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Head-on interaction of planar shock waves with ideal rigid open-cell porous materials. Analytical model
FLUID DYNAMICS
RESEARCH ELSEVIER
Fluid Dynamics Research 16 (1995) 203-215
Head-on interaction of planar shock waves with ideal rigid open-cell porous materials. Analytical model H. Li, A. Levy, G. B e n - D o r Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel
Received 28 March 1994
Abstract An analytical model for predicting the flow field which results immediately after the head-on collision of planar shock waves with ideal rigid porous materials in the vicinity of the porous material/air interface was developed. In addition, a simplifiedempirical method for readily estimating the enhanced pressure peak at the shock-tube end-wall is proposed. The predictions of the proposed analytical and empirical models werecompared to experimental results. The agreement was found to be good to excellent.
1. Introduction A continuous increasing interest in better understanding the head-on interaction of planar shock waves with porous material is evident in the scientific literature of the past decade [1-23]. This is due to its possible application both in civil and military industries. In general, two approaches have been employed by researchers for treating the above-mentioned interaction: the single-phase approach which is based on an oversimplified mixture theory, and the two-phase approach which is, in the authors' opinion, the better approach for analyzing transport p h e n o m e n a in fluid saturated porous media. In the two-phase approach, the fluid filling the pores and the porous medium are considered as two phases which interact with each other. A detailed description of this approach was given by Ref. [24]. A one-dimensional two-phase analysis of air as the fluid phase was presented in Ref. [25]. In a recent study [26] an analytical model for solving the flow field associated with regular reflections of planar shock waves over rigid porous layers was developed. The governing equations of the gaseous phase inside the porous material were obtained by simplifying the general macroscopic balance equations which were obtained by an averaging process over a representative elementary volume (REV) of the microscopic balance equations as was originally done in Ref. [27]. 0169-5983/95 / $4.25 © 1995 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved SSDI 0 1 6 9 - 5 9 8 3 ( 9 5 ) 0 0 0 0 8 - 9
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The analytical predictions of the model developed in Ref. [26] were compared to experimental results and very good to excellent agreement was evident. Based on these facts, it was decided to adopt a similar approach and develop an analytical model for predicting various properties related to the head-on interaction of planar shock waves with ideal rigid porous material. By the term ideal rigid porous material we mean that the solid particles are completely stationary. This is in contrast to the rigid porous material investigated by van Dongen's research group (see Refs. [4, 6 8]) in which the solid particles (sand particles) could move. While for movable solid particles m o m e n t u m and energy exchanges are significant and hence should be accounted for, in the case of non-movable solid particles, which is considered in the present study, the exchange of momentum and energy could be neglected as was shown in Ref. [26]. In the case of flexible porous materials m o m e n t u m and energy exchanges between the two phases become even more important and as a result the analytical model developed in the following cannot be applied to them. It should also be noted here that while in m a n y investigations [4, 6-8, 13, 14, 17-22] most of the attention was paid to the waves propagating inside the porous material and the fluid occupying its pores, in the present study we investigate the flow field only in the region adjacent to the front edge of the porous material, i.e., a region near to interface separating the gas and the porous material.
2. Present study Based on Ref. [27], the one-dimensional conservation equations of the gaseous phase which flows through the pores of the porous medium are given by 8U
8U + A~-- = 0,
(1)
ox
where U, the dependent variables vector, is
,:Eil
(1.1)
and A, the coefficients matrix, is
A =
o1
u
z/p
[1 + ( 7 - 1)~]p
u
.
(1.2)
In the above equations u is the gaseous phase velocity, p is its density, p is its pressure, 7 is its specific heat capacities ratio and T is its tortuosity. The tortuosity 1 is a geometric parameter which 1The tortuosity of either the fluid or the solid expresses the total static momentum of the oriented elementary surface which comprises the surface between the gas and the solid phases, with respect to the plane passing through the centroid of the representative elementary volume (REV) per unit volume of the fluid or the solid phase within the REV. For more details see Ref. [27].
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
205
accounts for the fact that the average pressure gradient and the average gas velocity are not in the same direction. The tortuosity for gases is, in general, smaller than unity. It was found by us that by redefining the thermodynamic properties of the gaseous phase, both the dependent variables vector [Eq. (1.1)] and the coefficients matrix [Eq. (1.2)] can be transformed to obtain the forms appropriate to a pure gas, i.e.,
U*=
A*=
u* p*
,
i
(1.3)
~* 0
p* u*
0 1 , 1/p*
0
7" P*
u* _]
(1.4)
where the properties with an asterisk are transformed properties. The advantage of having the forms given by (1.3) and (1.4) over those given by (1.1) and (1.2) is that the forms given by (1.3) and (1.4) have analytical solutions which can readily be adopted for the present case. The above-mentioned redefinition (transformation) is given in the following. Let us redefine the gaseous phase dynamic and thermodynamic properties as follows
p*=p,
e* .
p*=~p,
.
1
.
p*
.
7" - 1 p*
.
T * = T,
.
1
.
p
e,
7 - lp
u*=u,
7"= 1 +(7-1)~,
7*
h* - - -
P*
7" - 1 p*
-
7* 7
R*=vR,
h.
(2)
Based on the above redefined gaseous phase properties it could be further shown that
C* = ~~e* T , v* =
Oh*
O~TTv = Cv,
7* Oh p
7*
ca=er, =7ar =Tc.
(3.1)
(3.2)
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
(recall that the tortuosity, • is a constant parameter). Combining the above two expressions results in: 7* ?
7" 7
C ~ - C~ = - - C p -
c*
(7" /7 )c~
Cv*
Cv
Cv = - - C p
1 7" - 1 Cp Cp--- 7 7
= T,
7-1 -
-
Cp =
zR
= R*,
(4.1)
7
- Y*.
(4.2)
These two equations indicate that the proposed redefinition [see Eqs. (2)] maintains the properties of a perfect gas. Substituting the redefined properties into the conservation Eqs. (1) yields a set of equations identical to that of a pure gas, i.e., 8p*
,8p*
p , 8u*
--St + u -~--x + 1 8p*
8u*
8x =0,
8u*
8u*
1 8p*
-~+u*-~--x + p , - - - 0 , S x
u* 8p*
p~ %T + 7" -~x + p* ~x - 0.
(5)
The eigenvalues of these equations are
u* and u* + ~/ -~- . Consequently, the disturbance propagation speed, a*, is clearly
a * = ~/ )-~ .
(6.1)
In consistence with that obtained for a perfect gas. Note that with the aid of the definitions given by (2) (a*) 2 -- Y'P* -- 7* P* 7
za 2 .
