Laminar sidewall boundary layer in a dusty-gas shock tube

LAMINAR SIDEWALL BOUNDARY LAYER IN A DUSTY-GAS SHOCK TUBE

October 1986

by

B. Y. Wang and I. I. Glass

TECHNISCHE UNIVERSITEIT DELFT

LUCHTVAART- EN RUlMlEVil\AHliECHNIEK

BIBLIOTHEEK

Kluyverweg 1 - 2629 HS DELFT

UTIAS Report No. 312

CN ISSN 0082-5255

LAMINAR SIDEWALL BOUNDARY LAYER IN A DUSTY-GAS SHOCK TUBE

Submitted April, 1986

October 1986

by

B. Y. Wang and I. I. Glass

UTIAS Report No. 312

CN ISSN 0082-5255

Acknowl edgements

One of us (B. Y. Wang) is grateful to the Institute of Mechanics, Academia Sinica, Beijing, China, and to UTIAS for the opportunity to do

research work during the period 1984-1986.

The financial support received from the Natural Science and Engineering Research Counci 1 of Canada under grant No. A1647, the U.S. Ai r Force under grant AF -AFOSR-82-0096, the U.S. Defence Nuclear Agency under DNA Contract 001-85-0368, and the Defence Research Establishment Suffield (DRES) is acknowledged with thanks.

J

Abstract

Details are given of an implicit finite-difference scheme to study the difficult problem of sidewall boundary-layer flows induced by a shock wave in a shock-tube channel containing air laden with suspended small solid particles. The numeri cal solution to the laminar sidewall boundary-layer equations describe the development of the sidewall boundary layer, give the boundary-layer characteristic parameters, and show the effects of the pressure ratio across the shock front, the mass loading ratio of the particles and the particle size on the flow properties for the two phases. The limitations of the present solutions are discussed and future improvements are recommended.

Contents

Acknowledgements

Abstract

Contents

Notati on

1.

I NTRODUCTI ON

11. BASIC EQUATIONS

111. BOUNDARY AND INITIAL CONDITIONS

IV. FINITE-DIFFERENCE PROCEDURE

V. BOUNDARY-LAYER PARAMETERS

VI. RESULTS AND DISCUSSIONS

VII. CONCLUSIONS

REFERENCES

FIGURES

APPENDIX - COMPUTER PROGRAM EDBLES

Page

i i

i i i

iv

v 1 3 9 15 22 24 28 29

Ai

n

A2l

a

an

aw

Bi

n

bn

bw

ei

n

cn

cp

Cs

Cv

Cw

D

Di

n

d

dn

dw

Ec

K

k

M J

Notation

coefficients of finite-difference equations

ratio of sound speeds, A21

=

ä*/ä*

2 1

sound speed

grid parameters for six-point difference scheme, Eq.

(4.11)

coefficient for fitting a polynomial to the gas velocity or

temperature near the wall, Eq. (!:>.1O)

coefficients of finite-difference equations

grid parameters for six-point difference scheme, Eq. (4.11)

coefficient for fitting a polynomial tothe gas velocity of

temperature near the wall, Eq. (!:>.1O)

coefficients of finite-difference equations

grid parameters for six-point difference scheme, Eq.

(4.11)

specific heat at constant pressure for gas phase

specific heat for particle phase

specific heat at constant volume for gas phase

coefficient for fitting a polynomial to the gas velocity or

temperature near the wall, Eq. (5.10)

normalized drag coefficient based on Stokes' drag coefficient

coefficients of finite

-

difference equations

diameter of partieles

grid parameters for six-point difference scheme, Eq.

(4.11)

coefficient for fitting a polynomial to the gas velocity or

temperature near the wall, Eq.

(5.10)

gas Eckert number based on freestream temperature, Ec

= u*2/ c*pT*

00 ex>

ratio of consecutive step sizes in y-direction, K

=

~n/~n-l

heat conductivity for gas phase

gas Mach number

m N

Nu

n p • q R T u v

w

x y

grid line in y-direction

9 ri d poi nt at outer edge of boundary 1 ayer

Nusselt number based on particle diameter

grid line in x-direction

pressure ratio,

P21

=

p*/p*

2 1

pressure ratio,

P41

=

p*/p*

4 1

gas Prandtl number, Pr

=

c*~*/k* _p

gas pressure

heat-transfer rate

gas constant

particle Reynolds number based on freestream values,

Re

_p

=

p*u*d* / ~*

CD CD CD

slip Reynolds number based on slip velocity,

Re s

=

p*{up-u* )d* /

~*

flow Reynolds number based on particle equilibrium length,

Re

(X) =

p*U*>'*/II*

Q) Q)'~ r-oo

temperature

temperature rati

0,

T

21

=

T*/T*

2 1

tangential velocity in x-direction

normal velocity in y-direction

function representing any flow property, u, v, p or

T

coordinate along the wall

coordinate normal to the wall

Greek Symbo

1

s

ex

ratio of specific heats for two phases,

ex

=

Cp/cs

mass loading ratio of particles in freestream,

~ =

P"P.!P:'

density rti

0,

T

21

=

p*/ p*

2 1

ó

e

À*

_S ~. p 't* V w

Subscripts

m n p s w 1 3 4

displacement thickness of boundary layer

tolerance for desired accuracy

weighting factor for finite-difference schemes

particle velocity-equilibrium length based on shock speed,

À*

_S

=

p*d*2u*/U~II* _{S S ~oo}

dynamic viscosity for gas

derivative of gas viscosity with respect to gas temperature,

~.

=

d~/dT

dens i ty

density of particle material

thermal equilibrium time,

't~

=

psd*2cs/l~k*

velocity equilibrium time,

't~

=

psd*2/H~~*

shear stress at wall

power index for gas vi scos ity,

w =

u.

/I

grid line in y-direction

grid line in x-direction

parti cl e

shock wave

wall conditions

freestream conditions for si dewa

11

boundary

1

ayer

index for state

( 1)

in shock tube

index for state

(~)

in shock tube

index for state

(3)

in shock tube

index for state

(4)

in shock tube

Superscripts

*

dimensional quantities

i

index for dependent variables

modified quantities

I. INTRODUCTION , '.

A boundary-1ayer flow a10ng the wall of a shock tube is induced behind a movi ng shock (or rarefaction) wave. The boundary 1ayer introduces the fo11owing effects on shock-tube f10ws: shock-wave dece1eration and attenuation, contact-surface acce1eration, test-time reduction, flow non-uniformities and shock-wave curvature. These effects must be taken into account when estimating faci1ity performance and when eva1uating experimental data from shock tubes. Consequent1y, an understanding of the boundary-layer flow prob1em is essentia1 for shock-tube flow studies. Many such investigations were conducted for the case of pure-gas shock-tube f10ws since the 1950's [1-52]. Critica1 reviews of the ear1y work up to 1969 were given by Glass [1] and Davis [2].

