Transcript Production of Monodisperse Spherical Colloids

Lattice Boltzmann
Karin Erbertseder
Ferienakademie 2007
Outline
Introduction
 Origin of the Lattice Boltzmann Method
Lattice Gas Automata Method
Boltzmann Equation
 Explanation of the Lattice Boltzmann
Method
 Comparison between Lattice Boltzmann
Method and Navier-Stokes-Equations
 Applications

Introduction
Computational Fluid Dynamics
(CFD):
 solution of transport equations
 simulation of mass, momentum
and energy transport processes
Applications:
automotive, ship and aerospace
industry, material science, …
Advantage:
prediction of flow, heat and mass
transport
fundamental physical
understanding
optimization of machines,
processes, …
source: www.ansys.com
Introduction
Experiment vs. Simulation
Experiment




measurement often
difficult or impossible
expensive and time
consuming
parameter variations
extremely expensive
measurement of only
a few quantities at
predefined locations
Numerical Simulation




compliance of
similarity rules is no
problem
less expensive and
faster
easy parameter
variation
provides detailed
information on the
entire flow field
Introduction
General Procedure
Solution
of the
Problem
Flow
Problem
mathematical model,
measured data
Visualization
Analysis
Interpretation
Conservation
Equations
discretization, grid
generation
Algebraic System
of
Equations
algorithms
software,
computer
Numerical
Solution
Introduction
Macroscopic Methods
e.g. Navier-Stokes fluid simulation (FDM, FVM)
Mesoscopic Methods
e.g. Lattice Boltzmann method
Microscopic Methods
e.g. molecular dynamics
Lattice Gas Automata (LGA)
Cellular Automata (CA):
 idealized system where space and time
are discrete
 regular lattice of cells characterized by a
set of boolean state variables
1 or 0
particle at a lattice node
Lattice Gas Automata (LGA):
 special class of CA
 description of the dynamics of point
particles moving and colliding in a discrete
space-time universe
Lattice Gas Automata (LGA)
streaming step
flow simulation by moving
representative particles
one node per time step
collision step
source: www.cmmfa.mmu.ac.uk
Lattice Gas Automata (LGA)



advantages
stability
easy introduction of
boundary conditions
high performance
computing due to the
intrinsic parallel
structure



disadvantages
statistical noise
lack of Galilean
invariance
velocity dependent
pressure
motivation for the transition from LGA to LBM:
removal of the statistical noise by replacing the
Boolean particle number in a lattice direction with its
ensemble average
density distribution function
[all disadvantages are improved or vanish]
Boltzmann Equation (BE)
definition:
description of the evolution of the single particle
distribution f in the phase space by a partial
differential equation (PDE)
particle distribution function f (x,ξ,t):
probability for particles to be located within a phase
space control element dxdξ
about x and ξ at time t where x and ξ are the spatial
position vector and the particle velocity vector
macroscopic quantities, like density or
momentum, by evaluation the first moments of the
distribution function
Boltzmann Equation (BE)
f
 
t
time
variation
f
x
 G 
spatial
variation
f

 Q f , f
effect of a
force acting on
the particle
velocity vector
of a molecule position of
the
molecule
force per unit
mass acting on
the particle

collision term
interaction
between the
molecules
f = f (x, ξ, t)
distribution
function
The collision term is quadratic in f and has a complex integrodifferential
expression
simplification of the collision term with the
Bhatnagar-Gross-Krook (BGK) model
Q f , f


f
e
 f

Lattice Boltzmann Method (LBM)
assumptions: - neglect of external forces
- BGK model (SRT = single-relaxation-time
approximation)
- velocity discretization
using a finite set of
velocity vectors ei
- movement of the particles only along the lattice
vectors
- modeling of the fluid by many cells of the same type
- update of all cells each time step
- storage of the number of particles that move along
each of the lattice vectors
particle
function f
distribution
f i
t
 ei 
f i
x
 
1


 fi  fi
eq

velocity
discrete
Boltzmann
equation
Lattice Boltzmann Method (LBM)
common lattice nomination:
DXQY
number of
dimensions
number of
distinct lattice
velocities
model for two dimensions:
f4
4
3 f3
2
f2
D2Q9
e4
e3
- 9 discrete velocity directions
e2
e5
f5 5
1 f1
e1
e6
f6
6
e7
e8
7 f7
source: J.Götz 2006
- most common model in 2D
- eight distribution functions with
the particles moving to the
neighboring cells
- one distribution function
according to the resting particle
8
f8
Lattice Boltzmann Method (LBM)
models for three dimensions:
D3Q15
D3Q19
D3Q27
small range of
good compromise
highest
stability
between the two
computational
models
effort
19 distribution functions
12 velocities combining
two coordinate directions
one stationary velocity in
the center for the
particles at rest
resting particles don`t
move in the following
time step, but: changing
amount of resting
particles due to collisions
6 velocity directions
along the Cartesian axes
source: J.Götz 2006
Lattice Boltzmann Method (LBM)
next step: calculation of the density and momentum fluxes in the
discrete velocity space
starting point: velocity discrete BE
f i
t
 ei 
f i
x
 
