Shock wave structure in a binary mixture of broadwell discrete velocity gases

MECHANICS RESEARCH COMMUNICATIONS Vol. 14(5/6), 347-354, 1987. Printed in theUSA. 0093-6413/87 $3.00 + .00 Copyright (c) 1987 Pergamon Journals Ltd.

SHOCK WAVE

STRUCTURE

IN A B I N A R Y M I X T U R E

OF B R O A D W E L L D I S C R E T E V E L O C I T Y

GASES

T.Platkowski Dept.of

Mathematics

University

and M e c h a n i c s

of W a r s a w , 0 0 - 1 0 9

Warsaw,PKiN,Poland

(Received 21 February 1987; accepted for print 5 June 1987)

Introduction

The s t u d y of the s h o c k w a v e one

of

the c e n t r a l

(see refs.

problems

[1]--[4]

and

In this n o t e we c o n s i d e r two

Broadwell

structures

the

shock wave

the

the s h o c k w a v e

Our c h o i c e number

generalization wave

method

of s o l u t i o n

feature find

in

results

gases

applied

of o b t a i n i n g

to

solution

velocity

gas

ODE

scheme.

of

the

[5].

Our

two

gives An

the

important

it a l l o w s

the a p p r o x i m a t i o n s in

is a

and u p s t r e a m

set

is that

Mach

here

with

of the c o m p o n e n t s .

the s h o c k p r o f i l e s

We

infinite

considered

of t h i s

in this n o t e

347

problem

polynomials.

to the d o w n s t r e a m

solution

without

the

of s o l u t i o n w a s

to the

three

of

m a s s e s m I and m 2.

of p r o p a g a t i o n

discrete

profiles

shock profiles

other methods

model

in a set of

Numerical

of the m e t h o d

Legendre

s e n s e the m o d e l

correspond

and t e m p e r a t u r e

the

in

correspond

a one-component

states.

is

in a m i x t u r e

[7]. The m e t h o d

by use of a n u m e r i c a l

In this

points which

density

in

of the B r o a d w e l l ' s

shock

equilibrium

profiles

[6] and a p p l i e d

expansion

of l i m i t c o n d i t i o n s

s h o c k wave.

singular

in

structure

same p r o b l e m

gases

of r a r e f i e d

gases with different molecular

b a s e d on the t r u n c a t e d solve

in the d y n a m i c s

of

lit.cit.therein).

Such a model was constructed of

in m i x t u r e s

present

binary

to in

gas

348

T. PLATKOWSK I

mixtures e.g.

(

[i]

moment

- [4] note

results

;

profiles

of the

Statement

We

our m a i n we

we

= -cej,

a

the

Vj

velocity

of

shock with

to

with

number

match

etc.

- see

experimental

the question

shock wave

parameters

of

used

masses

in

the

vectors

momentum

during

modelling

procedure

the

look

, how

the

llke

and

of the p r o b l e m .

first

equations + Uj.~Nj

a model

collisions ). Let Nj

of m o l e c u l e s

per

six-velocity

(Uj)

and

unit

be

the

the

which (

of an

(Mj)

unit

orthogonal modulus

the

for d e t a i l s

with

denote

velocity

can be w r i t t e n

and

of the

Uj

the

(Vj).The

as f o l l o w s :

= 2cSII(Nj+INj+4+Nj+2Nj+5-2NjNj+3)/3 +

of

molecular

both masses

, j = 1,2,...6

volume

for Nj and Mj

[6]

gas

of the a d m i s s i b l e

conserves

see

(Vj)

velocity

choice

Uj+ 3

molecular

second

vectors

, c is t h e

This

sequel

Let Uj = cej,

gas and ~ = m l / m 2 is

components.

gives

[7].

, j= 1 , 2 , 3

the

O ( X l , X 2 , X 3)

of t w o

m I and m 2. In the

[6] a n d

first

ej d e n o t e

of the of

in a m i x t u r e

molecular

references

~Nj/~t

Mach

, Vj+ 3 = -~cej

velocity

kinetic

concerned

wave

notation

where

ratio

numbers

is not

on t h e p h y s i c a l

vectors

the m o l e c u l e s mass

rather

= ~cej

respectively, frame

objectlve

infinite

gases

follow

ansatz

the p r o b l e m

consider

Broadwell

are

depend

of

, the M o t t - S m i t h

).