(6.2)
Since a* can be considered as the local speed of sound of the gaseous phase occupying the pores of the porous medium, the gas Mach number inside the porous m e d i u m can be defined as U*
M* = -a*" -
(7)
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
(I) [~Mi
207
(0)
I I I
(a)
P
S
(hi
Fig. 1. Schematic illustration of the waves pattern and definition of flow states (a) prior to the head-on reflection, (b) followingthe head-on reflection.
If we limit our analysis to a short time immediately after the head-on interaction, during which all the discontinuities can be assumed to move with constant velocity, the Galilean transformation could be performed on the set of the governing equations (5), and they could be simplified from a set of time-dependent equations to the following set of time independent equations.
u* op*
+
p, Ou*
~x = 0 '
u*
Ou*
1 ~p*
---0, + p* ~x
~*
~u* ~
u* Op + p* ~x
-0.
(8)
Since both the unsteady (5) and steady (8) sets of governing equations of the gaseous phase occupying the pores of the porous medium are identical to those of pure gas, analytical solutions which were developed for pure gases e.g., the Rankine Hugoniot relations for calculating the j u m p conditions across shock waves, can readily be used for the gaseous phase occupying the pores of the porous medium with the appropriate (above-mentioned) transformation of both the dynamic and the thermodynamic properties of the gas. Consider Fig. la in which a constant velocity planar shock wave is seen to propagate from left to right towards a porous material. The flow states ahead and behind the incident shock waves are (0) and (1), respectively. The flow state inside the porous medium is (2). The front edge of the porous material is marked by P. The incident shock wave Mach number is Mi. The wave pattern which is obtained following the head-on reflection of the incident shock wave from the front edge of the rigid porous material is shown in Fig. lb. The head-on collision results in
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H. Li et al. /Fluid Dynamics Research 16 (1995) 203-215
~t P
reflected
S (4) ///13)
/
transmitted
Fig. 2. (x, t)-diagram of the wave pattern shown in Fig. lb.
a reflected shock wave, having a Mach number Mr, and a transmitted compaction wave, having a Mach number M*. The reflected shock wave propagates inside the gaseous phase from right to left, into state (1). The flow state obtained behind it is (5). The transmitted compaction wave propagates inside the porous material, from left to right. The flow state behind it is (3). Since the porous material is ideally rigid, its front edge, P, remains at its original position. Owing to the high pressure behind the reflected shock wave, in region (5), the gas flows into the porous material through it open cells. The contact surface separating the gases, which were originally inside the porous material and which penetrated into the porous material, is marked by S. The above-described process is shown in Fig. 2 in the (x,t)-plane. The front edge of the porous material is located at x = 0. The solid lines indicate the trajectories of the incident, the reflected, and the transmitted shock waves and the dashed line describes the trajectory of the contact surface, S. Note that based on the experimental finding of Refs. [-10,19] the transmitted shock wave propagates with a constant velocity while the reflected shock wave accelerates and the contact surface attenuates. Consequently only the trajectories of the incident and the transmitted shock waves are straight. However, if the discussion is limited to a very narrow domain near the front edge of the porous material, the reflected shock wave and the contact surface can be assumed to propagate with constant velocities.
3. The governing equations In the following the governing equations of the flow field shown in Fig. l b are listed. Across the incident shock wave: U1 --
2ao(')
y+l
Mi-
(9.1)
H. Li et al. /Fluid Dynamics Research 16 (1995) 203 215
P,
=
Po
M2
-- -
-
[27M~ - (7 - 1)3 '/2 [-(7 - 1)M~ + 23 '/2 a l = ao
209
(9.3)
(7 + 1)Mi
(7 + 1) M2 P, = Po (7 - 1)M2 + 2"
(9.4)
Similarly, across the transmitted c o m p a c t i o n wave:
u~
-
~* + ~ 1, 2a* ( M * )
[27* P* = P* L ~ -~--~ (M*)2
, 7"-~] 7* + '
[27"(M*) 2 - (7* - 1)]1/2[-(7" - 1) ( M r * ) 2
a* = a*
(10.1)
(7* + 1)M,*
(7* + 1)(M*) 2 P~ - P~ (7* - 1) (Mr*)2 -F 2"
(10.2) q-- 2 ] 1/2
(lO.3)
(10.4)
Across the reflected shock wave: us=u,---
P5 = Pl
7+1
Mr-
(11.1)
Mr - - -
.12,
[27Mr 2 - (7 - 1) 1/2 [(7 -- 1) Mr2 + 2]'/2 a5 = a I
(7 + 1)Mr
(7 + 1)M~ ps = p, (7 - 1) Mff + 2"
(11.3)
(11.4)
The similarity between the c o r r e s p o n d i n g e q u a t i o n s of the sets m a r k e d by (9), (10) and (11) is n o t e d here. The difference in (11.1) c o m p a r e d with (9.1) and (10.1) arises from the fact that while Uo = 0 and u~ = 0 (this is implied by the initial condition), u, # 0 [the velocity u, is induced by the incident shock wave, as can be seen from Eq. (9.1)]. It should also be n o t e d here that since the transmitted c o m p a c t i o n w a v e p r o p a g a t e s in a gaseous phase, which originally occupied the pores of the p o r o u s material, its d y n a m i c and t h e r m o d y n a m i c properties are transformed in a c c o r d a n c e with the t r a n s f o r m a t i o n presented earlier by Eqs. (2)-(4), (6) and (7), and hence they are m a r k e d with an asterisk.
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
The governing equations across the interface P are p s u s = ckp* u~,
(12.1)
~bp5 -~- 105u2 = -~) , 4,2 , -g p4~ -t- c~p4u
(12.2)
7 P5 + 1 2 _ o / - 1 P5 ~us 7 " - 1
(12.3)
P* + 1 u .2, p~
and the matching conditions across the contact surface S are p~ = p*,
(13.1)
u* = u~.
(13.2)
The above set of governing equations, which contains 17 equations, is solvable provided states (0) and (2) are given together with 7, ~b, r and Mi as initial conditions. For this situation, the 17 unknown variables are ul, Pl, al, Pl, u~, p~, a~, p~ us, Ps, as, Ps, u~, p*, p~, Mr and M*. 3.1.
