Basics of Lattice Boltzmann Method for CFD modeling

Introduction to Methods of Flow Simulations

• Computational flow dynamic (CFD) simulations using the finite volume method have been used for many industrial applications as discussed in the post.
• However, nowadays lattice Boltzmann method (LBM) has been widely used in micro fluidics, biomedical, and multiphase flow applications.

What are the Lattice Boltzmann Method Simulations?

• The Lattice Boltzmann Method (LBM) is a computational fluid dynamics (CFD) technique used for simulating fluid flow and solving fluid dynamics problems.
• It is an alternative approach to traditional CFD methods like Finite volume methods (FVM) to handle complex fluid behaviors and geometries in many mesoscopic applications.
• Lattice Boltzmann Method (LBM)  is also another dynamic method of fluid simulations at the macroscopic behavior using a simple mesoscopic model.
• It inherited the main principles of Lattice.
• Gas Automaton (LGA) and made improvements. From lattice gas automaton, it is possible to derive the macroscopic Navier-Stokes equations.
•  Lattice Boltzmann Method and its difference from the traditional macroscopic methods for numerical calculations are given below:
  • It is based on and starts from Non-equilibrium statistical mechanics and Discrete model
  •  It connected the dynamic lattice model, whose time, space, and velocity phase space are fully
    discrete, with the Boltzmann equation.
  •  The implementation of this method can describe the law of fluid motion without
    Solving Navier-Stokes equations

 

Comparison of LBM with Finite Volume Methods  and Molecular Modeling

• Three commonly used methods finite volume methods lattice Boltzmann methods and molecular dynamics are given  in the following tables for advantages and disadvantages
• Each simulation technique has a different level of time, length, and velocity scales.
• Finite Volume Method
  • CFD solvers like ANSYS FLUENT and OpenFOAM are based on the Finite volume method (FVM) as presented in the post.
  • In this method, conservation equations of mass, momentum, and energy are solved numerically at macroscopic scales. This is based on the continuum hypothesis (Eulerian approach). Molecular level properties like pressure and viscosity are modeled as molecular fluid properties like viscosity, and conductivity.
  • This method is suitable for large-scale geometries (macro-scale)  with high velocity of bulk fluid flows in engineering applications
  • The ratio of molecular to geometry size (Knudesn number) is less than 0.001

• Lattice Boltzmann Methods

  • LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice
  • This method is suitable for micro-scale (mesoscale) geometries with small velocities of bulk fluid flows at a mesoscopic level in biomedical applications

• Molecular dynamics (MD) simulations
  • Molecular dynamics (MD) simulations is used to predict every atom in a protein or other molecular system.
  • This method is  based on a general model of the physics governing interatomic interactions
  • This method is suitable for micro and nano-scale geometries with less velocity of molecular flows

 

 

 

Overview of LBM

• Here’s a brief overview of the Lattice Boltzmann Method:

• Discretization of Space and Time:
  • LBM operates on a discrete lattice grid, typically in a regular arrangement. Each lattice node represents a discrete point in space.
  • Time is also discretized into discrete time steps.
  • For the  Lattice Boltzmann Method (LBM), the lattice grid plays a central role as it discretizes the space where fluid flow (refer to the following figure (a, b))
  • This method used for simulations  at mesoscopic levels

 

 

• Particle Distribution Functions:
  • LBM models the fluid as a collection of fictitious particles, and it uses a set of distribution functions to represent the probability of particles moving in specific directions at each lattice node.
  • In a D2Q9 lattice (common in 2D simulations), there are nine distribution functions at each node, corresponding to nine discrete velocity directions.

