Normal reflection of a shock wave at rigid wall in a dusty gas

March

1984

\

NORMAL REFLECTION OF A SHOCK WAVE

AT A RIGID WALL IN A DUSTY GAS

by

H. Miura, T. Saito and I. I. Glass

6 JUl# 1984

TECHrJISCHE HOGESCHOOL

DELFT

~

LUCHTVAART-

ErJ RUIMTEVAARTIeCHNIEK

BIBLIOTHEEK

Kluyverweg 1 - DELFT

UTIAS Report No. 274

CN ISSN 0082

-

5255

•

NORMAL REFLECTION OF A SHOCK WAVE AT A RIGID WALL IN A DUSTY GAS

by

H. Miura, T. Saito and 1. 1. G1ass

Subrnitted January, 1984

Mareh, 1984 UTIAS Report No. 274

Acknow1edgements

The financia1 assistance received fr om the Natura1 Science and Engineering Research Counci1 of Canada under grant No. A1647, the U.S. Defence Nuc1ear Agency under DNA Contract 001-83-C-0266, and fr om the U.S. Air Force under grant AF-AFOSR-82-0096 is acknow1edged with thanks.

Summary

The flow occurring when a fully developed shock wave in a dusty gas reflects from a wall is studied analytically. By applying an idealized equilibrium-gas analysis, it is shown that there exist three types of shock reflection. The incident shock wave and the reflected shock wave are partially dispersed if PI/PO > P~, the former is partially dispersed but the latter is fully dispersed if P; ~ PI/PO >

pi

and both of them are fully dispersed if

pi

~ PI/PO > 1. Here PI/PO is the incident shock pressure ratio and P~ and Pi are critical values depending on the physical properties of the dusty gas.

The equations of motion are also solved numerically by means of a modified random-choice method and an operator-splitting technique in order to study the time-dependent nonequilibrium flow. The results demonstrate the details of the format ion of the reflected shock wave for the three types noted in the foregoing.

-~ J Contents Acknow1edgements Summary Notation 1. INTRODUCTION

2. REFLECTION OF A SHOCK WAVE AT A WALL FOR EQUILIBRIUM FLOWS

3. TIME-DEPENDENT NONEQUILIBRIUM REFLECTION OF A SHOCK WAVE AT A WALL 4. CONCLUSIONS RE FE REN CES FIGURES iv iii v 2 3 4

a _e af C m C p C _v D d m Pi, P* _r Pr p Q R Re T t u u . sl. U sr v x X Ct i3 Y Ye 9 ).J p 0 'w T Notation

equilibrium speed of sound frozen speed of sound

specific heat of solid material gas specific heat at constant pressure gas specific heat at constant volume drag force acting on a particle particle diameter

mass of a particle critical pressure ratios gas Prandtl number gas pressure

rate of heat transfer to a particle gas constant

particle Reynolds number gas temperature

time

gas velocity

propagation velocity of incident shock wave propagation velocity of reflected shock wave velocity of particles

space coordinate

2 -1 normalized space coordinate x(8m/npd ) mass concentration ratio olp

specific heat ratio of two phases Cm/C v

gas specific heat ratio C _p IC

v

specific heat ratio of idealized equilibrium gas temperature of particles

gas viscosity gas density

mass concentration of particles normalized time t(8m/npd2aOf)-1

v

1. INTRODUCTION

Compressible flows of a dusty gas have recently become of considerable interest in many branches of engineering and science. It is weIl known that dusty-gas shock-wave flows offer important examples

of two-phase relaxation phenomena. Rudinger (Ref. 1)

and Outa et al (Refs. 2, 3) carried out experimental studies using a dusty-gas shock tube for under-standing the interaction between a gas and a number of solid particles. On the analytical side, the format ion of a shock wave in a dusty-gas shock tube was studied numerically by Otterman and Levine

(Ref. 4), Marconi et al (Ref. 5) and Miura and

Glass (Refs. 6, 7). Unfortunately, a straight-forward comparison of the theoretical and the experimental results was not possible owing to the assumption that the distributions of the number and si ze of particles are uniform did not apply in the experiments. However, the analyses provided a good estimate of the relaxation time and length of a shock wave in a dusty-gas flow.

A situation in which a shock wave impacts on

an end-wall may take place in the dusty-gas shock-tube study. Marconi et al (Ref. 5) made a numer-ical study of the flow resulting from the reflection of a shock wave. They considered the case of a strong incident shock wave, in which both the incident and reflected shock waves were partially dispersed. Transitional flow in a case when the incident and/or the reflected shock waves are

fully dispersed has as yet not been analysed.

In the present work, we clarify the basic nature of shock-wave reflection in a dusty gas by means of an equilibrium-gas analysis. Numerical solutions of the equations of motion are also presented to show the detailed change of the flow structure with time.