It is well-known that, in a pure-gas shock-tube, after the diaphragm ruptures, a shock wave travels into the low pressure gas in the channe1 whi1e an expansion or rarefaction wave trave1s into the high pressure gas in the driver. The quasi-steady flow regions induced behind these waves are separated by a contact surface across whi ch the pressure and velocityare equa1 but density, entropy and temperature are, in general, different. A boundary layer exists between the heads of the shock front and the rarefaction wave. It starts growing at these two faces, which effectively are 1eading edges for the boundary-layer flow. The boundary 1ayer increases in thickness until it reaches a maximum at the contact region. A schematic diagram of a basic shock-tube flow is shown in Fig. 1. The idealized flow structures (Le., without boundary-1ayer effects ) i n a constant-area shock tube are shown in Fi g. 2. These flow parameters can be cons i dered as the freestream conditions at the outer edge of the boundary layer. It is seen th at the sidewall boundary 1ayer consists of three main regions. In region I between the shock front and the contact surface and between the contact surface and the diaphragm station, i.e., essentially in state (2), the boundary 1ayer is hot with respect to the tube wa11 and its growth occurs in a uniform unacce1erated region provided the boundary layer is relatively thi n. In reg i on 11 between the contact surface and the tail of the rarefaction wave, i.e., in state (3), the boundary layer is cold with respect to the tube wall and grows in a similar way to that in state (2). These two mai n boundary 1 ayers meet at the contact surface where they overlap between the contact surface and the diaphragm station and the flow is more complex. In region III, i.e., the rarefaction wave zone, the boundary layer is subjected to a monotonically decreasing pressure and increasing velocity from the head to the tail of the wave. As time passes, the 1 eadi ng edges have raced al ong the tube in oppos i te di rect i ons, the boundary 1ayer becomes thicker and transition to a turbulent boundary 1ayer occurs. Fi na11y, dependi ng on the tube di ameter and af ter a long enough time has e1apsed, the boundary 1ayer might fil1 the tube and the flow consists predominant1y of a turbulent pipe flow, and boundary-layer closure occurs.

Recently, the fluid dynOO1ics of a dusty gas has becane increasingly significant with appl ications to many branches of engineering and science [53-58J. lhe dusty-gas shock tube is an important research fac il ity for investigating various physical phenomena related to a tv.o-phase flow system [59-75J. Especially, the experimental studies by means of dusty-gas shock tubes provide useful infonnation about the rel axation process or the interaction relations between the gas and particles. In a dusty-gas shock tube, the driver contains high-pressure air and the channel contains low-pressure air laden with suspended 9llall sol id particles or a dusty gas. lhe flow structure in a dusty-gas shock tube is schematically shown in Fig. 3. It is assumed that there are no particles in the cold driver gas itself behind the contact surface. lhis impl ies that the boundary-layer flows in regions Il and III in a dusty-gas shock tube are similar to a pure-gas shock tube. lherefore, to investigate the sidewall boundary-layer flow in a dusty-gas shock tube, we can simpl ify the probl en by d irecting our attention to the boundary-layer developnent in state (2) only behind the shock front. Another simpl ification is to treat the extensive shock structure as a planar shock front and in this paper the compressible laminar sidewall boundary-layer flow is induced behind this shock wave vklich is moving with unifonn speed. To illustrate the physical features of the sidewall boundary layer in a dusty-gas shock tube, the structure of a stationary shock wave in a dusty gas is needed to spec ify the freestrean cond itions. lhe structure is derived from the idealized equilibrium-flow limit for a shock wave in a dusty gas.

It is possib1e to put the nonstationary boundary-1ayer flow with its mov ing 1 ead ing edge at the shock front into a steady-flow frane of reference by fix ing a set of coord inate axes to the wave front. Figure 4 gives same flow characteristics of the shock-tube sidewall boundary layer behind a constant velocity shock wave moving to the left into a stationary state (1). lhe shock wave s becanes fixed at x

=

0 by mathematica11y superposing a counterflow

Us

equa1 to the shock velocity. As aresult, the tube wall moves to the nght at the shock speed ~. At the wa 11, the gas has the same vel oc ity and temperature as the wa 11 , since there is no vel oc ity 51 ip or temperature jump at the surface of the \\611, due to the gas viscosity. However, for the partic1 es, a vel oc ity sl ip and a temperature jump at the wal 1 surface do occur. The partic1e velocity and the temperature at the wal1 depend on the interaction between the gas and the partic1 es. Typ ic al vel oc ity and temperature profil es across the sidewall boundary for the tv.o

phases are shown in Fig. 4.

In the present report, laninar sidewa11 boundary-layer flows in a dusty-gas shock tube are solved numerically by an impl icit finite-difference scheme vklich is similar to that used for the laninar boundary layer of a

dusty gas over a semi-infinite flat p1ate [76J. lhe main difference between ..; the analyses for flat-plate and shock-tube sidewa11 boundary layers is as

follows. lhe tangential velocity for the flat-plate case increases from zero at the wa 11 to the freestrean val ue at the outer edge of the boundary

,.

.,;

layer. For the sidewall boundary layer, however, the tangential velocity of

the gas decreases from the shock speed at the wall to the freestream value at the outer edg e, in the steady -flow coordi nates attached to the shock

wave. Dwing to this shape of velocity the profile, the displacement

thickness as well as the normal velocity for the sidewall boundary layer are negative, as expected. In addition, for shock-tube sidewall boundary-layer flows, it is usually required to use a non-Stokes relation since the shock

speed is always supersonic. Under this condition, the assumption that the

particle Reynolds number is of order unity is, in general, erroneous because

it can be much greater than one, especially in the neighbourhood of the tube

wall. In other words, Stokes' relation is of ten invalid in the case of

sidewall boundary-layer flows in a dusty-gas shock tube. Therefore, in this analysis, we employ an appropriate non-Stokes relation to describe the interaction of the two phases. The resulting velocity and temperature profiles across the boundary layer are given for the gas and particle phases respectively. The boundary-layer characteristic quantities such as the

fri cti on coeffi ci ent, heat transfer rate and di spl acement thi ckness are

ca 1 cul ated and compared with the correspondi ng rsults for the fully frozen and equilibrium flows. The effects of the loading ratio, the particle size and the diaphragm-pressure ratio upon the flow characteristics are discussed.

11. BASIC EQUATIONS

The present research deals with compressible, laminar, shock-tube sidewall boundary layer flows behind a constant-speed shock wave advancing into a stationary dusty gas. The basic assumptions are as follows:

(i) The gas phase i s a perfect gas with constant specifi c heats. The Prandtl number of the gas is assumed constant.

(ii) The particle phase consists of spherical particles of uniform size.

The volume occupied by the particles can be neglected. There is no

temperature gradient inside a particle.

(iii) The number density of particles is large enough to treat the particle

phase as a continuum while it is also small enough to neglect the

mutual colli si ons and other interactions among the particles.

(iv) The particles have no Brownian motion and there is no analog of pressure for the particle phase.

(v) Only the processes of drag and heat transfer couple the particles

with the gas. The drag coefficient and the Nusselt number for a single sphere in a viscous flow are assumed valid for the particle cl oud.

(vi) The shock-tube wall temperature remains constant. The assumption of constant wall temperature has been substanti ated as a good approximation ex cept for very strong shocks.

(viii) The interactions of the sidewall boundary layer with the shock-tube flow are neglected. Then the boundary conditions at the outer edge of the boundary layer can be determined from the idealized shock-tube flow properties resulting from the assumption that there is no boundary layer near the wall. In this paper, the dusty-gas shock -wave parameters in the equil i bri um-fl ow 1 imit are used as the freestream conditions for the boundary-layer flow. Therefore, there is no pressure gradient along the tube.

In the shock-fixed coordinates, the steady sidewall boundary-layer equations for a dilute gas-particle mixture can be written as [77]:

Gas continuity:

-~- p*u* + -~- p*v* = 0

~x* ~y*

(2.1)

Gas x-momentum:

u* - u* p* (u* ~u* + v* ~u*)

=

~ (~* ou*) + p* P D

~x* ~y* ~y* ~y* P 't'~

(2.2)

Gas energy:

p*c*(u* ~T* + v* ~T*)

=

Ö (k* öT*) + ~*(~u*)2

P öx* ~y* öy* ~y* ~y*

(u* - u*)2 + (v* - v*)2 (T*p - T) _Nu

+ p* P P D + p* c* - ( 2 • 3 )

P 't'* p s 't'* 2

v

T

Gas state: p*

=

p* R* T*

_(2.4)

Parti cle conti nuity:

~ ei!:.

u*p +

-~

- p* v*

=

0 ~x* ... ~y* p p (2.5) Particle x-momentum:

(

~up

v*

~uP)

p* u* +

=

p p ~x* p ~y* (u* - u*) -p*

P

D P 1:* V

(2.6)

,.

