-"
March 1984
FORMATION OF A SHOCK WAVE IN A DUSTY GAS
BY A MOVING PISTON
by
H. Miura and I. I. Glass
TECHt~rsCHE HOGESCHOOL DELFT
LUCHTVAART -E
RUIMTEVAARTIECHNIEK
BIBLIOTHEEK
Kluyverweg 1 - DELFT
t
7 JUNI 19
84
UTIAS Report No. 275
CN ISSN 0082-5255
1
.
/
BY A MOVING PISTON
by
H. Miura and I. I. G1ass
Subrnitted January, 1984
March, 1984
UTIAS Report No. 275 CN ISSN 0082-5255
The financial assistance received from the Natural Science and Engineering Research Council of Canada under grant No. Al647, the U.S. Defence Nuclear Agency under DNA Contract 00l-83-C-0266, and from the U.S. Air Force under grant AF-AFOSR-82-0096 is acknowledged with thanks.
The nonstationary flow induced by a piston moving impulsively at a constant ~peed in a dusty gas is studied analytically. The effect of suspended partieles on a fully developed shock wave is discussed on the basis of an idealized equilibrium-gas approximation. The equations of motion are also solved numerically by means of a modified random-choice method and an operator-splitting technique for the cases when the final shock structure is partially or fully dispersed. The numerical results show the detailed behaviour of the flow in a transitional nonequilibrium state.
The case when the partieles that collide with the surface of the piston reflect elastically is investigated together with the case when they stick to it. When the partieles are reflected from the piston at high speed, they move faster than the shock front for a while. It is shown that the shock front in this case propagates initially at a larger velocity than in the pure gas case.
Page
Acknowledgements ii
J Summary iii
Notation v
l. I NTRODUCTI ON 1
2. IDEALIZED EQUILIBRIUM-GAS ANALYSIS 1 3. TlME-DEPENDENT NONEQUILIBRIUM FLOW 1
4. CONCLUSIONS 3
REFERENCES 4
FlGURES
J
.1 a e d M* P M pe M se m p Pr Q R Re T t u v x
x
B y 9 p aequilibrium 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
critical piston Mach number
piston Mach number based on equilibrium speed of sound
piston Mach number based on frozen speed of sound
shock Mach number based on equilibrium speed of sound
shock Mach number based on frozen speed of sound
mass of a particle
gas pressure
gas Prandtl number
rate of heat transfer to a particle
gas constant
particle Reynolds number
gas temperature time gas velocity particle velocity space coordinate 2 -1 normalized space coordinate x(8m/npd )
mass concentration ratio alp
specific heat ratio of two phases Cm/Cv
gas specific heat ratio C p IC v
specific heat ratio of idealized equilibrium gas
particle temperature
gas viscosity
gas density
mass concentration of particles
•
-
J
When a large nwnber of small particles are sus-pended in a gas, th~y cause a significant effect on the gas flow througp the transfer of momentum and heat. The use of a' dusty-gas shock tube made it possible to study experimentally the relaxation properties of dusty-gas flows (Refs. 1-3). Analyt-ical studies of the nonstationary flow of a dusty gas with shock waves have also been made· by many authors (Refs. 4-9). Those results led to a quan-titative understanding of the basic properties of dusty-gas flows.
Among these analyses, the format ion of a shock wave in a dusty gas by an impulsively moving piston was discussed in Refs. 4, 5 and 7. In this problem, a shock discontinuity in the gas phase is produced at first just as in the case of a pure gas. The presence of the particles change the flow from this frozen state to a fully developed state in which the gas and the particles are in equilibrium except for a shock transition zone. Using the method of char-acteristics, Rudinger and Chang (Ref. 4) and Marconi et al (Ref. 7) studied numerically a case when the final structure of the shock wave is partially dis-persed. A case when the shock wave becomes fully dispersed was analysed by Miura (Ref. 5) using a perturbation method.
All of these studies assumed that the particles colliding with the piston stick to its surface. However, there is another possibility that the particles reflect from the solid wall aft er colli-sion (Ref. 10). The state of the particles af ter their impact must depend on the physical properties of the particles and the solid wall.
In the present paper, we consider both types of particle collisions with the wall for the piston problem. Numerical solutions of the equations of mot ion are obtained to demonstrate the development of the flow aft er the piston moves impulsively at a constant speed. An idealized equilibrium-gas analy-sis is also made in order to understand the effect of the particles on the shock wave which is finally formed in the dusty gas.
2. IDEALIZED EQUILIBRIUM-GAS ANALYSIS
An idealized equilibrium-gas approximation is useful for predicting the flow which is fully devel-oped af ter the piston begins to move. A stationary shock-wave structure forms in time in the dusty gas. At this stage, the gas and the particles are in an equilibrium state except for the relaxation region of the shock wave.' If we neglect the thickness of the shock wave, the mixture can be regarded effec-tively as a perfect gas. lts spec.ific heat ratio and speed of sound are given by (Refs.ll, 12 and 13)
{ y+aS ae
=
-'("-l-+a'!"')"":(7'l:""+-a-=S7') } 1/2.E.
PUsing perfect-gas theory, we can express shock Mach number Mse in terms of the piston number Mpe: (1) (2) the Mach 1
where Mse and Mpe are referred to the equilibrium speed of sound ahead of the shock wave. Flow quan-tities behind the shock wave are obtained fr om the Rankine-Hugoniot relations as follows:
PI 1 + 2Ye (M2 -1) Po Y e +1 se PI al (y e +I)M2 se Po cr 0 (y e -l)Mse +2 2 2(Y e -1) ( +1)2M2 Ye se 1 +
where the subscripts refer to the uniform flow regions shown in Fig. 1.
