several proposed modifications intended to improve accuracy and Galilean invariance.
B. Galilean invariant momentum exchange method
The conventional method implicitly treats the boundary as motionless during momentum transfer, violating Galilean invariance. GME replaces lattice-frame velocities with boundary-relative velocities and is reported to provide high computational accuracy.
- B. Galilean invariant momentum exchange method: Using lattice-frame momentum components makes the conventional force evaluation depend on the reference frame rather than boundary motion.
- B. Galilean invariant momentum exchange method: The conventional method assumes the boundary is motionless during momentum transfer regardless of reference-frame or actual-boundary speed, causing a divergent difference.
- B. Galilean invariant momentum exchange method: GME uses the relative velocity e_i - v for momentum increments and decrements when distributions cross the moving boundary.
- B. Galilean invariant momentum exchange method: The Galilean invariant momentum exchange method is defined by replacing the conventional lattice velocities in the interfacial force calculation with boundary-relative velocities.
- B. Galilean invariant momentum exchange method: Total force and torque are obtained by summing link forces and their moment arms around the solid particle’s mass center.
- B. Galilean invariant momentum exchange method: GME reduces to the conventional method for a motionless boundary and does not directly alter moving-boundary forcing terms.
- B. Galilean invariant momentum exchange method: The method works on moving-boundary motion states but has no direct influence on the surrounding fluid’s motion.
C. Comparisons in equilibrium state
The conventional fluid-solid force equation depends on reference-frame velocity, whereas the proposed GME uses velocity-set symmetry to eliminate that dependence and achieve Galilean-invariant force evaluation in equilibrium.
- The hydrodynamic force remains constant under arbitrary uniform reference velocity, proving GME completely Galilean invariant for the equilibrium system.
- The conventional equation produces a force that changes with reference-frame speed, with discrepancies proportional to the squared reference velocity.
- Because a single discrete velocity cannot satisfy Galilean invariance on one fluid-solid link, the method relies on the symmetry of the full discrete velocity set.
- The proposed local force evaluation eliminates the reference-velocity term through vector cancellation over the symmetric velocity set.
- For one-sided pressure on a vertical thin plate, GME satisfies Galilean invariance while the conventional equation violates it.
IV. SIMULATION RESULTS AND DISCUSSION
The simulations assess GME accuracy, robustness, and Galilean invariance using freely moving particles in two-dimensional cylinder sedimentation and three-dimensional sphere migration.
- Particle suspension is used to investigate computational accuracy because freely moving particles accumulate force errors that become apparent in their motion.
- The test cases include two-dimensional cylinder sedimentation and three-dimensional rigid-sphere lateral migration in Poiseuille flow.
- The study evaluates GME through direct numerical simulations targeting accuracy, robustness, and Galilean invariance.
- Simulations use second-order interpolation with an SRT model and multireflection boundaries with an MRT model.
- The reported SRT implementation uses relaxation time τ=6.0, while the MRT implementation specifies a diagonal relaxation matrix.
A. Galilean invariance in dynamic system
Dynamic sedimentation tests compare GME with the conventional equation and ALE across uniformly moving reference frames. GME agrees with benchmarks independently of reference-frame speed, while the conventional formulation deviates increasingly as that speed grows.
- The dynamic study examines cylinder sedimentation in a vertical channel, where an initially off-center cylinder rotates and translates before reaching steady centerline descent.
- The simulations vary uniform reference velocities assigned to the fluid, particle, and channel, with fluid values 0, 0.001, 0.002 and latter values 0, 0.01, 0.02.
- The comparisons report trajectories, angular velocities, and horizontal and vertical velocities relative to the channel against conventional-equation and ALE results.
- The conventional equation differs substantially from benchmarks even in a stationary frame, with deviations increasing as reference velocity increases.
- GME remains in excellent agreement with benchmarks across reference speeds, supporting full Galilean invariance in these numerical simulations.
B. Accuracy of hydrodynamic force
GME computes hydrodynamic forces with much higher agreement to ALE benchmarks and substantially smaller fluctuations than ALD and LME, making time averaging unnecessary.
- GME hydrodynamic forces extremely agree with ALE benchmarks, while ALD and LME exhibit large fluctuations.GME data are raw; ALD and LME data are smoothed by adjacent averaging.
- The force fluctuation of GME is very small, so time averaging becomes unnecessary.
- The relative L2-norm error compares an LBM result f(t) with an ALE result F(t).
- Relative errors of GME rapidly decrease as lattice scale increases, whereas ALD and LME errors remain more than one order larger.
- Second-order extrapolation for newborn fluid nodes can remarkably reduce fluctuations, and the A1 algorithm is used in the other simulations.
C. Fluctuations of velocity and angle velocity
Across the tested sedimentation conditions, GME produces smooth velocities in excellent agreement with ALE benchmarks, while ALD and LME fluctuate and deviate.
- GME horizontal, vertical, and angular velocities are very smooth and in excellent agreement with ALE benchmarks.
- ALD and LME velocities clearly fluctuate and show deviations from the ALE benchmarks.
- The simulations vary particle densities from 1.02 to 1.22 g/cm3, with Reynolds numbers growing from 6.13 to 34.75.
- GME results are more accurate and far steadier than ALD and LME results, making velocity time averaging unnecessary.
D. Three dimensional numerical simulation
Three-dimensional GME simulations reproduce lateral migration of neutrally buoyant spheres in tube Poiseuille flow and agree highly with experiments.
- The simulated system contains a neutrally buoyant rigid sphere migrating laterally in tube Poiseuille flow.
- GME simulations of the three-dimensional Segré-Silberberg effect produce equilibrium sphere positions far from the tube centerline.
- The trajectories begin at dimensionless radial positions r*/R=0.21 and 0.66.
- GME numerical results are highly consistent with Karnis et al.'s experiments, verifying competence for three-dimensional dynamic simulations.
V. CONCLUSION
The paper proposes a Galilean-invariant momentum exchange equation using relative velocity for fluid-solid interfacial momentum transfer, and validates it in dynamic simulations.
- The proposed GME equation introduces relative velocity to compute hydrodynamic force during interfacial momentum transfer.
- Numerical cases strongly support that GME meets full Galilean invariance and improves lattice Boltzmann numerical accuracy.
- Cylinder sedimentation and the three-dimensional Segré-Silberberg effect show that GME exactly depicts suspension-particle behavior and has excellent stability.
- The algorithm uses only local data and is independent of boundary geometries.
- GME can be combined with curved boundary conditions and adopted in SRT, MRT, TRT, and ELBE lattice Boltzmann models.