Physics of polyatomic gases in non- equilibrium based on ...acml.gnu.ac.kr/download/Conference/RGD31-Myong-180727.pdf · 2nd-order Boltzmann-type kinetic theory: Boltzmann-based gas

Physics of polyatomic gases in non-

equilibrium based on the second-order

Boltzmann-type kinetic theory

July 27th Fri, 2018 (8:30~9:10PM)

Rho Shin Myong

Department of Aerospace and Software Engineering

Research Center for Aircraft Core Technology

Gyeongsang National University

South Korea

Presented at 31st International Symposium on Rarefied Gas Dynamics

University of Strathclyde, Glasgow, UK

Talk 1/27 31st International Symposium on Rarefied Gas Dynamics R. S. Myong, Gyeongsang National University, South Korea July 23 – 27, 2018 – University of Strathclyde, Glasgow, UK

Outline

Part I

2nd-order Boltzmann-type kinetic theory: Boltzmann-based

gas dynamics (BGD):

Part II

2nd-order constitutive laws for polyatomic gases

Physics of polyatomic gases in non-equilibrium via

- Topological representation of constitutive laws

- Multi-dimensional flow fields obtained by CFD based on

discontinuous Galerkin method


2nd-order Boltzmann-type kinetic theory

Basic information can be found in the Youtube of an Indian GIAN (Global Initiative

Academic Networks) Lecture (2017; IIT Kanpur; 15 Lectures)

Rarefied & Microscale Gases and Viscoelastic Fluids: A Unified Framework

https://www.youtube.com/ and search “Rarefied & Microscale Gases”

PoF 1999, JCP 2001, JCP 2004: Eu’s generalized

hydrodynamics

PoF 2014, PoF 2016: Balanced closure & validation

via MD

PoF 2018: Polyatomic gases (shock-vortex interaction)

Cf. NCCR: Nonlinear Coupled Constitutive Relation

Conceptual revision

New closure theory

Physical insight

More validation

Discontinuous

Galerkin

Other independent NCCR works: Multi-species extension by Ahn & Kim (SNU, Korea, JCP09)

Implicit-FVM implementation by Jiang & Zhao (Zhejiang Univ., China, 2017)

https://www.youtube.com/


Closure-last moment method

Molecular Continuum Molecular Continuum

Pure (or Semi-) Simulation PDE-based Approach

MD DSMC Gas-

Kinetic

Scheme

LBM Liouville

Equation

Boltzmann

and

Simplified

Boltzmann

Method

of

Moments

Chapman-

Enskog

and

Burnett

NSF

Microscopic Mesoscopic Macroscopic Microscopic Mesoscopic Macroscopic

( , , ) x y zm f t dv dv dv u v r v

Breakdown of moment method: 1) when the statistical average is meaningless due to too

few particles; 2) when thermodynamics is not definable.

Closure-first approach: Grad’s 13 moment method (1949) based on polynomial expansion

Levermore method (1996) based on Gaussian (exponential) expansion

Regularized-13 moment method (2003)

Closure-last approach: Eu’s generalized hydrodynamics (1980) based on

Eu’s closure & equation of transfer

NCCR (2014) based on balanced closure & equation of transfer


Boltzmann transport equation (BTE)

A first-order partial differential equation with an integral collision term

2,

Collision (o

1( , , )

r Interaction)

Dissipation

Movement

Kinematic

f tt Kn

C f f

v r v

2 2 2 2

* *

2 ( )

(scattered into) - (scattered out)

,

Gain ss =Lo

C f f f ff d

f f

t

f

t

v v v

Maxwell’s equation of transfer for molecular expression

( ) ( ) ( ) ( ) ( ) ( )2[ , ]n n n n n nd

h f h f h f f h f h h C f ft dt

u c c

( )nh


Complexity out of simplicity: conservation laws

2

2

( , , )

the statistical definition ( , , ) and

with BTE ( , , are independent and

,

)

Here

[ , ]

f tt

Differentiating m f t with time

then combining t

fm f

C f f

C f fm m f mt t

m f

r v

u v r v

r v v u c

v v v v v

v

v v

(2)

2

pressure viscous shear stAfter the decomposition of the stress into and

and using the collisional invariance of the momentum

where Tr( ) / 3 ,

[ ,

ress

[

,

] ,

m f m f

m f p p m f

m

m f

C f f

vv uu cc

P cc cI Π cc

v

Π c

0,

we ha

]

0

ve

( ).p

t

I Π

uuu Exact consequence of the original BTE!