In this

h o w do t h e y

methods

+

(I+~)cSI2(Nj+4Mj+I+Mj+4Nj+I+Nj+5Mj+2

+Mj+BNj+2+2MjNj+B-6NjMj+3)/6

+

+

+~cSI2[Mj(Nj+I+Nj+2+Nj+4+Nj+5)+ -Nj(Mj+I+Mj+2+Mj+4+Mj+5)] (i)

~Mj/~t

+ Vj'~Mj +

= 2~cS22(Mj+IMj+4+Mj+2Mj+B-2MjMj+3)/3 (I+~)cSI2(Mj+4Nj+I+Nj+4Mj+I+Mj+5Nj+2+

+Nj+5Mj+2+2NjMj+3-6MjNj+3)/6

+

+~cSI2[Nj(Mj+I+Mj+2+Mj+4+Mj+5 -Mj(Nj+I+Nj+2+Nj+4+Nj+5 where for

j+n>6

the

The shock

=>

three

problem wave

can

j+n->j+n-6 different of

+

, Sli,S12,S22

types

stated

)]

are

the

cross

sections

of c o l l i s i o n s .

the propagation

be t h e n

)+

of t h e

as f o l l o w s :

infinite

Mach

number

SHOCK WAVES IN GAS MIXTURES

We

look

for

of

a wave

solutions

travelling

(2) j =

1,2.,,,6

of

in

the

the

set

= Nj(Xl+~t)

Mj

= Mj(Xl+~t) that

equations

positive

Nj

, such

of

the

349

direction

following

(i) of

limit

in

the

the

form

xl-axis:

conditions

are

satisfied: a)

for

(3) b)

Xl-->

- co :

N1 = NO

, N2 = N3 =

...

= N 6.

M1

, M2

...

= M6 "

= Mo

for

Xl-->

+ ~

N i = N oo where

N~

and

downstream These

shock

limit

wave

=

:

,

M~are

the

= M3

M i = M~

for

i = 1,2,..6.

constants

determining

and

the

~

conditions

is

define

speed

the of

the

infinite

the

set

number

densities

the

shock

wave.

Mach

number

shock

[5].

Having

obtained

macroscopic

the

solution

of

characteristics

of

the

(1)

flow

we

:the

6

can

evaluate

number

the

densities

6 %

N

=

>

Nk

,

M

=

Mk

/

/

k:t the

mass

velocities

k=t

:

6 - - ~~I- -

u and

the

6

temperatures

of

ukNk

k=l the

__I L V k N

v

,

k=l :

components

6 T1

In

the

=

m~C

6 Nk(U_Uk) 2

3kN sequel

k=l we

(4)

k

use N2

=

the N3

; T2

=

I~C .

N5

=

N6

Mk(v_V k

/

k=i symmetry

following =

\

=

)2

.

conditions:

N2

M2 = M3 = M5 = M6 = M2 To

solve

(4) w e

the

problem

introduce

new

(5)

above

variables

[nl,n2,n3,n4,n5,n symmetry

equations

conditions to

dimensionless

6

6] (4)

equations

variables

as:

=

with

the

symmetry

conditions

:

z = NoSllC(Xl/C

(6) The

stated

+~t)/3

, ~ = ~/c

[NI,N2,N4,MI,M2,M4]/N

reduce which

the can

original be written

, o set in

of the

12

:

350

T. PLATKOWSKI dn-

I

(7)

= F1

, i = 1,...6.

dz where be

F i are

now

given

written

a)

for

(8)

as

for

- oO

and

~

n~

; n4 = ~

z -->

+ oo

= M o / N o is

= N~

densities. sides

of

(9)

These

; n2 = n3 = n5 = n6 =

0.

