Propagation and Reflection of Shock Waves

0

-95(^)1' The values of to and the rate of diaphragm opening are in good agreement with experiment. The flow area at the diaphragm station varies slowly at initial instants and then reaches its maximum value very rapidly. The diaphragm opening process influences the shock formation in the shock tube. The formation

Structure and basic properties...

23

distance is proportional to the product of the maximum velocity of the shock by the opening time of the diaphragm. The third group of experimental studies is devoted to the distribution of flow parameters behind the shock wave. Considerable violation of the uniform­ ity of the flow is observed within the distance of shock formation due to the growth of shock velocity. The gas is influenced by longitudinal compression waves as well as transverse shock waves arising as a result of the reflection of the main shock from the walls of the shock tube. It is found that the density increases in the direction from the shock wave to the contact discontinuity. Faizullov (Faizullov 1962) measured the temperature distribution behind the shock wave in the shock tube when the shock velocity is constant. It was found that the temperature is practically constant. But there is a decrease of temperature by 12% in the region close to the contact discontinuity (N2 as the test gas; p0 = 1.3kPa, M = 9). Dam et al. (1995 a, 1995 b) studied perturbations superimposed to the pressure plateau following the incident shock wave. The origin of perturbations is as follows. The Mach reflection appears at initial instants after the burst of the diaphragm. A transverse shock wave is created which moves towards the axis of the shock tube, the main shock wave becoming plane in a short time. The transverse wave converges at the centre of the tube, giving rise to a reflected wave which is a spherical wave expanding towards the wall of the tube. A periodic process arises and this configuration moves approximately at the gas velocity. There is a substantial reduction of the perturbations if the velocity of sound of the heated gas is greater than that of the expanded driver gas. A number of papers are devoted to the distribution of flow parameters behind the shock wave when its velocity decreases. In this region the variation of flow parameters is defined mainly by the influence of the boundary layer. The shock and the contact discontinuity travel at the same velocity because the amount of driven gas between them remains the same. The mass flow across the shock is equal to the mass of gas that flows into the boundary layer at the contact discontinuity. The formation of the boundary layer behind the incident shock wave was investigated by many authors (Demyanov 1957; Mirels 1957; Duff 1959). Their results agree well with experimental data. Below we consider the effect that the diaphragm opening process has on the shock formation. It was investigated experimentally by Shtemenko (1967, 1968, 1972, 1974, 1982). Experiments were fulfilled in a shock tube of a rectangular cross-section of 34 x 72 mm 2 area. Brass, aluminium and copper foil diaphragms 0.1 mm thick were used. The diaphragm was made to open in a definite manner by weakening it by means of a special wire stamp. Two types of diaphragms

24

Propagation

and reflection

of shock

waves

were used. The diaphragms of the first type opened in the shape of triangular leaves (diaphragm 1). The diaphragms of the second type opened in the shape of two rectangular folds (diaphragm 2). In the first case the flow had two planes of symmetry. In the second case the flow was close to a two-dimensional one. The shock Mach number was equal to 2.1-2.7. Air was chosen as a test gas. The driver gas was a helium-air mixture. Shadowgraphs and streak-camera records were taken. The velocity of waves was measured from streak-camera records and with the aid of pressure transducers. Measurements were made at various distances / from the diaphragm, 0 < / < 22£), D = 46 m m , where D is the reduced tube diameter, D — ^p^, S\ being the area of the cross-section of the tube and P being its perimeter. The opening time of the diaphragm was determined from the motion of the leaf of d i a p h r a g m while bursting. This motion was fixed at the streak-camera record. Fig. 8 presents the ratio S/So against time, S being the flow area at the d i a p h r a g m station and So is the area of fully opened diaphragm. Left curve relates to the diaphragm £, right one to the diaphragm 1.

Fig. 8. Ratio S/So against time, S is the flow area at the d i a p h r a g m station, So is the area of fully opened diaphragm. As one may suppose, the time opening rate influences the test flow. A distribution was measured of the shock velocity along the shock tube at vari­ ous time dependants of the flow area at the d i a p h r a g m station. The various time dependencies were obtained by artificially slowing the movement of the diaphragm leaves either at the initial stage (diaphragm la) or at the final stage (diaphragm lb). Fig. 9 presents the shock velocity along the shock tube.

Structure and basic properties...