1


 fi  fi
eq

equilibrium distribution function for D2Q9 model:
fi
eq
3
9
3


2
   w i  1  2  e i  u 
 (ei  u )  2  u  u 
4
2c
2c
 c

weighting factor
wi
=
4/9
i=0
1/9
i = 1, 3, 5, 7
1/36 i = 2, 4, 6, 8
discrete particle
velocity vector
c
x
t
lattice speed
with the lattice
cell size
x
and the lattice
time step t
Lattice Boltzmann Method (LBM)
calculation of the density and the momentum:

 
density

fd  
momentum
 u 


N

fi 
i0


N

eq
i0
N
fd  

fi
N
ei  f i 
i0

ei f i
eq
i0
Discretization:
discretization in time and space leads to the lattice BGK equation
f i x  ei  t, t  t   f i x, t   
point in the discretized
physical space
1

 
 f x, t  
i

t
fi
eq
 x , t 
dimensionless relaxation
time
Lattice Boltzmann Method (LBM)
lattice BGK equation is solved in two steps:
out
collision step:
fi
streaming step:
fi
in
x, t  
fi
in
x, t  
1

f
in
i
x  ei  t , t  t  
x, t  
fi
out
fi
eq
 x , t 
x, t 
values after collision and propagation, values entering
the neighboring cell = data for the next time step
distribution values
after collision
collision step:
• interpretation as many particle collisions
•calculation of the equilibrium distribution function for each cell and at
each time step from the local density ρ and the local macroscopic flow
velocity u using the equations of the slide before
Lattice Boltzmann Method (LBM)
streaming step:
 streaming of the particles to their neighboring cells according to
their velocity directions
 lattice vector 0
no change of its particle distribution function
in the streaming step
particle distribution
before stream step
particle distribution
after stream step
source: J.Götz 2006
LBM Parametrization
standard parameters describing a given fluid flow problem:
 size of a LBM cell ∆x [m]
 fluid density ρ [kg/m3]
 fluid viscosity ν [m2/s]
 fluid velocity u [m/s]
 strength of the external force g [m/s2]
lattice time step ∆t*, lattice density ρ*, lattice cell size ∆x*
constant during simulation
x

t

x
*


1
* 
1
 t* 
1
x

t
no multiplications with real world values of the time
step, the density, the lattice size are necessary
LBM Parametrization
calculation of the dimensionless lattice values:
lattice viscosity:
lattice velocity:
lattice gravity:
*  
u*  u
g*  g 
t
x
2
t
x
t
2
lattice viscosity,
lattice velocity,
lattice gravity are
dimensionless
lattice
velocity of 0.3
means that the fluid
moves 0.3 lattice
cells per time step
x
relation of all lattice values to the physical ones:
calculation of the physical time step
restricted
time step depending on the maximal lattice velocity
LBM Parametrization
calculation of the lattice viscosity ν*:
1 
2   1

 *  c  
 
2
6

2
s
relaxation time
speed of sound = 1/√3
Calculation of the relaxation time:
fluid velocity v is given
calculation of the relaxation
time needed for a simulation with the formula above
 
6  *  1
2
due to stability reasons:
0.51    2.5
0.67   *  3.3  10
3
LBM Boundary Treatment

no-slip:
no movement of the fluid close to the boundary
each
cell next to a boundary has the same amount of particles
moving into the boundary as moving into the opposite
direction
zero velocity (along the wall and in wall
direction)
reflection of all distribution functions at the wall in the
opposite direction
source:N.Thürey 2005
LBM Boundary Treatment

free slip:
reflection of the velocities normal to the boundary
boundaries with no friction (zero velocity only in wall direction)

inflow:
given velocities
calculation of the distribution function
based on the equilibrium function (only on special type)

outflow
several different types
source:N.Thürey 2005
LBM Boundary Treatment

periodic
particles that leave the domain through the periodic wall reenter
the domain at the corresponding periodic wall
copying the
PDFs leaving the domain to the corresponding cells during the
streaming step
source: C.Feichtinger 2006
Navier-Stokes Equations (NSE)
description of the macroscopic behavior of an isothermal
fluid:
 u i

conservation of mass:

 0

t

xi
density
incompressible fluid (ρ = constant):
velocity in i-direction
(i = 1,2,3 for x,y,z)
u i
x i
 0
viscous stress tensor
u j
 u j
 ui
momentum equation   
x i
 t
advection
 ij

P

 x  x    g
j
i

j
pressure momentum
forces acting
due to molecule
upon the fluid
movement
Comparison between LBM and NSE
Navier-Stokes Equations




Lattice Boltzmann Method
second order partial
differential equations
non-linearity
quadratic velocity terms

need to solve the Poisson
equation for pressure
calculation
global solution for all
lattice cells
grid
generation needs longer
than simulation





set of first order partial
differential equations
linear
non linear
convective term becomes a
simple advection
pressure through an
equation of state
regular square grids
kinetic-based
easy
application to micro-scale
fluid flow problems
complicate simulation of
stationary flow problems
Applications
Java-Simulation
Applications
source: N.Thürey
Applications
metal foam simulation:
source:N.Thürey 2005
 D3Q19 model
 free-surface model
filled with fluid
interface: contains both liquid and gas
gas: not considered in fluid simulation
 computation of the fill level of a cell by dividing by the density of this
cell (0 = empty cell; ρ = filled cell)
 transformation of fluid and gas cells into interface cells and vice versa
Applications
source: N.Thürey
Thanks For Your Attention
any
questions ?