• Collision and Streaming Steps:
  • LBM consists of two main steps: the collision step and the streaming step.
  • During the collision step, the distribution functions are modified based on collision rules. These rules account for particle interactions and interactions between neighboring nodes.
  • In the streaming step, particles are moved to neighboring lattice nodes based on their velocity. This step simulates the advection of particles in the fluid.
• Macroscopic Properties:
  • From the distribution functions, macroscopic fluid properties such as density and velocity are calculated at each lattice node.
  • These properties can be used to visualize and analyze fluid flow and other fluid characteristics.
• Boundary Conditions:
  • LBM handles boundary conditions differently from traditional methods. Various techniques, such as bounce-back rules, are used to account for solid boundaries and other boundary conditions within the lattice framework.

Lattice Grid for Lattice Boltzmann Method

• In the Lattice Boltzmann Method (LBM), the lattice grid plays a central role as it discretizes the space where fluid flow and other phenomena are simulated.
• The choice of lattice grid is an important consideration in LBM simulations, and different lattice structures can be used depending on the specific problem being studied. Here are some common lattice grids used in LBM:

1. D2Q9 Lattice (Two Dimensions, Nine Velocities):
   • Dimensions: 2D grid.
   • Velocities: Nine discrete velocity vectors. The lattice structure often looks like a square grid with a central node and eight surrounding nodes.
   • Commonly used for 2D simulations of fluid flow.
2. D3Q19 Lattice (Three Dimensions, Nineteen Velocities):
   • Dimensions: 3D grid.
   • Velocities: Nineteen discrete velocity vectors. The lattice structure is more complex than D2Q9 and is often cubic or cubic-centered.
   • Suitable for 3D simulations of fluid flow.
3. D3Q27 Lattice (Three Dimensions, Twenty-Seven Velocities):
   • Dimensions: 3D grid.
   • Velocities: Twenty-seven discrete velocity vectors. This lattice structure provides higher resolution and accuracy for 3D simulations.
   • Used when finer resolution is needed in 3D simulations, particularly for complex geometries.
4. Hexagonal Lattice:
   • Dimensions: 2D grid.
   • Velocities: Six discrete velocity vectors arranged in a hexagonal pattern.
   • Commonly used for simulating flow in hexagonal-packed porous media and other problems with hexagonal symmetry.
5. Triangular Lattice:
   • Dimensions: 2D grid.
   • Velocities: Three discrete velocity vectors arranged in a triangular pattern.
   • Used in simulations involving triangular-packed porous media and other applications with triangular symmetry.
6. Body-Centered Cubic (BCC) Lattice:
   • Dimensions: 3D grid.
   • Velocities: Thirteen discrete velocity vectors arranged in a body-centered cubic lattice.
   • Suitable for simulating multiphase flows and other problems with cubic symmetry.
7. Face-Centered Cubic (FCC) Lattice:
   • Dimensions: 3D grid.
   • Velocities: Fifteen discrete velocity vectors arranged in a face-centered cubic lattice.
   • Useful for multiphase flow simulations and problems with cubic symmetry.
8. Lattices for Multiphase and Multi-component Flows:
   • Specialized lattices may be designed for multiphase or multicomponent flow simulations, allowing for the modeling of interfacial phenomena, phase separation, and mass transfer.

• The choice of lattice grid depends on several factors, including the dimensionality of the problem, the desired level of resolution, and the specific physical phenomena being simulated.
• In practice, researchers select a lattice structure that best matches the problem’s characteristics and computational requirements.
• The lattice structure and the discrete velocity vectors associated with it are essential components of LBM, as they determine the numerical accuracy and stability of the simulation.
• Researchers often choose lattice structures that strike a balance between computational efficiency and accuracy for their specific application.