Particle y-momentum:

Parti cle energy:

'oV* p* (u*

.:.:..e.

+ p p 'OX* 'OT* p* c*(u* ~ + p s p 'OX* v* 'OV

p)

P 'Oy* (v* - v*) = - p* P D P 't* _V 'OT*

(Tp -

T*) Nu v*p

J )

= -

p* c~ - -'oy* P

-r:r

2

where the velocity equilibrium time is p* d*2

~ _ --;.5...,.-_ V 18j..1.*

and the thermal equilibrium time is p* d*2 5 c* 12k* 5 (2.7) (2.8) (2.9)

(2.10)

The relation between 't~ and 'tI can be derived from the definitions (2.9) and

(2.10)

where the Prandtl number is

't* = 3Pr 't*

T 2a: v

c* j..I.*

P r

=

---,-P_-k*

and the ratio of the specific heats for the two phases is c* a:

=

..:e.

c*

_s

(2.11)

(2.12)

(2.13)

Therefore, there are eight equati ons to sol ve for the eight unknowns: u*, v*, T*, p*, up' vp '

Tp

and

Pp'

As mentioned before, gas pressure p* is known as it is constant across the boundary layer and equal to the freestream value p!. In order to close the system of equations, it is necessary to know the expressions for j..I.*, k*, D and Nu. Usually, the heat co nduct i vi ty k* can be exp ressed in terms of j..I.* and cp' Pr. From Eq.

(2.12),

c* k*

=

..:e.

j..I.*

Pr

(2.14)

where cp and Pr for air are assumed constant. In this analysis, the following relations are employed for the other three quantities:

(i)

Viscosity coefficient 1-1* for air [78]:

*

=

1 719xlO-

5(T*

)0.77

N /

2

I J . . 273

s m

(ii) Normalized drag coefficient

D [79]:

D = CD =

1-

Re

s

+ 7₆

Re~·15

24/Re_p 50

(iii) Nusselt number Nu

[80]:

1/2

Nu

=

2.0 + 0.6

Pr

1/3 _Re

s

(2.15)

(2.16)

(2.17)

where the slip Reynolds number Re s is based on the particle diameter and the

relative (or slip) velocity of the particle to the gas:

=

p*l{u

p

2 +

v

_p

2)1/2 - {u*2

+

v*2)1/2Id*

IJ.*

(2.18)

For numeri cal computation, it is convenient to introduce nondimensional

quantities defined in such a way so that all the quantities are of the same

magnitude. So the dimensionless variables used in this analysis are defined

as follows:

X

=

-,

x*

À* S U

=

- ,

u*

u*

CD

=

r..

Re 1/2 y À* CD

s

_ v*

R

1/2 V - - e ,

*

CD uCD

k

=

k*

= L

c* 1-1*

_p

Pr

CD

(2.19)

T

_ T*

_{- T*'}

CD

*

p

=

ie

p!,

(2.20)

T*

Tp =..:2.

_T*'

CD

(2.21)

(2.22)

where the particle velocity equilibrium length

À~

based on the shock

velocity is chosen as the characteristic length of the problem

p* d*2 "A.*

=

s

U*

s

18~!.

s

and the flow Reynolds number Re~ is defined as

Then the velocity and thermal equilibrium time can be expressed as "A.*

_s

u*

_s

* "A.*

* _

3Pr ~~ s ' t -T 2a ~* u~

(2.23)

(2.24)

(2.25)

(2.26)

Using the nondimensional quantities given by (2.19)-(2.22), the boundary-layer equations can be written in the following nondimensional form:

Gas continu i ty: _~

pU + ~ pv

=

0 ox oy

(2.27)

Gas x-momentum: (2.28) Gas energy: pU oT + pv oT

=

L

[0 (

oT)] oX oy P r oy

~

oy Us [ + - Pp (T p - T) ~Nu ] 3Pr (2.29)

Gas state:

_ 1

p

-T

Parti cle conti nuity:

.Q... u +.Q... o....v

=

0 êx

Pp

P êy ' fJ P

Particle x-momentum:

êU_n êU_n u - - L + V - - L

=

P êx P êy

Particle y-momentum:

Parti cle energy:

êT êT a U

up ..::...E..

+

_{vp ..::...E..}

= -

_{_ s [(Tp - T) IJ.Nu]}

êx êy

3Pr

where Ec is the Eckert number for the gas:

U*2

Ec

= Cl)

c* T*

_P

00

and Us

=

u;/u! is the nondimensional shock velocity.

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

Finally, with the nondimensional variables

(2.20)-(2.22),

the particle

slip Reynolds number Re s becomes

pl(u p2

+

vp2/Reoo) 1/2 - (u 2

+

_{v2/Re oo) 1/21}

Re

s

=

Rep

----'=:...-_~ _ _ _ _ _ _ _ _ _ _ _ _

IJ.

(2.36)

where Rep is the particle Reynolds number based on the freestream values:

P* u* d*

R

00 00

e

_p =

'ti

For boundary-layer flows the flow Reynolds number is usually ITlJch greater than unity ( i . e . , ReCX»> 1). This means that u2 » v2/ReCX) and

U_p2 _{» v p}2/ReCX). Sometimes, when these two inequalities are satisfied, the slip Reynolds number Re_s' Eq. (2.36) reduces to

(2.38)

and the energy equation for the gas, Eq. (2.29), becomes

(2.39)

111. BOUNDARY AND INITIAL CONDITIONS

In order to obtain a unique solution to the partial differential equations of the boundary-layer flow (2.27)-(2.34), it is necessary to satisfy the boundary conditions of the problem under consideration and to know the initial conditions applicable to it. For boundary-layer problems, the velocity and temperature at the outer edge and at the wall are set up for the boundary conditions, and the velocity and temperature profiles at the leadingt edge (or a certain specified position near the leading edge) for the initial conditions.

The velocity and temperatu re at the ou ter edge are determi ned from the flow properties in state (2) behind the shock front. In the present paper, it is assumed that everywhere in state (2), the partieles reach the equilibrium-flow limit of the gas velocity and temperature, i.e., u*p

=

u*

2 2 and T*p

=

T*. The dusty gas in this limit behaves effectively as a perfect

2 2

gas with modified thermodynamic properties [81]:

c* + _~c; 1 +

ê/

a: -*

₌

E

₌

c* cp 1 + _~ P 1 + ~ (3.1) c* + _~c; 1 + ~y./ a: -*

₌

v

₌

_c* Cv 1 + _~ v 1 + ~ (3.2)

ë* c* + ~c*s y=~=

p

=

-* c* + ~c* Cv v s R* R* = -1 + ~ (ex +

êh

ex + ~y

(3.3)

(3.4)

where ~ is the loading ratio. Then the equilibrium sound speed can be found as

= [ (ex + ê)y R*T*]1/2 =

(1 + ~)(ex + ~y)

[ (ex+ ê)y ~]l/2

( 1 + ~ )( ex + ~ ) p* ( 3 • 5 )

With these modified quantities in the equilibrium limit, the flow parameters i n state (2) behi nd the shock wave, whi ch are consi dered as the freestream values for the sidewall boundary layer, satisfy the classical shock-tube equat i ons [1]: p* p*

=

If

=

2 p* p* 1 1 ä* (rif - 1) _1 p* a* 1 -

(--.3. -

1) _ _ _ _ _ _ If _ _ : - - - _ p* p* 1 {2

Y

1 [ (

Y

1 -1) + (y 1 + 1) p:]} 1/2 1 p* p* (

h

-1) + (y 1 + 1)

p:

r

21

=

_p* 2

= ________

_p* 1_ 1 (Yl+1 ) + (Yl- 1)

p:

1

(3.6)

(3.7) ....

where

.~

p*

T*

(Y1+1) + (Yl-l)

--1

p* p*

T

=

~

=

1 2 21

T*

p* p*

u*

MS

=

s

ä* 1

u*

M ₂

=-1.