25

Curves 1-3 relate to the pressure ratio p^/pi — 45 and the velocity of sound c4 = 565 m/s. Curves 1^-6 relate to the pressure ratio p±jp\ = 115 and the velocity of sound C4 = 735 m/s. Curves 1, 4 correspond to the diaphragm 1, curve 2 corresponds to the diaphragm lb, curves 3, 5, 6 correspond to the diaphragm la. The opening time is equal to 420//s (curve 1), 680/is (curves 2 and 5), 750/is (curves 3 and 6), 540/zs (curve 4)- As one can see from Fig. 9, the maximum value of shock velocity depends on the rate of diaphragm opening at initial instants, the less the rate of opening the less is the maximum value of shock velocity and the greater is the formation distance. This effect depends on the shock strength. The decrease of the rate of diaphragm opening at the final stage of the process has small influence on the maximum value of shock velo­ city, only increasing the formation distance. Streak-camera records are shown in Figs. 10, 11. Flow patterns are presented near diaphragm 1 and la, initial conditions being the same.

Fig. 9. The distribution of shock velocity along the shock tube.

26

Propagation and reflection of shock waves

Fig. 10. Streak-camera record of the flow near the diaphragm, diaphragm 1. As one can see from streak-camera records, the rate of the flow area at the diaphragm station at the initial stage of the process influences to a great extent the flow. Namely, the velocity of disturbances decreases and the dis­ tance between the successive disturbances increases as the rate of the flow area decreases at the diaphragm station. The disturbances coalesce and form a shock wave. If the rate of the flow area at the diaphragm station decreases then the maximum velocity of the shock wave also decreases. The greater the interval between two successive disturbances the greater is the shock formation distance. The flow of the driver gas was studied experimentally near the diaphragm. Figs. 12 and 13 demonstrate streak-camera records of this flow. A jet appears after the burst of the diaphragm (Fig. 12). Successive disturbances are seen that arise inside the jet. They reflect from the walls of the shock tube. The

Structure cmd basic properties...

27

Fig. 11. Streak-camera record of the flow near the diaphragm, diaphragm 2. jet and the main shock ahead of it are presented in Fig. 13. An additional bell-shaped shock wave appears due to the fact that the velocity of the jet is supersonic. It propagates in the opposite direction as compared with the main shock wave. Fig. 14 shows a streak-camera record of the flow. A trace is seen of the second shock moving through the driver gas. At first its velocity grows and then decreases in the laboratory frame. Afterwards the second shock stops at a distance lm from the diaphragm, the value of lm being equal to lm = kd(p4/Pl)i., k = 0.55 (He); 0.65 (N 2 ). Here d is the reduced diameter for the area of fully opened diaphragm. The value of lm is less than the corresponding value in the case of a stationary jet. The second shock reflects from the walls of the shock tube (Fig. 15, dia­ phragm 2). A complex wave configuration arises and several shocks appear as a result. One of them coalesces with the second shock giving rise to two new

28

Propagation and reflection of shock waves

Fig. 12. Schlieren pictures of the low of the dri¥er gas, a - dia­ phragm 1, b - diaphragm 2; p\ = 105 Pa, time inter¥al between the frames is 8 jus. shocks that propagate in opposite sides. The second shock appears anew after its break-up due to the fact that the pressure ratio across the diaphragm area exceeds its critical Yalue. Fig. 16 shows the path 1(f) of the second shock against time. The greater the pressure ratio .across the diaphragm the greater is the velocity of the second shock. Its velocity increases .as the velocity of sound of the driver gas and the rate of the diaphragm opening time increase. Experimental values of l(t) can be approximated by the formula (air as a driver gas)

'»-'(£)'£)■ to being the opening time of the diaphragm. We measured the Mach number ahead of the second shock wave (N2 as a driver gas). For this purpose a slender cylindrical body was placed inside the shock tube at a distance of 0.29cf and 0.54(1 from the diaphragm. Fig. 17 demonstrates the second shock and a bow shock past the slender body. We determined the Mach number of the jet from the shape of the bow shock. The

Structure and basic properties..,

29

Fig. 13. The second shock wave near the diaphragm l,pi = 6.7 kPa, the time interval between the frames is 8 j*s. values of the Mach number of the flow against time are presented in Fig. 18. Curves 1, 2, and 5 relate to the values of initial pressure equal to 35 kPa, 12kPa, and 7kPa 5 respectively; / = 0.29d; curve 4 relates to the pressure equal t o 7 k P a ( f = 0.54cf). The Mach number of the jet increases against time till the instant when its boundary meets the walls of the shock tube. In the sequel the Mach number of the jet is described by the following empirical formula ±

\I=(lA

+ ZxlO-3?±\(^Y

i.