(4)
(5)
(6)
A shock wave in a dusty gas is partially or fully dispersed according to whether or not its propagation veiocity is larger or smaller than the frozen speed of sound af = (yp/p)1/2. If we use the shock Mach number Msf and the piston Mach number Mpf based on the frozen speed of sound, Eq.
(3) is rewritten in the form: Ye+l { Ye [ Y
e4+l Mpf)2}1/2 Msf
=
-4- Mpf + y(l+a) + (7)Therefore, the shock wave is partially or fully dispersed if Msf ~ 1 respectiveIy, or
M* > M* M* _ 2a(y+Sy-f3+af3y)
Pf< p' p-y(I+a) (y+I+2af3) (8) Figure 2 shows the variations of the shock Mach numbers Mse and Msf with the piston Mach number Mpf for several a's for the mixture of air (y = 1.4) and glass particies (13
=
1). At a fixed Mpf, the effec-tive shock Mach number Mse becomes greater with a. Thus strong shock waves are produced for larger mass concentration of particles. On the other hand, the shock Mach number based on the frozen speed of sound Msf reduces with a at a fixed Mpf' that is, the prop-agation veiocity of the shock wave becomes smaller for larger a. This is because the energy removed by the particles from the gas increases as the mass concentration of the particles increases. The domain of fully dispersed shock waves is below the dashed line in Fig. 2.3. TlME-DEPENDENT NONEQUILIBRIUM FLOW
The detailed development of the flow can be obtained onIy by sol ving the equations of motion for the dusty gas. We assume th at the mixture consists of a perfect gas and a lot of solid spherical par-ticles of uniform size. Let p, P, T, u be the pressure, density, temperature and velocity of the gas, and a,
e,
.
v be the mass concentration, temper-ature and velocity of the particles, respectively. The equations of motion are written in the form(Refs. 8 and li): (lp 0
dO - +
a
t
ax
a
(av)a
a
2at
(av) +a
x
(av ) ~ (vD+Q) ~ (vD+Q) m p pRTo
- ~ 0 a 0 m m (10) (11) (12) (13) (14) (15)where (Cp, Cv) and Cm are the specific heats of the gas and the particles, m is the mass of a particle and R is the gas constant. In the above equations, the viscosity and thermal conductivity of the gas are taken into account through the drag force D and the heat transfer rate Q for a particie. These quantities are assumed to depend on the flow quan-tities as follows:
o
Q lTdjJC Pr -1 (T -6) (2.0+0.6 Pr 1/3 1/2 Re ) p (16) (17)where Pr is the Prandtl number of the gas and Re is the Reynolds number based on the diameter d of the particles , .
Re plu-vld/jJ (18)
The viscosity and thermal conductivity of the gas vary with its temperature. For air considered in the present paper, we take
(19)
Pr 0.75 (20)
We use a modified random-choice method with an operator-splitting technique for sol ving the above set of equations. The numerical technique was successfully applied to the problem of the transient flow in a dusty-gas shock tube (Ref. 8) and also to the problem of the shock-wave passage through a
2
are omitted in this paper.
The boundary condition for the gas is that its velocity at the piston surface is the same as the speed of the piston. The particles are at rest initially. Due to their inertia, some particles must collide with the surface of the moving piston. We assume first that these particles stick to the surface. Since the volume fraction of the particles is very small in the case considered, the thickening of the surface due to the sticking particles is negligible.
Numerical computations were done for the mixture of air and glass particles of 10 jJm diameter with a
=
1. The mixture is assumed to be initially at room conditions. Flow structures at several values of time af ter the piston motion are shown in Figs. 3-8 for the case of Mpf=
2. The position of the piston surface is always set at x = 0 in the figures.Immediately af ter the piston begins to move, a shock discontinuity forms only in the gas phase. The particles behind this frozen shock front are accelerated and heated gradually as time elapses. It is seen from Figs. 3-5 that the strength and the propagation velocity of the shock front decrease with time as a result of the removal of energy by the particles. The flow structure in the pure-gas case is also shown by dashed lines in the figures for comparison. The force which the particles of low velocity act on the gas leads to a compression of the gas at some distance behind the shock front.
lts pressure increases monotonically with time.
The gas temperature rises at first as a result of compression. However, the temperature of the gas in a region close to the piston begins af ter a short time to dec rea se as can be seen by comparing Fig. 4 with Fig. 5. This change occurs because the effect of heat absorption by the particles predominate over the compression effect.
Subsequently, the shock front decays further and the magnitude of its discontinuity approaches a finite value. Figure 8 for t
=
1.25 ms shows th at an almost uniform flow region in an equilibrium state forms behind a nonequilibrium region following this shock front. The final shock structure in this case is shown in Fig. 9, which is obtained by solv-ing the set of ordinary differential equations of motion for a coordinate system moving steadily with the shock wave. The shock structure found in the numerical results for nonstationary flow (Fig. 8)agrees weIl with Fig. 9. For the mixture considered, Eq. (8) gives the critical piston Mach number
Mp
=
0.5195. Therefore, the shock wave for Mpf=
2 is partially dispersed as shown in Fig. 8. Flow quantities except the gas temperature increase monotonically across the thick relaxation reg ion of this strong shock wave. The temperature of the gas attains a maximum in the middle of the transition region. The compression effect produced by the drag force dominates over the effect of heat absorp-tion by the particles ahead of the point of maximum temperature, but the latter dominates over the former behind this point. In a region close to the piston surface, the temperature and the density of the gas are not uniform, while the pressure and the velocity are uniform. The temperature of the gas in this region has become so high at an initial stage as shown in Fig. 4 that it remains higher than that in the uniform region even aft er a long time. It is seen from Fig. 8 that this non-uniform\.1
,.)
•
...
/sition reg ion .