( , , )

where x y z

m f

dv dv dv

t

u v r v

(2) : Traceless symmetric

part of ten

[

]

sor

A

A

Π


Further complexity: constitutive equations

For arbitrary molecular expressions of general moment

(viscous shear stress and heat flux Q)

(2)

22

(2) (2

2

)(2)

22

/ Tr( ) / 3 0

[ ,

:/ 2 ( )/ 2

]

[ ,

/

]

22

/ 2

p

pp p

C f f

m f m f m fD

m fDt mc f C T pmc f

mp

DpC TC T mc mC C fT

tf

D

Π cc ccc ccc I

ccc ucc I ΠQ c

Π u ccu

uΠ Q u Π c

Π

No approximation so far: exact consequence of the original BTE via

Maxwell’s equation of transfer


Balanced closure: constitutive equations

(2)

22

(2) (2

2

)(2)

22

/ Tr( ) / 3 0

[ ,

:/ 2 ( )/ 2

]

[ ,

/

]

22

/ 2

p

pp p

C f f

m f m f m fD

m fDt mc f C T pmc f

mp

DpC TC T mc mC C fT

tf

D

Π cc ccc ccc I

ccc ucc I ΠQ c

Π u ccu

uΠ Q u Π c

2nd-order closure

2nd-order closure

Conceptual inconsistency of Eu’s closure (1992)

Tr( ) / 3 0 Vanishing heat flux

0 Cannot be zero in general

m f m f

m f

ccc ccc I

ccc

New balanced closure with closure-last approach (2014)

=2nd-order for kinematic LH 2nd-order for collsion RH


Conservation laws (exact

consequence of BTE) 1 / 0

t

Dp

DtE p

u

u I Π 0

u Π u Q

in conjunction with the 2nd-order constitutive relations (CR)

(2

1/2

(2

2 1

2 1

12 1

1

1/4

1

) )( )

( )

si

2( / )

,

( / ),

,nh :

2

( )

NS

p

p

nd

nd

n

NS N

S

S

N

p

d

pp

pCC

D

Dt

D

Dt

T T

k

DC T

Dt

q

qp Tk

qp

Π Π Q Q

u Π

Q

Π u

uΠ Q u Π

Π

Q

/

Non-hyperbolic

Implicit

Hyperbolic (inviscid)

Navier 1st law

Fourier 1st law

Conservation laws + 2nd-order constitutive relations

2nd-order coupling

2nd-order coupling


2nd-order theory in elementary flows

Compression

(expansion) Shear Decomposition

Critical role of term in compression case

Negligible role of term in expansion & velocity shear

(simple shear thinning – decreasing viscosity)

2 1( )ndq

2 1( )ndq

Navier-Fourier laws inclusive

like onion! 1st-order theory

2nd-order theory


A model for 1-D compression & expansion

1

1

(2)

1/21/4

1

(2) sinh :( / )2 2 ,

NS SNS N

D T

Dt k

T

p

pp

Π Π Q Qu

/uΠ Π

Π

(2) 1

1/24

1

1

(2)1/

(( / )

(sinh

( )) 2 ) 2 ,:

NS

NS

SNS N

D

Dtp p

p

T

uu

ΠΠ ΠΠΠ

(2)