(LY.

/ > 100/is.

(29)

We measured the velocity of disturbances ahead of and behind the second shock wave with the aid of streak-camera records. Then we calculated the sum (112+^2) behind the second shock by using Eq. (29). We compared the measured values of (V2 + C2) with the calculated ones. The data are given in Table 1. The values of the pressure ratio across the diaphragm are also presented and so are the positions that relate to measured values and calculated ones. The calculated values of («2 + C2) agree well with the measured ones. Fig. 19 shows the flow parameters inside the jet against time ahead of the second shock and behind it. The pressure increases as the diaphragm opens. The time-derivative ~£ increases by 6-8%. Compression waves arise as a result. They appear behind the second shock. One can see them at shadowgraphs. If the pressure ratio across the diaphragm is less than the critical value the second shock does

VI

Propagation and reflection of shock waves

V

Fig. 14. Streak-camera record of the flow, the second shock wa¥e is seen, pi = 8.7 kPa.

not appear. Table 2 gi¥es ¥alues of the velocity of disturbances (V), in successive se­ quence as the diaphragm bursts, for ¥arious initial conditions. Within the dotted frames are gi¥en the ¥alues of the shock wa¥e velocity U if one of the disturbances is assumed to be the shock wave and also calculated ¥alues of the velocity of acoustic disturbances behind the shock wa¥e (tf2 + £2)- As is e¥ident from Table 2, the velocity of the disturbances emerging into the driven gas directly behind the presumed shock wa¥e is in good agreement with the cal­ culated velocity of acoustic disturbances behind the shock wave. Hence, some of the disturbances are weak shock waves (M = 1.03-1.4) and the others are acoustic waves propagating behind the shock wave. Analysis of the data shows that after the diaphragm has burst, the strength and frequency with which the shock waves appear increase as the bursting time and initial pressure in the

Structure and basic properties...

31

Fig. 15. The reflection of the second shock waYe from the walls of the shock tube (diaphragm 2). Table 1. 1 mm

46 52 57 61

PA/PI

40 65 90 110

(V2 + Co) calc

(v2 + c2) exp

370 380 400 400

340 ± 30 380 ± 40 400 ± 40 410 ±40

\

driYen section decrease and as velocity of sound in the driver section increases. Note that the wave system adopted was constructed under the assumption that the change in flow parameters due to the weak disturbances is sufficiently small, and may be neglected. Within the limits of measurement error, this condition was satisfied: the velocity of each disturbance did not change at a distance of 0.5D-2D. How the disturbances observed in the experiment could have arisen will now be considered. As shown by analyzing shadowgraphs of the flow, a set of weak disturbances exists ahead of the second shock wave. These are formed when the jet of driver gas flows past the sharp edges of the leaves of the bursting diaphragm. As is known, weak disturbances propagating counter to the shock front are amplified after interaction with it. The intensity of the acoustic waves that have passed it may be increased a few times. In the driven gas, they must propagate at the local velocity of sound, as observed experimentally. The appearance of shock waves may be represented as follows. As shown by several authors (see e.g. Shtemenko 1972), directly behind the front of the

t,

Propagation and reflection of shock waves

PS

Fig. 16. The position of the second shock wave against time (dia­ phragm 1); t0 = 440/is; 1 - p4/Pl = 230; 2 - 110; 3 - 40; ^ 20. second shock the values of all the flow parameters of the driver gas increase a few times at the instant of diaphragm bursting; i.e., in the jet of driver gas, there exists a region of large gradients of density, velocity and pressure, increasing over time. At any instant, the density or pressure gradient may become infinite, with the appearance of a shock wave. The large-gradient region moves in the jet of driver gas. Thus, the problem is analogous to the problem of the formation of a shock wave from compression waves of finite amplitude when they propagate. The time for such a formation process to occur will now be calculated in one-dimensional flow. Since the compression wave propagates in a non-uniform driver gas, the estimate is performed using a relation between change in density gradient in the compression wave j^{p2x) and the density gradients ahead of (plx) and behind (p2x) the shock wave. This expression is obtained from the compatibility conditions (Shugaev 1983) l_d_ c\ dt

P2x Pi

7+ 1 2 ■

2f)2

1-1

Plx , PlxPlx

P2x~—

3 /pix "2 \pi

Pi~ Pix \ i cit cu c

+

A v\x \

Ii

da f gig

Pi ) V l ci ) d dt V Pi Here clt is the time derivative of the velocity of sound; vlx is the velocity gradient ahead of wave front. Integration of this expression gives the following value for p2x/pi

Structure and basic properties...

