Sixth International ASME Conference onNanochannels, Microchannels and Minichannels
23-25 June 2008, Darmstadt, Germany
DSMC Solution of Supersonic ScaleDSMC Solution of Supersonic Scaleto Choked Subsonic Flow in Microto Choked Subsonic Flow in Microto Choked Subsonic Flow in Microto Choked Subsonic Flow in Micro
to Nano Channelsto Nano Channels
Ehsan RoohiEhsan Roohi
to Nano Channelsto Nano Channels
PhD StudentDepartment of Aerospace Engineering
Sharif University of Technology
MasoudMasoud DarbandiDarbandiProfessor
Department of Aerospace Engineering
VahidVahid MirjaliliMirjaliliGraduate Student
Department of Aerospace EngineeringSharif University of Technology Sharif University of Technology
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
OutlineOutline IntroductionIntroduction
OutlineOutline
Flow RegimesBoltzmann Equation
N i l S hN i l S hNumerical SchemeNumerical SchemeDSMC MethodImplicit Boundary Treatment
Results and DiscussionResults and DiscussionSupersonic Flow Mixed Supersonic/Subsonic RegimeMixed Supersonic/Subsonic RegimeChocked Subsonic Flow
ConclusionConclusion2
ConclusionConclusion
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Flow RegimesFlow RegimesFlow RegimesFlow Regimes
• Classification:K < 0 01 C ti
LKn /– Kn < 0.01 Continuum– Kn < 0. 1 Slip– Kn < 1 Transition
3
Kn 1 Transition– Kn > 10 Free Molecular
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Boltzmann EquationBoltzmann EquationBoltzmann EquationBoltzmann Equation**
42 )()()()( dcdcffffnnfFnfcnf
Local rate of change
1110
)()(.)(.)( dcdcffffnnfc
Fnfr
cnft r
Binary Collision termInflux of molecules due
to external forceof number of molecules
to external force
Influx of molecules due
to convection
• Boltzmann Equation: temporal-spatial changes of number of molecules of class c
• Assumptions– i) Dilute Gas: Binary Collision
ii) Molecular Chaos: colliding particles are uncorrelated4
– ii) Molecular Chaos: colliding particles are uncorrelated
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)Direct Simulation Monte Carlo (DSMC)
Direct simulation of physical behavior of Direct simulation of physical behavior of molecules Particles move in physical space; Particles move in physical space; Particles possess microscopic properties, Collisions handled statistically; Collisions handled statistically; Movement handled deterministically. C ll i f th {u’, v’, w’
x y z Cell size ~ mean free path ; Time step ~ collision time;
{x, y, zerot, evib
5
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
DSMC Method Developed by Bird (1970’s)
DSMC Method Developed by Bird (1970’s) First applications: external hypersonic flow 1990s: MEMS, supersonic internal flow 2000s: MEMS, subsonic internal flow 2000s: Extension to very low speed
flow Information Preservation (IP) methodflow, Information Preservation (IP) method 2002: Developing hybrid continuum-
molecular schemes Current work: New Physical Simulation
6
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Current Work: Physical StudyCurrent Work: Physical Study Supersonic flow with vacuum/specified back Supersonic flow with vacuum/specified back pressure pressure
Current Work: Physical StudyCurrent Work: Physical Studypressure pressure
Mixed flow regime Study of the choked subsonic flowStudy of the choked subsonic flow
Implicit I/O Boundary conditions: Wang et. al [2004]Role of Buffer zone
InletSymmetry
OutletGeometry with buffer zoneH/2