; n 4 = n 5 = n 6 = moo

,

the

, m~=

The

following

number

M~

density

/n o

are

three

ratio

the

combinations

of

(~-l)dn3/dz

+ 4~dn2/dz

=

0

(~+~)dn4/dz

+

(~-~)dn6/dz

+ 4~dn5/dz

=

0

(9+l)dnl/dZ

-

(~-l)dn3/dz

+

equations

of

the

conservation

correspond

individual

of

momentum

conditions

(8)

(~+~)dn4/dz to

the

the

shock

number

the

right

hand

obtain

(~-~)dn6/dz

of

and

respectively.

we

-

law

components

=

conservation to

the

Integrating

law

(9)

0 of

of

with

the

:

(~+l)n l(z)+(~-l)n3(z)+4~n2(z)=

(10)

can

vanish: +

masses

(3)

upstream

normalized

(~+l)dnl/dZ

three

limit

conditions

:

/N o

(7)

boundary

:

n I = n 2 = n 3 = n~ where

The

follows:

z -->

nI = 1 b)

in A p p e n d i x .

#+l=6~n~

(~+~)n 4(z)+(~-~)n6(z)+4~n5(z)=(~+~)~

=

=6~m~

.

(~+i)n I (z)-(~-i)n3(z)+(~+~)n4(z)-(~-~)n6(z)=

=(G+I)+(~+~)< (i0)

Eqs.

give

downstream

and

Inserting

(i0)

the

=2no~+2~moo

Rankine-Hugoniote

upstream

equilibrium

into

we

(7)

obtain

relations

between

the

states. finally

three

ODE:

dn~ (ii)

= Fi(nl,n2,n

4)

,

i = 1,2,4,

dz where

FI,F2,F 4 are

for

the

unknown

(ii)

has

two

Fi

=

unknown

0,

i =

given

functions

equilibrium 1,2,

densities

in A p p e n d i x

which

and

nl,n2,n 4 are points

in

the

correspond

to

downstream

and

the given

phase the

upstream

by

(8).

space

limit the

conditions

limit

The

(nl,n2,n4)

values

shock

system

of

wave.

,

the In

SHOCK ~%VES IN GAS MIXTURES

the

numerical

equilibrium

s c h e m e we i n t e g r a t e

351

f r o m the p e r t u r b e d

s t a t e up to the u p s t r e a m

equilibrium

we o b t a i n

expression

downstream

(see the

next

section). From

(i0)

the

following

on the

speed

of

the s h o c k wave:

Note that the

for ~

= 1 and

for

c a s e of the B r o a d w e l l

four i n d e p e n d e n t ratio

~

S12 and

,

physical

the m a s s

$22.

~

model

[5].

parameters

ratio ~

Solutions

= -->0 w e o b t a i n ~ = 1/3

of

In the set : the

(ii) t h e r e

downstream

and the n o r m a l i z e d

(ii) w i l l

, as

cross

be d i s c u s s e d

in are

density sections

below.

Results We have the

solved

the set of e q u a t i o n s

Runge-Kutta's

solutions the

procedure

are the i n t e g r a l

downstream

approximate

and the t e m p e r a t u r e s We

have

physical

In

the f i g u r e s

interesting Fig.l

Having

(ii) w e e v a l u a t e

obtained

the n u m b e r

and

S]2

: ~

of

, ~

the

,

for

large

S12

and

dowstream

= $22 = 0.i.

For ~

density

density

ratios

$22. density

the

,

It can be seen

and

most

of the d e n s i t y

profiles

5 we have o b s e r v e d

(~1) is on the

left of the h e a v y one

the

The t e m p e r a t u r e

succession

in

shock

heavier

(T 2) is a l w a y s

on the r i g h t

density profiles.

shock increase

the

; for ~

of the light

> 5 the

change

profile

of the

profiles

~

never

: the t e m p e r a t u r e

n o t e that the t e m p e r a t u r e

left of the c o r r e s p o n d i n g

: for s m a l l e r

profiles

fixed

the c h a n g e

component

changed.

with

that the

profiles

also

of

and the t e m p e r a t u r e

of the d e n s i t y

We

intervals

for

succession

gas

the

densities

the n o r m a l i z e d

components

(normalized)

and the s e p a r a t i o n

decreasing ~

is

and

cases.