Now, we consider the case when the partieles are allowed to reflect from the piston surface.
For the present problem, we may conjecture that the paths of the reflected partieles do not inter-seet since the difference in velocity between the colliding partieles and the piston decreases mono-tonically wit.h time. Then, a cloud of the reflected partieles can also be treated as a continuum. The equations of mot ion for the reflected partieles are similar to Eqs. (10), (12) and (14). Additional interaction terms due to the reflected partieles should be put on the right-hand sides of the equa-tions of momentum and energy (11) and (13) for the gas. We assume that the reflected partieles experi-ence a perfect elastic collision. If we use the subscript r to represent the flow quantities of the reflected partieles, the boundary condition at the surface of the piston is written as
2u -v
p. ' o e 0,
where up is the piston speed.
El (21)
Numerical results,taking the reflected partieles into account for the case of Mpf 2, are shown in Figs. 10-15. Flow structures at a short time af ter the piston begins to move are very different from those for the case without reflected partieles
(compare Figs. 3-5 and 10-12). The partieles that reflect from the piston surface move initially at 2up ' Their velocity is greater than the propagation velocity of the frozen shock wave if
or
1
M
pf > (3-y) -"2 'V 0.7906 (22)
Therefore, the reflected partieles produce a distur-bance first in the present case of Mpf
=
2, and th en the shock front propagates. A compression part yielded by the reflected partieles can be seen in Fig. 10 ahead of the discontinuous shock front. This precompression of the gas makes it possible for the shock wave to propagate faster than in the pure-gas case. The temperature of the gas behind the shock front is higher than in the pure-gas case (see Fig. lOc). This is because the work done by the force due to the reflected partieles dominates over both the non-reflected partieles and the heat absorbed by the partieles. The pressure of the gas behind the shock front is almost unifo~ in Fig. 10. This implies a balance of the forces due to the reflected and non-reflected partieles at this stage. As time elapses, the reflected partieles decelerate and the shock front gets ahead of them in time. It should be noted that the mass concentration of the reflected partieles R becomes very high (or/oo 'V 18, seeFig. llb) at the instant when they are passed over by the shock wave.
-At later stages, the transfer of momentum and heat between the gas and the non-reflected par-ticles mainly governs the motion of the dusty gas.
3
to approach an equilibrium value and the final stationary shock structure forms, as seen in Fig. 15. A difference in flow structure at a final stage between the cases with and without reflected partieles can be seen only in a non-uniform region close to the piston surface (compare Figs. 8 and 15). As a result of additional absorption of heat by the reflected partieles, the temperature of the gas in this region is lower than in the case of no reflected partieles and almost uniform .in Fig. l5c.
Next, we consider a case when the piston Mach number is sosmali th at the final shock structure may be fully dispersed (see Fig. 2). Figures 16-22 show the development of the flow in the case of Mpf
=
0,4 when the partieles are assumed to stick to the piston af ter collision. Owing to the inter-action between the gas and the partieles, the degree of discontinuity reduces monotonically as it propagates. As in the case of Mpf=
2, acompression of the gas in a reg ion close to the piston surface is produced by the force due to the partieles (see Fig. 16). In contrast to the case of a large piston Mach number, however, the temper-ature of the gas in this region is not raised beyond the value for the pure-gas case. We might say th at the absorption of heat by the partieles has an influence relatively larger than the com-pression effect in the case of a low-piston speed. It can be seen in Fig. 21 that the shock front has almost decayed. Further dispersion occurs subse-quently and a fully-dispersed shock structure of large thickness is formed at t = 5.46 ms (see Fig. 22). Solving a set of ordinary differential equa-tions of mot ion, we obtain the final shock structure for this case shown in Fig. 23. It is a little more dispersed than the flow structure shown in Fig. 22. The time required for the format ion of a fully-dispersed shock wave is thus much longer than that of a partially-dispersed shock wave. A non-uniform flow reg ion of high temperature at a final stage is generated in the neighbourhood of the piston surface similar to the case of Mpf = 2.
Numerical results for Mof = 0.4 when the partieles are allowed to reflect from the piston are shown in Figs, 24-28. In this case of a low piston speed, the reflected partieles cannot move faster than the shock front. Comparing Figs. 24-26 with Figs. 16-18, we see that the heat absorption by the reflected partieles very significantly reduces the temperature of the gas. The pressure of the gas varies little across the region where the reflected partieles are present. Flow struc-tures at subsequent stages shown in Figs. 27 and 28 are almost the same as those for the case of no reflected partieles except for a region close to the piston surface. In the region of the reflected partieles, the gas is at lower temper-ature and higher density than that just outside the region. Since the partieles in the region are almost in equilibrium with the gas in Fig. 28, this non-uniform flow structure will remain unchanged when a fully-dispersed stationary shock structure forms later.
4. CONCLUSIONS
Development of the flow induced by the impulsive motion of a piston in a dusty gas was analysed. Using an idealized equilibrium-gas analysis, it was
from detailed flow structure) of a stationary
sho~k wave that forms in time af ter the piston beg1ns to move. The propagation velocity of the shock wave decreases, but the effective shock Mach number increases as the mass concentration of the particles increases for a fixed piston speed. The shock wave is partially or fully dispersed accord-ing to whether the piston speed is larger or smaller than a critical speed which is given in terms of the physical proeprties of the dusty gas.