2. REFLECTION OF A SHOCK WAVE AT A WALL FOR EQUILIBRIUM FLOWS

An idealized equilibrium gas can be utilized in the present problem if we neglect, from a large-scale point of view, the transition region of a shock wave. In this limit, the mixture of a gas and particles is regarded effectively as a perfect gas. lts specific heat ratio and speed of sound are given by (Refs. 8, 9)

a _{e {} y+Ct6

(l+Ct) (1+Ct6) • E. p } 1/2

(1)

(2)

According to perfect-gas theory, the pressure ratio of a reflected shock wave is expressed in terms of the pressure ratio of an incident shock wave as follows (Ref. 10):

where the sub scripts refer to the uniform flow regions shown in Fig. 1. The variations of P21 = P2/PI with PlO = PI/PO for the mixture of

(3)

1

air (y = 1.4) and glass particles (6 = 1) are shown

in Fig. 2 for va lues of Ct over the range 0 $ a ~ 2.

The presence of the particles reduces significantly

the effective specific heat ratio fr om y. As a

result, the reflected shock wave becomes stronger

as the mass concentration of the particles increases for a given incident shock pressure ratio.

Two kinds of transition exist for a shock wave propagating in a dusty gas, that is, a partially dispersed structure and a fully dispersed one. The former, which arises when the velocity of propaga-tion is greater than the speed of sound af= (yp/p)1/2

in a pure gas, consists of a discontinuous jump in the gas phase followed by a relaxation region. The latter has only a smooth transition region and its velocity of propagation is smaller than af.

For the present problem, the propagation veloc-ity of the incident shock wave is given by

Therefore, the incident wave is partially or fully dispersed according to usi ~ aOf, respectively, or

p~

1

2Y(1+a)-Ye+ l

y e +1 (5)

Similarly, the propagation velocity of the reflected

shock wave relative to the uniform flow behind the incident shock wave is given by

The reflected shock wave is partially or ~ully dis-persed according te (Usr-Ul) ~ alf, respectively. We can express this condition from Eqs. (3) and (6) in the form:

P

1 > P*

iJ

< r'

o

The critical pressure ratio

P;

is always greater than

pi

because

P*-P~

r 1

(8)

Thus, we have three types of shock wave reflection as follows:

(i) For PI/PO > P;, both incident and reflected shock waves are partially dispersed.

(ii) For P; ~ PI/PO >

pi,

the incident shock wave is partially dispersed, but the reflected one is fully dispersed.

(iii) For

pi

~ PI/PO > 1, both incident and reflect-ed shock waves are fully dispersreflect-ed.

Figure 3 shows the domains of a different shock-wave reflection on the (a, PI/PO)-plane for a mixture of air and glass particles. It should be noted that P; becomes indefinitely large for some value of a

if the specific heat ratio of the two phases B is extremely small. For the mixture of He and solder partieles, for which

B

= 0.055, P; approaches infinity at a ~ 4.25. The three types (i), (ii) and (iii) can exist for a < 4.25, but .the reflected

shock wave is fully dispersed irrespective of the strength of the incident shock wave for a > 4.25 in this case.

3. TIME-DEPENDENT NONEQUILIBRIUM REFLECTION OF A SHOCK WAVE AT A WALL

The results of the idealized equilibrium-gas analysis are valid when the time aft er the shock-wave reflection is large enough. Details of the formation of the reflected shock wave can be c1ari -fied only by sol ving the time-dependent equations of motion. A numerical ana1ysis of the transient flow is given in this section.

We assume that the mixture consists of a perfect gas and a large number of solid spherical partieles of uniform size. The viscosity and thermal conductivity of the gas are taken into account only for the interaction between the gas and the partieles. Then, the equations governing the mot ion of the mixture are written as follows (Refs. 6-8): ap a

at

+ ax (pu)

o

aa

a

at

+

ax

(av)

o

a

(pu) +

a

(

2) ~ - _ ~ D

at

ax pu + ax - m

a

a

2 at (av) + ax (av ) a (vD+Q) m ~ (vD+Q) m P pRT ~ D m (9) (10) (11) (12) (13) (14) (15)

Here, p, p, T, u are the pressure, density, temper-ature and ve10city of the gas, cr, 8, vare the mass concentration, temperature and velocity of the partieles, Cp' Cv and Cm are the specific heats of the gas and the partieles and m is the mass of aparticle. The drag force D and the heat transfer rate Q acting on a partiele are expressed in terms of the flow quantities as

2

D

i

nd2p(u-v)lu-vl(0.48 + 28 Re-0.85) (16)

Q

where Pr is the Prandtl number of the gas and Re is the Reynolds number based on the diameter d of the partieles

Re plu-vld/)l (18)

For air considered in this paper, the viscosity )l and the Prandtl number are given by

(19)

Pr 0.75 (20)

Assuming that the viscosity and thermal conduc-tivity of the gas are constant, Marconi et al (Ref. 5) used the method of characteristics to solve numerically the system of equations (9)-(18) for the shock-wave reflection of the type (i) noted in the previous section. In the present paper, we use a modified random-choice method with an operator-splitting technique. The method was applied success-fully to the problems of the transient flow in a dusty-gas shock tube (Ref. 6) and of the shock-wave passage through a dusty-gas layer (Ref. 7). An advantage of the method is the capability of des-cribing precisely discontinuous jumps without dis-torting their positions. Details of the technique are given in Ref. 6.

