Lecture on Lattice Boltzmann Method for Hydrodynamics

Lecturer: Tim Kruger, Reader in Chemical Engineering, University of Edinburgh

Overview

• Introduction to the Lattice Boltzmann Method (LBM) for hydrodynamics
• Background on kinetic theory of gases
• Explanation of the Lattice Boltzmann Method, boundary conditions, and immersed boundary method
• Summary and resources for further reading

Introduction

• Goals: Understand what LBM is, its advantages and disadvantages
• Applications: Simulating flows in simple geometries and modeling fluid-structure interactions

Fluid Mechanics Scales

• Covers scales from microfluidics (bacteria swimming) to atmospheric phenomena (ocean currents)
• All scales governed by the Navier-Stokes equations

Navier-Stokes Equations

• Describe momentum conservation in a continuum (fluids)
• Consist of viscous terms, nonlinear velocity terms
• Complex PDEs often requiring computational models for solutions

Kinetic Theory of Gases

• Essential for understanding LBM
• Describes molecular/atomic movement and collisions
• Governing equations shift from Newton’s laws to Boltzmann equation as scale increases

Boltzmann Equation

• Governs molecular velocity distributions
• Incorporates collision operator for molecular interactions
• Simplified using BGK model which focuses on relaxation towards equilibrium

Lattice Boltzmann Method (LBM)

• Uses Boltzmann equation to solve Navier-Stokes equations
• Discretizes distribution functions into lattice nodes, time steps, and velocity space

Discretization

• Space is divided into a lattice
• Time steps are constant
• Velocity space is discretized into finite directions (e.g., 9 in 2D)

Lattice Boltzmann Equation

• Utilizes a simplified BGK model
• Consists of a collision step and propagation step

Advantages of LBM

• Fast and explicit method
• Easily parallelizable
• Handles complex geometries

Limitations of LBM

• Suitable for low Knudsen and Mach numbers
• Original form not suitable for high-speed flows or gases

Boundary Conditions

• Critical for accurately simulating fluid dynamics
• Various types: bounce-back, ghost methods, immersed boundary method

Bounce-Back Method

• Simulates no-slip boundary conditions by reversing velocity at boundaries
• Simple to implement but less accurate for moving boundaries

Immersed Boundary Method

• Coupled with LBM for simulating complex and moving boundaries
• Uses a separate mesh (Lagrangian system) on top of the lattice (Eulerian system)

Summary

• LBM offers a computational method for solving Navier-Stokes equations
• Suitable for high-performance computing
• Immersed boundary method complements LBM for complex boundary conditions

Further Reading

• Recommended textbook: "The Lattice Boltzmann Method: Principles and Practice"

Next Steps: Participate in a live Q&A session for further clarification.