In order to study the nonstationary flow struc-ture, the equations of motion were solved by making use of a modified random-choice method with an operator-splitting technique. Numerical results showed the details of the format ion of partially and fully-dispersed shock waves. Immediately af ter the piston begins to move, a shock discon-tinuity is generated in the gas phase while the particles remain at rest owing to their inert ia. The gas and the particles behind the shock front interact with each other through the transfer of momentum and heat. As a result of the absorption of energy from the gas by the particles, the degree of discontinuity at the shock front reduces with time to approach a final value. When the piston speed is sufficiently large, the shock front is followed by a stationary relaxation region, across
wh~ch the gas and the particles attain an equili-br1um state. In the case of a piston speed smaller than a critical value, the frozen shock front vanishes and a thick transition structure of a fully-dispersed shock wave forms aft er a long time.
The particles that we re situated initially in a domain close to the piston collide with it. Two types of collision, that is, a sticking and a perfect elastic collision, were considered in the present study. In the case when the particles stick to the surface of the piston aft er their impact, the strength and the propagation velocity of the frozen shock wave decrease monotonically as it propagates. The gas in a downstream reg ion behind the shock front becomes compressed gradually by the force due to the particles. In the case when the particles are allowed to reflect from the piston, on the other hand, they have a significant influence on the flow structure especially at an initial stage. Spatial change in pressure is small
i~ comparison with the case of no reflected par-t1cles. The reflected particles can move faster
~han th~ :hock front at first if the piston speed 1S suff1c1ently large. In this case, the shock front propagates at a velocity greater than that for the pure-gas case for a while as a re~ult of the compression produced by the reflected particles ahead of it. The reflected particles remain in a limited domain adjacent to the piston surface. Therefore, the flow structure aft er a long time elapses depends little on the type of partic1e collision at the piston surface. The flow struc-ture at a final stage is modified by the presence of the ref1ected part ic les on1y in a non-uniform reg ion close to the piston. Some ana10gy exists bet ween the piston prob1em and a steady supersonic
4
the flow structures due to the different types of partic1e collisions with the wa11, which was found in the present study for the former prob1em, may a1so arise in the latter prob1em.
It wou1d be usefu1 to check these resu1ts in the new UTIAS dusty-gas shock-tube faci1ity in the near future.
REFERENCES
1. Rudinger, G., "Effective Drag Coefficient for Gas-Partic1e Flow in Shock Tubes", Trans. ASME, J. Basic Eng., Vol. 92, 1970, pp. 165-172.
2. Outa, E., Taj ima, 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 Particles During Shock Process Propagating Through a Gas-Particle Mixture in a Shock Tube", Proc. l3th Int. Symp. Shock Tubes and Waves, 1981, pp. 655-663.
4. Rudinger, T., Chang, A., "Analysis of Nonsteady Two-Phase Flow", Phys. Fluids, Vol. 7, 1964, pp. 1747-1754.
5. Miura, H., "Weak Shock Waves in a Dusty Gas", J. Phys. Soc. Japan, Vol. 33, 1972, pp. 1688-1692.
6. Otterman, B., Levine, A. S., "Analysis of Gas-Solid Partic1e Flows in Shock Tubes", AlAA J., vol. 12, 1974, pp. 579-580.
7. Marconi, F., Rudman, S., Calia, V., "Numerical Study of One-Dimensiona1 Unsteady Particle-Laden Flows with Shocks", AlAA J., Vol. 19, 1981, pp. 1294-1301.
8. Miura, H., G1ass, I. I., "On a Dusty-Gas Shock Tube", Proc. R. Soc. Lond., A382, 1982, pp. 373-388.
9. 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.
10. Soo, S. L., "Fluid Dynamics of Multiphase Systems", B1aisdell, Waltham, 1967.
. 11. Rudinger, G., "Relaxation in Gas-Particle Flow", in P. P. Wegener , "Nonequilibrium Flows", Vol. 1, Marcel Dekker, N.Y., 1969, pp. 119-161.
12. Marbie, F. E., "Dynamics of Dusty Gases", Annua1 Review of F1uid Mechanics, Vol. 2, 1970, pp. 397-446.
13. Miura, H., Saito, T., and G1ass, I. I., "Normal Ref1ection of a Shock Wave at a Rigid Wal1 in a Dusty Gas", UTIAS Report No. 274, 1984.
•
•
t
+
P
o
x
FIG. 1 SCHEMATIC x,
tDIAGRAM OF PISTON PROBLEM.
s
'
r---.
12.00
1l. 00
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
l.00
O.OO~________________
~--__________
~______________
~_____
~2.0
1.
·
5
1.0
0.5
~a=O ~2.00.00
0.50
1.00
LSO
2.00
2.50
3.00
3.50
4.00
4.50
5.00
M
pf
FIG. 2 VARIATION OF SHOCK MACH NUMBER WITH PISTON MACH NUMBER.
Msf, SHOCK MACH NUMBER BASED ON FROZEN SPEED OF SOUND;
Mse, SHOCK MACH NUMBER BASED ON EQUILIBRIUM SPEED OF
SOUND.
•
16.00~---.13.00
lO.OO~
7.00
Î
4.00
1.00~---~---~0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
Ca)
R
(b)
7.00r---__________________________________________________
~5.60
FIG. 3 FLOW QUANTITIES AT
t =0.0195 MS IN PARTICLE-STICKING
CASE (Mpf
=
2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
--- PURE GAS CASE.
3.0~
________________________________________________________ --;
2.6
2.20
1.80
1.40
1.0C~----__
~~______ --__ --______ --__ --______ --______________
~0.00
0.10
0.20
0.30
0.40
0.50
x0.60
0.70
0.80
0.90
1.00
Cc)
UCd)
4.0Cr---.
3.20
2.40
G1.60
11
0.80
~!
O.
OO~---.::::..J..---I0.00
0.10
0.200.30
0.400.50
0.60
0.70
0.80
x
FIG. 3 - CONTINUED
0.90
1.00
FLOW QUANTITIES AT
t =0.0195 MS IN PARTICLE-STICKING
CASE (Mpf
=2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
Cd)
VELOCITY.
