Post on 28-Jul-2020
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