33

Fig. 17. Flow past a slender cylindrical body at a distance of 0.29d from the diaphragm 1, Pl = 3.3 kPa. The time interval between the frames is 12/is.

^ =^ +— Pl

Pl

}?

• (30)

\

J £i12+11 exp (j ( M i - » - » ^ _ s c u g , , ^ dt\ dt + ao

where

1 '

a0 =

'

.

(P3*/Pl)t=0 ~ (Pl*/Pl)t=0 When all the gradients in front of the compression wave are zero, Eq. (30) coincides with the formula proposed by Thomas (1957), which gives the value of p2x in a compression wave propagating in a uniform gas. The compression wave transforms into a shock wave at the instant when p2x - oo, that is

[^r^'-r-iir-^* /^-(/(^?-!?-W*)dt dt

-^-^-P

=-.

..° (P2,/Pl)t

i

.

= 0-(P1*/Pl)t=0'

.

( (

}

Thee time t will now be evaluated express To this end, the raluated from this expression. dependence ters of the gas on the time and distance is linearly ence of the flow parameters

Propagation and reflection of shock waves

34

Fig. 18. The Mach number of the jet against time; a - diaphragm 1, 1-P! = 34kPa; 2- 12kPa; 5, ^ - 6 . 7 k P a ; 1, 2, 3- l/d = Q.29\ 4~ 0.54; b - diaphragm %l,2-pi = 20 kPa; 5 - 6 . 7 kPa; 4 - 3.7 kPa; Z/d = 0.29. approximated: d = fc(x)t + cio(ar), c,-t = k(x), vi = a(s)t 4- vio(a), fft = a(^)> vi = £(t)x + A 0 (t), cix = £(*)• Then, taking the obvious relations p\xjp\ = -vit/cj, w v n —V2t/c2 — 2t/cl i ^° account, Eq. (31) takes the form

~

and pix/pi

=

(«2t/cl)*=0-(vlt/Ci)t=o"

In the conditions of the given experiment, f depends very weakly on t; assuming £0 =const, the integrand is written in dimensionless coordinates: 2T-1

(7+!)**§ J?v^

400

J \to L

--■-.J

t,,

kt0

10

(V2t)t=0 ~

(Vlt)t=0

2

a F

1 2

e x p ( | t,UfSo ^ ) r f ( ^

Structure and basic properties...

35

Fig. 19. Flow parameters of the jet (nitrogen) ahead of (a) and behind (6) the second shock wave, 1 - p; 2 - p\ 3 — v\ 4 ~ c; 5 dp/dt.

where to = d/cio, cio is the initial velocity of sound in the driven section. The integrand is expanded in Taylor series, retaining only the first term. Integration yields 2(kt0)

t\ (7 + l)c 10

2

(32) ({V2t)t=0 ~

(Vlt)t=0)

In our experiments, kt0 « 70m/s 2 ; a/k « 1; the mean value v2t « 4 • 10 6 m/s 2 ; vu « 10 6 m/s 2 , cio lies in the range 100-150 m/s. The value of the time t obtained from Eq. (32) lies in the range 20-50 ps. The time interval AT between the departure of one shock wave directly from the front of the second shock and the arrival of the subsequent shock wave in the driven gas from the driver gas was determined, using a linear approximation of the shock-wave path in the driver gas. As shown by analyzing shadowgraphs of the flow, on which waves moving both in the driven and in the driver gas are seen, this approximation changes the time interval by no more than 10%. It is obvious that AT is the maximum time during which shock waves may be formed in the region of large density and pressure gradients between the front of the second shock discontinuity and the contact surface, i.e., t < AT. Table 3 shows the relation between the shock-wave velocity and the time interval AT. AS is evident from Table 3, AT is always larger than the value of t calculated from Eq. (24). Note that, in calculating t, the value vit ~ the

00

Table 2. Initial conditions

Disturbance velocity V V±20

c4 = 350 m/s po = 1.01 x 105 Pa h = 470/zs

330 330330 360 370 360 370 360 350 370 360 370 3*50 390 390 390 400400 390 420420 410 1 1 I U = 360 U = 390 j ac = 402 |wae = 378j K±20