The boundary condition at the rigid wall is th at the gas is at rest. In order to satisfy this condi-tion, we take a fictitious grid point for the numerical calculation on the opposite side of the flow domain and assume a symmetrie flow condition with respect to the wall for the gas. On the other hand, the partieles continue to move owing to their inert ia and we assume that their state at the fictitious grid point is the same as that of the partieles at the grid point nearest to the wall. The particles may collide with the wall. Such particles are assumed to stick to the wall as in Refs. 5 and 11.

Numerical calculations are done for the reflec-tion of an incident shock wave which has propagated steadily in the mixture of air and glass part ic les of 10 )lm diameter with a mass concentration ratio a

=

1. For this mixture, we have

pi

=

2.455 and P;

=

2.655 from Eqs. (5) and (7).

First, we consider the case of PI/PO

=

5 as an example of a type (i) reflection. The stationary structure of the incident shock wave can be obtained by solving a set of ordinary differential equations for a coordinate system moving with the shock wave. Figure 4 shows it for PI/PO

=

5. The mixture ahead of the shock wave was assumed to be at room condi-tions. Taking the flow state shown in Fig. 4 as an

initial condition, the transient mot ion was calcul-ated af ter the impingement of the shock front on a rigid wall at x

=

O.

Flow structures at 0.39, 1.17 and 2.73 ms af ter

the shock-wave reflection are shown in Figs. 5-7

respectively. The frozen shock front is reflected first and the reflection of the compressive part in the relaxation region of the incident shock wave

follows. As the relaxation region of the incident

shock wave propagates toward the wall, the particles in the neighbourhoód of the wall experience initial-ly acceleration and heating. The reflected shock front passes through them before they attain the

equilibrium state behind the incident shock wave.

Af ter that, the particles decelerate but are heated further by the gas of low velocity and high

temper-ature. As time elapses, the differences in

temper-ature and velocity between the gas and the particles become small except for a reg ion just behind the

reflected shock front. A stationary structure of

the reflected shock wave, which is partially dis-persed in this case, forms finally as seen in Fig.

7. lts relaxation length is smaller than th at of

the incident shock wave. As Marconi et al (Ref. 5) noted, this is because the density of the gas ahead of the shock front is larger for the reflected shock wave than for the incident one 50 that the particles experience a larger drag force and heat transfer rate.

While the gas is always at rest at the wall, the particles are not, as they stop immediately on striking the wall (see Fig. 5). Figure 8 shows the variation with time of the impact velocity of the particles at the wall. It increases at first, reaches a maximum and then decreases gradually to zero. As seen from Fig. 7, the particles in a region close to the wall are not so compressed

as the gas. As aresult, the heat transferred

from the gas to the particles is 50 small that the temperature of the gas in this region remains higher than that in a uniform region formed behind

the reflected shock wave. This leads to a smaller

gas density there, since the pressure is uniform across the region.

Next, we consider the case of Pl/PO = 2.65 for

a type (ii) reflection. Figure 9 shows the initial

structure of the partially dispersed shock wave. Flow structures at 0.39, 1.17 and 2.73 ms af ter the shock reflection are shown in Figs. 10-12 respectively. The frozen shock front is reflected at first as in the type (i) reflection. It lies in the middle of the relaxation region of the incident shock wave in Fig. 10. A discontinuous jump in the gas phase still exists in this case af ter it has propagated behind the incident shock wave (see Fig. 11). The interaction between the gas and the particles reduces its strength with time to produce a fully-dispersed reflected shock wave as seen in Fig. 12. A non-uniform region of a high temperature is also produced in the neighbourhood of the wall at this final stage. The variation with time of the impact velocity of the particles is shown in Fig. 13. The time 'for the particles t.o strike the wall is longer than in the case of Po/PO

=

5, since the relaxation time is longer for weaker shock waves.

Finally, we consider the case of Pl/PO

=

2 for

a type (iii) refleétion. There is no discontinuous

front of the incident shock wave in this case, for

it is fully dispersed. We performed a numerical

3

calculation starting from an initial flow structure

shown in Fig. 14. Flow structures at 1.56,3.12

and 9.36 ms are shown in Figs. 15-17, respectively. A discontinuous jump in the gas phase never arises in this case. In Fig. 17, a fully-dispersed reflected shock wave is seen to form aft er a long time. The particles close to the wall collide with it in this case as well. The change in their impact

velocity is shown in Fig. 18. A smooth profile of

the incident shock structure results in an initial gradual increase in the impact velocity.

