lecture 6 gas kinetic theory and boltzmann...

GIAN Course on

Rarefied & Microscale Gases and Viscoelastic Fluids:

a Unified Framework

Lecture 6

Gas Kinetic Theory and Boltzmann Equation

R. S. Myong

Gyeongsang National University

South Korea

Feb. 23rd ~ March 2nd, 2017

GIAN Lecture 6-1 Rarefied & Microscale Gases and Viscoelastic Fluids: a Unified Framework R. S. Myong, Gyeongsang National University, South Korea Feb. 23 – March 2, 2017 – IIT Kanpur, India

Content

I. Gas kinetic theory

II. Boltzmann equation

III. V & V of DSMC



Hypersonic rarefied regime: Kn=O(10-3 ~ 10-1), Nδ=O(10-1 ~ 1)

Low-speed microscale creeping regime: Kn=O(10-2 ~1), Nδ=O(10-5 ~ 10-2)

High-speed free-molecular regime: Kn=O(1 ~ 10), Nδ=O(1 ~ 10)

Local Equilibrium

• Global thermal equilibrium exists when all parts of a system are at the same

temperature and this temperature is the same as that of the surroundings.

• Global thermodynamic equilibrium (GTE) means that the intensive parameters

are homogeneous throughout the whole system

• Local thermodynamic equilibrium (LTE) means that the intensive

parameters are varying in space and time, but are varying so slowly that for

any point, one can assume thermodynamic equilibrium in some neighborhood

about that point.

• Translational, rotational, vibrational modes



The gas kinetic theory aims to explain and compute the macroscopic

properties of gases such as conserved and non-conserved variables and

transport coefficients from the properties of their microscopic constituents.

There are about 1020 molecules in a cubic meter at the altitude of 100 km.


I. Gas kinetic theory: Distribution function

Let us consider the number of cars whose weights lie within a certain interval

∆w. Then the fraction of cars per unit interval of weight can be written as

When ∆w becomes infinitesimally small, the function becomes a continuous

function, referring to as the normalized distribution of car weights

This distribution function can be interpreted as a probability density function.

One can calculate the average weight of a car in the community

where the number of cars with weights in intervalkk k

Nf N k

N w

00 0 00

( ) lim lim and ( ) lim 1kk k

w w wk

Nf w f f w dw f w

N w

1

0 000

lim ( ) ( ) ( )2

k k k

wk

w w Nw wf w dw Q Q w f w dw

N



The normalized phase space distribution function is the probability density of

finding a particle at the velocity space point v at the configuration space

location r.

Macroscopic physical parameters are averages of quantities that depend on

molecular velocities (velocity space averages).

J. C. Maxwell derived a velocity distribution for spatially uniformly distributed

molecules in the gas

( , )( , ) where ( ) ( , )

( )x y z

Ff n F dv dv dv

n

r vr v r r v

r

( ) ( , ) or simply ( ) ( , )x y zQ Q f dv dv dv Q f d

v r v v r v v

3/2

2( )2

f e

v v

v



Until this stage, no explicit mention of what the distribution function actually

means is given; anything like molecules in a gas or probability density

function of agents with wealth in open market economy is possible in principle.

In fact, it turned out that the determination of another unknown β provides the

connection necessary to assign a physical meaning to the distribution

function, otherwise, a pure mathematical object.

In gas kinetic theory, thermodynamics plays such a role.

Local equilibrium Maxwell-Boltzmann distribution

B

m

k T

23/2 ( )

2( )

2B

m

k T

B

mf e

k T

v u

v



Worldwide poll among mathematicians: What do you think is the most

beautiful mathematical equation?

Worldwide poll among every human being: What do you think is the most

beautiful or amazing number?


Memo



Let us assume that the gas density is low enough to ensure that the mean free

path is large in comparison with the effective range of the intermolecular forces.

The velocities of the colliding particles are uncorrelated, that is, particles which

have already collided with each other will have many encounters with other

particles before they meet again. (Molecular chaos and break of time reversal)

Assume time, location and particle velocity are independent variables of the

phase space distribution function. We have a first-order partial differential

equation in space and time.