C4 = 735 m/s Po = 3.47 x 104 Pa
V±25

340 350 340 340 390 430 420 480 480 470 510 490 560 530 560 535 540 580 600 600 630 650 1 1 | I I | U = 390 l | U = 470 | \U = 530 1 \U = 580 C/ = 650 | ; Vae = 434 jj Vac = 5031 V±10

c4 = 350 m/s p 0 = 3.47 x 105 Pa ti = 425 ps

V±30

! Vac = 555 {

V±15

I 3'

|Vac = 603] ^±20 as

350 370400400 420425 470 470 480 500515 530 550 610 1 1 1 1 1 j U = 375 | U = 435 j \U = 480 j j u = 530 j (7 = 610 j

3* CO

j«ac = 407|

|Uac = 4K > 1 ; \vac = 498] jVac = 059j

o

Co

Table 2 (continued). Disturbance velocity V

Initial conditions

V±20 c4 = 350 m/s r~ Po = 3.47 x 105 Pa

360 370 380 400 425 430 410 470460480470 490 495 490 530 550 1 1 l l [/ = 370 U = 415 | 17 = 460 | | 17 = 460 Vac = 4321 I'vac = 485; Vac = 398 boc = 485

Vac =

V±15

3

1

V±20

350 400 410 450 520 520 520 540 550 540 630 1 1 1 1 1 U == 360 \U = 400 \U = 450 | £/■ = 505 | 17 = 630 Vac z = 378 \Vac = 421 Vac = 476 ): V±10

c4 = 350 m/s po = 8 x 103 Pa ii = 390//s

V±30

^±25

V±10 c4 = 350 m/s p 0 = 1.33 x 104 Pa

*8

405 1 U == 405 Vac : = 460

V±15

K±20

7 ±25

455 475 570 620 640 655 660 620 1 I £7 = 810 tf = 560 Vac ==

bi.9

V2+C2

The values of V, U, vac are given in m/s GO

38

Propagation and reflection of shock wanes

U. m/s AT.

fis

390 415 460 530 435 480 530 610 630 810 60 120 80 90 120 120 110 100 170 140

acceleration of the driver gas directly behind the front of the second shock - was obtained by graphical differentiation of the curve of V2{t). For each shock wave, the value (v2t)t=o corresponds to the instant immediately after the departure of the preceding shock wave from the front of the second shock wave. A linear approximation of vi(t) was used to determine t'n.

Fig. 20. Streak-camera record of the flow at a distance of 7D from the diaphragm. Thus, it may be concluded that the hypothesis regarding the mechanism of the appearance of shock waves in the region of large unsteady pressure gradients behind the front of the second shock in the jet of driver gas is correct. Shock waves arise there analogously to the shock wave from compression waves of finite amplitude. The discreteness of the process by which shock waves appear in the shock tube may be explained as follows. After a shock wave is formed, a rarefaction wave propagates in the opposite direction, thus reducing the pressure gradient in the flow behind the shock wave. It takes some time to form a new compression wave in the flow. Thus the process of shock formation in a shock tube is as follows. A nonstationary supersonic jet of driver gas flows into a driven section. A second shock appears inside the jet. This shock propagates upstream the jet. Initially

Structure and basic properties...

•-1

D-5

Fig. 21. Velocity of flow and velocity of sound against distance, 1 - P! = 12kPa; 2- 18kPa; 3- 35kPa; ^ - 6 . 7 k P a ; 5 - 8 k P a ; 68kPa; 7- 6.7 kPa; 1-5 - c4 = 735 m/s; 6, 7 - c4 = 350 m/s. the strength of the second shock exceeds that one of the main shock, and it increases up to the instant when the second shock reflects from the walls of the shock tube. The pressure increases behind the second shock, so that com­ pression waves arise. These turn into weak shock waves which catch up with one another and form the main shock wave. The strength of the second shock depends on the pressure ratio across the diaphragm and the rate of flow area at the diaphragm station at the initial stage. In this way the initial stage of the flow at the diaphragm station influences the formation of the shock wave, i.e., the change of the shock velocity along the shock tube, shock formation distance and the maximum value of velocity of the shock wave. Non-uniformities of the flow behind the shock wave arise due to the process of its formation. Weak shock waves catch up with the main shock wave along the formation distance. The Mach number of the shock waves is equal to 1.01-1.5. Contact discontinuities appear as a result of the interaction of weak shocks with the main shock. Fig. 20 shows a streak-camera of the flow (/ = ID). Contact discontinuities are seen. They enable us to determine the velocity of the gas flow. This value increases in the direction towards the contact discontinuity that separates the driven gas from the driver one. We measured the velocity of propagation of weak disturbances. Thus we determined the distribution of the velocity of sound behind the shock wave. This quantity decreases slowly in the direction towards the contact discontinuity. Fig. 21 presents the velocity of the

40

Propagation

and reflection

of shock waves

flow and the local velocity of sound against distance. Below we consider the propagation of shock waves of an arbitrary shape through a homogeneous and inhomogeneous gas and their reflection from rigid boundaries.