4. CONCLUSIONS

The reflection of a shock wave at a rigid wall in a dusty gas was studied on the basis of an

idealized equilibrium-gas analysis. A stronger

shock wave is shown to reflect since the mixture contains particles of higher mass concentration. It was also shown that three types of reflection exist. For a strong incident shock wave, both the incident and reflected shock waves are partially dispersed. When the incident shock wave is of moderate strength, the incident shock wave is partially dispersed but the reflected one becomes

fully dispersed. For a weak incident shock wave,

both the incident and reflected shock waves are fully dispersed.

Numerical analyses of the transient nonequili-brium flow were also made to demonstrate the three types of reflection. When the incident shock wave is partially dispersed, its frozen shock front reflects abruptly from the wall as a discontinuity in the gas phase. The compressive part of the incident relaxation region interacts with the reflected shock front subsequently. Af ter the reflected shock front has passed, the particles are surrounded by a gas of low velocity and high temperature which decelerates and raises their temperature. A stationary structure of the re-flected shock wave is formed in time. The frozen shock front survives or decays depending on the incident shock-wave strength. When the incident shock wave is fully dispersed, it is reflected smoothly and a fully dispersed reflected shock wave forms af ter a long time.

The particles in a region close to the wall collide with it in any case. We assumed that those particles stick to the wall. A thickening of the wall must occur, but its effect is considered to be negligibly small since the volume fraction of the particles is very small.

The mixture at a distance from the wall experi-ences a complete passage of the incident and re-flected shock waves that are fully developed.

Therefore, their final state becomes uniform. On

the other hand, the incident shock wave interacts with a transient reflected shock wave in a reg ion

near the wall. As aresult, the mixture in this

region cannot attain a uniform state even af ter a long time, although it reaches an equilibrium state. The gas and the particles in the non-uniform region have lower mass concentrations and a higher temper-ature than those in the uniform region at a distance from the wall.

It would be useful to check these results in the new UTIAS dusty-gas shock-tube facility in the near future.

REFERENCES

1. Rudinger, G., "Effecti ve Drag Coefficient for Gas-Particle Flow in Shock Tubes", Trans. ASME, J. Basic Eng., Vol. 92, 1970, pp. 165-172. 2. Outa, E., Tajima, K., Morii, H., "Experiments

and Analyses on Shock Waves Propagating through a Gas-Particle Mixture", Bulletin of JSME, Vol. 19, 1976, pp. 384-394.

3. Outa, E., Tajima, K., Suzuki, S., "Cross-Sectional Concentration of Partieles during Shock Process Propagating through a Gas-Particle Mixture in a Shock Tube", Proc. 13th Int. Symp. Shock Tubes and Waves, 1981, pp. 655-663. 4. Otterman, B., Levine, A. S., "Ana1ysis of

Gas-Solid Partiele Flows in Shock Tubes", AlM J.,

Vol. 12, 1974, pp. 579-580.

5. Marconi, F., Rudman, S., Calia, V., "Numerical Study of One-Dimensional Unsteady Partiele-Laden Flows with Shocks", AIAA J., Vol. 19, 1981, pp. 1294-1301.

4

6. Miura, H., G1ass, l.I., "On a Dusty-Gas Shock Tube", Proc. R. Soc. Lond., A382, 1982, pp. 373-388.

7. Miura, H., G1ass, I. I., "On the Passage of a Shock Wave through a Dusty-Gas Layer", Proc. R. Soc. Lond., A385, 1983, pp. 85-105.

8. Rudinger, G., "Relaxation in Gas-Partic1e Flow", in P. P. Wegener, "Nonequilibrium F1ows", Vol. 1, Marcel Dekker, N.Y., 1969, pp. 119-161. 9. Marbie, F. E., "Dynamics of Dusty Gases",

Annual Review of Fluid Mechanics, Vol. 2, 1970,

pp. 397-446.

10. Courant, R., Friedrichs, K.O., "Supersonic Flow and Shock Waves", Interscience, N.Y., 1948.

11. Rudinger, G., Chang, A., "Analysis of Nonsteady Two-Phase Flow", Phys. Fluids, Vol. 7, 1964, pp. 1747-1754.

12. Miura, H. and Glass, I. I., "Formation of a Shock-Wave in a Dusty Gas by a Moving Piston", UTIAS Report No. 275, 1984.

I

•

t +

S

r .

o

x 1

o

FIG. 1 SCHEMATIC x,

t

DIAGRAM OF SHOCK-WAVE REFLECTION AT A RIGID

WALL.

Si, INCIDENT SHOCK WAVE; Sr, REFLECTED SHOCK WAVE.

