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![Page 1: Simulation of Micro Flows by Taylor Series Expansion- and Least Square- based Lattice Boltzmann Method Simulation of Micro Flows by Taylor Series Expansion-](https://reader035.fdocuments.us/reader035/viewer/2022062408/56649efd5503460f94c10b02/html5/thumbnails/1.jpg)
Simulation of Micro Flows by Taylor Simulation of Micro Flows by Taylor Series Expansion- and Least Square-Series Expansion- and Least Square-
based Lattice Boltzmann Methodbased Lattice Boltzmann Method
C. Shu, X. D. Niu, Y. T. Chew and Y. Peng
Department of Mechanical EngineeringFaculty of Engineering
National University of Singapore
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OutlineOutlineIntroductionStandard Lattice Boltzmann
method (LBM)TLLBM for Non-uniform Mesh
and Complex geometriesTLLBM for Micro FlowsNumerical ExamplesConclusions
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1. Introduction1. Introduction Micro Flows in MEMS Classification of Flow Regimes according
to Knudsen number
Re2
Ma
LKn
0.001 0.1 3 (10) 100
Continuum Flow
(Ordinary Density Levels)Slip-Flow Regime
(Slightly Rarefied)
Transition Regime
(Moderately Rarefied)Free-Molecule Flow
(Highly rarefied)
22
1
n
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Studies of Micro Flows– Experimental– Numerical and Theoretical
Navier-Stokes Solver with Slip Condition– Only applicable to slip flow regime
Particle-based Methods– MD (Molecular Dynamics)– DSMC (Direct Simulation Monte Carlo)– Applicable to slip flow and transition regimes– Huge computational effort
Demanding More Efficient Solver for Micro Flow Simulation
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• Does LBM have Potential to be an Does LBM have Potential to be an Efficient Method?Efficient Method?
LBM is also a particle-based method Features of LBM and MD, DSMC
Micromechanics of real particles by MD and DSMC; micromechanics of fictitious particles by LBM
Local conservation laws are satisfied by all
Tracking particles by MD and DSMC (Lagrangian solver); Dynamics of particles at a physical position and a given velocity direction by LBM (Eulerian solver)
LBM More efficient than MD and DSMC
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2. Standard Lattice Boltzmann 2. Standard Lattice Boltzmann Method (LBM)Method (LBM)
Particle-based Method (streaming & collision)
Streaming process Collision process
D2Q9Model
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/)],,(),,([
),,(),,(
tyxftyxf
tyxfttteytexf
eq
yx
2
2
2242
6
1
2
1
cccf eq
2UUeUe
Nf
0
eU
Nf
0
2/2cP tc 2
8
)12(
Isothermal Flows:
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Thermal Flowso Difficulty (stability and fixed Pr)o A number of thermal models
availableMulti-speed model; Hybrid model; two distributions model (passive scalar; IEDDF)
o We prefer to use IEDDF for the thermal flows (He et al. 1998)
f (mass) is used for velocity field
g (energy) is used for temperature field
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Features of Standard LBM
o Particle-based methodo Density distribution function as
dependent variableo Explicit updating; Algebraic
operation; Easy implementationNo solution of differential equations and resultant algebraic equations is involved
o Natural for parallel computing
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Limitation----Difficult for complex geometry and non-uniform mesh
Mesh points
Positions from streaming
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3. TLLBM for Non-uniform 3. TLLBM for Non-uniform Mesh and Complex GeometryMesh and Complex Geometry