C H A P T E R 2. SHOCK WAVE PROPAGATION THROUGH A GAS

4. B a s i c n o t i o n s A shock wave is a nonlinear disturbance propagating through a medium and having a sharp front (discontinuity surface). The velocity of propagation of the shock depends on its strength. Flow parameters (the pressure, density, entropy and particle velocity) are all discontinuous over the front. The motion of a shock wave can be described by nonlinear equations of gas dynamics. Generally there are no analytical solutions. T h a t is why numerical methods are used widely. They are based on difference approximations of the equations of gas dynamics. Nevertheless, we can obtain some qualitative information without numerical solution. Compatibility conditions m a y be an i m p o r t a n t tool for this goal. The compatibility conditions are discussed by many authors (Hadamard 1903; Kotchine 1926; Predvoditelev 1959; T h o m a s 1961; Chen et al. 1971; Nunziato et al. 1972). Their physical meaning is as follows. Let there be a discontinuity surface in a medium. If the j u m p s of physical quantities across the discontinuity surface are arbitrary then the discontinuity breaks u p (Kotchine 1926; Rozhdestvensky et al. 1969). Two waves arise that propagate in opposite directions. Let the fluid parameters have some spatial distribution in front of and behind the singular surface. If the functions which give the distribution of physical quantities ahead of and behind the discontinuity surface are connected with one another in a definite manner then the discontinuity propagates without break-up. The connection between the distributions of the flow parameters is given by the compatibility conditions. As will be shown below, the variation of the shock velocity in a single manner depends on the state of the medium ahead of the wave front at a given instant of time, on the geometry of the front and on the distribution for one of the parameters behind the front. The distribution of the other quantities behind the front can be derived from the values for this

41

Propagation and reflection of shock waves

42

parameter provided that all the quantities ahead of the wave front and behind it are continuous and differentiable. The compatibility conditions are deduced from the fact that the discon­ tinuity front remains a single one during some interval of time. There are three kinds of compatibility conditions, namely, geometrical ones, kinematical and dynamical. The first two of them are derived by using geometrical and kinematical considerations only, without taking into account the equations of motion. Dynamical compatibility conditions make use of the momentum and continuity equations together with geometrical and kinematical considerations. The geometrical compatibility conditions and the kinematical ones are de­ rived below. We use the Lagrangian variables a1, a2, a3. These variables were used by Hadamard (1903). He obtained the compatibility conditions up to the third order supposing the function itself to be continuous and its derivatives to be discontinuous across the wave front, i.e. [$] = 0, [d$/dal] ^ 0, [d$/dt] ^ 0. Here [ ] denotes the difference in the values of the quantity across the wave front: [<£] = $2 — $ 1 , $ being a flow parameter. The indices 1,2 refer to the values ahead of and behind the wave front, respectively. We proceed from the considerations used by Thomas (1961), but in contrast with him a dependence on time of the metric tensor is taken into account in the space a1 a2a3. The wave front is a singular surface which propagates through a medium. Besides, there are discontinuity surfaces that are at rest relative to the medium (contact discontinuities). We denote as v\ contravariant components of the external unit normal to the wave front in the space a1 a2a3 ahead of the front (the subscript 1 will be omitted in the sequel). We define as compatibility conditions of the rath order the relations which connect the jumps of the derivatives of the 772th order. The motion of the wave front may be represented in the following manner a

\ =a',(u 1 ,u 2 ,*)-

Here u a , a — 1,2 are the curvilinear coordinates on the wave front. We determine the velocity of propagation G as follows. We take the state of the medium at the instant t a s a reference point. Otherwise, x[(al,a2,a3,t) = a1. We denote two positions of the wave front in the space a1 a2a3: one at the instant t and another at the instant t + At. The latter coincides with the posi­ tion at the instant t of the particles which are on the wave front at the instant t + At. We take the point N on the wave front whose position refers to the in­ stant t and we draw the external normal. This normal intersects at the point N' the wave front which refers to the instant t + At. The limit Al/At at At -> 0 is

Shock wave propagation through a gas

43

equal to G, A/ being the length of the segment NN'. one of a ray. The equation of the ray is as follows