G, GAS; P, PARTICLES.
--- PURE GAS CASE.
•
•
P
·1.0(~---L---~----~--~0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
(a)R
(b)
7.00
~---~~---~5.80
4.60
G
3.40
2.20
P
1.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
FIG.
4
FLOW QUANTITIES AT
t =0.039
MS IN PARTICLE-STICKING
CASE (Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
Cc)
TEMPERATURE;
3.00
r---~
2.60
2.20
1.80
1.40
1.00
~________ --____
--~~--____ --____ --____ --____ --__________
~0.00
0.10
0.20
0.30
3.20
2.40
G1.60
ü.80~
0.00
~
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
(c)
(d)0.00
0.00
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
FIG. 4 - CONTINUED
FLOW QUANTITIES AT
t =0.039 MS IN PARTICLE-STICKING
CASE (Mpf
=2).
(a)
PRESSURE; eb) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
, -'
•
16.00r---~13.00
10.00
-
'
I
7.00
I
I
4.00
I
1.00~---~----~---~ R0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
Ca)
Cb)
7.00r---~5.60
4.60
3.40
2.20
P
---,
I
I
1.00
L-__________
~
__
================~~~-1
____
~
____ --____
~
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
FIG. 5 FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-STICKING CASE
C~f =
2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
3.00~---.
G
2.60
2.20
1.80
1.40
1.00
~__ - -______________________________
~~____
~________
~~0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
XCc)
UCd)
4.00~---.3.20
2.40
1.60
0.80
G
) - - =
-
=-- -=----==-~---.:::.::.:.=:..:...-=:-=--
=-
-
-
-=-=-~--.:.
-
.=
-
--
=---=-=
-
=-~----
-
--1
I
I
O.OO~____________________________________
~____
~__________
~0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
xFIG. 5 - CONTINUED
FLOW QUANTITIES AT
t=
0.078 MS IN PARTICLE-STICKING CASE
CMpf
=2).
Ca)
PRESSUREj
Cb)
MASS CONCENTRATIONj
Cc)
TEMPERATUREj
Cd)
VELOCITY.
G, GAS; P, PARTICLES.
---PURE GAS CASE.
•
•
16.00~---~-13.00
10.00
7.00
4.00
1.00~__________
-L
____________________________________________
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
R
7.00
5.80
4.60
3.40
2.20
Ca)
Cb)
1.00
~________
~~________________________________________
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
FIG. 6 FLOW QUANTITIES AT
t =0.312 MS IN PARTICLE-STICKING CASE
C~f =
2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
T
3.00~---~2.60
G
2.20
1.80
1.40
1.00~____________________________________
~__________
~______
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
x
8.00
9.00
10.00
Cc)
Cd)
3.20
2.40
G
1.60
0.80
0.00
~---__
--~__ - -____ - -____ - -__
~--____ - -__
~--____ - -____
~0.00
1.00
2.00
3.00
4.00
5.00
(x)
FIG. 6 - CONTINUED
6.00
7.00
8.00
9.00
10.00
FLOW QUANTITIES AT
t =0.312 MS IN PARTICLE-STICKING CASE
(Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
Cc)
TEMPERATURE;
Cd)
VELOCITY.
G, GAS; P, PARTICLES.
•
IJP
16.00r---~---~---~13.00
10.00
7.00
4.00
1.00~________________
~________
~____ - -______________
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
Ca)
R
(b)
7.00r---~5.80
4.60
3.40
2.20
1.00~______ - -______
--~______________ - -______________
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
FIG. 7 FLOW QUANTITIES AT
t =0.624 MS IN PARTICLE-STICKING CASE
(Mpf
=
2).
(a)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
T
3.00
2.60
2.20
1.80
1.40
G
1.00
~--~~~--~---~----______ - -________________ - -____
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
Cc)
U
Cd)
4.00r---~3.20
2.40
1.60
0.80
O.OO~__ - -____________
--~__ - -________
~~--____ - -____ - -____
~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
FIG. 7 - CONTINUED
FLOW QUANTITIES AT
t=
0.624 MS IN PARTICLE-STICKING CASE
(Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
Cc)
TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
•
J ,.) 16.00T---~13.00
10.00
7.00
4.00
1.00
~__________ - -__________ - -____________
~__________ - -__
~~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
R
7.00
5.80
4.60
3
.
40
2.20
1.00
x
Ca)
(b)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7
.
00
8
.
00
9.00
10.00
x
FIG.
8
FLOW QUANTITIES AT
t =1.248
MS IN PARTICLE-STICKING CASE
(Mpf
=2).
Ca) PRESSURE; (b) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
T
3.00~---'2.60
G
2.20
1.80
1.40
1.00~---~---~---~0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
U
4.00
3.20
2.40
1.60
0.80
0.00
x
Cc)
Cd)
r---==~~~----_G
P
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
FIG. 8 - CONTINUED
FLOW QUANTITIES AT
t=
1.248
MS IN PARTICLE-STICKING CASE
CMpf
=
2).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
P
16.00r---,
G
13.00
10.00
7.00
4.00
1.00
~__ ----__ - -____ - -____ - -____ - -____ - -____ - -____ - -____ - -____
~-10.00
-9.00 -8.00 -7
.
00
-6
.
00
-5.00
-
4.00
-3.00
-
2.00
-1.00 0.00
x
Ca)
eb)
G
5.80
4
.
60
3.40
2.20
1.00~__ - -____ - -____ - -____ - -____ - -____ - -____ - -____ - -____ - -____
~-10.00 -9.00 -8.00 -7.00 -6.00
-5.00 -4.00 -3.00 -2.00
-1.00 0.00
Jx
FIG. 9 STATIONARY STRUCTURE OF SHOCK WAVE CMpf
=
2).