ä* 2 1 _{(Yl-l) + (Y1+1)}

_-2

1

p*

1

p*

1/2

=

{~

[(Yl-l) + (Yl+l)

-2]}

2Yl

p*

1

p*

2 _ 1

p*

1

-

p*

{2:.!.

-2

[(Yl+l) + 2

p*

1

p*

1/2 (Yl-l)

...2]}

p*

1 [ (a + (3) Y ~ ] 1/2 (3.8) (3.9)

(3.10)

(3.11)

(3.12)

(3.13)

p*

J

( 1 + ~ )( a+ I3Y )

p*

1

(3.14)

(3.15)

a* = (Y4 R* T*)1/2 = (yR* T*)1/2

4 4 4 4

(3.16)

The flow parameters in state (2) can be found in terms of the known

quantities in the initial state (1) and (4). In the shock-fixed

coordinates, the flow quantities with subscript CD which reprsent the

freestream val ues of the boundary-layer flow are

p!,

=

p*, p!,

=

p* , T* CD

=

T*

,

u* a> = u* _S

-

u*

2 2 2 2

(3.17)

Pp!, = ~p!, T* _Pa>

=

T* _a>' u* _Pa> = u* _a>

(3.18)

The bounda ry condit i ons at the wa 11 are readily specifi ed for the gas phase since there is no slip, no suction or blowing and no temperature jump

at the wall. The gas velocity relative to the wal1 is zero. The third

boundary condition at the wal1 depends on the thermal state of the wall. For the shock-tube sidewa11, the wa11 temperature remains substantially

constant and is usual1y taken as T*. Consequently, T~

=

T*. However, for

1 1

the particle phase only one condition can be physica11y determined: the

normal velocity at the wal1 is equal to zero, because there is no mass

transfer (suction or blowing). In the shock-fixed coordinates, these

boundary conditions can be expressed as follows:

u:

=

u~,

v:

=

0, v* = 0

Pw

T*

=

T*

w

1 (3.19) (3.20)

The boundary conditions in nondimensional form can be obtained by

substituting Eqs. (2.20)-(2.21) and (3.7)-(3.11) into Eqs. (3.17)-(3.20),

u Cl) = 1, T Cl)

=

1 (3.21)

uPa> = 1, Tpa> = 1, p _Pa> = ~ (3.22)

0, 1

-'

"

(3.24)

Similarly, some other nondimensional quantities used in the basic

equations (2.27)-(2.34) can be expressed in the following form:

u*

Us

=

U!

=

r

2l a> U*2 Ec = _a>_

=

Rea>

=

c* T* P a> p* ä* d* 1 1 Re_pl

=

--\..1.-*--1 P* ä*d* P: u!

~

=

Re {s 1 \..1.: p \..1.* 1

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

Ms

}

18T~i77

(3.30)

In this analysis, a six-point difference scheme was employed for the

x-momentum and energy equations of the particle phase in order to reduce the

truncation error and then additional boundary conditions are required.

These bounda ry condi t i ons can be deri ved from the compat i bi 1 i ty condi t i ons

as follows.

At the wall, since the normal velocity vanishes, Eqs. (2.32)

and (2.34) become

~T

Pw ex Us ]

Up - -

= -

-3P

_{[(Tp - Twhlw Nuw}

w ~x r w (3.32)

Equations (3.31) and (3.32) are ordinary differential equations and can be solved numerically or even analytically. In the case of sidewall boundary layers and non-Stokes interaction relations, it is difficult to integrate Eqs. (3.31) and (3.32) analytically as in the case of flat-plate boundary layers and Stokes interaction relations [76J. However, they are readily solved numerically provided that the values of upw and Tpw at the initial position (say, the leading edge at the shock front) are known. Clearly, these two initial values can be set to the corresponding freestream values, that is, up (0)

=

1 and Tpw(O)

=

1. In this paper, the solutions to Eqs.

(3.~1)

and (3.32) were obtained using Gearls method [82J

and they give the particle velocity and temperature at the wall as a function of the distance x:

(3.33)

(3.34)

These two compatibility conditions supply additional boundary conditions at the wall for the particle velocity and temperature.

The initial flow profiles used in this paper are the so-called extended Wu-type of initial profil es at the leading edge. The extended Wu-type initial profiles have been proven as simple and reasonable initial conditions for dusty-gas boundary-layer flows provided a proper step size is chosen [76J. The extended Wu-tyipe profiles take the following form for the gas phase: u(O, y)

₌

{

Uw 1 y

=

0 y > 0 v(O, y)

₌

0 T(O, y)

{

Tw

=

1 y

=

0 (3.35) y > 0

{

l/Tw Y

=

0

p{O, y)

₌

1 _Y

>

°

and for the particle phase:

up(O, y)

=

1, vp(O, y)

=

0, Tp(O, y)

₌

1, pp(O, y)

=

_~ (3.36)

The nondimensional basic equations (2.27)-{2.34) with the boundary conditions (3.21)-(3.24) and (3.33)-(3.34), and the initial conditions (3.35)-(3.36) consist of the complete mathematical description for the sidewall boundary-layer flows in a dusty-gas shock tube. They can be solved numerically with a finite-difference procedure.

IV. FINITE-DIFFERENCE PROCEDURE

The problem of sidewall boundary-layer flows in a dusty-gas shock tube can be solved with a finite-difference procedure. This process is similar to that used for the flat-plate boundary-layer analysis [76J. The main difference between them is that the x-momentum and energy equations for the particle phase are discretized by a six-point scheme along the whole length from the leading edge to far downstream, in the sidewall case. By contrast, a four-point scheme and a six-point scheme are used respectively before and af ter the critical point for the flat-plate case, since af ter the critical point very simple compatibility conditions can be obtained. However, for the sidewall case, there is no critical point along the wall because the particles at the wall are accelerated. Besides, the wall boundary condition for the tangential velocity of the gas in the sidewall case is different from that of the flat-plate case in which the tangential velocity at the wall vanishes. This change in the boundary conditions results in some difficulties in the finite-difference procedure: a finer mesh and a proper ratio of step size are required in order to obtain reasonable solutions.

In the finite-difference method, the partial differential equations are replaced with finite-difference equations. The flow quantities are assumed known at the grid points in column (m) and unknown in column (m+l) (see Fig. 5). In the present paper, a six-point implicit scheme is used for the x-momentum and energy equat i ons of the two phases, whil e three different four-point implicit schemes are applied to the continuity and y-momentum equations respectively. In these implicit schemes, the derivatives are replaced with linear differnce quotients and the partial-diffrerence equations are evaluated at the mid-way point (m+9, n), where the parameter 9

unity. Of course, the value of e should be suitably adjusted to improve the convergence of the finite-difference schemes. In this analysis, e

=

0.75.