The p r e s s u r e at the s h o c k - t u b e e n d - w a l l
Based on the model shown schematically in Fig. lb, the transmitted-compaction-wave-induced flow [-i.e., the flow in state (3)] stagnates on the shock-tube end-wall. Consequently, it is proposed here to calculate the peak pressure at the shock-tube end-wall using an empirical modification of the following well-known relation (see Ref. [-28], p. 417). I 7 2?* P~ 7* 1 M~/?. - 1' (14) 2-g = 1 4 P3
3
where p~ is the total (stagnation) pressure of the flow in state (3). Owing to the fact that the gas which flows through the open-cell pores of the rigid porous material attenuates it is proposed to modify Eq. (14) which is an isentropic relation by adding to it a dissipation/dispersion factor ~ such that it will be changed to the following empirical relation 2-g-p~ P3
E
l+~7*-l~M~
1
?*-1"
(15)
It is clear from Eq. (15) that for the case of a pure gas ~ = 1. Furthermore, for the present case, in which the flow attenuates as it passes through the pores, ~ should be less than unity, i.e., ~ < 1.
4. Comparison with experimental results and discussions Predictions of the above-described analytical model were compared to the experimental results of Ref. [19] and to additional experiments which were conducted during the same study but were not published so far. Details of the experimental set-up and typical experimental results can be found in Ref. [19]. Eight experiments have been used for the comparison. Their initial conditions
H. Li et al. /Fluid Dynamics Research 16 (1995) 203-215
211
Table 1 Initial conditions of the various experiments Case
Material
pores per inch
porosity
tortuosity
Mi
1 2 3 4
SiC
10
0.728
0.7
20
0.745
0.7
1.38 1.54 1.38 1.53
5 6 7 8
A120 3
30
0.814
0.75
40
0.821
0.75
1.38 1.54 1.37 1.54
Remarks: SiC-silicon carbide, A1203-alumina, Mi-incident shock wave Mach number. For all the experiments p ~ 8 3 KPa, To-~288 K
Table 2 The reflected shock wave and the pressure behind it. Comparison between the analytical predictions and the experimental results Case
1 2 3 4 5 6 7 8
Mr analytical
Mr experimental
Mj~lr
1.20 1.31 1.21 1.31 1.18 1.30 1.18 1.30
1.23 1.32 1.25 1.33 1.20 1.31 1.22 1.33
2.47% 0.76% 3.25% 1.52% 1.68% 0.77% 3.33% 2.28%
P5 [KPa]
P5 [KPa]
analytical
experimental
259 403 260 400 250 392 250 386
270 409 280 415 255 405 268 405
Aps/p5
4.2% 1.5% 7.4% 3.7% 2.0% 3.3% 6.9% 4.8%
are given in Table 1. An experimental method of calculating the tortuosity, z, of the investigated porous materials was presented in Ref. [26]. As can be seen from Table 1, four different rigid samples were used. Two were made of silicon-carbide (SIC) and two of alumina (A1203). Each of these four samples was tested with two different incident shock waves, one with Mi ~ 1.38 and the other with Mi---- 1.54. The comparison between the analytical predictions and the experimental results is summarized in Table 2. The reflected shock wave Mach number, Mr, and the gas pressure behind it, P5, as measured experimentally, are compared to their corresponding predictions by the presently proposed analytical model. The analytical predictions regarding both the reflected shock wave Mach number, Mr, and the pressure behind it, P5, underestimate the experimentally obtained values. The underestimations of
H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
212
Table 3 The shock-tube end-wall pressure: comparison between the empirical predictions and the experimental results Case
p~ [KPa] empirical
p~ [KPa] experimental
A-*/~* Pt~/Pt3
1 2 5 6
357 616 341 590
340 565 358 523
4.9% 8.6% 4.9% 12.0%
the reflected shock wave Mach numbers for both the silicon carbide and the alumina models are within the range 0.76% ~< AMr/~lr <~3.33% Similarly the underestimations of the pressures behind the reflected shock waves are in the range 1.5% ~< Aps/~5 <~ 7.4%. A general inspection of Tables 1 and 2 indicates that there is no evidence of a better agreement with either of the four different samples, i.e., porosities, which were used in the experiments. In addition, it seems that the analytical predictions agree better with the higher Mach number experiments (Mi - 1.54) than with the lower ones (Mi ~ 1.38). In general, however, the overall agreement is very good in view of the fact that it is based on a simple analytical model rather than a tedious and expensive numerical simulation.
3.3. The end-wall pressure As discussed earlier [see Eqs. (14) and (15) and the relevant text] the pressure at the shock-tube end-wall can be correlated to the stagnation pressure of the transmitted-compaction-induced flow. In the following, the validity of our empirical proposed relation is checked. In order to use Eq. (15) there is a need to assign a value to the coefficient ~. Using a simple try and error procedure it was found that c~ = 0.5 results in the best fit with all the available experiments. Table 3 presents the comparison of the results predicted by Eq. (15), with 0~-- 0.5, and some of the experimental results which were available to us (Ref. [19] and the additional ones which were mentioned earlier). Note that in order to apply Eq. (15) both M* and p* had to be supplied by solving the presently proposed analytical model outlined by Eqs. (9)-(11). It is evident from Table 3 that the agreement between the empirically predicted and the experimentally measured values is fairly good. In contrast to the earlier discussed comparison regarding the reflected shock wave Mach number and the pressure developed behind it, here the agreement is clearly better for the weaker incident shock wave. For the two compared cases, for which Mi-= 1.38, (i.e., cases 1 and 5) Apt3/Pt 3_~5%. Fig. 3 is a reproduction of Fig. 3 of Ref. [19]. As can be seen, the pressure at the shock-tube end-wall is seen to be still increasing. Consequently, the experimentally measured values of p* which are given in Table 3 are actually smaller than those which would have been obtained, had p~ reached its final value. Consequently, for cases 1, 2 and 6 for which p~ (empirical)> p*
H. Li et al. /Fluid Dynamics Research 16 (1995) 203 215 600
PtkPol
'
'
.P,,
'
213
j
.l.-1-
500 400
300 SiC(lOppi) Lo = 81 mm IVli= 1.542
200 100
0
I
I
I
1.5
2
2.5
i 3
t [ m sec] i 3.5
Fig. 3. The pressure history of the gas occupying the pores of the porous material as recorded by a pressure transducer located at the shock-tube end-wall.