Ca) PRESSURE; Cb) MASS CONCENTRATION;
Cc) TEMPERATURE;
Cd) VELOCITY
.
G, GAS; P, PARTICLES.
T
3.00
~---~---I2.60
G
2.20
1.80
1.40
1.00
~---~---~
-10.00
-9.00
-8.00 -7.00 -6.00
-5.00
-4.00 -3.00 -2.00 -1.00 0.00
x
(c) U (d)4.00
~---.3.20
2.40
G
1.60
~
P
0.80
0.00
~---____ - -____ - -____ - -____ - -____ - -____
~-10.00 -9.00 -8.00
-7.00 -6.00
-5.00 -4.00 -3.00 -2.00 -1.00 0.00
x
FIG. 9 - CONTINUED
STATIONARY STRUCTURE OF SHOCK WAVE (Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)TEMPERATURE;
16.00
13.00
10.00
7.00
4.00
1.00
R
7.00
5.80
4.60
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
Ca)
Cb)
3.40
G
2.20
1.00
0.00
0.10
0.20
0.30
0.40
0.50
x
0.60
0.70
0.80
0.90
1.00
FIG. 10 FLOW QUANTITIES AT
t =0.0195 MS IN PARTICLE-REFLECTION
CASE
CMpf
=2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
Cd)
VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES; --- PURE GAS CASE.
3.00
~---.2.60
2.20
1.80
1.40
1.00
U0.00
0.10
0.20
0.30
0.40
0.50
x
(c)
(d)
0.60
0.70
0.80
0.90
1.0
4.00
r---~--_.3.20
2.40
G1.60
0.80
0.00
~---__
~~__
~__________________________ - -__
~--____
~0.00
0.10
0.20
0.30
0.40
0.50
x
FIG. 10 - CONTINUED
0.60
0.70
0.80
0.90
1.0
FLOW QUANTITIES AT
t =0.0195 MS IN PARTICLE-REFLECTION
CASE (Mpf
=
2).
(a)
PRESSUREj (b) MASS CONCENTRATIONj
(c)TEMPERATUREj
(d) VELOCITY.
G, GASj P, PARTICLESj R, REFLECTED
PARTICLESj --- PURE GAS CASE.
16
.
00
13.00
10.00
7.00
4.00
1.00
o
R7.00
5.89
4.60
3.40
2.20
1.00
G
. 00
0.10
0.20
G
R
Ir
0.20
O .
~O x
0.
,
50
0.60
0.70
0.80
0.90
1.
00
Ca)
Cb)
~..,.,.
.J
P
0.00
0.10
0.20
0.30
0.40
0.50
x
0.60
0.70
0.80
0.90
1.00
FIG
.
11
FLOW QUANTITIES AT
t =0.039 MS IN PARTICLE-REFLECTION
CASE
CMpf
= 2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
T
3.00
2.60
2.20
1.80
1.40
1.00
R
0.00
0.10
0.20
0.30
0.40
0.50
x
Cc)
Cd)
3.20
2.40
G
1.60
0.80
P
0.60
0.70
0.80
0.90
1.00
0.00
0.00
~---~~---~
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
x
FIG.
11 -CONTINUED
FLOW QUANTITIES AT
t =0.039 MS IN PARTICLE-REFLECTION
CASE CMpf
=
2).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES; --- PURE GAS CASE.
P
16.00
13.00
10.00
7.00
4.00
1.00
G - - --
---0.00
0.10
0.20
0.30
0.40
0.50
x
0.60
0.70
0.80
0.90
1.00
R7.00
5.80
4.60
3.40
2.20
1.00
Ca)
Cb)
G
F-~----~~~---~
-
~--~~~l
I
R
__ ---____ /
---
P
0.00
0.10
0.20
0.30
0.40
0.50
x
0.60
0.70
0.80
0.90
1.00
FIG. 12 FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-REFLECTION
CASE CMpf
=2).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
3.00
~---~--~2.60
-G
2.20
R
1.80
1.40
1.00
~---____ --__________________
~~____________
~ U4.00
3.20
2.40
1.60
0.80
0.00
0.00
0.10
0.20
0.30
0.40
0.50
x
Cc)
Cd)
R
0.60
0.70
0.80
0.90
1.00
G j-~~~~--=..
~..
=-~-=.
.
_-=----~~--~- -~_. ~-~~~P
I
I
I
0.00
0.10
0.20
0.30
0.40
0.50
x
0.60
0.70
0.80
0.90
1.00
FIG. 12 - CONTINUED
FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-REFLECTION
CASE (Mpf
=2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
Cd)
VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES; --- PURE GAS CASE.
16.00
13.00
10.00
7.00
4.00
1.00
R
7.00
5.80
4.60
3.40
2.20
1.00
G
0.00
1.00
2.00
3.00
4.00
5.00
x
Ca)
Cb)
6.00
7.00
8.00
9.00
10.00
0.00
1.00
2.00
3.00
4.00
5.00
x
6.00
7.00
8.00
9.00
10.00
FIG. l3 FLOW QUANTITIES AT
t =0.312 MS IN PARTICLE-REFLECTION
CASE CMpf
=2).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
3.00
2.60
2.20
1.80
1.40
1.00
U0.00
1.00
2.00
3.00
4.00
5.00
x
(c)
(d)
6.00
7.00
8.00
9.00
10.00
4.00
~---.3.20
2.40
1.60
0.80
0.00
R0.00
1.00
2.00
3.00
4.00
5.00
x
6.00
7.00
8.00
9.00
10.00
FIG. 13 - CONTINUED
FLOW QUANTITIES AT
t =0.312 MS IN PARTICLE-REFLECTION
CASE
(Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES .
16.00
13.00
10.00
7.00
4.00
R
7.00
5.80
4.60
3.40
2.20
1.00
2.00
3.00
4.00
5.00
x
Ca)
Cb)
6.00
7.00
8.00
9.00
10.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
x
FIG. 14 FLOW QUANTITIES AT
t =0.624 MS IN PARTICLE-REFLECTION
CASE CMpf
=2).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES.
3.00
2.60
2.20
1.80
1.30
1.00
U4.00
3.20
2.40
1.60
0.80
0.00
G R0.00
1.00
2.00
3.00
4.00
5.00
x
(c)
(d)R
6.00
7.00
8.00
9.00
10.00
0.00
1.00
2.00
3.00
4.00
5.00
x
6.00
7.00
8.00
9.00
10
:
00
FIG. 14 - CONTINUED
FLOW QUANTITIES AT
t =0.624 MS IN PARTICLE-REFLECTION
CASE (Mpf
=2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
Cc)
TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES.
16.00
13.00
10.00
7.00
4.00
1.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
R
7.00
5.80
4.60
3.40
2.20
1.00
x
(a) (b)0.00
1.00
2.00
3.00
4.00
5.00
x
6.00
7.00
8.00
9.00
10.00
FIG. 15
FLOW QUANTITIES AT
t =1.248 MS IN PARTICLE-REFLECTION
CASE
(~f =2).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)
TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES.
3.00
2.60
2.20
1.80
1.40
1. 00
U4.00
3.20
2.40
1.60
0.80
0
.
00
R0.00
1
.
00
2.00
3.00
4.00
5.00
x
Cc)
(d) R G6.00
7.00
8.00
9.00
10.00
G Po .
00
1. 00
2.00
3.00
4.00
5.00
x
6.00
7.00
8.00
9.00
10.00
FIG. IS - CONTINUED
FLOW QUANTITIES AT
t =1.248 MS IN PARTICLE-REFLECTION
CASE CMpf
=2).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION; Cc) TEMPERATURE;
Cd)
VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED
PARTICLES.
2.25
2.00
1. 75
1.50
1.25
1.00
R2.00
1.80
1.60
1.40
1.20
1.00
0.00
0.50
1.00
1.50
2.00
2.50
x
(a)
(b)
0.00
0.50
1.00
1.50
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
3
.
00
3.50
4.00
4.50
5.00
FIG. 16 FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-STICKING
CASE
(Mpf
=
0.4).
(~)
PRESSURE; (b) MASS CONCENTRATION;
(c)
TEMPERATURE;
1.20
1.16
1.12
1.08
1.04
1.00
-Gl
I
I
I
I
I
I
0.00
0.50
1.00
1.S0
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
U0.80
0.64
0.48
0.32
0.16
0.00
~
I
I
~I!
Cc)
(d)0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
x
FIG. 16 - CONTINUED
FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-STICKING
CASE CMpf
=0.4).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; --- PURE GAS CASE.
2.25
2.00
1. 75
1.50
1.25
1.00
R2.00
1.80
1.60
1.40
1.20
1.00
G
0.00
0.50
1.00
1.50
2.00
2.50
x
Ca)
Cb)
3.00
3.50
4.00
4.50
5.00
0.00
0.50
1.00
1.50
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
FIG.
17
FLOW QUANTITIES AT
t =0.156
MS IN PARTICLE-STICKING CASE
CMpf
=0.4).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd)
VELOCITY.
G,
.
GAS; P, PARTICLES.
T
1.20
~---~1.16
1.12
1.08
1.04
1.00
0.64
0.48
0.32
0.16
0.00
0.00
0.50
1.00
1.~02.00
2.50
x
(c)
(d)3.00
3.50
4.00
4.50
5.00
0.00
0.50
1.00
1.50
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
FIG. 17 - CONTINUED
FLOW QUANTITIES AT
t =0.156 MS IN PARTICLE-STICKING CASE
(Mpf
=0.4).
(a)
PRESSURE; eb) MASS CONCENTRATION;
(c)TEMPERATURE;
2.25
2.00
1. 75
1.50
1.25
1.00
R
2.00
1.80
1.60
1.40
1.20
I
I
I
0.00
0.50
1.00
1.50
2.00
2.50
x
Ca)
Cb)
I
I
I
I
3.00
3.50
4.00
4.50
5.00
1.00
~---~---__ - -____
~~~~________________________
~0.00
0.50
1.00
1.50
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
FIG.
18
FLOW QUANTITIES AT
t =0.234
MS IN PARTICLE-STICKING CASE
CMpf
=0.4).
Ca)PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
1. 20
1.16
1.12
1.08
1.04
1.00
0.64
0.48
0.32
0.16
0.00
-,
I
I
0.00
0.50
1.00
1.S0
2.00
2.50
x
Cc)
Cd)
- I
3.00
3.50
4.00
4.50
5.00
0.00
0.50
1.00
1.50
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
FIG. 18 - CONTINUED
FLOW QUANTITIES AT
t =0.234 MS IN PARTICLE-STICKING CASE
CMpf
=0.4).
Ca)
PRESSURE; (b) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
2.25
2.00
1. 75
1.50
1.25
1.00
R
2.00
1.80
1.60
1.40
1.20
1.00
G0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00
40.00
45.00 50.00
x
Ca)
Cb)
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35
.
00 40.00 45.00 50.00
x
FIG. 19
FLOW QUANTITIES AT
t =0.78 MS IN PARTICLE-STICKING CASE
CM
p
f=0.4).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
1.20
1.16
1.12
1.08
1.04
1.00
U0.80
0.64
0.48
0.32
0.16
0.00
5.00 10.00 15.00 20.00
25.00
30.00 35.00
40.00 45
.0
0 50
.