2( , , ) ,

Movement Collision (or Interaction)

Kinematic Dissipation

f t C f ft

v r v



Let us assume the molecules and their intermolecular forces are assumed to be

spherically symmetric. This assumption is sometimes questionable for real

gases.

Assume also effects of the external forces on the magnitude of the collision

cross section are neglected.

Boltzmann collision integral

2

* *

2 2 2 2

, Gain (scattered into) - Loss (scattered out)

=

( )

C f f

f f

t t

f f ff d

v v v


III. V & V of DSMC

Background: why we care about the verification of pure simulation methods

There always exist computational errors in any simulation methods so that we may

need a tool to verify them.

But, it is impossible to find exact solutions free from computational errors for pure

simulation methods like the DSMC, hindering further refinement of the method.


2

1 10 or 0 at steady-state

Re Sp dS

M

uu I Π F n

A verification is possible based on the following property [C&F 15]

A challenge is the statement of C. Villani (2004):

“The conservation laws should hold true when there are no boundaries.

In presence of boundaries, however, conservation laws may be

violated: momentum is not preserved by specular reflection.”

In conservative PDE-based schemes, the nearest cell to the wall

always satisfies the conservation laws via

but it may not be true in the case of non-PDE schemes.

41

, ,

1,

n n

i j i j k k

ki j

tL

A

U U F

III. V & V of DSMC


First benchmark problem: Gaseous compressive shock structure

Rarefied shock structure

High p

Low p

Gas particle

2 2

error

error

error

error

error ( ) ( )

error Round-off error

mass

x momentum xx xx

y momentum xy xy

z momentum xz xz

energy xx x xx x

EOS

u u

u p u p

E p u u Q E p u u Q

p RT

III. V & V of DSMC


High p

Low p

X/

Pe

rcen

tag

eo

fn

orm

alize

de

rro

r

-40 -20 0 2010

-9

10-7

10-5

10-3

10-1

101

Conservation of MassConservation of X-MomentumConservation of Y-MomentumConservation of Z-MomentumConservation of EnergyEquation of State

EnergyX-Momentum

y-Momentum

z-Momentum

Mass

EOS

8

2.0, Kn=1.0,

0.01 , 1/ 32 ,

320, 10S

M

t x

N N

III. V & V of DSMC


High p

Low p

Time step interval size

Pe

rce

nta

ge

of

err

or

10-2

10-1

100

101

10-2

10-1

100

101

102

Mass Conservation Eq.

Momentum Conservation Eq.

Energy Conservation Eq.

III. V & V of DSMC


Number of Particle per cell

Pe

rce

nta

ge

of

err

or

101

102

103

10-2

10-1

100




Cell size

Pe

rce

nta

ge

of

err

or

10-2

10-1

100

10-3

10-2

10-1

100




III. V & V of DSMC


error

error

error

error

error Round-off error

x momentum xx xx

y momentum xy xy

z momentum xz xz

energy xy x xy x

EOS

p P

v Q v Q

p RT

Second benchmark problem: Wall-driven Couette flow

III. V & V of DSMC


X/H

Pe

rce

nta

ge

of

no

rma

lize

de

rro

r

-1 -0.5 0 0.5 1

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Conservation of X-Momentum

Conservation of Y-Momentum

Conservation of Z-Momentum

Conservation of Energy

Equation of State

Energy

X-Momentum

y-Momentum

z-Momentum

EOS

8

2.0, Kn=1.0,

0.01 , 1/ 32 ,

320, 10S

M

t x

N N

III. V & V of DSMC


Convergence history of DSMC simulation

III. V & V of DSMC


NS

||E

||2

102

103

104

105

106

107

108

10910

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

EX-Momentum

EY-Momentum

EZ-Momentum

EEnergy

EEOS

N=40N=80N=160N=320

Energy

X-Momentum

y-Momentum

z-Momentum EOS

III. V & V of DSMC


Q & A

lecture 6 gas kinetic theory and boltzmann...

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