a computationally efficient framework for modeling...
Post on 18-Jul-2020
4 Views
Preview:
TRANSCRIPT
A Computationally Efficient Framework for Modeling Microscale and Rarefied Gas Flows
Based on New Constitutive Relations
Jan. 5, 2010
R. S. MyongDept. of Mechanical and Aerospace Engineering
Gyeongsang National UniversitySouth Korea
myong@gnu.ac.kr; http://acml.gnu.ac.kr
Presented at 48th AIAA Aerospace Sciences Meeting, Jan. 4-7, 2010, Orlando, Florida, U.S.
Rarefied and micro/nanoscale gasesIntermediate Experimental Vehicle
Compression-dominated
High M, low Kn
Shear-dominated
Low M, high Kn
Micro and nanoscale cylinder
An overview of rarefied and micro/nanoscale gases
• Rarefied (hypersonic) gasesGas flow + hypersonic vehicle flying at high altitude
• Micro/nano devices:Gas (liquid) flow + MN solid devices
1) Molecular interaction between gas (liquid) particles and solid atoms
2) Gas (liquid) flows in thermal nonequilbrium regimes3) Electrokinetics, surface tension etc.
MN solid + MN solid devices => Interface heat transfer etc.• Micro/nano particles:
MN particles in gas => Aerosol etc.MN particles in liquid => Suspension etc.MN gas in liquid => Micro bubble etc.Production of MN particles
Modeling micro and nanomechanics of fluids and rarefied gases
Top-down: the classical linear (fluid mechanics) theories can account for virtually everything about materials (fluids).
Bottom-up: only a molecular-statistical theory of the structure of fluids can provide understanding of their true behavior.
( )
( )
(2)
3 22
1/
,
Linear uncoupled constitutive relations
Example. 124
t
outin
D pDt
E p
k T
H Wpm pLRT
ρρ
η
η
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ + ∇ ⋅ + =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ + ⋅ +⎣ ⎦ ⎣ ⎦
= − ∇ = − ∇⎡ ⎤⎣ ⎦
= −
uu I Π 0
I Π u Q
Π u Q
&
Navier Fourier
A critical observation on how to combine two approaches: an efficient way to include the molecular nature of gases is to develop full (nonlinear coupled) constitutive relations but to retain the conservation laws.
Modelling of nonequilibrium gas system (I)
[ ]2,);,( ffCtft
=⎟⎠⎞
⎜⎝⎛ ∇⋅+∂∂ vrv
( ) ),(,,,,, rQu tT LΠρ
Molecular (Probabilistic) Phase Space Boltzmann
Continuum
(Hydrodynamic)
Thermodynamic
Space
Conservation Laws
Moment Equation
( ) 0=Π+⋅∇+ Iu pDtDρ
);,( vrtf
TkB
1=β
Thermodynamics
(Reduction of
Information)
Navier-Stokes-Fourier
Not far from LTE
∫∫∫=
=
=
zyx dvdvdv
tfm
tmf
LL
);,(
);,(
vrvu
vr
ρ
ρ
Modelling of nonequilibrium gas system (II): The moment method
( ) 0tρ ρ∂+∇ ⋅ =
∂u
[ ]2( , ; ) , ( , ; ) , ( , ; ) ,vmf t m f t f t C f ft
ρ ρ ∂⎛ ⎞= = + ⋅∇ + ⋅∇ =⎜ ⎟∂⎝ ⎠r v u v r v v a r v
[ ]
[ ]
( ) ( )
2
2
the statistical definition ( , ; ) and with the Boltzmann equation
( , ; ) ,
, 0
0
Differentiating mf t with timethen combining
fmf t m mC f f m ft t t
m f mC f ft
m f mf m ft t
t
ρ
ρ
ρ
ρ ρ
ρ
≡
∂ ∂ ∂= = = − ⋅∇
∂ ∂ ∂∂
+ ⋅∇ = =∂∂ ∂
+ ∇ ⋅ − ∇ ⋅ = + ∇ ⋅ =∂ ∂∂
+∇∂
r v
r v v
v
v v v
0mf⋅ =v
The moment method (I)
Λcollision) (Boltzmann termndissipatio
Zterm kinematic
variable
order-high
variableconserved-non
+
=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅∇+⎟⎟
⎠
⎞⎜⎜⎝
⎛DtDρ
[ ] ( ) [ ]( ) ( ) ( )
( ) ( ) ( )
, ,1, , / , ,//
TT Tt
Q Q Q
p pEDDt
ρ ρ ρ ρ ρρ ρ
ρ
Π Π Π
⎡ ⎤⎡ ⎤ ⎡ ⎤+ + ⋅ +⎡ ⎤ ⋅⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥+∇ ⋅ = +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥+⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
u I Π I Π u Qu 0 a a uΠ Ψ Z ΛQ Ψ Z Λ
Mp ⋅Π Kn~/Main parameter (not Kn alone)
( )( ) ( ) ( )
v [ ]k
k k kh fD Dh f f h h C f
Dt Dtρ
ρ
⎛ ⎞ ⎛ ⎞⎜ ⎟ + ∇ ⋅ = + ⋅∇ + ⋅∇ +⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠c c a
The moment method (II): Closureproblem
The mathematician plays a game in which he himself invents the rules, while the physicist plays a game in which the rules are provided by nature. [P. Dirac, 1939]
Physically motivated closure
[ ]
( )
( )
(2)( )
( ) 2
/ 0
/ 0
where 12
Q
Q
DDt
m f
mc f
ρρ
ρ
Π
Π
⎡ ⎤⎡ ⎤ ⎡ ⎤+∇ ⋅ =⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤⎢ ⎥⎡ ⎤
≡ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
Π ΨQ Ψ
cc cΨΨ cc
Nonlinear coupled constitutive relations (NCCR),
but algebraic unlike differential in other theories
Shear driving force
Stresses
Anti-symmetry
Symmetry
NSF
NCCR
A computational framework based on nonlinear coupled constitutive relations
( )
NSF NSF NSF NSF
1/
and( , , , ), ( , , , )
nonlinear coupled constitutive algebraic relations
t
Q
D pDt
E p
F p T F p T
ρρ
Π
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ + ∇ ⋅ + =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ + ⋅ +⎣ ⎦ ⎣ ⎦
= =
uu I Π 0
I Π u Q
Π Π Q Q Π Q
Compression
(expansion)
Shear
Computationally efficient at the same level of NS CFD solvers
−6 −4 −2 0 2 4 6−6
−4
−2
0
2
4
6
Navier−Stokes
NCCR (Monatomic)
NCCR (Diatomic)
Karlin EH (Monatomic)
Nonlinear coupled constitutive relations in shock wave (stresses vs strain rate/p)
velocity gradientdivided by pressure
du pdx
η−
xx pΠ
Non-Navier (viscoelastic) behavior!