The segment NN' is that

£ = <*■ to

The of aa fluid the wave wave front front along along the ray is is equal equal The derivative derivative of fluid parameter parameter on on the the ray

dd+ A

+{N',t ++ At)-4>(N,t) M)-+{N,t) HN',t

d± 0+

,j*

Gv>°*-{

dt At->o A* dt da4'' dt At->o A* dt da The symbol d/dt denotes the time-derivative as apparent to an observer .ving moving with the velocity G relative to the medium. The velocity of displacement U of the wave is defined in an analogous manner U = lim ^ , where Am is a distance along the segment of the normal between the two positions of the wave front: one refers to the instant t and the other to the positions of me wave froui: one refers 10 me instant i and une ouner to the instant t + At in the space x11x22x33, x{{ being the Cartesian coordinates (the instant t + At in the space x x x , x being the Cartesian coordinates (the Eulerian coordinates of a particle). There is a simple relation between G and Eulerian coordinates of a particle). There is a simple relation between G and U. At the instant t the wave front intersects the particle whose coordinates U. At the instant t the wave front intersects the particle whose coordinates x\{alll,a222, a33,t) ,i) the At it intersects with are are x[{a x\{a ,a , a 1,i) and and1l at at 2the2 instant instant tt 3+ + At it intersects the the particle particle with the the a2 + Gv22At,a At, a33 + Gv33At,t + At). The distance Am 1 +GvlAt, coordinates x^a At,a coordinates x\ (a +Gv At. a + Gv At. a + Gv At.t + At\. The distance Am is equal to is equal to

Am =

(dx\

Ur

+

r, ~

9

dx[

\ A UjG niAt

da- ) >

m m being being the the components components of of the the external external unit unit normal normal in in the the space space As

x 2 3 xx1xx2xx3..

then rr

dx\ xl

V = Cr -+- -^-

+ vln, U = G+ —Hi Hi = = u G+Vi n, «i„ being the normal component of the particle velocity ahead of the t t>i„ wave front. The vector ^ - is not orthogonal to the wave front. In fact, we have

dx\ ax\

dx[ ox\

(33)

Propagation and reflection of shock waves

44

Let ga/3 denote the covariant components of the surface metric tensor on the wave front dxl dx1 Multiplying Eq. (33) by the vector dx1

1

which is tangent to the wave front, we obtain 1

dx\ dx{ _ 1 dt dua /q~ x

dxj dx{ dt dua

via,

Qf is not summed.

This quantity is the velocity component tangent to the wave front ahead of it. Generally it is not equal to zero. If the velocity of propagation is a continuous and differentiable function of time then fluid parameters and their derivatives ahead of and behind the wave front must satisfy the so-called compatibility conditions. Geometrical compatibility conditions follow from the existence of a smooth discontinuity front. Kinematical compatibility conditions are derived from the fact that there is no break-up of discontinuity front. The compatibility conditions may be violated only at different instants of time. If the compatibility conditions are violated then the discontinuity front transforms into two fronts. We proceed to the derivation of the compatibility conditions. 5. Compatibility conditions To begin with, we consider geometrical compatibility conditions. The derivative of a fluid parameter (j> has the following values on both sides of a wave front di _ d<j>i da1

d(j)2 _ dfo

9al

Subtracting the former equation from the latter, we have

da1

di a

du

.da

Let us multiply this relation by the term <7a/3ff?-: d(j)

dua duP

da{

9

du« dur

m

Shock wave propagation

through a gas

45

We use the following identity (Thomas 1966) 9

af3

da1 dai

-•

to?w=9

• .

(35)

-"*•

Here gij are the components of the metric tensor in the space a1a2a3. tuting Eq. (35) into Eq. (34), we have n%3

3

Multiplying by gjki

du<* du?

[da\

-vj

Substi­

_dai\

we get

d<j>

= L^k + gjkg^dM^,L^ a

dak

du

=

duP'

d

(36)

dai

Eq. (36) is the geometrical compatibility condition of the first order in a n arbitrary reference frame. It is the same as that obtained by T h o m a s (1961). One can get the compatibility conditions of a higher order from Eq. (36) by substituting -^ instead of . In the particular case x\ — a\ it is convenient to use the following relation which is derived by induction L(*)=JL('+

+

1

)1/, »«i/?i

(s — m)\

dL^-^

-^U-^

da1

dm 1

da r(0

da{dai

-dak

--dak m = 0, 1, • • •, s.

i/J • • • v

(37)

Here 6a/g are the components of the second fundamental surface tensor on the discontinuity front. Let us deduce the compatibility conditions of the second order. Substituting -> §§r into Eq. (36), we get

d2 Oaidak

d = -W, Ly*k+gjkg°'dua

d
w-

dai

<38»