P

21

12.00 ---________________

~

11.00

10.00

9.00

8.00

7.00

6.00

5.00

4.00

3.00

2.00

1.00

0.00 2.00

6.00

10.00

14.00

18.00

22.00

2

26.00

PlO

30.00

,

_..

4.84

4.52

4.20

3.88

3.56

3.24

2.92

2.60

2.28

1.96

1.64

1. 32

1.00

.J

(i)

(iii)

0.00

0.20

0.40

0.60

0.80

1.00

1.20 1.40 1.60

1.80

2.00

Cl.

FIG. 3 DOMAlNS OF THREE TYPES OF SHOCK-WAVE REFLECTION:

(i)

INCIDENT AND REFLECTED SHOCK WAVES, BOTH PARTIALLY

DISPERSED.

(ii)

PARTIALLY DISPERSED INCIDENT SHOCK WAVE AND FULLY

DISPERSED REFLECTED SHOCK WAVE.

(iii)

INCIDENT AND REFLECTED SHOCK WAVES, BOTH FULLY

P

25.00

r---,

20.00

15.00

10.00

5.00/

0.00

+---_r----_+---~----+_----_r----~---~----4

0.00

3.00

6.00

9.00

R

x

(a) (b)

12.00

15.00

18.00

21.00

24.00

12.50~---,

10.00

7.50

5.00

G

2.50

V

0.00+---~---+---4r---+---r---+----~r---~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

2i.00

24.00

x

FIG. 4 INCIDENT SHOCK-WAVE STRUCTURE (PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

T

2.25

2.00

1. 75

1.50

1.25

f

G P

1.00

*---~----~----~~----~----_r----~---~--~

u

1. 25

1.00

0.75

0.50

0.25

0.00

0.00

0.00

3.00

6.00

9.00

12.00

15.00 18.00

21.00 24.00

3.00

x

(c) (d)

6.00

9.00

12.00

x

15.00

18.00

21.00

24.00

FIG. 4 - CONTINUED

INCIDENT SHOCK-WAVE STRUCTURE (PI/PO

=

5).

Ca)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

P

25.00

.---~

20.00

15.00

10.00

5.00

0.00

R

12.50

10.00

7.50

5.00

2.50

0.00

3.00

G

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(a) (b)

0.00

~----~----_+---~----+_----~----_+----~~--~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 5

FLOW QUANTITIES AT

t

=

0.39 MS (PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE

T

2.25

.-~---~

2.00

1. 75

1.50

.

'

1.

25

1.00

+---~----~---r_----~----_r----~---r_--~

0.00

3.00

6.00

9.00

12.00

15.00 18.00

21.00

24.00

x

Cc)

U

Cd)

1.

25

1.00

0.75

G

0.50

0.25

0.00

0.00

3.00

6.00

9.00

12.00 15.00

18.00

21.00

24.00

x

FIG. 5 - CONTINUED

FLOW QUANTITIES AT

t =

0.39 MS

(PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

Cc)

TEMPERATURE

P

25.00r---~---,

20.00

-r----__

15.00

10.00

5.00

0.00

+-~--~----_+---+_----4_----~----_;---+_----~

0.00

3.00

6.00

R

9.00

12.00

15.00 18.00

21.00 24.00

x

(a)

(b)

12.50~---,

10.00

7.50

5.00

2.50

O.oo+---~----_+---~----+_----~----~---~--~

o .

00

3 . 00

6.00

9.00

12. 00

x

15.00

18.00

21.00 24.00

FIG. 6

FLOW QUANTITIES AT

t

=

1.17 MS (PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE:

Cd)

VELOCITY.

GAS;

PARTICLES.

T

2.25

.---~

2.00

1. 75

_P

_G

1.50

1. 25

1.00

+---~----~---+_----~----_+----~---+_---.-0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(c) U (d)

1. 25

1.00

0.75

0.50

_G

0.25

0.000.00

3.00

_6.00

_9.00

_12.00

_15.00

_18.00

_21.00

_24.00

x

FIG. 6 - CONTINUED

FLOW QUANTITIES AT

t

= 1.17

MS

(PI/PO

= 5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

P

25.00~---.

20.00

+---__ __

15.00

10.00

5.00

0.00 .

R

12.50

10.00

7.50

5.00

2.50

0.00

0.00

3.00

6.00

9.00

12.00

15.00 18.00

21.00 24.00

x

(a) (b)

0.00

3.00

6.00

_9.00

_12.00

_15.00

_18.00

_21.00

_24.00

x

FIG. 7

FLOW QUANTITIES AT

t =

2.73 MS (PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

T

2.25

2.00

1. 75

1.

50

1.

25

~,---,---~-~~--~

P

G

1.00

+.---~----~----~----_+---~----~----_r----~

0.00

3.00

U

1.

