[Paper Review] Lattice Boltzmann scheme for hydrodynamic equation of phonon transport

Yangyu Guo, Moran Wang|arXiv (Cornell University)|Oct 5, 2020

Lattice Boltzmann Simulation Studies58 references4 citations

TL;DR

This paper proposes a lattice Boltzmann scheme to numerically solve the phonon Boltzmann equation under Callaway's dual relaxation model in the hydrodynamic limit. By applying a Chapman-Enskog expansion with a resistive scattering term as a source, the method recovers the Guyer-Krumhansl equation and Fourier's law in limiting cases, accurately simulating phonon hydrodynamic phenomena such as Poiseuille flow and second sound with good agreement to benchmark solutions.

ABSTRACT

In this work, a lattice Boltzmann scheme is developed for numerical solution of the phonon Boltzmann equation under Callaway's dual relaxation model in the hydrodynamic limit. Through a Chapman-Enskog expansion to the lattice Boltzmann equation with the resistive scattering term as an equivalent source term, we recover a phonon hydrodynamic equation which is reduced to the Guyer-Krumhansl heat transport equation and the Fourier's law in the limit of dominant normal scattering and dominant resistive scattering respectively. Several cases of heat transport from diffusive regime to hydrodynamic regime are modeled extensively by the present numerical scheme, which produces results in good agreement with the benchmark solutions. Two well-known phonon hydrodynamic phenomena including the phonon Poiseuille flow and second sound propagation are well captured by the lattice Boltzmann scheme. This work will promote the numerical modeling and deeper understanding of the non-Fourier heat transport induced by phonon normal scattering.

Motivation & Objective

• To develop a numerical scheme for solving the phonon Boltzmann equation in the hydrodynamic regime.
• To model heat transport across the transition from diffusive to hydrodynamic behavior.
• To capture non-Fourier heat transport phenomena driven by phonon normal scattering.
• To validate the scheme against benchmark solutions for phonon hydrodynamics.
• To enable efficient and stable simulation of phonon transport using lattice Boltzmann methodology.

Proposed method

• A lattice Boltzmann equation is formulated with a resistive scattering term treated as an equivalent source term.
• A Chapman-Enskog expansion is applied to the lattice Boltzmann equation to derive the macroscopic phonon hydrodynamic equation.
• The method recovers the Guyer-Krumhansl equation in the presence of normal scattering and Fourier's law under dominant resistive scattering.
• The scheme is validated across multiple regimes, from diffusive to hydrodynamic, using benchmark solutions.
• Phonon Poiseuille flow and second sound propagation are simulated to demonstrate the scheme's capability.
• The approach enables stable and accurate simulation of phonon transport in complex geometries.

Experimental results

Research questions

• RQ1 Can a lattice Boltzmann scheme accurately simulate the transition from diffusive to hydrodynamic phonon transport?
• RQ2 How well does the scheme capture the Guyer-Krumhansl equation and Fourier's law in limiting cases?
• RQ3 Can the scheme reproduce well-known phonon hydrodynamic phenomena such as Poiseuille flow and second sound?
• RQ4 What is the numerical stability and accuracy of the scheme across varying scattering regimes?
• RQ5 Can the resistive scattering term be effectively modeled as a source term in the lattice Boltzmann framework?

Key findings

• The lattice Boltzmann scheme successfully recovers the Guyer-Krumhansl equation in the hydrodynamic limit with normal scattering.
• In the limit of dominant resistive scattering, the scheme reduces to Fourier's law, confirming consistency with classical heat conduction.
• The scheme accurately captures phonon Poiseuille flow, demonstrating velocity profiles characteristic of hydrodynamic flow.
• Second sound propagation is well resolved, showing the expected wave-like thermal front propagation.
• Numerical results show good agreement with benchmark solutions across diffusive and hydrodynamic regimes.
• The method demonstrates robustness and accuracy in simulating non-Fourier heat transport phenomena.