Validation in compression-dominated flow
M=23.47 at altitude 105 km
(5 species)
(J. W. Ahn et. al, JCP 2009)
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mach number
Inve
rse
dens
ity th
ickn
ess
NS ( fb = 0.0 )
NS ( fb = 0.8 )
NCCR ( fb = 0.8 )
Shock structure
(Monatomic & diatomic)
(R. S. Myong, JCP 2004)
Nonlinear coupled constitutive relations in shear flow (stresses vs strain rate/p)
velocity (shear) gradientdivided by pressure
du pdy
η−
pxyxx ,Π
Shear-thinning non-Navier (viscoelastic) behavior! (cross fluid in rheology)
Coupled since normal stress is generated by shear velocity gradient
0 1 2 3 4 5 6−1.5
−1
−0.5
0
0.5
1
1.5
Shear stress (Navier−Stokes)
Normal stress (Navier−Stokes)
Shear stress (monatomic NCCR)
Normal stress (monatomic NCCR)
Cf. negative axis
−2
−1
0
1
2
−5
0
5−3
−2
−1
0
1
2
3
Qy0
Πxy0
Qy
Nonlinear coupled constitutive relations in force-driven shear gas flow (heat flux vs temp. gradient)
ˆ ˆ /xy xy pΠ ≡ Π
( )Non-Fourier
behavior!( )0 0
0
2
3 where is force.3 2y y xy
xy
Q Q a a= + Π+ Π
) ) )))
( ), , / /(2 Pr)x y x y pQ Q p C T≡)
Fourier law0y yQ Q=
) )
Force (gravity)-driven Poiseuille 1-d gas flow (I)
• Identified as one of three surprising hydrodynamic results discovered by DSMC (1994)
• Global failure of the NSF theory in predicting non-uniform pressure profile and the central minimum in the temperature profile Hydrodynamic theories in trouble
y
xp
TUniform
force a
Qy
0 ,xy
yy
xy y
ad pdy
auu Q
ρ
ρ
⎡ ⎤Π ⎡ ⎤⎢ ⎥ ⎢ ⎥+Π =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Π + ⎣ ⎦⎣ ⎦
0
0
0 0
0
( Π ) / /02 /(3 ) /0
0 / /(Pr ) /0 /( Π ) /
yy xy xy
xy xy yy
xy p y p y xy xx p x
p yyy p y xy
p p
p
C Q k C Q k a pC Q k
pC Q kp C Q k a
η ηη η
+ Π Π⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − Π Π Π⎢ ⎥ ⎢ ⎥⎢ ⎥ = − +⎢ ⎥ ⎢ ⎥⎢ ⎥ Π + Π + Π⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + Π⎣ ⎦ ⎣ ⎦⎣ ⎦
Qx
Force-driven Poiseuille flow (II): An analytical solution for constant force (Kn=0.1)
Temperature profile across channel
(○-DSMC, ●-NCCR, NSF)Normal and tangential heat flux profile across
channel
Not only confirming the temperature minimum due to non-Fourier relation,
but also showing a heat transfer from the cold region to the hot region near the centerline
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.94
0.95
0.96
0.97
0.98
0.99
1
1.01
Monatomic Diatomic
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Normal heat flux Qy
Tangential heat flux Qx
Force-driven Poiseuille flow (III): An analytical solution for constant force (Kn=0.1)
Pressure profile across channel Stress profile across channel
Not only confirming the non-uniform pressure and the non-zero normal stress due to non-Navier relations,
but also showing its reversal (from concave to convex) in case of diatomic gases
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Monatomic
Diatomic
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Shear stress
Normal stress
Summary• New constitutive relations (NCCR):
- multi-axial, viscoelastic flow in stress/pressure domain (similar to rheology) and in heat flux - mathematically coupled nonlinear (algebraic)- computationally efficient
• Solving challenging problems that render the classical hydrodynamic theories (NSF) a global failure.
• Describing how coupled and nonlinear relationship affects the prediction of gas flow and heat transfer in rarefied and micro/nano-system
Acknowledgements• Supported by Korean Research Foundation
( ), , / /(2 Pr)x y x y pQ Q p C T≡)
top related