25

1.00

0.75

0.50

0.25

6.00

9.00

12.00

15.00

18.00

2

1.00

24

.0

0

x

(c)

(d)

P

G

O.OO+---+---~----~----~~--~~---r---r----~

0.00

3.00

6.00

FIG. 7 - CONTINUED

9.QO

12.00

15.00

18.00

21.00

2

4.00

x

FLOW QUANTITIES AT

t

=

2.73

MS (PI/PO

=

5).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

u

0

:

50

0.40

0.30

0. 20 7

0.10

O.OO+.~--~~=-~r---~----+---~----~

0.00

10.00

20.00

30.00

40.00

50.00 60.00

T

FIG. 8 VARIATION OF PARTICLE IMPACT VELOCITY WITH TIME

P

8.50

7.00

5.50

4.00

2.50~

1.

00

R

6.00

5.00

4.00

3.00

2.00

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(a)

(b)

1.00~~~~----~----~----~---r---r---+----~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 9

INCIDENT SHOCK-WAVE STRUCTURE

(PI/PO

=

2.65).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

T 1.50~---~

1.40

1.30

1.20

1.10

1.00

~----~---+---~----+---~---+---~--~ U

1.00

0.80

0.60

0.40

0.20

0.00

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

0.00

3.00

6.00

Cc)

(d)

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 9 - CONTINUED

INCIDENT SHOCK-WAVE STRUCTURE (PI/PO

=

2.65).

(a)

PRESSURE;

Cb)

MASS CONCENTRATION;

Cc)

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

J

P

8.50

7.00

5.50

2.50

1.00

R

6.00

5.00

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(a) (b)

1.00

+---r----~---T_----~----_+----~---+_--~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 10 FLOW QUANTITIES AT

t

=

0.39 MS (PI/PO

=

2.65).

(a)

PRESSURE; (b) MASS CONCENTRATION; (c) TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

T

1.50

.---~

l.40

l.30

l.20

1.10

.

l.00

0.00

3.00

u

l.00

0.80

0.60

0.40

0.20

0.00

0.00

3.00

6.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(c)

(d)

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 10 - CONTINUED

FLOW QUANTITIES AT

t

=

0.39 MS (PI/PO

=

2.65).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

P

8.50

7.00

5.50

4.00

2.50

1.00

0.00

R

6.00

5.00

4.00

3.00

2.00

3.00

6.00

9.00

12.00

_

15.00

18.00

21.00

24.00

x

(a) (b)

1.00

+---~----~---~----~---+----~---~--~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 11

FLOW QUANTITIES AT

t

=

1.17 MS (PI/PO

=

2.65).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

T

l.50

l.40

l.30

l.20

l.10

l.00

U

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(c)

(d)

1.00.---~---~

0.80

0.60

0.40

0.20

O.OO~~~~--~---+----~----~----~----~--~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 11 - CONTINUED

FLOW QUANTITIES AT

t =

1.17 MS (PI/PO

=

2.65).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

---

---

--

---

---

----

--

--

---P

8.50

7

.

00

5.50

4

.

00

2.50

1.

00

0.00

3.00

6.00

9.00

12.00

15.00

18.00

2

1.00

24.

00

x

(a)

(b)

5

.

00

4.00

3.00

2.00

1 .00+---~----~---~----4---~----~---+---~

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

FIG. 12

FLOW QUANTITIES AT

t

=

2.73 MS (PI/PO

=

2.65).

(a)

PRESSURE ; (b) MASS CONCENTRA TI ON;

(c )

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

T

l.50

l.40

l.30

l. 20

l.10

1.00+---~---+---+---+---~---4---+---~

u

0.00

3.00

6.00

0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

x

(c) (d)

9.00

12

DO

15.00

18.00

21.00

24.00

X·

FIG. 12 - CONTINUED

FLOW QUANTITIES AT

t =

2.73 MS (PI/PO

=

2.65).

(a)

PRESSUREj (b) MASS CONCENTRATIONj

(c)

TEMPERATUREj

U

0.10

0.08

0.06

0.04

o.oo+---~----~--==~===-~----~----~

0.00

10.00

20.00

30.00

40.00

50.00

60.00

T

FIG. 13

VARIATION OF PARTICLE IMPACT VELOCITY WITH TIME

P

5.00

4.00

3.00

1.00

2.00

v

0.00

+---_r---+---~----~----_r----~---~--~

0.00

12.00

24.00

36.00

48.00

60.00 72.00

84.00 96.00

R

3.50

3.00

2.50

2.00

1.50

x

(a)

(b)

1.00

~~--~----+---~----~----~----~----_r----~

0.00

12.00

24.00

36.00

48.00 60.00

72.00

84.00 96.00

x

FIG. 14

INCIDENT SHOCK-WAVE STRUCTURE (PI/PO

=

2).