Table 4 Dependence of the pressure at the shock-tube end-wall on the length of the sample as predicted empirically and comparison with experiments Case
4
Model length [mm]
p~ [KPa] experimental
p~ [KPa] empirical
41 62 83
515 510 490
581 -
Pt~/Pt~
12.0% 13.0% 17.0%
(experimental) the actual agreement is probably better than that indicated in Table 3. For case 5 for which p~ (empirical) < p* (experimental), the actual agreement is probably worse. It should also be mentioned that owing to the flow attenuation the stagnation pressure should depend on the length of the porous material model. This dependence is shown in Table 4 for case 4 of Table 1. It is evident from Table 4 that p~ decreases as the model length increases. However, the stagnation pressure seems to only slightly depend on the length of the model. When the length of the model is doubled from 41 to 83 mm, the pressure at the shock-tube end-wall decreases by about 6% from 515 to 490 KPa. The empirical prediction, which is based on Eq. (15), is independent of the length of the model. It should also be noted that in accordance with the above-mentioned fact that the experimentally measured values of p~ are underestimated (see Fig. 3) a better agreement than that shown in Table 4 probably exists. The dependence of the ratio between the pressures at the shock-tube end-wall with a porous material, p~ [as calculated by Eq. (15)] and without it, Pw, as a function of the porosity, ~b, for Mi = 1.5 is shown in Fig. 4. The ratio P~/Pwis always greater than unity and hence the presence of a porous material results in an enhancement in the pressure at the shock-tube end-wall. Similar results were reported in Ref. [13] for flexible porous materials, e.g., polyurethane foams. However,
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
2.4
Pw
2.2
2.0
1.8
16
1.4
/x
1.2
+ 1_0 0
0.2
0.4
0.6
0.8
1.0
Fig. 4. The pressureenhancementat the shock-tubeend-wallas a functionof the porosityas predictedby the empirical relation given by Eq. (15) and comparison with experiments.
since the solid material particles of a flexible porous material are also accelerated towards the shock-tube end-wall, the pressure enhancement in the case of flexible porous materials is much larger than it is in the case of rigid porous materials. Two experimental points which are added to Fig. 4 reveal a good agreement with the empirical predictions in the region 0.7 < 4' < 0.8. However, more experiments are required outside this narrow range in order to have a more definite verification of the empirical relation given by Eq. (15) regarding the pressure enhancement at the shock-tube end-wall.
5. Conclusion
An analytical model describing the head-on interaction of planar shock waves with ideal rigid porous materials has been developed. Predictions of the analytical model were compared to experimental results and very good agreement was evident. A simplified empirical method of calculating the pressure at the shock-tube end-wall was presented. Predictions based on the proposed empirical method were compared to experimental results and good agreement was obtained. The good agreement between the experimental results regarding the pressure at the shock-tube end-wall and the empirical predictions provides a justification of the assumption that the pressure at the shock-tube end-wall is actually equal to the stagnation pressure of the transmitted compaction-wave induced flow.
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
215
In a d d i t i o n , since the e m p i r i c a l m o d e l is b a s e d on the general a n a l y t i c a l m o d e l which was d e v e l o p e d in the c o u r s e of the p r e s e n t study, its g o o d p r e d i c t i o n s p r o v i d e a f u r t h e r v a l i d a t i o n of the proposed analytical model.
References [1] B.E. Gelfand, A.V. Gubanov and E.I. Timofeev, Sov. Phys. Fluid Mech. 4 (1983) 79. [2] L.G. Gvozdeva, Yu.M. Faresov and V.P. Fokeev, Sov. Phys. Appl. Math. and Tech. Phys. 3 (1985) 111. [3] H.W. Sandusty and T.P. Liddiard, Dynamic compaction of porous beds, NSWC TR 83 256 Naval Surface Research Center, White Oak, MD, 1985. [4] J.G.M. van der Grinten, M.E.H. van Dongen and H.J. van der Kogel, Appl. Phys. 58 (1985) 2937. [-5] L.G. Gvozdeva, Yu.M. Faresov, J. Brossard and N. Charpentier, in: Dynamics of Explosions, eds. J.C. Brown, J.C. Leyer and R.I. Solukin, Prog. Aero. Astro. 106 (1986). [6] J.G.M. van der Grinten, M.E.H. van Dongen, and H. van der Kogel, J. App. Phys. 62 (1987) 4682. 1-7] J.G.M. van der Grinten. An experimental study of shock induced wave propagation in dry, water saturated and partially saturated porous media, Ph.D. Thesis (University of Eindhoven, The Netherlands, 1987). [8] J.G.M. van der Grinten, M.A. Smits, H. van der Kogel and M.E.H. van Dongen, in: Shock Tube and Wave, ed. H. Gronig (VCH Verlagsgesellschaft mbH, Weinheim, 1988) pp. 357 362. [9] V.P. Korobeinikov and P.A. Utriew, in: Unsteady Interaction of Shock and Detonation Waves in Gases (Hemisphere Publishing, 1989) pp. 48-61. [10] J.M. Powers, D.S. Stewart and H. Krier, ASME J. Appl. Mech. 56 (1989) 15. [-1I] L.F. Henderson, R.J. Virgona, J. Di and L.G. Gvozdeva, in: Current Topics in Shock Waves, ed. Y.W. Kim (American Institute of Physics, New York, 1990) pp. 814-818. [12] B.W. Skews, Shock Waves 1 (1991) 205. [13] G. Ben-Dor, G. Mazor, G. Cederbaum, O. Igra and S. Sorek, in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 279-282. [14] S. Sorek, J. Bear, G. Ben-Dor and G. Mazor, Transport in Porous Media 9. (1992) 3. [15] M.W. Seitz, and B.W. Skews, in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 517-520. [16] B.W. Skews, M.D. Atkins and M.W. Seitz in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 509 516. [17] M.R. Baer, Shock Waves 2 (1992) pp. 121. [18] A. Levy, G. Ben-Dor, S. Sorek and J. Bear, Shock Waves 3 (1993) 105. [19] A. Levy, G. Ben-Dor, B.W. Skews and S. Sorek, Experiments in Fluids 15 (1993) 183. [20] M. Olim, M.E.H. van Dongen, K. Kitamura and K. Takayama, Int. J. Multiphase Flow 20 (1994) 557. [-21] G. Mazor, G. Ben-Dor, O. Igra and S. Sorek, Shock Waves 3 (1994) 159. [22] G. Ben-Dor, G. Mazor, O. Igra, S. Sorek and H. Onodera, Shock Waves 3 (1994) 167. [23] G. Ben-Dor and E. Zaretsky, Arch. Appl. Mech. 64 (1994) 365. [24] M.R. Baer and J.W. Nunziato, Int. J. Multiphase Flow 12 (1986) 861. [25] M.R. Baer, ASME J. Appl. Mech. 55 (1988) 36. [26] H. Li, A. Levy, and G. Ben-Dor Analytical prediction of regular reflection over porous surfaces in pseudo-steady flows, J. Fluid Mech. (1994) to he published. [27] J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena (Kluwer, Dordrecht, 1990). [28] P.H. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1993).
