Evaluation of TwoPhase Lattice Boltzmann Models for Investigating Liquid BreakUp

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Evaluation of Two-Phase Lattice Boltzmann Models
for Investigating Liquid Break-Up
Michael McCracken School of Mechanical
Engineering Purdue University Maurice J. Zucrow
Laboratories
2
Motivation

• Multiphase Flows are found in many practical
  applications. These include
• Liquid Break-up (Sprays)
• Bubble Laden Flows (Boiling)
• Drop Collisions (Rain)
• Examples of Liquid Break-up Applications
• Paint Sprays
• Medicinal Applications inhalers
• Diesel Engines

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Background

• Experimental Approaches
• High number density of drops near the orifice.
• Data must be coupled with the governing
  equations.
• Computational Approaches
• Modeling multiple length and time scales.
• Difficulty representing surface tension and
  interfacial phenomena.
• Lattice Boltzmann Method
• Computational approach based upon particle
  movement and collisions.
• Additional information is available at the
  mesoscopic level to model molecular based
  properties.

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Overview of Presentation

• Introduction of the lattice Boltzmann method
  (LBM)
• Two-phase lattice Boltzmann model
• Test problems (both single and two phase)
• Preliminary computations of planar sprays
• Multiple Relaxation Time (MRT) model
• Two-phase test problems for MRT model
• Summary and Conclusions

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Front-Capturing Methods Marker and Cell (MAC)
and Volume of Fluid (VOF)

• MAC
• Marker particle are moved based upon N-S.
• The markers represent liquid.
• The interface is constructed based on the
  location of the markers.
• Physical properties for the fluid at each
  location is determined by the phase present.
• VOF
• Similar to MAC, but track fraction of fluid
  instead of marker particles.
• Interface still must be reconstructed in order
  to determine the surface tension force.

Schematic of MAC method
Schematic of VOF method and interface
reconstruction
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Front-Tracking Methods Boundary-fitted grid

• The interface between the two fluids is located
  between computational cells.
• Two sets of N-S equations are solved one for
  each fluid.
• Movement of the interface is determined by a
  force balance.
• Liquid break-up is difficult to resolve due to
  ad hoc criteria for interface rupture.

Schematic of boundary-fitted grid (Wadhwa 2003)
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Particle Based MethodsMolecular Dynamics (MD)

• In molecular dynamics, the movement of individual
  molecules are modeled using Newtons law
• F m a
• Two-phase flow can be simulated by using an
  interaction potential between molecules, e.g.
  Lennard-Jones.
• Current computational limitations, allow for the
  modeling of around 100,000 molecules. Therefore,
  this model is highly reliant on scaling from
  length scales of a few thousand molecular
  diameters to macroscopic length scales.

Illustration of Molecular Dynamics (Figure from
Shiladitya Mukherjee)
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The Lattice Boltzmann Method (LBM)A mesoscopic
approach

• Macroscopic Physics
• A result of collective behavior of many
  microscopic particles.
• Not sensitive to underlying microscopic
  dynamics.
• Mesoscopic Physics
• Remove unwanted details - use minimal set of
  velocities in phase space.
• Model just enough physics to obtain macroscopic
  behavior e.g. observe conservation laws.

N-S
LBM
MD
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The Lattice Boltzmann MethodFrom a lattice gas
perspective

• Lattice Gas
• Individual particles move (stream) along a
  lattice structure.
• Collisions are resolved using a set of
  collision rules.
• Large number of lattice nodes and results have
  statistical noise due to its Boolean nature.
• Lattice Boltzmann
• Instead of tracking individual particles, LBM
  tracks distribution function (the probability of
  finding a particle at a given location at a given
  time)
• This approach eliminates the statistical noise.

Example of lattice gas collision
LBM D2Q9 lattice structure indicating velocity
directions
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The Lattice Boltzmann MethodFrom the Boltzmann
equation

• The continuous Boltzmann equation
• The Maxwell-Boltzmann equilibrium distribution
  function
• The BGK model for collisions
• where ? is the relaxation time.

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The Lattice Boltzmann MethodFrom the Boltzmann
equation (continued)

• The lattice Boltzmann equation can be rigorously
  derived from the continuous Boltzmann equation
  (He and Luo 1997) and is given by
• where t is the dimensionless relaxation time (t
  ?/dt), a indicates the discrete velocity
  direction, and the equilibrium distribution
  function, f eq, is given by
• where c is the speed at which a distribution
  function moves, i.e. dx/dt.

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The Lattice Boltzmann MethodDetermining
macroscopic properties

• Continuous Boltzmann
• Density
• Momentum

Lattice Boltzmann Density Momentum
The N-S equations can be derived from the lattice
Boltzmann equation using a Chapman-Enskog
expansion (He and Luo 1997). From the expansion,
the macroscopic property of viscosity is found
from the following relationship for the D2Q9
model
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Two-Phase Lattice Boltzmann Models

• The continuous Boltzmann equation
• where F is an additional force. This term is
  used to model the intermolecular attraction based
  on mean field theory by van der Waals (Rowlinson
  1979)
• Force term used to account for increased
  probability of collisions due to high density
  (Chapman and Cowling 1953)
• where ? is the non-ideal part of the equation of
  state. In this work, the Carnahan-Starling
  equation of state is employed.

