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≡)