Ca)

pRESSURE;

Cb)

MASS CONCENTRATION;

Cc)

TEMPERATURE;

T

1.37

1.30

1.22

1.15

1.07

1.00

~~--~----~---~----~---+----~---r---~

u

0.50

0.40

0.30

0.20

0.10

0.00

12.00

24.00

36.00

48.00

60.00 72.00

8

4

.00

96.

00

x

(c)

(d)

o.oo~~--~----~---~----~---+----~---r---~

0

.

00

12.00

24.00

36.00

48.00 60.00

72.00 84.00

96.0

0

x

FIG. 14 - CONTINUED

INCIDENT SHOCK-WAVE STRUCTURE (PI/PO

=

2).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

P

5.00.---

ï

4.00

3.00

2.00

1.00

0.00

+---_+----~---r_----~----_+----~---+_----~

0.00

12.00

24.00

36.00

48.00 60.00

72.00

84.00

96.00

x

(a)

R

(b)

3.50

3.00

2.50

2.00

1.50

1.00

+-____

~----_r----~----~---r---+---~----~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00

96.00

x

FIG. 15

FLOW QUANTITIES AT

t

=

1.36 MS

(p

1

/P

O

=

2).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

T

1.37 ,---,

1.30

1.22

1.15

1.07

1.00

u

0.50

0.40

0.30

0.20

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00

96.00

x

-

Cc)

(d)

0.10

0.00

~----4---_+----~---+_----~----_+---~--~

0.00

12.00

24.00

36.00 48.00

60.00

72.00

84.00 96.00

x

FIG. 15 - CONTINUED

FLOW QUANTITIES AT

t

=

1.56 MS (PI/PO

=

2).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

P

5.00.---~

4.00

~

3.00

2.00

1.00

o.oo+---~----~---~----~----~----~---+_--~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

Ca)

R

(b)

3.50

3.00

2.50

2.00

1.50

1.00+---~---+---~----+---~----~---+---~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

FIG. 16

FLOW QUANTITIES AT

t =

3.12 MS (PI/PO

=

2).

Ca)

PRESSURE;

Cb)

MASS CONCENTRATION;

Cc)

TEMPERATURE;

T

1.37

1.30

1.22

.

P

1.15

1.07

1.00+---~---+----~~----+---~----~---~--~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

Cc)

u

Cd)

0.50

0.40

0.30

0.20

0.10

0.00

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

FIG. 16 - CONTINUED

FLOW QUANTITIES AT

t

= 3.12 MS (PI/PO

= 2).

Ca)

PRESSURE; (b) MASS CONCENTRATION;

Cc)

TEMPERATURE;

P

5.00

4.00

3.00

2.00

1.00

0.00

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

Ca)

R

Cb)

3.50

3.00

2.50

2.00

o

1.50

:

o 1.00:+---~----_+---~----+_----~----_+----~~--~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

FIG. 17 FLOW QUANTITIES AT

t =

9.36 MS (PI/PO

=

2).

Ca)

PRESSURE; (b) MASS CONCENTRATION;

Cc)

TEMPERATURE;

(d) VELOCITY.

GAS;

PARTICLES.

•

T

1.37

r---~

1.30

1. 22

1.15

1.07

1.00 -~-~---~----~---+----~----~~----+---+---~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

U

0.50

I

O.4ot

0.30

0.20

0.10

(c) (d)

o.oo~----~----~----~--~~~--~~----+---+---~

0.00

12.00

24.00

36.00

48.00

60.00

72.00

84.00 96.00

x

FIG. 17 - CONTINUED

FLOW QUANTITIES AT

t =

9.36 MS (PI/PO

=

2).

(a)

PRESSURE; (b) MASS CONCENTRATION;

(c)

TEMPERATURE;

u

0.10

0.08

0.06,

0.04 '

0.02

O.OO~----~--~-T----~---+---~

____

~

0.00

30.00

60.00

90.00 120.00

150.00 180.00

"[

FIG. 18 VARIATION OF PARTICLE IMPACT WITH TIME

~

,..

UTIAS Report No. 274

University of Tarantc. Institute for Aerospace Studies (lfI'IAS)

4925 Dufferin Street, Downsview • Ontario, Canada. M3H ST6

NORMAL REFLECTION OF A SHOCK WAVE AT A RIGID WALL IN A DUSTY GAS H. Miura. T. Saito and I. I. Glass

1. Dusty-gas flows

2. Normal-shock-wave reflections 3. Relaxation effects

I. Miura. H., Saito, T" Glass, J. I.

4. Nonequilibrium flows S. Shock structure

6. Rémdom-choice appl ications 11. UTIAS Report No. 274

~

The flow occurring when a fully developed shock wave in a dusty gas reflects from a wall is studied

analytically. 8y applying an idealized equilibrium-gas analysis, it is shown that there exist three types of shock reflectien. The incident shock wave and the reflected shock wave are partially dispersed

if PI/PO > p;. the farmer is partially dispersed but the latter is fuUy dispersed if P; ~ PI/PO> pi

and bath of them are fuUy dispersed if pi ~ Pl/PO > 1. Here PI/PO is the incident shock pressure ratio

and P~ and

pi

are critical values depending on the physical properties of the dusty gas.