 

Lattice Functions

• These are fundamental equations and concepts used in the Lattice Boltzmann Method.
• The specific details and formulations can vary depending on the specific LBM variant used and the problem being simulated (e.g., incompressible flow, thermal LBM, multiphase LBM, etc.).
• The choice of lattice structure, collision model, and boundary conditions can also influence the equations used in LBM simulations.
• In LBM, the Boltzmann equation is discretized to model fluid flow on a lattice grid.
• You can refer books for more details of mathematical functions for this methods. Here are some key formulas and concepts used in the Lattice Boltzmann Method:

Distribution Functions:

• LBM represents the probability distribution of particles at discrete velocity vectors in each lattice direction.
• These distribution functions are typically denoted function represents a specific velocity direction.
• In a D2Q9 lattice, there are nine distribution functions

Equilibrium Distribution Functions:

• Equilibrium distribution functions (fieq​) are calculated based on local macroscopic properties, such as density ($\rho$) and velocity ($u$).
• They are used to model the particles’ relaxation toward equilibrium in the collision step of LBM.

Streaming Step:

•  In the streaming step, particles are moved to neighboring lattice nodes based on their velocity vectors:

Macroscopic Properties:

• Macroscopic properties, such as density ($\rho$) and velocity ($u$), are calculated from the distribution functions

Boundary Conditions:

• Boundary conditions are applied at the domain boundaries to ensure that particles interact correctly with solid walls or other boundary types.
• Common boundary conditions include the bounce-back rule for no-slip walls

Lattice Boltzmann Equation (Boltzmann BGK Equation):

• The Lattice Boltzmann equation is a simplified form of the Boltzmann equation:

Steps in Lattice Boltzmann Method (LBM) 

The Lattice Boltzmann Method (LBM) is a computational fluid dynamics (CFD) technique that simulates fluid flow by modeling the behavior of particles on a lattice grid. Here are the main steps involved in performing CFD simulations using LBM:

Initialization:

• Define the computational domain: Specify the physical dimensions of the simulation domain, including its size and geometry.
• Set up the lattice grid: Choose a lattice structure (e.g., D2Q9 for 2D or D3Q27 for 3D simulations) and establish the grid resolution (spacing between lattice nodes).
• Initialize fluid properties: Assign initial values for fluid density and velocity at each lattice node.
• You can also set the initial conditions for other scalar quantities like temperature or concentration if they are part of the simulation.
• Specify boundary conditions: Define the boundary conditions at the domain boundaries, which can include no-slip walls, inflow, outflow, and other conditions relevant to the problem.

Time Stepping:

• Select a time step size: Determine an appropriate time step for advancing the simulation in discrete time increments.
• The choice of time step affects stability and accuracy.
• Set the total simulation time: Define the duration for which the simulation will run or specify stopping criteria.

Collision Step:

• Calculate equilibrium distribution functions: Based on the macroscopic properties (density, velocity, temperature, etc.) at each lattice node, compute the equilibrium distribution functions using a suitable collision model.
• Perform collision: Update the distribution functions by applying collision processes. The collision step incorporates relaxation towards the local equilibrium distribution functions.

Streaming Step:

• Move particles: Displace particles by streaming them to neighboring lattice nodes based on their velocity. This step represents the advection of particles in the fluid.

Boundary Conditions:

• Apply boundary conditions: Implement the defined boundary conditions at the domain boundaries.
• Commonly used boundary conditions include bounce-back (no-slip), inflow/outflow conditions, and periodic boundaries.
• Handle special boundaries: Adapt the boundary treatment for specific scenarios or complex geometries.
• For instance, modeling porous media may require special boundary conditions.

Macroscopic Property Calculation:

• Calculate macroscopic properties
• Compute macroscopic properties such as fluid density, velocity, and temperature at each lattice node based on the updated distribution functions.

Time Advancement:

• Advance the simulation time: Move to the next time step by repeating steps 3 to 6.
• Continue iterating until the simulation reaches the desired time or satisfies predefined convergence criteria.

Visualization and Analysis:

• Visualize results: Use the computed macroscopic properties to visualize and analyze fluid flow patterns, temperature distribution, pressure, or other relevant data.
• Post-processing: Perform additional post-processing and analysis, such as calculating forces, heat transfer rates, or other quantities of interest.