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Two-Phase Lattice Boltzmann ModelIndex Function
Model (He et al. 1999)

• Two distribution functions
• One to track an indexing parameter similar to VOF
  or level set methods.
• Another to determine the pressure from the zeroth
  moment.
• The equation of state is used to separate the two
  fluids by determining the gradient of ?.

p-v curve for Carnahan-Starling equation of state
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Channel FlowUsing extrapolation and non-equil.
bounce-back
Uniform spacing (10 x 100)
Uniform spacing (25 x 100)
Velocity profiles for channel flow using
different lattice spacings
Non-uniform spacing (16 x 100)
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Parallel Performance
Schematic for domain decomposition
Speed-up Curve
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Binary Diffusion
Diffusion of two fluids with the same molecular
weight (N2 N2)
Diffusion of two fluids with different molecular
weights (N2 He)
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Single-Phase Mixing Layer?1/?2 1
Schematic of mixing layer
Spreading rate of mixing layers compared to
predictions by Brown and Roshko (1974)
Density profile for gas 1
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Oscillating Liquid Cylinders
t 0
t .1
Amplitude of oscillation with time
t .2
t .3
t .4
t 1.55
Frequency of oscillation compared to analytical
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Capillary Waves
t 0
t 28.9
t 57.8
t 86.7
Amplitude of oscillation with time
t 115.6
t 404.5
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Capillary Waves (continued)
Amplitude of oscillation for computations with
two different resolutions
Error in decay rate and oscillation frequency for
different ratios of Reynolds to Capillary
number (Chandrasekhar 1961)
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Planar Sprays
t U / d (?l/?g)1/2 10.83
t U / d (?l/?g)1/2 75.78
t U / d (?l/?g)1/2 32.48
t U / d (?l/?g)1/2 97.43
t U / d (?l/?g)1/2 54.13
t U / d (?l/?g)1/2 119.08
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Planar SpraysVelocity vectors
Velocity vectors for planar spray computation
with ?l/?g 3 and U .1
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Planar SpraysEffects of density ratio and
injection velocity
After 18,000 lattice time steps...
With const. inject. velocity, U .1
With const. density ratio, ?l / ?g 10
?l / ?g 3
U .05
?l / ?g 5
U .075
?l / ?g 10
U .1
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Multi-Relaxation Time (MRT) Model(Lallemand and
Luo 2000)

• Distribution functions are mapped to variables
  in Moment space.
• Variables in the velocity space are based upon
  physical quantities.
• Some new variables are relaxed to their
  equilibrium values based upon different
  relaxation parameters.
• Density and momentum in the x and y directions
  are conserved.
• Tuning of multiple relaxation times provides
  increased numerical stability.

Example of relaxation of Moment variable
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Two-phase, Index function MRT Model
The index function LBM equation can be written in
matrix form
By multiplying by a transformation matrix, the
distribution functions are transformed to moment
space
The equilibrium values of the moment variables
are
The relaxation matrix for this model is diagonal
in the moment space, and has 6 adjustable
relaxation times
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Test Problems for MRT model
Oscillating Liquid Cylinder
Capillary Wave
Amplitude of oscillations vs. time (?
0.00675). 0.5 error in frequency 1.5 error
in decay rate compared to Chandrasekhar (1961)
Amplitude of oscillations vs. time (?
0.00675). 0.7 error in frequency compared to
Lamb (1932)
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Summary

• The lattice Boltzmann method was employed to
  study several single-phase and two-phase test
  problems.
• The results from single-phase test problems of
  Couette flow, channel flow, lid-driven cavity,
  and binary diffusion were within a few percentage
  points of their corresponding analytical
  solutions
• The results from the single-phase mixing layer
  computations agreed with the experiments.
• The oscillation frequency for both the
  oscillating liquid cylinders and the capillary
  waves matched their corresponding analytical
  solutions within 10.

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Summary (continued)

• The error in decay rate for the capillary waves
  showed an increasing trend with increasing ratio
  of Reynolds to Capillary number when using
  standard central difference schemes to
  calculate gradients. A hybrid scheme for
  calculating the gradients has shown some success
  in obtaining more accurate results.
• Preliminary work on studying planar liquid jets
  was presented. This work illustrated the
  inherently viscous nature of the LBM.
• A novel two-phase MRT model has been developed,
  implemented and evaluated on a couple of test
  problems. The results agree with the
  corresponding analytical solutions within a few
  percent.

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Future Work

• Finish evaluating the two-phase MRT model.
• Employ this model to study the effects of density
  and viscosity ratios on liquid break-up.