Current LBM Methods for Complex Problemso Interpolation-Supplemented LBM
(ISLBM)
He et al. (1996), JCPFeatures of ISLBM Large computational effort May not satisfy conservation Laws at mesh points Upwind interpolation is needed for stability
Mesh points
Positions from streaming
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o Differential LBMTaylor series expansion to 1st order
derivatives
Features:Wave-like equationSolved by FD and FV methodsArtificial viscosity is too large at high ReLose primary advantage of standard LBM
(solve PDE and resultant algebraic equations)
t
tyxftyxf
y
fe
x
fe
t
f eq
yx
),,(),,(
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• Development of TLLBMDevelopment of TLLBM
o Taylor series expansion
P-----Green (objective point) Drawback: valuation A----Red (neighboring point) of Derivatives
])(,)[(),(
),()(
2
1),()(
2
1
),(),(),(),(
332
2
22
2
22
AAAA
AA
AA
yxOyx
ttPfyx
y
ttPfy
x
ttPfx
y
ttPfy
x
ttPfxttPfttAf
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o Idea of Runge-Kutta Method (RKM)
Taylor series method:
Runge-Kutta method:
0 , ),,( 0 twhenuutufdt
du
thdt
udh
dt
udh
dt
duhuu nn ,
6
1
2
13
33
2
221
n n+1
n+1n
Need to evaluate high order derivatives
Apply Taylor series expansion atPoints to form an equation system
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o Taylor series expansion is applied at 6 neighbouring points to form an algebraic equation system
A matrix formulation obtained:
[S] is a 6x6 matrix and only depends the geometriccoordinates (calculated in advance in programming)
}{}]{[ gVS TyxfyfxfyfxffV }/,/,/,/,/,{}{ 22222
/),,(),,(),,( tyxftyxftyxfg iiiieq
iiiT
igg }{}{
(*)
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o Least Square OptimizationEquation system (*) may be ill-conditioned or singular (e.g. Point coincide)
Square sum of errors
M is the number of neighbouring points usedMinimize error:
)2 5(,...,2,1,0 DforMMi
M
i jjjii
M
ii VsgerrE
0
26
1,
0
2
6,...,2,1,0/ kVE k
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Least Square Method (continue)The final matrix form:
[A] is a 6(M+1) matrix
The final explicit algebraic form:
}]{[}{][][][}{1
gAgSSSV TT
1
1
1,100 ),,(
k
M
kk gattyxf
ka ,1 are the elements of the first row of the matrix [A] (pre-computed in program)
(Shu et al. 2002 PRE, Vol 65)
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Features of TLLBM
o Keep all advantages of standard LBM
o Mesh-free
o Applicable to any complex geometry
o Easy application to different lattice models
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Flow Chart of ComputationFlow Chart of Computation
Input
Calculating Geometric Parameterand physical parameters ( N=0 )
Calculating eqf
,,1 ka
N=N+1
1
1
1,100 ),,(
k
M
kk gattyxf
Mf
0
eU
Mf
0
Convergence ?
No
OUTPUT
YES
eU Re,,,
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Boundary TreatmentBoundary Treatment
Fluid Field
Stream from fluid field
Bounce back from the wall
ff
Non-slip condition is exactly satisfied
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Square Driven Cavity (Re=10,000, Non-uniform mesh 145x145)
Streamlines (right) and Vorticity contour (left)
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4. TLLBM for Micro Flows4. TLLBM for Micro Flows LBM for macro flows
is related to viscosity () through Chapman-Enskog expansion
Bounce back rule (non-slip condition) Features of micro flows
Knudsen number (Kn) dominantVelocity slip at wallTemperature jump at wall
LBM for micro flows relationship not valid. kn ?Bounce back rule not valid. New B. C. ?
