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Introduction to the lattice Boltzmann method

Física dos Meios Contínuos – Faculdade de Ciências da Universidade de Lisboa

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Computational fluid dynamics

● Solve the Navier-Stokes equation in the macroscopic limit;

● Finite differences, finite volumes, lattice-Boltzmann (LBM), smoothed particles hydrodynamics (SPH), …

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Finite differences

● Regular grids;● Easy to implement;● Direct implementation of the differential equations;● Runge-Kutta metods;● Lack of stability;● Difficult to treat complex boundaries.

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Finite volumes

● Irregular grids;● Professional software;● Fine grids close to corners;● Complicated for moving parts, complex geometries and interfaces.

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Example of application: Gearbox

● Moving parts● Complex geometry● Free surface● Fluid-structure

interaction● Turbulence

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Applications

Porous media Hydraulic pumps

Multiphase fluids Medical research

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Introduction

● Do not discretize the Navier-Stokes directly;

● Modeling a gas from a mesoscopic point of view;

● High performance in parallel architectures;

● Simple mesh generation: cartesian grid;

● Many physical models;

Ludwig Eduard Boltzmann

(1844-1906)

The lattice Boltzmann method

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Introduction

Molecular dynamics Lattice Boltzmann Finite volumes

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Introduction

Lattice Boltzmann

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Introduction

Lattice Boltzmann

Gaussian quadrature

Numerical method to calculate integrals

Ex.: density

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Introduction

Lattice Boltzmann

Gaussian quadrature Boltzmann equation

Numerical method to calculate integrals

Gives the time evolution of the

distribution function

Ex.: density

Important concepts: Distribution function, Boltzmann equation, Gaussian quadrature

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Distribution function

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Distribution function

Total number of particles Distribution of number of particles

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Distribution function

Total number of particles Distribution of number of particles

Total mass

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Distribution function

Total number of particles Distribution of number of particles

Total mass

= (⍴( x,t)

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Macroscopic fields (moments)

Density

Momentum -> velocity

Energy -> temperature

Connection between microscopic and macroscopic

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Equilibrium distribution

Maxwell-Boltzmann distribution function: used in the classical LBM

Probability density to find a particle with velocity u in a gas with density ρ and temperature θ.

Fields at equilibrium

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Equilibrium distribution

Maxwell-Boltzmann distribution function: used in the classical LBM

Probability density to find a particle with velocity u in a gas with density ρ and temperature θ.

Fields at equilibrium

Second order expansion in Hermite polynomials (to use the Gauss-Hermite quadrature)

Approximation: small Mach number

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Hermite polynomials

Hermite weight function

Orthogonalization

Rodrigues’ formula

The Hermite polynomials are the mathematical basis of the LBM: Gauss-Hermite quadrature and expansion of the equilibrium distribution function. In D dimensions they are:

where

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Expansion of the EDF

Maxwell-Boltzmann distribution

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Expansion of the EDF

Expansion in Hermite polynomials

Maxwell-Boltzmann distribution

Projections of the distribution function on the Hermite polynomials

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Expansion of the EDF

Expansion in Hermite polynomials

Maxwell-Boltzmann distribution

Hermite polynomials

Hermite weight function

Order of the expansion

Projections of the distribution function on the Hermite polynomials

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Expansion of the EDF

Expansion in Hermite polynomials

Maxwell-Boltzmann distribution

Hermite polynomials

Hermite weight function

Order of the expansion

Projections of the distribution function on the Hermite polynomials

Ex.: expansion up to second order (K=2)

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Time evolution

Without collisions and external force

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Time evolution

Without collisions

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Time evolution

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Time evolution

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Boltzmann equation

Now considering collisions...

All the physics of the scattering process is contained in the collision operator

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Collision operator

Two-body collision term

● Tricky to calculate computationally● Not efficient for simulating macroscopic fluids● Unnecessary complexity from the macroscopic point of view

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Collision operator

Bhatnagar-Gross-Krook (BGK) collision operator: The distribution function f tends exponentially to the equilibrium distribution feq with a characteristic time 𝜏

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Collision operator

Lattice-BGK equation

Bhatnagar-Gross-Krook (BGK) collision operator: The distribution function f tends exponentially to the equilibrium distribution feq with a characteristic time 𝜏

Related to the kinematic viscosity

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Collision operator

Lattice-BGK equation

Does it describe the macroscopic equations?

Bhatnagar-Gross-Krook (BGK) collision operator: The distribution function f tends exponentially to the equilibrium distribution feq with a characteristic time 𝜏

Related to the kinematic viscosity

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Macroscopic equations

Ex.: Equation for density (mass conservation). Integration of the Boltzmann equation

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Macroscopic equations

Ex.: Equation for density (mass conservation). Integration of the Boltzmann equation

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Macroscopic equations

General procedure: Chapman-Enskog method.Recovers continuity, Navier-Stokes and energy equations.