For the six-point scheme, with the function W(x,y) representing the dependent variables, the difference quotients are written as,

W

=

_{9Wm+1,n + (1-e)Wm,n}

öW

=

1 (W W )

öx !sJ< m+1,n - m,n

+ 2(1-e)K _{[Wm,n+1 - (K+1)Wm,n + KWm,n_d}

(K+l)AY n2

where K is the ratio of step size in the y-direction:

K

=

AYn AYn-1 (4.1) (4.2) (4.3) (4.4) (4.5)

When these difference quotients and expressions, Eqs. (4.1)-(4.5), are used for the partial-differential equations (2.28)-(2.29), (2.32) and (2.34), the finite-difference equations become the simultaneous algebraic equations:

(4.6)

where n

=

2, 3, ••• , N-1 and N represents the gri d poi nt at the outer edge of the boundary layer. The superscript i in Eq. (4.6) is the index for dependent variables:

Wl

=

u, W2

=

T, W3

=

up' and

W4

=

_{Tp •}

The difference equation (4.6) is a system of linear algebraic equations of the tridiagonal type. It can be solved with the Thomas algorithm provided the coefficient matrix is known [83]. The coefficients in Eq. (4.6) are given below:

(i) For the gas x-momentum equation (i = 1, WI = u)

AnI

=

a n (pv - 11 I _{,... Y m+S,n} T ) - C _nrllJTS,n LL,

(4.7)

(ii) For the gas energy equation (i = 2, W2 = T):

I

An2

=

an(pv - L T ) - c

(L)

Pr y m+S,n n Pr m+S,n

D~

=

[(pU)m+9,n - (I_S)6X(u's"

o...~u)

_{]Tm n - bh(pv _}.

L

_{Ty )}

~Tm

n 3Pr . fJ m,n ' Pr m+S,n '

+ d

(L)

~2T

+. 6X{EcIJ.U2. + Ec u o...[(u -u)2 + _1_ (v

-v)2]~

n Pr m+S,n m,n Y S" fJ P Rea> p

u

+ _5_ P T IlNu}

3P _r P P _{m ,n} +S

(4.8)

(iii) For the partiele x-momentum equation (i

=

3

W

3

₌

_Up):

A3 = a v n n Pm+s,n Bn3 = U + a (K2_1)v + Sóx(u lill) 1 Pm+s,n n . Pm+s,n 5 m+ ,n

c

3 _{= -a K}2_v n n Pm+s,n

(4.9)

(iv) The partiele energy equation (i

=

4, W

4

= T

p): A4

=

a v n n Pm+s, n a U B4

=

U + a (K2-1)v + SÓX( _ _ 5 IlNu) n Pm+s,n n Pm+S,n 3Pr m+1,n

c

4 ₌

_-a

_K2

_v

n n Pm+S,n

(4.10)

a U D 4

=

[u - (1- s ) óx ( _ _ 5 IlNu) ]T P - b v ~ T n Pm+S,n 3Pr m,n m,n n Pm+S,n Pm,n where

a

=

S~x n (K+1)

_AY

n' b =

(l-s)l\)(

n (K+1)

_AY

n' dn

=

2(1-S)Kóx

.

.

(4.12)

For the gas continuity equation, the difference quotients take the

form,

(4.13)

aW = e ( W W )

_{m+1,n -}

_m+1,n-1

+ 1- e ( W_{m , n -} W_{m, n} -1 )

ay

AY n -1

AY n-1

Then the corresponding finite-difference equation is:

AY n-1 [ ( )

( )

( )

( ) ]

- 296X

pu m+1,n -

pu m,n + pu m+1,n-1 -

pu m,n-1

(4.14)

where n

= 2, 3, ••• , N.

Equation (4.14) is used to obtain the normal

velocity of the gas.

The partiele continuity equation is used to obtain the partiele

density. It is discretized with the following four-point scheme:

W

=

SW

_{m+1,n + (1-e)wm,n}

aW -

1 (W W )

-

- -

m+ 1 n -

m, n

ax

6X '

öW -

e

(W W ) + 1-

e

(W W )

öy - AY n

m+ 1 , n + 1 -

m+ 1 , n

AY

n

m, n + 1 -

m, n

The resulting difference equation can be written as

A5 P

+

SS P

=

CS

n Pm+1,n+1

n Pm+1,n

n

(4.16)

where n

=

1,

2, ••• ,

N-l.

The coefficients are given by

A5

=

eill< v

n

AYm Pm+e,n

+ e(l-e)ill< (v

- 2v

)

!::if

_n

Pm,n+l

Pm,n

CS

=

_[(I-e)ill< v

]p

_{+ [(2e-1)u p}

_{+ 2(I-e)u p}

n

AYn

Pm+e,n P

m

,n+l

m+l,n

m,n

- e(l-e)ill< (v

- 2v

) - (l-e)26)( (v

- 2v

_p

)]0...

AYn

Pm+l,n+l

Pm+1,n

!::if

_n

Pm,n+l

m,n

'Pm,n

(4.17)

The partiele y-momentum equation is then used to find vp and its

four-point difference representation is,

A6 V + S6 V

=

C6

n Pm+l,n

n Pm+l,n-l

n

where n

=

2,3, ••• ,

N,

and

A6

=

n u

_Pm+e,n

+ S 6

= _

etsx

n !::if

n-l

v

Pm+e,n

(4.18)

c~

=

[Up _{- (1-9)6>< Vp - (1-9)6>« us lill)m,n ]} VPm,n

m+9,n AYn-1 m+9,n

+ [(l-9)ill( v ] V + 6>«u vlill)

Av Pm+9 n Pm n-1 s m+9,n

'-\1n-1 ' ,

(4.19)

For this y-momentum equation, the following quotients are employed:

W

=

_{9Wm+1,n + (l-9)Wm,n} oW - 1 (W W ) - - - m+ 1

n -

m, n

ox 6>< ' oW

=

9 oy AYn-1 ( _{Wm+1,n - Wm+1,n-1 +} ) 1-9 ( _{Wm,n - Wm,n-1} ) AYn-1 (4.20)

The finite-difference equations presented above are solved stepwise from the leading edge and downstream, <7iIing to the parabolic character of the boundary-layer equations. The details of the computer program are given

in the Appendix. This program includes three different techniques of

solution: linearization, predictor-corrector and iteration. In the

linearization procedure, the elements of the coefficient matrix are

evaluated with the flow parameters at the previous grid line (m) unless the

flow properties at the present grid line (m+1) are known. In the iteration

procedure, the first Iguessl _{solution is obtained with the linearized values}

[i.e., the values at the point (m, n)], and then the successive solutions are obtai ned with the weighti ng averaged values [i .e., the val ues at the

point (m+9, n)] until a desired accuracy is achieved. In the

predictor-corrector procedure, only one iteration is employed. In general ,

the iteration technique is more accurate but time-consuming, while the

linearization technique is simple but less accurate. In those aspects

concerning the computation accuracy and time, the predictor-corrector

technique is a compromise. From the experience in this analysis, 5 ~ 10

iterations are required for the tolerance € of 10-5, and the solutions by

using the linearization technique within 1%. Therefore, for practical

purposes, the linearization procedure provides finite-difference solutions with acceptable accuracy.

v.

BOUNDARY-LAYER PARAMETERS

The foll owi n9 three parameters are the boundary-l ayer characteri sti cs of interest: the wa" shear stress, the rate of heat transfer at the wa 11 , and the displacement thickness. These quantities are given by:

1:.~

=

u*:(ou*) -w rw oy* W ci~

=

-k* (oT*) 'w w oy* w ö*

=

f

(1

o

*u*

e

)dy* P* u* Cl) Cl)

(5.1)

(5.2)

(5.3)

Similarly, it is convenient to employ nondimensional parameters which are defined as can

~

R

1/2 'tw

=

eCl) p* u*2 Cl) Cl) .*

R

1/2 •

₌

qw qw

p* u*3 _{ex> ex>} eCl) ö* R 1/2 ö

= -

_À* e _Cl)

s

With Eqs.