The equations of mot ion are álso solved numerically by means of a modified random-choice method and an operator-splitting technique in order to study the time-dependent nonequilibrium flow. The results

demonstrate the details of the format ion of the reflected shock wave for the three types noted in the foregoing.

...

_•

UTIAS Report No. 274

University of Toronto, lnstitute for Aerospace Studies (UTlAS)

4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6

NOR.'lAl REFLECTlON OF A SHOCK WAVE AT A RIG ID WAll IN A DUSTY GAS H. Miura, T. Saito and I. I. Glass

I. Dusty-gas flows

2. Normal-shock-wave reflections 4. Nonequi librium flows 5. Shock structure

3. Relaxation effects 6. Random-choice applications 1. Miura, H .• Saito, T .• Glass, I. I. I I. UTIAS Report No. 274

~

The flow occurring when a fuUy developed shock wave in a dusty gas reflects from a waU is studied

analytically. Sy applying an idealized equilibrium-gas analysis, it is shown that there exist three

types of shock reflection. The incident shock wave and the reflected shock wave are partially dispersed

if PI/PO > P;, the farmer is partially dispersed but the latter is fully dispersed if P; ;:: PI/PO > pi

and bath of them are fully dispersed if pi ~ PI/PO > I. Here PI/PO is the incident shock pressure ratio

and P; and

pi

are critical va lues depending on the physical properties of the dusty gas.

The equations of mot ion are á-lso solved numerically by means of a modified random-choice methad and

an operator-splitting technique in order to study the time-dependent nonequilibrium flow. The results demonstrate the details of the formation of the reflected shock wave for the three types noted in the foregoing.

Available copies of this report are limited. Return this card to UTIAS, if you require a copy. Available copies of this report are limited. Return this card to UTIAS, if you require a copy.

UTIAS Report No. 274

University of Toronto, 1nstitute for Aerospace Studies (UTlAS)

4925 Dufferin Street. Downsview , Ontario, Canada, M3H 5T6

NORMAL REFlECTION OF A SHOCK WAVE AT A RIGID WAll IN A DUSTY GAS H. Miura, T. Saito and I. I. Glass

I. Ousty-gas flows

2. Normal-shock-wave reflections 4. Nonequilibrium flows 5. Shock structure 3. Relaxation effects 6. Random-choice appl ications

I. Miura, H .• Saito, T., Glass, 1. I. I I. UTIAS Report No. 274

~

The flow occurring when a fully developed shock wave in a dusty gas reflects from a wall is studied analytically. Sy applying an idealized equilibrium-gas analysis, it is shown that there exist th ree

types of shock reflection. The incident shock wave and the reflected shock wave are partially dispersed

if PI/PO > P;, the former is partially dispersed but the latter is fully dispersed if P; i::: PI/PO > pi

and both of them are fully dispersed if

pi

~ PI/PO > 1. Here PI/PO is the incident shock pressure ratio and P; and

pi

are critical values depending on the physical properties of the dusty gas.

The equations of mot ion are áIso solved numerically by means of a modified random-choice method and an operator-splitting technique in order to study the time-dependent nonequilibrium flow. The results

demonstrate the details of the formation of the reflected shock wave for the three types noted in the foregoing.

UTlAS Report No. 274

University of Toronto, Institute for Aerospace Studies (UTlAS)

4925 Oufferin Street. Downsview, Ontario, Canada, M3H 5T6

NOR.'IAL REFlECTION OF A SHOCK WAVE AT A RIGID WALL IN A DUSTY GAS

11. ~1iura. T. Saito and I. I. Glass

1. Dusty-gas flows 4. Nonequi librium flows

2. Normal-shock-wave reflections 5. Shock structure

3. Relaxation effects 6. Random-choice appl ications I. Miura. H., Saito. T., Glass, I. I. 1I. UTIAS Report No. 274

~

The flow occurring when a fully developed shock wave in a dusty gas reflects from a wall is studied analytically. By applying an idealized equilibrium-gas analysis, it is shown that there exist th ree types of shock reflection. The incident shock wave and the reflected shock wave are partially dispersed

if PI/PO > P;. the former is partially dispersed but the Iatter is fully dispersed if P; ~ PI/PO > pi

and both of them are fully dispersed if

pi

~ PI/PO > 1. Here PI/PO is the incident shock pressure ratio

and P; and pi are critical values depending on the physical properties of the dusty gas.

The equations of motion are also solved numerically by means of a modified random-choice method and

an operator-splitting technique in order to study the time-dependent nonequilibrium flow. The results demonstrate the details of the format ion of the reflected shock wave for the three types noted in the foregoing.