Ex.: Equation for density (mass conservation). Integration of the Boltzmann equation

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Macroscopic equations

Perturbation theory: Expansion for small Knudsen number

Distribution function

Chapman-Enskog method

In the limit of small Knudsen number, the statistics of a fluid is described by the equilibrium distribution function. Off-equilibrium effects (viscosity, heat flux)

appear at non-vanishing Knudsen number.

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Macroscopic equations

Mass conservation (continuity): recovered with a first order expansion

Momentum conservation (Navier-Stokes): recovered with a second order expansion

Viscosity stress tensor

Energy conservation: recovered with a fourth order expansion (thermal models)

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Gauss-Hermite quadrature

How to calculate the integrals?

Density Momentum

Lattice Boltzmann:● The equilibrium distribution function (Maxwell-

Boltzmann) is expanded in Hermite polynomials up to order N (usually N=2);

● The phase (velocity and position) space is discretized.

● Regular lattices

Gaussian quadrature: ● Approximate integrals by sums:

● The integrand is needed just at few points ξ𝛼;● The method is exact if the function g(ξ) is polynomial up to a maximum order given by the

quadrature (e.g.: for the D2Q9 lattice, the integration is exact for monomials up to fifth order).

where

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D2Q9 lattice

Nomenclature DdQq: “d” spatial dimensions and “q” lattice vectorsEx.: D2Q9 - two dimensions and nine velocity vectors

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D2Q9 lattice

Quadrature equations

where

Nomenclature DdQq: “d” spatial dimensions and “q” lattice vectorsEx.: D2Q9 - two dimensions and nine velocity vectors

For the D2Q9 lattice

where

Up to the fifth moment of the weight function

The MB distribution is expanded up to second order

Solution

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3D lattice

3D lattices

D3Q19 D3Q27

D3Q27

D3Q19

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Discrete space

Discrete distribution function

Macroscopic quantities (density and velocity) D2Q9 lattice

Boltzmann equation

Collision step

Streaming step

Δxx

where

Lattice-Boltzmann: Quadrature + Boltzmann Eq.

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Algorithm

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Boundary conditions

Bounce back BC

● Used for complex and static obstacles (e.g., porous media);

● No-slip condition;● Do not treat moving boundaries.

● Initial and BC conditions are necessary to solve partial differential equations (e.g., N-S);

● Problem: Calculate the distributions from the macroscopic fields.

Periodic BC● Simulates an infinite periodic system;● Conserves the macroscopic quantities.

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Forcing scheme

Continuum Boltzmann equation

Discrete form

● External forces (e.g., gravity)● Boundary conditions (e.g., IBM)● Multiphase methods (e.g., Shan-

Chen)

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Forcing scheme

Method 1 (Guo): calculate the forcing term explicitly (second order accurate)

Method 2 (Shan-Chen): shift the macroscopic velocity in the equilibrium distribution(first order accurate)

Newton’s law

These two methods are equivalent up to first order in Δxt

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Multiphase model

Interaction potential between the particles

Strength of the interaction (repulsive if G is positive)

Force implemented through the velocity:

Multicomponent fluid: more distributions

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Conclusion

• Lattice-Boltzmann = Gauss-Hermite quadrature + Boltzmann equation;

• Mesoscopic scale: the LBM numerically solves the Boltzmann equation and “extracts” the macroscopic fields from the distribution functions;

• LBM solves the Navier-Stokes equations (simple mesh generation);

• Naturally treats complex geometries;• Many physical models: multiphase, supersonic, relativistic ...

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• Kruger et al, The Lattice Boltzmann Method. Springer (2017);

• Sukop and Torme, Lattice Boltzmann Modeling. Springer (2006);

• Landau and Lifshitz, Fluid Mechanics. Elsevier (2013).

Units conversion

Three independent conversion factors are required to generate the dimension of mechanical quantities:

Law of similarity: two incompressible flow systems are dynamically similar if they have the same Reynolds number and geometry

Units conversion

Lattice units● Time step (for one LBM cycle): Δxt = 1● Lattice spacing: Δxx = 1● Fluid density: = 1⍴(

Thermal models

Energy conservation● Fourth order expansion of the equilibrium distribution function;● Higher order lattices (e.g., D2Q37);● Usually less stable than the isothermal models due to the complexity of the lattices.

Advection-diffusion model: treats temperature as a concentration.

Ex.: smoke

Advection Diffusion Advection-Diffusion(convection)

Advection-diffusion model

Advection-diffusion equation for a concentration C (similar to the NS equation)

The concentration is calculated through the distribution and the velocity is taken from another solver

Evolution

Boussinesq approximation: the effect of a small density change creates a buoyancy force density

The density is changed only in the force term, not in the equilibrium distribution

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