(2.19)-(2.22)

and

(5.4)-(5.6),

be written in nondimensional form as,

qw

= -

~

(oT) EcPr oy w

(5.4)

(5.5)

(5.6)

the expressions

(5.1)-(5.3)

(5.7)

(5.8)

CD

ö

=

f (

1 -

pIJ)

dy

o

(5.9)

In the above expressions, the nondimensional variables u, Tand pare known

at the discrete grid points af ter solving the boundary-layer equations using

a finite-difference method.

In order to obtain the boundary-layer

characteristics, it is necessary to calculate the derivatives and the

integration in Eqs. (5.7)-(5.9).

In order to evaluate the derivatives, the gas velocity u and

temperature T near the wall can be approximated in terms of an interpolation

polynomi al

(5.10)

where w represents u or T. The derivative at the wall can be obtained,

e~W) =

b

oy w w (5.11)

With the values of w at the first four grid points from the wall, the

coefficient bw can be determined,

(5.12)

where Wl' W

_{2 '}

W

₃

and Wit are the values of W at the first four grid points.

Therefore, the nondimensional shear stress and heat transfer rate at the

wall are given by

=

1+K+K

2

!lw

{(u

-u

) _

1

(u

3

-u

l)

+

1

(ul.-uI)}

~w

_K2

_~ A\/I

2

I

K(1+K)

_K(1+K+K2)2

~

1+K+K2

K2

(5.13)

(5.14)

The integration in Eq. (5.9) is eva1uated by a three-point difference formu1a and it resu1ts in

ö

=

~

6Yn-1 [2+3K (1- u)

~ 6 1 +K P n-1 +

1~3K

(l-pu)n -

-K(--=-~-+K-)

(l-pU)n+l

J

(5.15) With the ca1cu1ated profiles for the gas velocity and temperature, the boundary-1ayer parameters can be obtained by using Eqs. (5.13)-(5.15).

VI. RESULTS AND DISCUSSIONS

The basic equations for 1aminar sidewall boundary-1ayer f10ws in a dusty-gas shock tube, Eqs. (2.27)-(2.34), were solved numerically using an imp1icit finite-difference scheme presented in this paper. In the computation, the conditions in stat es (1) and (4) are specified as

and p*

=

1.01325x105 Pa, 1 . T*

=

T* It 1 T*

=

300K 1

The wall temperature can be taken as T* or T*. Besides, the following thermodynamic parameters are chosen in

thi~ ana1~is:

Pr

=

0.75, y

=

1.4, a

=

1.0, P~ = 2.5x103 kg/m3

The flow parameters in state (2) are required for they give the boundary conditions at the outer edge. In the equilibrium flow limit, the stati onary-f1 ow parameters in the dusty-gas shock tube are readily determined from Eqs. (3.6)-(3.16). These shock re1ations are the same as those for the pure-gas case except the modifi ed thermodynami c properties (3.1)- (3.5) shou1d be emp1oyed. The pressure ratio across the shock front P₂₁ as a function of Plt1 is shown in Fig. 6, where the solid 1ine is for ~ = 1.0 and the dashed 1i ne is for ~ = 2.0. It is found that with increasing mass-1oading ratio ~ of the partieles, the pressure ratio P₂₁ increases when P₄₁ keeps the same va1ue. With the va1ue of P₂₁ known, the shock Mach number and the other flow parameters in state (2) can be ca1cu1ated. They are 1isted in Tab1e 1.

Table 1

Dusty-Gas Shock Parameters in Equilibrium-Flow Limit

P 41 ~ MS Mco _M2

r

₂₁ T 21 A21

1.0 1.1985 0.8385 0.3268 1.3898 1.0577 1.0285 2.0

2.0 1. 2189 0.8236 0.3710 1.4505 1.0410 1.0202 4.0 1.0 1.4283 0.7137 0.6344 1.8889 1.1224 1.0594

From these shock parameters, the boundary conditions for the sidewall

boundary layer can be derived:

T

=

_l_

w T ' 21

Then the Eckert number, flow Reynolds number and particle Reynolds number

can be calculated.

For the two Reynolds numbers, the diameter of the

particle should be specified. They are given in Table 2.

Table 2

Values of Flow Similarity Parameters

P

_{41 ~}

d*

Ec

Re

co

Rep

( ~) 10 0.1172 3.1653x106 _1.6371x102 1.0 2.0 40 0.1172 5.0646x107 _{6. 5482x10 2} 2.0 10 0.0714 2.1191 x106 _{1.3394 xlO 2} 4.0 1.0 10 0.0849 4.1030x106 _{1. 864 xlO2}

From the values listed above, it is seen that the particle Reynolds number is quite high in dusty-gas shock-tube studies. It means that the Stokes relations are no longer valid for shock-tube sidewall boundary-layer flows. Therefore, only non-Stokes I rel at i ons for the interaction terms between the

two phases, say, Eqs. (2.16)-(2.17), are considered in this research. Numeri cal results give the flow tructures of the dusty-gas sidewall boundary layer along the whole length from the leading edge fixed at the shock front to far downstream. The effects of the pressure ratio, loading ratio and particle size on the flow properties of sidewall boundary layers are discussed.

The detailed flow structures for the two phases in the sidewall boundary layer with Plt1

=

2.0 and ~

=

1.0 are shown in Figs. 7 to 12. They are, respectively, for x

=

0.05, 0.10, 0.5, 1.0, 5.0 and 10.0. From these figures, it is seen that, similar to the case of flat-plate boundary-layer flows, there exist three different flow regimes, which are characterized by large, moderate and small slip between the gas and particles. In fact, when x ;> 5.0, a quasi-equilibrium state between the two phases is already

achieved. By contrast to the flat-plate case, the tangential velocity has its maximum value at the wall instead of zero velocity for the flat-plate case. Therefore, the particles are accelerated in the sidewall case and the normal velocity has a negative value as well as for the displacement thickness. Nevertheless, the existence of three distinct flow regimes is the common feature for dusty-gas boundary-layer flows.

Figures 13 to 18 give the flow profiles for the case of P41 = 4.0 and

~

=

1.0, with the same loading ratio and higher pressure ratio compared with the previous case. They are also for x = 0.05, 0.1, 0.5, 1.0, 5.0 and 10.0. In general, they show a similar behaviour as for the case of P₄₁ = 2.0, that is, large, moderate and small slip successively appear with increasing distance x from the leading edge. However, because of a greater pressure ratio the shock speed u~ and then the nondimensional wall velocity Uw = Us in the shock-fixed coordinates increase. Hence, there is larger slip between the two phases in the nea r 1 eadi ng edge reg i on. Consequent ly, greater interaction terms arise. As aresult, much more significant changes in the flow properties of the particles occur at the same distance (compare Fig. 7 with Fig. 13) and rruch larger normal velocity is induced (for example, compare Fig. 9 with Fig. 13). However, as the particle velocity increases, the slip between the gas and particles decreases, and the particle slip Reynolds number decreases. Far from the leading edge, the slip Reynolds number becomes less than one and then the interaction terms become the same. Therefore in the far-downstream reg i on, two-phase flow structures are similar for the two cases: they both achieve a quasi-equilibrium state at x

=

10.0.

Boundary-layer characteristics ('tw' qw and 6) for these two cases are shown in Fig. 19. All three quantities have negative values, as expected

for the sidewall boundary layers. With increasing pressure ratio P41' the

three quantities increase in magnitude. Dwing to the presence of particles, the shear stress and heat-transfer rate increase more than in the corresponding pure-gas case, especially in the leading-edge region.