RESEARCH ELSEVIER
Fluid Dynamics Research 16 (1995) 203-215
Head-on interaction of planar shock waves with ideal rigid open-cell porous materials. Analytical model H. Li, A. Levy, G. B e n - D o r Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel
Received 28 March 1994
Abstract An analytical model for predicting the flow field which results immediately after the head-on collision of planar shock waves with ideal rigid porous materials in the vicinity of the porous material/air interface was developed. In addition, a simplifiedempirical method for readily estimating the enhanced pressure peak at the shock-tube end-wall is proposed. The predictions of the proposed analytical and empirical models werecompared to experimental results. The agreement was found to be good to excellent.
1. Introduction A continuous increasing interest in better understanding the head-on interaction of planar shock waves with porous material is evident in the scientific literature of the past decade [1-23]. This is due to its possible application both in civil and military industries. In general, two approaches have been employed by researchers for treating the above-mentioned interaction: the single-phase approach which is based on an oversimplified mixture theory, and the two-phase approach which is, in the authors' opinion, the better approach for analyzing transport p h e n o m e n a in fluid saturated porous media. In the two-phase approach, the fluid filling the pores and the porous medium are considered as two phases which interact with each other. A detailed description of this approach was given by Ref. [24]. A one-dimensional two-phase analysis of air as the fluid phase was presented in Ref. [25]. In a recent study [26] an analytical model for solving the flow field associated with regular reflections of planar shock waves over rigid porous layers was developed. The governing equations of the gaseous phase inside the porous material were obtained by simplifying the general macroscopic balance equations which were obtained by an averaging process over a representative elementary volume (REV) of the microscopic balance equations as was originally done in Ref. [27]. 0169-5983/95 / $4.25 © 1995 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved SSDI 0 1 6 9 - 5 9 8 3 ( 9 5 ) 0 0 0 0 8 - 9
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The analytical predictions of the model developed in Ref. [26] were compared to experimental results and very good to excellent agreement was evident. Based on these facts, it was decided to adopt a similar approach and develop an analytical model for predicting various properties related to the head-on interaction of planar shock waves with ideal rigid porous material. By the term ideal rigid porous material we mean that the solid particles are completely stationary. This is in contrast to the rigid porous material investigated by van Dongen's research group (see Refs. [4, 6 8]) in which the solid particles (sand particles) could move. While for movable solid particles m o m e n t u m and energy exchanges are significant and hence should be accounted for, in the case of non-movable solid particles, which is considered in the present study, the exchange of momentum and energy could be neglected as was shown in Ref. [26]. In the case of flexible porous materials m o m e n t u m and energy exchanges between the two phases become even more important and as a result the analytical model developed in the following cannot be applied to them. It should also be noted here that while in m a n y investigations [4, 6-8, 13, 14, 17-22] most of the attention was paid to the waves propagating inside the porous material and the fluid occupying its pores, in the present study we investigate the flow field only in the region adjacent to the front edge of the porous material, i.e., a region near to interface separating the gas and the porous material.
2. Present study Based on Ref. [27], the one-dimensional conservation equations of the gaseous phase which flows through the pores of the porous medium are given by 8U
8U + A~-- = 0,
(1)
ox
where U, the dependent variables vector, is
,:Eil
(1.1)
and A, the coefficients matrix, is
A =
o1
u
z/p
[1 + ( 7 - 1)~]p
u
.
(1.2)
In the above equations u is the gaseous phase velocity, p is its density, p is its pressure, 7 is its specific heat capacities ratio and T is its tortuosity. The tortuosity 1 is a geometric parameter which 1The tortuosity of either the fluid or the solid expresses the total static momentum of the oriented elementary surface which comprises the surface between the gas and the solid phases, with respect to the plane passing through the centroid of the representative elementary volume (REV) per unit volume of the fluid or the solid phase within the REV. For more details see Ref. [27].
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
205
accounts for the fact that the average pressure gradient and the average gas velocity are not in the same direction. The tortuosity for gases is, in general, smaller than unity. It was found by us that by redefining the thermodynamic properties of the gaseous phase, both the dependent variables vector [Eq. (1.1)] and the coefficients matrix [Eq. (1.2)] can be transformed to obtain the forms appropriate to a pure gas, i.e.,
U*=
A*=
u* p*
,
i
(1.3)
~* 0
p* u*
0 1 , 1/p*
0
7" P*
u* _]
(1.4)
where the properties with an asterisk are transformed properties. The advantage of having the forms given by (1.3) and (1.4) over those given by (1.1) and (1.2) is that the forms given by (1.3) and (1.4) have analytical solutions which can readily be adopted for the present case. The above-mentioned redefinition (transformation) is given in the following. Let us redefine the gaseous phase dynamic and thermodynamic properties as follows
p*=p,
e* .
p*=~p,
.
1
.
p*
.
7" - 1 p*
.
T * = T,
.
1
.
p
e,
7 - lp
u*=u,
7"= 1 +(7-1)~,
7*
h* - - -
P*
7" - 1 p*
-
7* 7
R*=vR,
h.
(2)
Based on the above redefined gaseous phase properties it could be further shown that
C* = ~~e* T , v* =
Oh*
O~TTv = Cv,
7* Oh p
7*
ca=er, =7ar =Tc.
(3.1)
(3.2)
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
(recall that the tortuosity, • is a constant parameter). Combining the above two expressions results in: 7* ?
7" 7
C ~ - C~ = - - C p -
c*
(7" /7 )c~
Cv*
Cv
Cv = - - C p
1 7" - 1 Cp Cp--- 7 7
= T,
7-1 -
-
Cp =
zR
= R*,
(4.1)
7
- Y*.
(4.2)
These two equations indicate that the proposed redefinition [see Eqs. (2)] maintains the properties of a perfect gas. Substituting the redefined properties into the conservation Eqs. (1) yields a set of equations identical to that of a pure gas, i.e., 8p*
,8p*
p , 8u*
--St + u -~--x + 1 8p*
8u*
8x =0,
8u*
8u*
1 8p*
-~+u*-~--x + p , - - - 0 , S x
u* 8p*
p~ %T + 7" -~x + p* ~x - 0.
(5)
The eigenvalues of these equations are
u* and u* + ~/ -~- . Consequently, the disturbance propagation speed, a*, is clearly
a * = ~/ )-~ .
(6.1)
In consistence with that obtained for a perfect gas. Note that with the aid of the definitions given by (2) (a*) 2 -- Y'P* -- 7* P* 7
za 2 .