2NS NS Π u

No heat flux

One-dimension

ˆˆ ˆΠ Π ˆ ˆ ˆ( ) ( ) sinΠ̂

, where ˆ

hˆ /ˆΠΠNS NS q qD

Dt

Πˆ ,4

ˆ , Π/ 3

NS NSNS

t ut

p p x

Or steady-state 8(10 sec)stress flowt t

ˆˆ ˆΠΠ + ˆΠ Π ( )ˆNSNS q

2nd-order 1st-order 1st-order x 2nd-order Kinematic motion Collision


2nd-order constitutive laws in compression/expansion

Grad non-classical

model (1952)

Myong non-classical

model (1999, 2014)

Navier classical model (1822)

Velocity gradient /pressure

Singularity

Viscous stress /pressure

Π /NS p

/ p

by assuming

Maxwellian mol

1

ecule

q

sinh pq

p

ˆ ˆ+Πˆ ˆΠΠ )Π ˆ(ΠNNS S q

1-D compression and expansion

ˆ ˆ+ΠΠ 1ˆ Π̂ ΠNSNS

Π+ ˆ ˆΠ0 1NS

x

q(x)

q(0)=1


1-D shock result: inverse shock thickness

Shock density thickness of argon gas Shock temperature-density distance

(Karchani, PhD Thesis, GNU, 2017)

New 2nd-order & 2nd-order: Best

Navier 1st-order & 1st-order: Reference

Quasi-linear 1st-order & 2nd-order: Worse than Navier

Grad 2nd-order & 1st-order: Singularity

Importance of

balanced closure

Experimental

data

ˆ ˆΠ Π Π̂)0 + (NS q

DSMC

Navier-Fourier

Quasi-linear

Quasi-linear

Navier-Fourier


Visco-elastic

2nd-order constitutive laws in velocity shear

(2)

2 1

2 1

(2)( / ),

( / )

(

(

2 )

)

2NS

p

p dp

d

NS

n

n

DC T

D

q

q

D

Dt

D

Dt

pp

p

t

CC p T

k

Π u

uΠ Q

Πu Π

Q uQ

Π

1-D velocity shear

2 1

01

2 /( )

31

NS

NS

NS

yy

xy xyNS N

xy xy

yy xyS Sn

Nd

p

pq

2 3ˆ ˆ ˆ12

xy yy yy

Kinematic stress

constraint due to 2nd-

order coupling!

No such thing in 1st-order

Navier law!

No heat flux

MD: Molecular

Dynamics(2014)

NCCR: Nonlinear Coupled

Constitutive Relation

NSF: Navier-Stokes-Fourier

Newtonian

0D

Dt t

u


Multi-dimensional CFD based on DG

Acknowledgements

Former PhDs & Postdocs:

A. Karchani (ANSYS, US), S. Singh (NTU, Singapore), H. Xiao (NWPU, China)

PhD Students:

P. Raj, O. Ejtehadi, A. Rahimi, T. Chourushi

Supported by

National Research Foundation in Korea

Number of iterations

Log(

||

hn+

1-

hn||

)

0 1000 2000 3000 400010

-7

10-6

10-5

10-4

10-3

10-2

10-1

NSF Solver NCCR solver

Super-parallel performance (via the

rate of cost reduction) (C&F, 2017)

Log (number of processors)

Log

(tp/t s)

100

101

10210

-4

10-3

10-2

10-1

100

NSF- Kn=0.5NCCR - Kn=0.5

Convergence property

NCCR: Nonlinear

Coupled

Constitutive

Relation

Linear

reduction

Exponential

reduction

ˆ ˆ+Πˆ ˆΠΠ )Π ˆ(ΠNNS S q


Two-dimensional shock structure around cylinder

(Karchani, PhD Thesis, GNU, 2017) NCCR: Nonlinear Coupled Constitutive Relation


Monatomic, diatomic, and (linear) polyatomic gases

Argon Nitrogen Carbon dioxide

Translational degree of freedom

x y

z

Rotational degree of freedom

n

t2

Monatomic

Diatomic

Polyatomic

Once the Stokes assumption is abandoned, an additional constitutive law of the

excess normal stress ∆ related to the bulk viscosity will appear

∆= 𝜇𝑏 ∆ ∙ 𝒖 .