Termination and Output:

• Stop the simulation: Terminate the simulation when it reaches the desired time or satisfies convergence criteria.
• Save results: Save simulation results and data for further analysis or documentation.

Parallelization (Optional):

• Implement parallel computing techniques to distribute the computational workload across multiple processors or nodes, enhancing simulation efficiency for large-scale problems.

Adaptation and Optimization (Optional):

• Continuously adjust simulation parameters (e.g., relaxation time, grid resolution) to optimize accuracy and efficiency based on problem-specific requirements.

These steps provide a general outline of the Lattice Boltzmann Method for CFD simulations. The details and complexities of the implementation may vary depending on the specific problem being solved and the software or code used for the simulation.

 

Applications of Lattice Boltzmann Method (LBM)

The Lattice Boltzmann Method (LBM) is a computational fluid dynamics (CFD) technique used for simulating fluid flow and solving fluid dynamics problems.

The Lattice Boltzmann Method (LBM) has found applications in a wide range of fields due to its versatility and ability to handle complex fluid behaviors and geometries. Some of the notable applications of LBM include:

1. Fluid Flow Simulations
   • LBM can be used for fluid flows of l fluid dynamics, including sediment transport in rivers, pollutant dispersion in air and water, and ocean circulation
   • Incompressible and Compressible Flows:
2. Heat Transfer simulations
3. Micro fluid flow
   • LBM is used to model fluid flow in micro-channels and microdevices, which is important in fields like lab-on-a-chip technology, micro-fluidic mixing, and biofluidics.
4. Porosity and Permeability:
   • LBM is applied to study fluid flow through porous media, making it useful in hydrogeology, petroleum engineering, and geophysics for modeling groundwater flow, oil reservoirs, and subsurface processes.
5. Blood Flow and Cardiovascular Modeling:
   • LBM is used to simulate blood flow in arteries and veins, aiding in the understanding of hemodynamics, the study of blood vessel diseases, and the design of medical devices such as stents.
6. Multiphase flows:.
   • Bubble flow modeling
   • Particle-Laden Flows
7. Bio-fluid Dynamics:
   • LBM is used in biomechanics and bioengineering to simulate the flow of fluids in biological systems, such as airflow in the respiratory system, cerebrospinal fluid flow in the brain, and blood flow in the heart.
8. Material Science:
   • LBM can simulate the flow of complex fluids, including colloidal suspensions and polymer melts, aiding in the design of materials with desired properties.
9. Non-Newtonian Flows:
   • LBM can model non-Newtonian fluids like viscoelastic fluids, which are encountered in applications such as polymer processing and food industry processes.
10. Environmental Impact Studies:
    • LBM is used to assess the environmental impact of various projects, including the dispersion of pollutants in urban areas and the modeling of water quality in lakes and rivers..
11. Nano-fluidics:
    • LBM is applied to study fluid flow at the nanoscale, which is relevant in nanotechnology and the design of nanofluidic devices.

Summary

• Overall, LBM is a versatile and powerful tool for simulating fluid flows, particularly in scenarios where traditional methods face limitations or computational challenges.
• It has contributed to advancements in the understanding of fluid dynamics in various scientific and engineering applications.
• LBM is a valuable tool for simulating and understanding fluid flow and transport phenomena in microfluidic systems.
• Its ability to handle small-length scales, complex geometries, and multiphase flows makes it well-suited for studying and optimizing microfluidic devices and processes across various scientific and engineering disciplines.

References

1. Guo, Zhaoli, and Shu, Chien-Ting., Lattice Boltzmann Method and Its Applications in Engineering,  (2013)
2. Rothman, Daniel H. and Zaleski, Stéphane, Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. (1997)
3. Succi, Sauro, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,. (2001)
4. Mohamad, A. A, Lattice Boltzmann Method and Its Applications, (2013)
5. M.E. Kutay, LBM Basics lecture notes
6. Comsol User Guide for Molecular Dynamics Solver, 2020
7. Comsol for LBM Simulation, 2022