(6.2)
Since a* can be considered as the local speed of sound of the gaseous phase occupying the pores of the porous medium, the gas Mach number inside the porous m e d i u m can be defined as U*
M* = -a*" -
(7)
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
(I) [~Mi
207
(0)
I I I
(a)
P
S
(hi
Fig. 1. Schematic illustration of the waves pattern and definition of flow states (a) prior to the head-on reflection, (b) followingthe head-on reflection.
If we limit our analysis to a short time immediately after the head-on interaction, during which all the discontinuities can be assumed to move with constant velocity, the Galilean transformation could be performed on the set of the governing equations (5), and they could be simplified from a set of time-dependent equations to the following set of time independent equations.
u* op*
+
p, Ou*
~x = 0 '
u*
Ou*
1 ~p*
---0, + p* ~x
~*
~u* ~
u* Op + p* ~x
-0.
(8)
Since both the unsteady (5) and steady (8) sets of governing equations of the gaseous phase occupying the pores of the porous medium are identical to those of pure gas, analytical solutions which were developed for pure gases e.g., the Rankine Hugoniot relations for calculating the j u m p conditions across shock waves, can readily be used for the gaseous phase occupying the pores of the porous medium with the appropriate (above-mentioned) transformation of both the dynamic and the thermodynamic properties of the gas. Consider Fig. la in which a constant velocity planar shock wave is seen to propagate from left to right towards a porous material. The flow states ahead and behind the incident shock waves are (0) and (1), respectively. The flow state inside the porous medium is (2). The front edge of the porous material is marked by P. The incident shock wave Mach number is Mi. The wave pattern which is obtained following the head-on reflection of the incident shock wave from the front edge of the rigid porous material is shown in Fig. lb. The head-on collision results in
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H. Li et al. /Fluid Dynamics Research 16 (1995) 203-215
~t P
reflected
S (4) ///13)
/
transmitted
Fig. 2. (x, t)-diagram of the wave pattern shown in Fig. lb.
a reflected shock wave, having a Mach number Mr, and a transmitted compaction wave, having a Mach number M*. The reflected shock wave propagates inside the gaseous phase from right to left, into state (1). The flow state obtained behind it is (5). The transmitted compaction wave propagates inside the porous material, from left to right. The flow state behind it is (3). Since the porous material is ideally rigid, its front edge, P, remains at its original position. Owing to the high pressure behind the reflected shock wave, in region (5), the gas flows into the porous material through it open cells. The contact surface separating the gases, which were originally inside the porous material and which penetrated into the porous material, is marked by S. The above-described process is shown in Fig. 2 in the (x,t)-plane. The front edge of the porous material is located at x = 0. The solid lines indicate the trajectories of the incident, the reflected, and the transmitted shock waves and the dashed line describes the trajectory of the contact surface, S. Note that based on the experimental finding of Refs. [-10,19] the transmitted shock wave propagates with a constant velocity while the reflected shock wave accelerates and the contact surface attenuates. Consequently only the trajectories of the incident and the transmitted shock waves are straight. However, if the discussion is limited to a very narrow domain near the front edge of the porous material, the reflected shock wave and the contact surface can be assumed to propagate with constant velocities.
3. The governing equations In the following the governing equations of the flow field shown in Fig. l b are listed. Across the incident shock wave: U1 --
2ao(')
y+l
Mi-
(9.1)
H. Li et al. /Fluid Dynamics Research 16 (1995) 203 215
P,
=
Po
M2
-- -
-
[27M~ - (7 - 1)3 '/2 [-(7 - 1)M~ + 23 '/2 a l = ao
209
(9.3)
(7 + 1)Mi
(7 + 1) M2 P, = Po (7 - 1)M2 + 2"
(9.4)
Similarly, across the transmitted c o m p a c t i o n wave:
u~
-
~* + ~ 1, 2a* ( M * )
[27* P* = P* L ~ -~--~ (M*)2
, 7"-~] 7* + '
[27"(M*) 2 - (7* - 1)]1/2[-(7" - 1) ( M r * ) 2
a* = a*
(10.1)
(7* + 1)M,*
(7* + 1)(M*) 2 P~ - P~ (7* - 1) (Mr*)2 -F 2"
(10.2) q-- 2 ] 1/2
(lO.3)
(10.4)
Across the reflected shock wave: us=u,---
P5 = Pl
7+1
Mr-
(11.1)
Mr - - -
.12,
[27Mr 2 - (7 - 1) 1/2 [(7 -- 1) Mr2 + 2]'/2 a5 = a I
(7 + 1)Mr
(7 + 1)M~ ps = p, (7 - 1) Mff + 2"
(11.3)
(11.4)
The similarity between the c o r r e s p o n d i n g e q u a t i o n s of the sets m a r k e d by (9), (10) and (11) is n o t e d here. The difference in (11.1) c o m p a r e d with (9.1) and (10.1) arises from the fact that while Uo = 0 and u~ = 0 (this is implied by the initial condition), u, # 0 [the velocity u, is induced by the incident shock wave, as can be seen from Eq. (9.1)]. It should also be n o t e d here that since the transmitted c o m p a c t i o n w a v e p r o p a g a t e s in a gaseous phase, which originally occupied the pores of the p o r o u s material, its d y n a m i c and t h e r m o d y n a m i c properties are transformed in a c c o r d a n c e with the t r a n s f o r m a t i o n presented earlier by Eqs. (2)-(4), (6) and (7), and hence they are m a r k e d with an asterisk.
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
The governing equations across the interface P are p s u s = ckp* u~,
(12.1)
~bp5 -~- 105u2 = -~) , 4,2 , -g p4~ -t- c~p4u
(12.2)
7 P5 + 1 2 _ o / - 1 P5 ~us 7 " - 1
(12.3)
P* + 1 u .2, p~
and the matching conditions across the contact surface S are p~ = p*,
(13.1)
u* = u~.
(13.2)
The above set of governing equations, which contains 17 equations, is solvable provided states (0) and (2) are given together with 7, ~b, r and Mi as initial conditions. For this situation, the 17 unknown variables are ul, Pl, al, Pl, u~, p~, a~, p~ us, Ps, as, Ps, u~, p*, p~, Mr and M*. 3.1.
The p r e s s u r e at the s h o c k - t u b e e n d - w a l l
Based on the model shown schematically in Fig. lb, the transmitted-compaction-wave-induced flow [-i.e., the flow in state (3)] stagnates on the shock-tube end-wall. Consequently, it is proposed here to calculate the peak pressure at the shock-tube end-wall using an empirical modification of the following well-known relation (see Ref. [-28], p. 417). I 7 2?* P~ 7* 1 M~/?. - 1' (14) 2-g = 1 4 P3
3
where p~ is the total (stagnation) pressure of the flow in state (3). Owing to the fact that the gas which flows through the open-cell pores of the rigid porous material attenuates it is proposed to modify Eq. (14) which is an isentropic relation by adding to it a dissipation/dispersion factor ~ such that it will be changed to the following empirical relation 2-g-p~ P3
E
l+~7*-l~M~
1
?*-1"
(15)
It is clear from Eq. (15) that for the case of a pure gas ~ = 1. Furthermore, for the present case, in which the flow attenuates as it passes through the pores, ~ should be less than unity, i.e., ~ < 1.