Wall
Geometry with buffer zone
7L
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Simulated CasesSimulated CasesSimulated CasesSimulated Cases Ch l A t R ti 5 Channel Aspect Ratio: 5 Buffer: Either at the inlet or Outlet Inlet Mach: 4.15 Grid: 120-60 (W. Buffer), 100-60 (W.O. Buffer)Grid: 120 60 (W. Buffer), 100 60 (W.O. Buffer) Kn: Slip (0.062)/Transition (0.43) Mi d S i S b i Fl Mixed Supersonic-Subsonic Flow Subsonic Choked Flow
8
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Convergence HistoryConvergence HistoryConvergence HistoryConvergence History250 4
Inlet Mass Fluxm
s)
200
3
3.5Outlet Mass FluxInlet PressureOutlet Pressure
Rat
e(K
g/m
Pou
t
100
150
2
2.5
ass
Flow
R
P/P
50
100
1.5
2
M
00.5
1
9Print Cycle0 1000 2000 3000 4000
-50 0
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Role of Knudsen NumberRole of Knudsen Number Kn=0.062: Oblique Shock,
Mach: 0 5 1 1 5 2 2 5 3 3 5 4 Kn=0.43: Normal Shocks, Mixed Regime
Mach: 0.5 1 1.5 2 2.5 3 3.5 4
4
X2E-05 4E-05 6E-05
3
3.5
Mach: 0.2 0.6 1 1.4 1.8 2.2 2.6 3
Mac
h
2
2.5
Kn=0.062
X2E-06 4E-06 6E-06
0 5
1
1.5 Kn=0.35
Kn=0.74
10X/L0 0.2 0.4 0.6 0.8
0.5
Balancing Effects of Shear stress/ Heat Transfer
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Temperature FieldTemperature FieldTemperature FieldTemperature FieldT : 5 0 1 5 0 2 5 0 3 5 0 4 5 0 5 5 0
2 E -0 5 4 E -0 5 6 E -0 5
T : 5 0 1 5 0 2 5 0 3 5 0 4 5 0 5 5 0
Kn=0.062, Heating
X2 E -0 5 4 E -0 5 6 E -0 5
T: 450 650 850 1050
Kn=0.43, Cooling
X1E-06 2E-06
, g
11
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Back Pressure EffectsBack Pressure EffectsBack Pressure EffectsBack Pressure Effects0.7
0.5
0.6
0.7
Current
M ach: 0.2 0.6 1 1.4 1.8 2.2 2.6 3
ress
ure
(Mpa
)
0.3
0.4
CurrentLe et. al.
X2E-06 4E-06
M=3.39, Vacuum
Pr
0.1
0.2 M ach : 0 .2 0 .6 1 1 .4 1 .8 2 .2 2 .6 3
X/L0.2 0.4 0.6 0.8 1
X2 E -0 6 4 E -0 6
12Validation M=3.39, Pb=0.5 Mpa
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Choked Flow: Role of Buffer ZoneChoked Flow: Role of Buffer Zone
Mach: 0 1 0 25 0 4 0 55 0 7 0 85 1 1 15Mach: 0.1 0.25 0.4 0.55 0.7 0.85 1 1.15
a) Mach contours, without buffer zone, Non-physical solution (Mout > 1)
X5E-07 1E-06 1.5E-06
Mach: 0.1 0.25 0.4 0.55 0.7 1
b) Mach contours, with bufferzone, Correct physical
X5E-07 1E-06 1.5E-06
zone, Correct physical simulation
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Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Role of Buffer ZoneRole of Buffer ZoneRole of Buffer ZoneRole of Buffer Zone0.2
KnGLL D it with Buffer
0.15
KnGLL-Density with BufferKnGLL-Density without Buffer Once choking occurs, exit
pressure does not drop more,
Applying a lower back pressure
n GLL
-Den
sity
0.1
Applying a lower back pressure at the outlet results in physically incorrect prediction, (Wavy local Knudsen)
Kn
0.05
Fix pressure at the end of buffer and let the outlet pressure found during Simulation (Smooth local
0
during Simulation (Smooth local Knudsen)
14X/L
0 0.2 0.4 0.6 0.8 10
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Velocity ProfilesVelocity ProfilesVelocity ProfilesVelocity ProfilesDSMCAnalytical (1st order)
d
0.2
0.4
X/L=0.3Kn=0.056 0.2
0.4
X/L=0.6Kn=0.0715 0.2
0.4 Analytical (2ndorder)Analytical (unified model)
X/L=0.9Kn=0.111
Y/H 0 Y/H 0 Y/H 0
-0.4
-0.2
-0.4
-0.2
-0.4
-0.2
U*0.4 0.6 0.8 1 1.2 1.4
U*0.4 0.6 0.8 1 1.2 1.4
U*0.4 0.6 0.8 1 1.2 1.4
Why DSMC predicts lower normalized velocity?
