Lattice Boltzmann Method and Its Applications in Multiphase Flows

1
Lattice Boltzmann Method and Its Applications in
Multiphase Flows

• Xiaoyi He
• Air Products and Chemicals, Inc.
• April 21, 2004

2
Outline

• Lattice Boltzmann method
• Kinetic theory for multiphase flow
• Lattice Boltzmann multiphase models
• Applications
• Conclusions

3
A Brief History of Lattice Boltzmann Method

• Lattice Gas Automaton (Frisch, Hasslacher,
  Pomeau,, 1987)
• Lattice Boltzmann model (McNamara and Zanetti
  (1988)
• Lattice Boltzmann BGK model (Chen et al 1992 and
  Qian et al 1992)
• Relation to kinetic theory (He and Luo, 1997)

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Lattice Boltzmann BGK Model

• fa density distribution function
• t relaxation parameter
• f eq equilibrium distribution

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Kinetic Theory of Multiphase Flow

• BBGKY hierarchy

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Intermolecular Interaction

• Lennard-Jones potential

Interaction models
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Model for Intermolecular Repulsion
For D1 (repulsion core)
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Model for Intermolecular Attraction
For D2 (attraction tail), by assuming
We have
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Model for Intermolecular Attraction
Vm is the mean-field potential of intermolecular
attraction
For small density variation
where
Control phase transition
Control surface tension
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Kinetic Model for Multiphase Flow
Boltzmann equation for non-ideal gas / dense fluid
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Kinetic Model for Multiphase Flow
Chapman-Enskog expansion leads to the following
macroscopic transport equations
Mass transport equation
Momentum transport equation
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Kinetic Model for Multiphase Flow
Comments on momentum transport equation 1.
Correct equation of state 2. Thermodynamically
consistent surface tension
3. Thermodynamically consistent free energy
(Cahn and Hillary, 1958)
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Kinetic Model for Multiphase Flow
Energy transport equation
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Kinetic Model for Multiphase Flow
Comments on energy transport equation 1.Total
energy needs include both kinetic and potential
energies, otherwise the pressure work becomes
2. Last term is due to surface tension and it is
consistent with existing literature (Irving and
Kirkwood, 1950)
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LBM Multiphase Model Based on Kinetic Theory

• Temperature variations in lattice Boltzmann
  models
• Discretization of velocity space
• Discretization of physical space
• Discretization of temporal space.

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Temperature in Lattice Boltzmann Method

• Non-isothermal model model is still a challenge
• Small temperature variations can be modeled
• Need for high-order velocity discretization
• Isothermal model is well developed

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Isothermal Boltzmann Equation for Multiphase Flow
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Discretization in Velocity Space

• Constraint for velocity stencil
• Further expansion of f eq

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Discretization in Velocity Space
wa weight coefficients
9-speed model
7-speed model
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Discretization in Physical and Temporal Spaces

• Integrate Boltzmann equation

• Discretizations in velocity, physical and
  temporal spaces are
• independent in principle
• Synchronization simplifies computation but
  requires
• Regular lattice
• Time-step constraint

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Further Simplification for Nearly Incompressible
Flow

• Introduce an index function f

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Applications

• Phase Separation
• Rayleigh-Taylor instability
• Kelvin-Helmholtz instability

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Phase Separation
Van der Waals fluid T/Tc 0.9
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Rayleigh-Taylor Instability (2D)
Re 1024 single mode
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RT instability (2D) Single mode Density ratio
31 Re 2048
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RT instability (2D) Multiple mode Density ratio
31
hB /Agt2 0.04
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RT instability (3D) single mode Density ratio
31 Re 1024
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RT instability (3D) single mode Density ratio
31 Re 1024 Cuts through spike
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RT instability (3D) single mode Density ratio
31 Re 1024 Cuts through bubble
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KH instability Effect of surface tension Re
250 d1/d2 1
Ca 0.29
Ca 2.9
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Other Applications

• Multiphase flow in porous media (Rothman 1990,
  Gunstensen and Rothman 1993)
• Amphiphilic fluids (Chen et al, 2000)
• Bubbly flows (Sankaranarayanan et al, 2001)
• Hele-Shaw flow (Langaas and Yeomans, 2000).
• Boiling flows (Kato et al, 1997)
• Drop break-up (Halliday et al 1996)

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Challenges in Lattice Boltzmann Method

• Need for better thermal models
• Need for better model for multiphase flow with
  high density ratio
• Need for better mode for highly compressible
  flows
• Engineering applications

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Conclusions

• Lattice Boltzmann method is a useful tool for
  studying multiphase flows
• Lattice Boltzmann model can be derived form
  kinetic theory
• It is easy to incorporate microscopic physics in
  lattice Boltzmann models
• Lattice Boltzmann method is easy to program for
  parallel computing.

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Thank You!
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Acknowledgement

• Raoyang Zhang, ShiyiChen, Gary Doolen
• Xiaowen Shan