4. Comparison with experimental results and discussions Predictions of the above-described analytical model were compared to the experimental results of Ref. [19] and to additional experiments which were conducted during the same study but were not published so far. Details of the experimental set-up and typical experimental results can be found in Ref. [19]. Eight experiments have been used for the comparison. Their initial conditions
H. Li et al. /Fluid Dynamics Research 16 (1995) 203-215
211
Table 1 Initial conditions of the various experiments Case
Material
pores per inch
porosity
tortuosity
Mi
1 2 3 4
SiC
10
0.728
0.7
20
0.745
0.7
1.38 1.54 1.38 1.53
5 6 7 8
A120 3
30
0.814
0.75
40
0.821
0.75
1.38 1.54 1.37 1.54
Remarks: SiC-silicon carbide, A1203-alumina, Mi-incident shock wave Mach number. For all the experiments p ~ 8 3 KPa, To-~288 K
Table 2 The reflected shock wave and the pressure behind it. Comparison between the analytical predictions and the experimental results Case
1 2 3 4 5 6 7 8
Mr analytical
Mr experimental
Mj~lr
1.20 1.31 1.21 1.31 1.18 1.30 1.18 1.30
1.23 1.32 1.25 1.33 1.20 1.31 1.22 1.33
2.47% 0.76% 3.25% 1.52% 1.68% 0.77% 3.33% 2.28%
P5 [KPa]
P5 [KPa]
analytical
experimental
259 403 260 400 250 392 250 386
270 409 280 415 255 405 268 405
Aps/p5
4.2% 1.5% 7.4% 3.7% 2.0% 3.3% 6.9% 4.8%
are given in Table 1. An experimental method of calculating the tortuosity, z, of the investigated porous materials was presented in Ref. [26]. As can be seen from Table 1, four different rigid samples were used. Two were made of silicon-carbide (SIC) and two of alumina (A1203). Each of these four samples was tested with two different incident shock waves, one with Mi ~ 1.38 and the other with Mi---- 1.54. The comparison between the analytical predictions and the experimental results is summarized in Table 2. The reflected shock wave Mach number, Mr, and the gas pressure behind it, P5, as measured experimentally, are compared to their corresponding predictions by the presently proposed analytical model. The analytical predictions regarding both the reflected shock wave Mach number, Mr, and the pressure behind it, P5, underestimate the experimentally obtained values. The underestimations of
H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
212
Table 3 The shock-tube end-wall pressure: comparison between the empirical predictions and the experimental results Case
p~ [KPa] empirical
p~ [KPa] experimental
A-*/~* Pt~/Pt3
1 2 5 6
357 616 341 590
340 565 358 523
4.9% 8.6% 4.9% 12.0%
the reflected shock wave Mach numbers for both the silicon carbide and the alumina models are within the range 0.76% ~< AMr/~lr <~3.33% Similarly the underestimations of the pressures behind the reflected shock waves are in the range 1.5% ~< Aps/~5 <~ 7.4%. A general inspection of Tables 1 and 2 indicates that there is no evidence of a better agreement with either of the four different samples, i.e., porosities, which were used in the experiments. In addition, it seems that the analytical predictions agree better with the higher Mach number experiments (Mi - 1.54) than with the lower ones (Mi ~ 1.38). In general, however, the overall agreement is very good in view of the fact that it is based on a simple analytical model rather than a tedious and expensive numerical simulation.
3.3. The end-wall pressure As discussed earlier [see Eqs. (14) and (15) and the relevant text] the pressure at the shock-tube end-wall can be correlated to the stagnation pressure of the transmitted-compaction-induced flow. In the following, the validity of our empirical proposed relation is checked. In order to use Eq. (15) there is a need to assign a value to the coefficient ~. Using a simple try and error procedure it was found that c~ = 0.5 results in the best fit with all the available experiments. Table 3 presents the comparison of the results predicted by Eq. (15), with 0~-- 0.5, and some of the experimental results which were available to us (Ref. [19] and the additional ones which were mentioned earlier). Note that in order to apply Eq. (15) both M* and p* had to be supplied by solving the presently proposed analytical model outlined by Eqs. (9)-(11). It is evident from Table 3 that the agreement between the empirically predicted and the experimentally measured values is fairly good. In contrast to the earlier discussed comparison regarding the reflected shock wave Mach number and the pressure developed behind it, here the agreement is clearly better for the weaker incident shock wave. For the two compared cases, for which Mi-= 1.38, (i.e., cases 1 and 5) Apt3/Pt 3_~5%. Fig. 3 is a reproduction of Fig. 3 of Ref. [19]. As can be seen, the pressure at the shock-tube end-wall is seen to be still increasing. Consequently, the experimentally measured values of p* which are given in Table 3 are actually smaller than those which would have been obtained, had p~ reached its final value. Consequently, for cases 1, 2 and 6 for which p~ (empirical)> p*
H. Li et al. /Fluid Dynamics Research 16 (1995) 203 215 600
PtkPol
'
'
.P,,
'
213
j
.l.-1-
500 400
300 SiC(lOppi) Lo = 81 mm IVli= 1.542
200 100
0
I
I
I
1.5
2
2.5
i 3
t [ m sec] i 3.5
Fig. 3. The pressure history of the gas occupying the pores of the porous material as recorded by a pressure transducer located at the shock-tube end-wall.