=

order

of

The s h o c k w a v e

joint the u p s t r e a m

the s h o c k p r o f i l e s

profiles

for v a r i o u s

with

by use

of the c o m p o n e n t s .

we show the

thickness

states.

below we have plotted

profiles 0.03

4th order.

curves which

parameters

the t e m p e r a t u r e

In

of

investigated

the

of the

equilibrium

solutions

(ii) n u m e r i c a l l y

their

of the

light

are a l w a y s

one(~l). on the

352

T. PLATKOWSKI

For the

comparable

Fig.l]

we

note

downstream

the

densities

relaxation

tail

light

component

(n I) as a r e s u l t

heavy

component

upon

The

shapes

~12

and ~ 2 2

, as can be to

the

of the

contrary

to

gas

seems

however

of d i f f e r e n t

could

a role.

decreasing

changes the

[

cf.

profiles

(i.e.small amount

that gas

the

:

larger show

increases

distinguish

two

if ~ 1 2

we

profiles

infinite

of

the

Broadwell

profiles The

the

case also

of

not

give

light

Xenon

on the when

note

overshoot

density

have

this that of

exist in t h i s the

heavy

,

in

[i]

-

It

between

direction)

profiles

in Fig.2 the

, which

~

= ~

shows

components is in t r a c e

on the

the

shock

wave

light

amount

component

is

, found

light

one

give

mixture of

the

the

model.

on the

density

[7]

for

seems

the

[i]

the

to be a

rather

than

celebrated

component

the

can ~J n I.

temperature

solution

(see e.g. no

the

profile

of

in

also

note

sensitivity

overshoot of

We

in the

overshoot

not

of the

there

and

parameters

in the m e t h o d

b)

density

density

; this

of

case

investigated

does

= 0.3.

profiles

, reported

Helium

model

which

considered

same

d)

between

the d e n s i t y

in t r a c e

one

density

; in the

physical

profile

,

collisions

the

for

number

phenomena. The model

a)

question.

in the

component

sections

results

of the

]. D i a g r a m

obtained

We

the

, the d e n s i t y

light

open

on the d e n s i t y

Mach

of

scale.

component with

the

cross

decreases

of t h e

of a p p r o x i m a t i o n s

real

"hump"

does

of the

mixture result

gases.

to c h a n g e s

model

profile

have

balls

theoretical

of

of e v o l u t i o n

Conclusions In this note

of

is an

profiles

tail

scales

of the

that

travelling

heavy

shock

relaxation

of h a r d

deficiencies

relaxation

the

case

coupling

ofthe

such mixtures.

In the

in F i g . 2

the

influence

Fig.2.

other

in

profile

cross

succession

"very w e a k "

S12)

we

(~I)

the

b),c),d)

for

, has

In F i g . 3

S12

1.0

on the

of the

gases

=

also

phenomena

absence

particles

For

and

physical

( in p a r t i c u l a r

play

precedes

of t h i s that

from

~

density

larger for

mixture

experimental

Explanation

of

one

e.g.

on t h e

depend

seen

Xe/He

heavy

[4].

model

light

of the p r o f i l e s

correspond profile

the

[

in -

the

[4]).

We

temperature

in m o r e

complicated

SHOCK ~%VES IN GAS MIXTURES

models

of the B o l t z m a n n

velocities some

[2]

interest

profiles

- [4].

vectors

Abbreviated

In t h i s

to o b t a i n

for d i s c r e t e

velocity

equation

per

Paper

with

models

each

- For

the

respect

the d e n s i t y

and

with

353

more

continuum

it c o u l d

spectrum

possibly

be of

the

temperature

shock

than

one m o d u l u s

of the

component.

further

information

,

please

contact

the a u t h o r .

References [i] A . G m u r c z y k , M . T a r c z y n s k i , Z . W a l e n t a

et al.