The inner structure of strong shock waves (Emanuel and Argrow 1994)

Dilatational waves in transition in hypersonic boundary layers (Zhu et al. 2016)


Monatomic, diatomic, and (linear) polyatomic gases

Argon Nitrogen Carbon dioxide

Translational degree of freedom

x y

z

Rotational degree of freedom

n

t2

Monatomic

Diatomic

Polyatomic

Once the Stokes assumption is abandoned, an additional constitutive law of the

excess normal stress ∆ related to the bulk viscosity will appear

∆= 𝜇𝑏 ∆ ∙ 𝒖 .

The inner structure of strong shock waves (Emanuel and Argrow 1994)

Dilatational waves in transition in hypersonic boundary layers (Zhu et al. 2016)


Boltzmann-Curtiss equation for polyatomic gases

2, , , , ,j

f t C f ft I

v v r j

, , , ,m f tu v v r j

1/ 0 0

0 .

0t

dp

dtE p

u

u I I

u ( I) u + Q

, , , ,m f tv r j

21, , , ,

2rotE mc H f tv r j

.... ... x y zdv dv dv jdjd d

(2)

: normalstress tensor = 2

:excess normalstress =

:heat flux vector

NS

b

k T

u

u

Q

Rotational energy

(0) (0) = Tr( ) / 3 Tr( ) / 3 , = Tr( ) / 3m f m f p m f cc cc cc

I = moment of inertia

j = angular momentum

ψ= azimuthal angle


2nd-order constitutive relations for polyatomic gases

(1)

(2)

(3)

h f

h f

h f

Q

(2)(1)

(2) 2

(3) 2

/ 3 /

ˆ/ 3 rot

h m

h mc p n

h mc H mh

cc

c

(

2

2

2

2) ˆˆ ˆ[ ]

3 ˆˆ ˆ

ˆ ˆ1

ˆ ˆ

ˆ

ˆ ˆ1

( )

:2

ˆ

ˆ( )

ˆ( )ˆˆ

NF

NF

NF

b

b b

nd

nd

ndNS b

q cR

q cR

q

f

f f

cRf

+

Q Q

=

u

I u

Q

(2)ˆ 2

ˆ

ˆ

NS

NS b

NS k T

u

u

Q

1st-order

Linearization

2nd-order

Topological representation of constitutive relations is

possible!


Topology of 2nd-order CR (compression/expansion)

N2 μbulk=μshear

Vis

cous

shear

stre

ss

Argon μbulk=0

2nd-

order

1st-

order

CO2 μbulk=2,000μshear

𝛱 𝛱 𝛱

𝛱 𝛱 𝛱


N2 μbulk=μshear

Heat

flux

Argon μbulk=0

2nd-

order

1st-

order

CO2 μbulk=2,000μshear

𝑄 𝑄

𝑄

𝑄 𝑄

𝑄

Topology of 2nd-order CR (compression/expansion)


Trajectory of shock structure in topology of CR

Mach number = 5.0


Inverse density thickness of shock wave (nitrogen)

Symbols-experiment

NSF (fb=0)

NCCR (fb=0.8)

NF (fb=0.8)

Bulk viscosity effect

2nd-order effect


Multi-dimensional case: shock-vortex interaction

2

A

Enstrophy , , ,zt x y t dxdy

Vorticity z

v u

x y

PoF 2018



Sound pressure

Nitrogen

0.75

1.4

bf

Methane

1.33

1.289

bf

Argon

0

1.667

bf

Shock Mach 2.0

Vortex Mach = 0.8

Knudsen =0.001

Vorticity



Macro

SVI

Micro

SVI


Dusty gas flows in Lunar landing

Plume-

Surface

Simulation in

Rarefied

Condition

Erosion

Modeling

Dust-gas

Interaction

Physics of polyatomic gases in non- equilibrium based on ...acml.gnu.ac.kr/download/Conference/RGD31-Myong-180727.pdf · 2nd-order Boltzmann-type kinetic theory: Boltzmann-based gas

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