To study the effects of the mass loading ratio on the boundary-layer flows, the numeri cal results for the case of P₄₁ = 2.0 and ~ = 2.0 are given in Figs. 20 to 25, which are, respectively, for x

=

0.05, 0.1, 0.5, 1.0, 5.0 and 10.0. Compared with the results for the case of P41 = 2.0 and ~

=

1.0 (i.e., Figs. 7 to 12), there are no significant changes concerning the flow features. That is, qualitatively, there are three distinct flow regimes and similar trends in the flow profiles. Besides, quantitatively, the flow properties of the particles do not change very much with the loading ratio, especially in the large slip region. This phenomenon can be explained as follows. The particle density does not appear in the momentum and energy equations for the particle phase, Eqs. (2.32)-(2.34). Physically, it means that for each particle, the force exerted by the gas (or the heat transfer from the gas) depends on the particle velocity (or temperature) and the gas flow parameters, but not on the particle density directly. Of course, it should be pointed out that the particle density can have some effects on the particle motion but it just goes through the gas velocity and temperature. It is found in the basic equations for the gas phase that the particle density appears in Eqs. (2.28) and (2.29) as an influencing factor in addition to the gas viscosity. Then the gas flow profiles can be modified by the particle density, although this influence may not be important in the large or small slip region. Moreover, for sidewall boundary layers, the increase in loading ratio of the particles results in a stronger shock wave. In other words, the pressure ratio P₂₁ increases ànd then the shock speed u~ (on the nondimensional wa" velocity of the gas uw) increases too. This causes some changes in the flow profiles of the gas, as found in Figs. 20 to 25. Consequent ly, in the mode rate and sma 11 s 1 i p reg i ons, there are some changes in the flow profiles of the particles. As a result of the changes in the gas flow properties, the shear stress and heat-transfer at the wall increase while the displacement thickness decreases. This variation of boundary-layer characteristics with the loading ratio can be seen in Fig. 26.

To discuss the effects of the particle size on the boundary-layer flows, some comparative computations were done. All the numeri cal results given previously are for the case of particle diameter d* = 10~. The flow properties for the case of d*

=

40 ~, P 41

=

2.0 and ~ = 1.0 are presented in Figs. 27-32, for x

=

0.05, 0.1, 0.5, 1.0, 5.0 and 10.0. Comparing Figs. 27-32 with Figs. 7-12, it is seen that the change in particle size has a 1 most no i nfl uence on the gas fl ow propert i es. The reason is th at the particle diameter appears as a parameter only in the expressions for Reynolds number and equilibrium length. In this analysis, the computation was performed in a nondimensional form, where the equilibrium length is

effect on the f1 ow parameters of the gas. from the results for the boundary-layer However, for the particles, their size has

with ,increasing particle diameter, the

decreases.

VII. CONCLUSIONS

The same conclusion can be drawn characteristics (see Fig. 33). some influence on their motion:

slip between the two phases

The implicit four- and six-point schemes presented in this analysis can be used for laminar sidwall boundary-layuer flows in a dusty-gas shock tube.

The numerical solutions were obtained for several examples. They describe

the development of the sidewall boundary layer only between the shock front and the free-stream contact surface and give the boundary-layer

characteristic parameters and show the effects of the pressure ratio P4b

mass loading ratio ~, and particle diameter d* on the flow properties for

the two phases have also been shown. In the dusty-gas sidewall boundary

layers, there exist three distinct flow regimes (large, moderate and small

slip), similar to the case of flat-plate boundary layers. This is a major

feature common to both types of dusty-gas boundary-layer flows.

Major simplifications were made in the present analysis in order to

solve this difficult problem. For example, the dusty-gas shock structure

was assumed as a plane. The dusty-gas shock-wave relations in the

equilibrium-flow limit were used to calculate the free-stream conditions. In the future, the development of the boundary layer through the lengthy shock-wave-structure should be considered, as well as the overlap of the hot and cold gas in the boundary layer between the diaphragm station and the

free-stream contact surface. Finally, the rarefaction wave induced boundary

layer will have to be solved in order to obtain all the effects on the freestream properties in states (2) and (3), which may no longer be uniform.

The foregoing are difficult tasks that await resolution. In addition, since

the boundary layer undergoes transition from laminar to turbulent flow, the foregoing wil 1 have to be reconsidered for turbulent boundary layers as well.

REFERENCES

1. Gl ass, 1. 1. and Hall, J. G., Handbook of Supersoni c Aerodynami cs, Sec. 18: Shock Tubes. NAVORD Report 1488 (Vol. 6), Dec. 1959.

2. Davi s, L., "Vi scous Effects on Shock Tube Fl ows", VKI Short Course on Advanced Tube Techniques, Vol. 2, 1969.

3. Fuehrer, R. G., "Measurements of Incident-Shock Test Time and Reflected Shock Pressure at Fully Tu rbul ent Boundary -Layer Test Condit i ons" , Proc. 7th Inter. Shock Tube Symp., University of Toronto, Toronto, June 1969, pp. 31-59.

4. Bazhenova, T. V., Naboko, I. M. and Nemkov, R. G., "Experimental Study of the Boundary Layer Effects on the Di stri buti on of Flow Parameters Behind a Shock Wave in a Shock Tube", Proc. 7th Inter. Shock Tube Symp., University of Toronto, Toronto, June 1969, pp. 60-68.

5. Hubbard, E. W. and deBoer, P. C. T., "Flow Field Behind a Shock Wave in a Low Pressure Test Gas", Proc. lth Inter. Shock Tube Symp., University of Toronto, Toronto, June 1969, pp. 109-125.

6. Lacey, J. Jr., "Experimental Shock Tube Test Time - Turbulent Regime", Proc. 7th Inter. Shock Tube Symp., University of Toronto, Toronto, June 1969, pp. 126-142.

7. Dumitrescu, L. Z., Popescu, C. and Brun, R., "Experimental Studies of the Shock Reflection and Interaction in a Shock Tube", Proc. 7th Inter.

Shock Tube Symp., University of Toronto, Toronto, June 1969, pp.

751-770.

8. Ackroyd, J. A.D., "Laminar Boundary Layers Behind Strong Moving Shock Waves", Aeronautical Journal, Vol. 74, 1970, pp. 155-159.

9. Belles, F. Land Brabbs, T. A., "Experimental Verification of Effects of Turbulent Boundary Layers on Chemical-Kinetic Measurements in a Shock Tube", NASA-TM-X-52761, 1970.

10. Brabbs, T. A. and Belles, F. L, "Experimental Study of Effects of Laminar Boundary Lasyers on Chemical-Kinetic Measurements in a Shock Tube", NASA-TM-X-=52950, 1970.

11. Murdock, J. W., "A Solution of Shock-Induced Boundary-Layer Interaction Problems by an Integral Method", AS ME Paper 71-APM-21, 1971.

12. Akamatsu, T., "On Some Aspects of Unsteady Boundary Layers Induced by Shock Waves", Proc. Symp. on Unsteady Boundary Layer, Quebec, May 1971,

13. Mirels, H., "Boundary Layer Growth Effects in Shock Tubes", Proc. 8th Inter. Shock Tube Symp., Imperial College, London, July 1971, pp. 6/1-6/30.

14. Light, G. C., "Effects of Shock Wave Deceleration on Ionization Level

in a Shock Tube", AD-725733, 1971.

15. Morkovin, M. V., "Lessons from Transition of Shock Tube Boundary

Layers", AD-738355, 1971.

16. Walker, J. D. A. and Dennis, S. C. R., "The Boundary Layer in a Shock Tube", Journalof Fluid Mechanics, Vol. 56, 1972, pp. 19-47.

17. Dumitrescu,

L.

Z., "An Attenuation-Free Shock-Tube", Physics of Fluids,

Vol. 15, 1972, pp. 207-209.