- Proc.of

the

llth

Symp.on

Rarefied

Dynamics,Paris:CEA,vol.l(1979)p.333. [2] R . F e r n a n d e z

Feria,J.Fernandez

de

la M o r a

- subm.to

J.Fluid

Mech. [3] M . B r a t o s , R . H e r c z y n s k i

- IFRT Reports,19/1983.

[4]

- Arch.of

T.Platkowski

[5] J . B r o a d w e l l [6] N . B e l l o m o , L . d e

-

Socio

[7] R . M o n a c o

Mechanics,33(1981)p.785.

Phys.Fluids,Vol.7,No.8(1964)p.1243.

- M e c h .

Res. C o m m . 1 0 ( 1 9 8 3 ) p . 2 3 3 .

- Acta

Mechanica

also

Proc.of

Grado

(1986)

55(1985)p.239 the

15th

Italy

Symp.on

].

Appendix F1 =

[4(n22-nln3)+(l+~)S12(2n2n5+n3n4-3nln6) +12~)~12(n2n4-nln5]/(l+

+

~)

F 2 = -[2(n22-nln3)+(l+~)S12(n2n5-nln6/2-n3n4/2)

+

~(l+~2)S~12(n2n4+n2n6-nln5-n3n5)]/~ F3 =

[4(n22-nln3)+(l+~)S12(2n2n5+nln6-3n3n4)

+

+12(~+~2)~12(n2n6-n5n5)]/(~-l) F4 =

[4 ~ 2 2 ( n ~ 2 - n 4 n 6 ) + ( l + ~ ) ~ 1 2 ( 2 n 2 n 5 + n l n 6 - 3 n 3 n 4 +12

2)S12(nln5-n2n4)]/(~+

) +

~)

F 5 = -[2~S22(n52-n4n6)+(l+~)~12(n2n5-n3n4/2-nln6/2)

+

+3 l + ~ + ~ 2 ) ~ 1 2 ( n 3 n 5 + n l n 5 - n 2 n 6 - n 2 n 4 ) ] / ~ F6 =

[4~SP'22(n52-n4n6)+(l+~)~12(2n2n5+n3n4-3nln6 +12~2)~12(n3n5-n2n4)]/(~-~)

where

S12

= S12/SII

, ~22

= $22/Sli

of

) +

[ see RGD

354

T. PLATKOWSKI

l~Ot

I

,--""

',

'

;

/I ,, ../

,,

I!

,,,

- ~ . Z , > ' * " - ~ * ' ~~

-'

."

~

.:~';~"

~ ';I

....

i "'e

//,/I ,.,//,_I /

0.001

Fig. 1

~

V/

i•~ ., .~
".-"

.'/,"

:b

IIi" ///

///

. ....

~if!"/ ,i!1 ..'_,.~

.

= 1.0

..............T2 ,

,

Z

,

)

oc = 30.0

Density and temperature profiles for various density ratios ~ w i t h

~-

0.03 , ~12 = ~22 = 0.1

1.0

/

/o,. ,,'/ // /,, / , , ../ ..//,,./..," ,,/

0,5

/;..

f"~

> Fig.2 Density profiles for different cross sections ~12 ' 722.

a: S12 c: $12

b: 212 ~ = 1.0 , ~'. 2 " 0.1 d: $12 0.05 = 0.1 = ' 022

: 5.62,~^2 = 10.25 ~ = 1.0 0.1 , 22

! 1o0 ,,.,.. .....i:-/...- ,

.,..

/ //,,'"

...

,.

/~ " 2 ' ' .-'i___ .

0.5 J

/

..

/

/

:1

: !

•

s"

.'I t .'//

,'1

...'/.-

#

/

l

/ ,"

../

."1

.'2' . p ,';

," .,;# . . j /,'/ ,,"

i

I

.........

~_ T. '

............T

"2

Fig. 5 Density and temperature profiles for different cross sections g12 for ~ = ~

= 0.3 and $22 = 0.1

a: $12 . 0.1

b: $12 = 0.05