00
x
(c)
(d)0.00
~---~---~---~
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
FIG. 19 - CONTINUED
FLOW QUANTITIES AT
t =0.78 MS IN PARTICLE-STICKING CASE
(Mpf
=
0.4).
(a)
PRESSURE; eb) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
P
2.25
~---.2.00
1.75
1.50
1.25
1.00
R
0.00
5.00 10.00 15.00 20
.
00
25.00
30.00 35.00 40.00 45.00 50.00
x
(a)eb)
2.00
r---~1.80
1.60
1.40
1.20
1.00
0.00
5
.
00
10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
FIG. 20 FLOW QUANTITIES AT
t =1.56 MS IN PARTICLE-STICKING CASE
(Mpf
=0.4).
(a)
PRESSURE;(b) MASS CONCENTRATION;
(c)TEMPERATURE;
T
1.20
1.16
1.12
1.08
1.04
G
1.00
~---~---~
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
(c)U
(d)0.80
~---~0.64
0.48
0.32
0.16
0.00
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
FIG. 20 - CONTINUED
FLOW QUANTITIES AT
t =1.56 MS IN PARTICLE-STICKING CASE
(Mpf
=0.4).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)
TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
2.25
2.00
1. 75
1.50
1.25
1.00
~---~---~---~---~---~
0.00
5.00 10.00 15.00 20.00
25.00
30.00 35.00 40.00 45.00 50.00
x
Ca)
R
Cb)
2.00
~---~---.1.
80
t::::::::::::::::::_~--1.60
1.40
1.20
1.00
.
0.00
5.00 10.00 15.00 20.00
25.00
30.00
35.00 40.00 45.00 50.00
x
FIG.
21
FLOW QUANTITIES AT
t =3.12
MS IN PARTICLE-STICKING CASE
(Mpf
=0.4).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
T
1.20
r---1.16
1.12
1.08
1.04
1.00
~---~~---
____ - -____
_J
0.64
0.48
0.32
0.16
0.00
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
(c)
(d)0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
FIG. 21 - CONTINUED
FLOW QUANTITIES AT
t =3.12 MS IN PARTICLE-STICKING CASE
(~f =
0.4).
(a)
PRESSURE; (b) MASS CONCENTRATION;
(c)TEMPERATURE;
(d) VELOCITY.
G, GAS; P, PARTICLES.
2.25
2.00
1.75
1.50
1.25
1.00
R
0.00
5.00 10.00 15.00 20.00
25.00
30.00 35.00 40.00 45.00 50.00
x
Ca)
Cb)
2.00~---~---~ 1.80t=====~~=-===---____
== ____ _
1.60
1.40
1.20
1.00
~---~~---~
0.00
5.00 10.00 15.00 20.00
25.00
30.00 35.00 40.00 45.00 50.00
x
FIG. 22
FLOW QUANTITIES AT
t =5.46 MS IN PARTICLE-STICKING CASE
CMpf
=
0.4).
Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
1.20
~---~1.16
1.12
1.08
1.04
1.00
U1.80
0.64
0.48
0.32
0.16
0.00
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
x
Cc)
Cd)
0.00
5.00 10.00 15.00 20.00 25.00
30.00 35.00 40.00 45.00 50.00
xFIG. 22 - CONTINUED
FLOW QUANTITIES AT
t=
5.46 MS IN PARTICLE-STICKING CASE
CMpf
= 0.4).Ca)
PRESSURE;
Cb)
MASS CONCENTRATION;
Cc)
TEMPERATURE;
2.25
2.00
1. 75
1.50
1.25
1.00
-50.00 -45.00 -40.00 -35.00
-30.00 -25.00
-20.00 -15.00 -10.00
-5.00 0.00
x
R
2.00
1.80
1.60
1.40
1.20
1.00
(a) (b)-50.00 -45.00 -40.00 -35.00 -30.00 -25.00 -20.00 -15.00 -10.00
-5.00 0.00
x
FIG.
23
STATIONARY STRUCTURE OF SHOCK WAVE
(Mpf
=0.4).
(a)
PRESSURE; (b) MASS CONCENTRATION;
Cc)TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
1. 20
1.16
1.12
1.08
1.04
1.00
-50.00 -45.00 -40.00 -35.00 -30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00
x U0.80
0.64
0.48
0.32
0.16
0.00
(c) (d)-50.00 -45.00 -40.00 -35.00 -30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00
x
FIG. 23 - CONTINUED
STATIONARY STRUCTURE OF SHOCK WAVE CMpf
=0.4).
Ca) PRESSURE;
eb)
MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES.
'
.
2.25
2.00
1. 75
1.50
1.25
-1.00
0.00
1.80
1.60
1.40
1.20
R1.00
0.00
G
-1
I
I
I
I
I10.50
1.00
'
:]
I
I
I
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
x
Ca)
Cb)
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
x
FIG. 24 FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-REFLECTION CASE
CMpf
=
0.4).
Ca) PRESSURE; Cb) MASS CONCENTRATION; Cc) TEMPERATURE;
Cd) VELOCITY.
G, GAS; P, PARTICLES; R, REFLECTED PARTICLES;
1. 20
1.16
1.12
1.08
1.04
1.00
U0.80
0.64
0.48
0.32
--~I
I
I
R
I
I
P
0.00
0.50
1.00
1.50
2.00
2.50
x
(c)
(d)R
l---~
I
3.00
3.50
4.00
4.50
5.00
0.16
I
~I
0.00
0.00
0.50
1.00
1.50
FIG. 24 - CONTINUED
2.00
2.50
x
3.00
3.50
4.00
4.50
5.00
FLOW QUANTITIES AT
t =0.078 MS IN PARTICLE-REFLECTION CASE
(Mpf