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• Relationship of Relationship of -Kn-Kn
Re2
Ma
LKn
Pf
Pg Pr
)3/5
In Kinetic Theory:
;/c(Pr p );1/(c Rp
Relaxation time in velocity field:
Relaxation time in thermal field:
Knudsen number:
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• Relationship of Relationship of -Kn-Kn
s
ref
c
UMa
HU refRe
)3/1( sc
KnHf
Pr
KnHg
In LBM:
Mach number and Reynolds number:
We derived:
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• Boundary Conditions For Micro FlowsBoundary Conditions For Micro Flows Specular boundary condition
Diffuse-Scattering Boundary Condition (DSBC)
Kinetic Theory:
Maxwell Kernel:
Normalized condition:
'''
0)(
' )()()()()('
ξξξξnuξξnuξnuξ
dffw
ww
w
eqw
w
D
www
w
w
w f
RTRTRTRT uu
ξnuξuξnuξ
ξξ
)())((
)2(
1)
2
)(exp(
)(2
)()(
2
2
2
2'
Fluid
Wall
(incoming mass fluxEquals to outgoingMass flux)
)0n)u((1)( '
0n)u(
'
w
wd
n
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LBM VersionReplace integral by summation
0)(
')()()(nue
''
'
eenuenuew
ff fww
w
eqw
w
Nf f
Auu' nueee
))(()(
)92(36 QDBA NN
0)(
')()()(nue
''
'
eenuenuew
gg gww
Ww
eqw
w
Ng g
B
,
))(()(uu' nueee
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Theoretical analysis of DSBC--a 2D constant density flow along an infinite plate with a moving velocity and a temperature
2
2
0 2 n
uKn
n
uKnUuU wS
2
2
0 Pr2Pr n
TKn
n
TKnTTT wS
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0
0.1
0.2
0.3
0
0.25
0.5
0.75
1
-0.5
-0.25
0
0.25
0.5
Pressure-Driven, Isothermal Micro Channel Flow
YX
Kn=0.05Pr=2.0
U*
5. Numerical Examples5. Numerical Examples
High pressure Low pressure
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-0.002
-0.001
0
0.001
0.002
0
0.2
0.4
0.6
0.8
1-0.4
-0.2
0
0.2
0.4
Pressure-Driven, Isothermal Micro Channel Flow
Y
X
Kn=0.05Pr=2.0
V*
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0 0.25 0.5 0.75 10
0.025
0.05
ArkilicSpec.U. Ext.UCLA
P'
X
Kn=0.056, Pr=1.88
Non-linear Pressure Distribution along the Channel
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0 0.25 0.5 0.75 1
0
0.025
0.05
ArkilicSpec.U. Ext.UCLA
P'
X
Kn=0.155, Pr=2.05
Non-linear Pressure Distribution along the Channel
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Pressure-Driven Microchannel FlowsPressure-Driven Microchannel Flows
Kn=0.053 Kn=0.165
0
0.2
0.4
0.6
0.8
1
1.2
1 1.5 2 2.5 3 3.5 4 4.5 5
Exe. data, Argon [23]
No-Slip
Present LBM
Analytical [23]
Mas
s F
low
(Kg/
s)
Pressure ratio
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1 1.5 2 2.5 3
Exe. data, Helium [24]
No-Slip
Analytical [23]
Present LBM
Mas
s F
low
(Kg/
s)
Pressure ratio
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Comparison of slip magnitude at wall in the Comparison of slip magnitude at wall in the slip flow regime (shear-driven flow)slip flow regime (shear-driven flow)
Kn Analytic LBM
0.001 0.000998 0.0008628
0.00167 0.001667 0.0015939
0.002 0.001990 0.0019277
0.004 0.003968 0.0039492
0.005 0.004950 0.0049355
0.01 0.009800 0.0097963
0.02 0.019231 0.0192269
0.025 0.023800 0.0238064
0.03 0.028302 0.0280455
0.04 0.037037 0.0370351
0.05 0.045400 0.0454530
0.1 0.083300 0.0833333
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Thermal Couette FlowThermal Couette Flow
Velocity Profile Temperature Profile (Ec =10)
-1
-0.5
0
0.5
1
-0.5 -0.25 0 0.25 0.5
No-Slip
LBM(Kn=0.05)
LBM(Kn=0.1)
LBM(Kn=0.3)
Y
U
0
0.5
1
1.5
2
-0.5 -0.25 0 0.25 0.5
No-Slip
Kn=0.05
Kn=0.1Kn=0.3
Y
T
High T
Low T