15
Why DSMC predicts lower normalized velocity?Each velocity is normalized with its respective average
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Computed DSMC mass flow rates and Computed DSMC mass flow rates and comparison with the analytical solutionscomparison with the analytical solutions
m0 (×105) (DSMC) 13.1
m0 (×105) Ref. [14] 1.42
m0 (×105) Ref. [16] 1.27
m0 (×105) Ref. [17] 1.33
DSMC gives much more mass flow rate
16
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Comparison of DSMC and analytical Comparison of DSMC and analytical p yp ypredictions for average velocitypredictions for average velocity
X/LX/L UUAnalytical Analytical (m/s)(m/s) UUDSMC DSMC (m/s)(m/s) UUDSMCDSMC /U/UAnalyticalAnalytical
0 30 3 69 769 7 98 4698 46 1 411 410.30.3 69.769.7 98.4698.46 1.411.41
0.60.6 90.990.9 125.4125.4 1.371.37
0.90.9 147.4147.4 205.7205.7 1.391.39
DSMC velocity (pressure gradient) is more than analytical’sDSMC velocity (pressure gradient) is more than analytical s,
Analytical solutions fail to predict correct properties of chokedflow due to high compressibility effects, not considered in derivation.
17
flow due to high compressibility effects, not considered in derivation.
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Average Mach along the Centerline: Average Mach along the Centerline:
0 7
e age ac a o g t e Ce te ee age ac a o g t e Ce te eComparison with Fanno TheoryComparison with Fanno Theory
0.6
0.7
DSMCFanno
h
0.5 Good AgreementGood Agreement
Mac
h
0.4
0.3
18
X/L0.2 0.4 0.6 0.8 1
0.2
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Conclusion
Knudsen Number greatly affects the flow behavior Knudsen Number greatly affects the flow behavior,
Imposing Back pressure leads to mixed regime,
Expansion cooling occurs while flow approaches vacuumed ambient,
Wrong solution for choked flow without buffer zone,
DSMC gives more mass flow rate, pressure gradient and DSMC gives more mass flow rate, pressure gradient and velocity for choked flow,
DSMC agrees with Fanno Theory.
19
g y
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Further Work Simulating Nozzle Flow IP scheme for long channels
(L/H=100) (L/H=100)
)/PO
ut
0.04
M h N b 0 5 1 5 2 5 3 5
(P-P
Line
ar)
0.02
IP
Mach Number: 0.5 1.5 2.5 3.5
0 0 2 0 4 0 6 0 8 10
IPAnalytical, 1storderAnalytical, 2ndorder
5E-06 1E-05
20X/L
0 0.2 0.4 0.6 0.8 1
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
StepFlowp
Mach: 0.05 0.2 0.35 0.5 0.65 0.8 0.95
Y/S 1
2Mach: 0.05 0.2 0.35 0.5 0.65 0.8 0.95
Kn=0.01
X/S
Y
-2 0 2 4 6
Mach: 0.02 0.12 0.22 0.32 0.42 0.52
Y/S 1
2
Kn=0.10
21X/S
-2 0 2 4 6
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
StepFlowp
Mach: 0.05 0.15 0.25 0.35 0.45
Kn=1.0
Y/S 1
2Mach: 0.05 0.15 0.25 0.35 0.45
X/S
Y
-2 0 2 4 6
2Mach: 0.05 0.15 0.25 0.35 0.45
Y/S 1Kn=10.
22X/S-2 0 2 4 6
Sixth International ASME Conference on
Nanochannels, Microchannels and Minichannels
Validation by Analytical/NS solutionsValidation by Analytical/NS solutions
1.51.5
Kn = 0.5Current studyDSMC
Kn = 0.3
Uav
g
11
Uav
g
1
U/U
Analytic Sol.Navier-Stokes
Kn = 0.2
U/
0.5Navier StokesDSMC (coarse)DSMC (fine)
0.50.5
23y*0 0.1 0.2 0.3 0.4 0.5 y*
0 0.5y*
0 0.5