Table 4 Dependence of the pressure at the shock-tube end-wall on the length of the sample as predicted empirically and comparison with experiments Case
4
Model length [mm]
p~ [KPa] experimental
p~ [KPa] empirical
41 62 83
515 510 490
581 -
Pt~/Pt~
12.0% 13.0% 17.0%
(experimental) the actual agreement is probably better than that indicated in Table 3. For case 5 for which p~ (empirical) < p* (experimental), the actual agreement is probably worse. It should also be mentioned that owing to the flow attenuation the stagnation pressure should depend on the length of the porous material model. This dependence is shown in Table 4 for case 4 of Table 1. It is evident from Table 4 that p~ decreases as the model length increases. However, the stagnation pressure seems to only slightly depend on the length of the model. When the length of the model is doubled from 41 to 83 mm, the pressure at the shock-tube end-wall decreases by about 6% from 515 to 490 KPa. The empirical prediction, which is based on Eq. (15), is independent of the length of the model. It should also be noted that in accordance with the above-mentioned fact that the experimentally measured values of p~ are underestimated (see Fig. 3) a better agreement than that shown in Table 4 probably exists. The dependence of the ratio between the pressures at the shock-tube end-wall with a porous material, p~ [as calculated by Eq. (15)] and without it, Pw, as a function of the porosity, ~b, for Mi = 1.5 is shown in Fig. 4. The ratio P~/Pwis always greater than unity and hence the presence of a porous material results in an enhancement in the pressure at the shock-tube end-wall. Similar results were reported in Ref. [13] for flexible porous materials, e.g., polyurethane foams. However,
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H. Li et al. / Fluid Dynamics Research 16 (1995) 203 215
2.4
Pw
2.2
2.0
1.8
16
1.4
/x
1.2
+ 1_0 0
0.2
0.4
0.6
0.8
1.0
Fig. 4. The pressureenhancementat the shock-tubeend-wallas a functionof the porosityas predictedby the empirical relation given by Eq. (15) and comparison with experiments.
since the solid material particles of a flexible porous material are also accelerated towards the shock-tube end-wall, the pressure enhancement in the case of flexible porous materials is much larger than it is in the case of rigid porous materials. Two experimental points which are added to Fig. 4 reveal a good agreement with the empirical predictions in the region 0.7 < 4' < 0.8. However, more experiments are required outside this narrow range in order to have a more definite verification of the empirical relation given by Eq. (15) regarding the pressure enhancement at the shock-tube end-wall.
5. Conclusion
An analytical model describing the head-on interaction of planar shock waves with ideal rigid porous materials has been developed. Predictions of the analytical model were compared to experimental results and very good agreement was evident. A simplified empirical method of calculating the pressure at the shock-tube end-wall was presented. Predictions based on the proposed empirical method were compared to experimental results and good agreement was obtained. The good agreement between the experimental results regarding the pressure at the shock-tube end-wall and the empirical predictions provides a justification of the assumption that the pressure at the shock-tube end-wall is actually equal to the stagnation pressure of the transmitted compaction-wave induced flow.
H. Li et al. / Fluid Dynamics Research 16 (1995) 203-215
215
In a d d i t i o n , since the e m p i r i c a l m o d e l is b a s e d on the general a n a l y t i c a l m o d e l which was d e v e l o p e d in the c o u r s e of the p r e s e n t study, its g o o d p r e d i c t i o n s p r o v i d e a f u r t h e r v a l i d a t i o n of the proposed analytical model.
References [1] B.E. Gelfand, A.V. Gubanov and E.I. Timofeev, Sov. Phys. Fluid Mech. 4 (1983) 79. [2] L.G. Gvozdeva, Yu.M. Faresov and V.P. Fokeev, Sov. Phys. Appl. Math. and Tech. Phys. 3 (1985) 111. [3] H.W. Sandusty and T.P. Liddiard, Dynamic compaction of porous beds, NSWC TR 83 256 Naval Surface Research Center, White Oak, MD, 1985. [4] J.G.M. van der Grinten, M.E.H. van Dongen and H.J. van der Kogel, Appl. Phys. 58 (1985) 2937. [-5] L.G. Gvozdeva, Yu.M. Faresov, J. Brossard and N. Charpentier, in: Dynamics of Explosions, eds. J.C. Brown, J.C. Leyer and R.I. Solukin, Prog. Aero. Astro. 106 (1986). [6] J.G.M. van der Grinten, M.E.H. van Dongen, and H. van der Kogel, J. App. Phys. 62 (1987) 4682. 1-7] J.G.M. van der Grinten. An experimental study of shock induced wave propagation in dry, water saturated and partially saturated porous media, Ph.D. Thesis (University of Eindhoven, The Netherlands, 1987). [8] J.G.M. van der Grinten, M.A. Smits, H. van der Kogel and M.E.H. van Dongen, in: Shock Tube and Wave, ed. H. Gronig (VCH Verlagsgesellschaft mbH, Weinheim, 1988) pp. 357 362. [9] V.P. Korobeinikov and P.A. Utriew, in: Unsteady Interaction of Shock and Detonation Waves in Gases (Hemisphere Publishing, 1989) pp. 48-61. [10] J.M. Powers, D.S. Stewart and H. Krier, ASME J. Appl. Mech. 56 (1989) 15. [-1I] L.F. Henderson, R.J. Virgona, J. Di and L.G. Gvozdeva, in: Current Topics in Shock Waves, ed. Y.W. Kim (American Institute of Physics, New York, 1990) pp. 814-818. [12] B.W. Skews, Shock Waves 1 (1991) 205. [13] G. Ben-Dor, G. Mazor, G. Cederbaum, O. Igra and S. Sorek, in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 279-282. [14] S. Sorek, J. Bear, G. Ben-Dor and G. Mazor, Transport in Porous Media 9. (1992) 3. [15] M.W. Seitz, and B.W. Skews, in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 517-520. [16] B.W. Skews, M.D. Atkins and M.W. Seitz in: Shock Waves, ed. K. Takayama (Springer, 1992) pp. 509 516. [17] M.R. Baer, Shock Waves 2 (1992) pp. 121. [18] A. Levy, G. Ben-Dor, S. Sorek and J. Bear, Shock Waves 3 (1993) 105. [19] A. Levy, G. Ben-Dor, B.W. Skews and S. Sorek, Experiments in Fluids 15 (1993) 183. [20] M. Olim, M.E.H. van Dongen, K. Kitamura and K. Takayama, Int. J. Multiphase Flow 20 (1994) 557. [-21] G. Mazor, G. Ben-Dor, O. Igra and S. Sorek, Shock Waves 3 (1994) 159. [22] G. Ben-Dor, G. Mazor, O. Igra, S. Sorek and H. Onodera, Shock Waves 3 (1994) 167. [23] G. Ben-Dor and E. Zaretsky, Arch. Appl. Mech. 64 (1994) 365. [24] M.R. Baer and J.W. Nunziato, Int. J. Multiphase Flow 12 (1986) 861. [25] M.R. Baer, ASME J. Appl. Mech. 55 (1988) 36. [26] H. Li, A. Levy, and G. Ben-Dor Analytical prediction of regular reflection over porous surfaces in pseudo-steady flows, J. Fluid Mech. (1994) to he published. [27] J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena (Kluwer, Dordrecht, 1990). [28] P.H. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1993).