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Slip length Temperature Jump
0
0.4
0.8
1.2
0.01 0.1 1
Present LBMMaxwel [19]MD [20]DSMC [20]
Ls
Kn0
0.5
1
1.5
2
2.5
0.01 0.1 1
Present LBM
Maxwell [19]
Lj
Kn
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Thermal Developing FlowThermal Developing Flow
(Constant Temperature on the walls Tw=10Tin)
0.5
1
1.5
0
5
10
15
20
-0.5
0
0.5
XY
Z
U
velocity
5
10
0
5
10
15
20
-0.5-0.250
0.250.5
YX
ZTemperature
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Thermal Developing FlowThermal Developing Flow
Thermal developing Channel flow---Wall coefficients along channel
15
25
35
45
55
0.1 1 10
Kn=0.015
Kn=0.03
Kn=0.046
Cf*Re
x/H6
9
12
15
18
0.1 1 10
Kn=0.015
Kn=0.03
Kn=0.046
x/H
Nu
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Thin Film Gas Slider Bearing Thin Film Gas Slider Bearing LubricationLubrication
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1
2nd-order model
stress-density ratio model
LBMP
X/L
Knudsen Number, Kn
Inverse Knudsen Number, D
Loa
dCarryingCap
acity
,W
10-1100101
10-1 100
10-3
10-2
10-1
100
Continuumstress-density ratio modelsecond-order modelLBM
Boltzmann
=10
x
y
x=0 x=L
Kn=1.24, =123.2
200/6 HPLU w
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Three-dimensional Thermal Three-dimensional Thermal Developing Flow in DuctsDeveloping Flow in Ducts
H
YX
Z
Flow
Constant Heat flux at the wall;Initial flow is assumed static with a constant temperature; Re=0.1, Pr=0.7
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Three-dimensional Thermal Three-dimensional Thermal Developing Flow in DuctsDeveloping Flow in Ducts
Y
X
Z
Y
X
Z
1.04621.02401.01101.00551.00371.00100.99980.99890.9979
Velocity distribution along duct
Thermal distribution along duct
5.006.0Kn
.
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Velocity profile at different cross Velocity profile at different cross sectionssections
0.5
1
1.5
u/U
0 -0.5-0.25
00.25
0.5
-0.25-0.125
00.125
0.25
X Y
Z
0.5
1
1.5
u/U
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
0.5
1
1.5
u/U
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
0.5
1
1.5
u/U
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Zx=0.0 x=0.3442
x=0.7826 x=1.230
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0.5
1
1.5
T/T
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
Temperature profile at different Temperature profile at different cross sectionscross sections
x=0.0 x=0.3442
x=1.230
0.5
1
1.5
T/T
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
0.5
1
1.5
T/T
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
0.5
1
1.5
T/T
0 -0.5-0.25
00.25
0.5
y/w
-0.25-0.125
00.125
0.25
z/w
X Y
Z
x=0.7826
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Three-dimensional Thermal Three-dimensional Thermal Developing Flow in Ducts: Developing Flow in Ducts:
ComparisonComparison
Slip model Present Slip model Present
1 0.21 0.25 2.85 3.16
0.75 0.25 0.27 2.81 3.06
0.5 0.32 0.30 2.71 2.92
1 0.28 0.31 2.69 2.87
0.75 0.33 0.33 2.62 2.79
0.5 0.41 0.36 2.48 2.66
1 0.34 0.37 2.53 2.60
0.75 0.39 0.38 2.44 2.54
0.5 0.48 0.41 2.26 2.44
NusU
04.0Kn
06.0Kn
08.0Kn
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6. Conclusions6. Conclusions TLLBM is an efficient method for
simulation of different micro flows New relationship of -Kn is valid Specular and diffuse-scattering
boundary conditions can capture velocity slip and temperature jump
Not accurate for cases with high pressure ratio (an incompressible model)