18. Connor, J. G., Boundary Layer Closure in the Conical Shock Tube", Physics of Fluids, Vol. 15,1972, pp. 1403-1408.

19. Macken, N. A., Besse, A.

L.

and Levine, M. A., "Shoek Tube Boundary

Layers in Ionized Argon-Helium Mixtures", AIAA Journal, Vol. 10, 1972, pp. 1129-1130.

20. Lida, S. and Higashida, A., "Shoek Attenuation in a Shock Tube due to

Boundary Layer", Japan Society for Aeronautical and Space Sciences, Transactions, Vol. 15, No. 30, 1973, pp. 193-207.

21. Gupta, R. N., "Nonlinear Problem of a Shock-Tube Interaction-Region

Boundary Layer", AIAA Journal, Vol. 11, 1973, pp. 548-551.

22. Dumit rescu,

L.

B. , Bru n, R. and Si des, J. , "Computat i on of Gas

Parameters and Compensation of Attenuation Effects on Real Shock-Tube Fl ows", Proc. 9th Inter. Shock Tube Symp., Stanford University, 1973, pp. 439-448.

23. Golobic, R. A., Measurements of Boundary Layer Transition on a Shock

Tube Wall for Wall Cooling Ratios Between 0.39 and 0.255", AD-A021406, 1975.

24. Srinibasan, G. and Hall, J. G., "Heat Transfer in Laminar Wall Boundary

Layer Within Noncentered Unsteady Expansion Wave", Proc. lOth Inter. Shock Tube Symp., International Conference Hall, Kyoto, Japan, July 1975, pp. 102-110.

25. Boison, J. C., "Highly Cooled Boundary Layer Transition Data in a Shock Tube", Proc. 10th Inter. Shock Tube Symp., International Conference Hall, Kyoto, Japan, July 195, pp. 127-140.

26. Bertin, J. J., Ehrhardt, E. S., Gardiner, W. C. Jr. and Tanzawa, T.,

Chemically Reactive Test Gas", Proc. 10th Inter. Shock Tube Symp., International Conference Hall, Kyoto, Japan, July 1975, pp. 595-604. 27. deBoer, P. C. T. and Miller, J. A., "Boundary Layer Effects on Density

Measurement in Shock Tubes", Journalof Chemical Physics, Vol. 64, 1976, pp. 4233-4234.

28. Bander, J. A. and Sanzone, G., "Shock Tube Chemistry 1 - The Laminar-to-Turbulent Boundary Layer Transition", Journalof Physical Chemi stry, Vol. 81, 1977, pp. 1-3.

29. Van Dongen, M. E. H., "Light Reflection as a Simple Experimental Method in Compressible Boundary-Layer Studies", AIAA Journal, Vol. 15, 1977, pp. 1210-1212.

30. Demmig, F., "Ionization Relaxation in Shock Tubes under the Influence of a Weakly Unsteady Shock Front and Wall Boundary Layer Effects", Proc. 11th Inter. Shock Tube Symp., Seattle, July 1977, pp. 119-126. 31. Takano, Y., Shimomura, Y. and Akamatsu,

T.,

"Ionizing Argon Flows in

Boundary Layer and in Free Stream in a Shock Tube", Proc. llth Inter. Shock Tube Symp., Seattle, July 1977, pp. 313-320.

32. Ando, H. and Asaba,

T.,

"Evaluation of Boundary Layer Effect on Particle Time of Flight and lts Application to Study on Reaction Kinetics of Nitric Oxide and Hydrogen System", Proc. llth Inter. Shock Tube Symp., Seattle, July 1977, pp. 321-328.

33. Bracht, K. and Merzkirch, W., "Schlieren Visualization of the Laminar-to-Turbulent Transition in a Shock-Tube Boundary Layer", Proc. Inter. Symp. on Flow Visualization, Tokyo, Japan, Oct. 1977, pp. 241-246. 34. Novgorodov, M. A., Poliakov, L. A., Tishchenko, V. A., Khandurov, N. V.

and Chekalin, E. K., "Boundary Layer in a Shock Tube and lts Effect on Current Flow", Soviet Physics-Technical Physics, Vol. 23, 1978, pp. 662-664.

35. Brun, R., Auberger, P. and Qué, N. V., "Shock Tube Study of Boundary Layer Instability", Acta Astronautica, Vol. 5, 1978, pp. 1145-1152. 36. Hall, J. G. and Amr, Y. M., "Stability and Transition of Wall Boundary

Layers Induced by Moving Waves", AD-A082534, 1979.

37. Zeitoun, D., Brun, R. and Valletta, M. J., "Shock-Tube Flow Computation Includi ng the Di aphragm and Boundary Layer Effects", Proc. 12th Inter. Symp. on Shock Tubes and Shock Waves, Jerusalem, July 1979, pp. 180-186.

38. Liu, W. S., Glass, 1. I. and Takayama, K., "Coupled Interactions of Shock -Wave Structure with Lami nar Boundary Layers in Ioni zi ng-Argon Flows", Journalof Fluid Mechanics, Vol. 97, 1980, pp. 513-530.

39. Oeckker, B. L., "Boundary Layer on a Shock Tube Wall and at a Leading Edge USing Schlieren", Proc. Inter. Symp. on Flow Visualization, Bochum, Sept. 1980, pp. 555-559.

40. Reese, T., Demmig, F. and Bc)ttichcer, W., "Determination of Shock Tube Boundary Layer Parameter Utilizing Flow Marking", Proc. 13th Inter. Symp. on Shock Tubes and Shock Waves, Niagara Falls, Canada, July 1981, pp. 271-279.

41. Seil er, F., "Boundary Layer Infl uenced Shock Structure", Proc. 13th Inter. Symp. on Shock Tubes and Shock Waves", Niagara Falls, Canada, July 1981, pp. 317-325.

42. Smeets, G. and George, A., "Investigation of Shock Boundary Layers with a Laser Interferometer", Proc. 13th Inter. Symp. on Shock Tubes and Shock Waves., Niagara Falls, Canada, July 1981, pp. 429-435.

43. Dumitrescu, L. Z., Brun, R. and Sides, J., "Computation of Gas Parameters and Compensat i on of Attenuati on Effects of Rea 1 Shock -Tube Flows", Proc. 13th Inter. Symp. on Shock Tubes and Shock Waves", Niagara Falls, Canada, July 1981, pp. 439-448.

44. Knapp, K., Schneider, C. P. and Hindelang, F., "Performance of LOl in Predicting Density Profiles of Compressible Boundary Layers", Proc. Inter. Congress on Instrumentation in Aerospace Simu1ation Faci1ities, Oayton, Ohio Sept. 1981, pp. 261-270.

45. Di11on, R. E. Jr. and Nagamatsu, H. T., "Heat Transfer Rate for Laminar, Transition and Turbulent Boundary Layer and Transition Phenomenon on Shock Tube WallIl

, AIAA Paper 82-0032.

46. Cook, W. J. and Chaney, M. J., "Further Experiments on Shock Tube Wall Boundary-Layer Transition", AIAA Journa1, Vol. 21, 1983, pp. 1046-1048.

47. Shirai, H. and Tabei, K., "Ionization Re1axation of Argon in Shock Tubes, Inc1 udi n9 the Combi ned Effects of Wa 11 Boundary Layers and Non-Maxwell E1ectronsll

, Japan "Society for Aeronautica1 and Space

Sciences, Transactions, Vol. 26, 1983, pp. 55-64.

48. Tabei, K. and Shirai, H., "Ionization Re1axation Behind Xe and Ne Shock Waves in Shock Tubes - Effects of Non-Maxwell E1ectrons and Wa11 Boundary Layers" JSME Bulletin, Vol. 26, 1983, pp. 1575-1582.