In order to find the quantity L[aJ, let us multiply Eq. (38) by i / \ We get

d

dai

i<J> = i">.» + «W£(i<'>)&-wf

d da1'

du? dua'

(39)

Propagation and reflection of shock waves

46

The quantity J ^ can be determined from the formula of the covariant deriv­ ative (Thomas 1966) dv

da1

jda

r ^ =^ duc + ^ du

^ = - ^ < w du*'

T%lrn being Christoffel's symbols which relate to the space a1 a2a3. The notation v%a indicates covariant surface differentiation. Consequently, ^

_

dai

J>*h

du

„idamTi

(40)

dua

du°

Then 8 da{

du^--g

K

9ik9

? d ^ -

L

^-du^T™1

n d[] dak dal-ni *** ■vT Lml ' du- dui du°" -

(41)

Taking into account Eqs. (35), (36), (41), we have instead of (39)

4 ? - ^-*Vr<, £<",, + *,?*<££;

Vk

l ( i ) , , . + g g. A „ 3 M tor + vJ-ri T\k [L^Vi ri dux du°

+ 9jk9

9L^

a/3

, ~

Xo.

+g

d[fi]\ dai

(42)

* 6 < * A ^ TuJ-

Substituting Eq. (42) into Eq. (38) and using the change of indices, we get d2 daldas

= U ( 3 > - * V l t , [LWVk+grkff Xa<m<w_ +

9jig

ap

3lJ^_daP_ ,x„h du<* du?+ 9

dux du° d[+]da?\ b <*\ ^du-x r duP) ^77T

v\vs

+ i(Vi{+^ £|^), du" du x

+

The quantity ^ written as follows d dua

da1

d da1

"*"'*?

d da1

j

da dvfi'

(43)

which is on the right-hand side of Eq. (43) can be

= Vf

dLW „ d2i] dam + L{1)-^z a + 9mlg£ a du du du(dua du"

Shock wave propagation

through a gas

9mig

47

d2«m

in M

,

dut du«du*>

9ml

dg*" 9[] da™

du« dut du*

in d9mi d[<j>] da" (44) + 9' dua dut du7* ' The derivative with respect to ua for the covariant component of the unit external normal can be determined from the following relation dvi oua

d> t( dua v

{

f.i\x

dv "is

idgu

'

oua

oua

As (Kotchine 1965)

Q^=(9ikTls+glkTis)

— ,

then, with Eq. (40) substituted, dv\

du°-"=K*lr"krt

dar

da1 du°

l3a 9il9 bvap at ll

8u«

»»

Let us examine the fourth term on the right-hand side of Eq. (44). Using the expression of the covariant derivative nm

_

m

L

we obtain d2an duadu^

da"

Y° a

das

dak

sk " • a.+ " r du° " du* du« da™ _ flo^do* (45) bVaVm + dvP du<* sk' du° "« Therefore the fourth term on the right-hand side of Eq. (44) is equal to

9

V "a dvf

l + 9ml

V erf du° "a du<> du« dutl

sk

The derivative - ^ r on the right-hand side of Eq. (44) can be expressed by Christoffel's symbols on the front of the discontinuity (Kotchine 1965). There­ fore

d dua

9[] da1

(dL(l)

M+9 (»h

= {-^r + gmigt" + 9m39

d

^ \

^^rh

d^_ ( a 2 M du$

\dundu<*

d\4>] dvF

du* dw> du« '"'

"a

(46)

Propagation

48

and reflection

of shock

waves

Finally we get, taking into account Eq. (46),

= (L^-^nJ(L^k +

d2<)) daldas

+ \L a0

V}+gt,g

9r^§§^n^

duaduP)hs

(dL^

t

„

0[<j>]\ (

L

da''

dai

Let us proceed to the geometrical compatibility condition of the third order. We use a rectangular reference frame for the sake of simplicity, i.e. x\ = a \ Besides, we assume [] to be equal to zero. First of all we must find the second derivative for the unit normal along the front (see Eq. (39)) d2vj duadua

~

d_ du°

«"•«£)

= w ( ^ ^ -9'n S?) - ^ ^ -

(47)

Substituting d<j)/dal instead of into Eq. (65) and taking into account Eq. (47), we get

d34> da*daidak (da''

dai

- \L^9^9"Tbaa

X

Qak

\

+ L^{1H9aP9irbai

9 ^9 dai

aj

dak da*

V{

Kg^)

■)}

duadu" dak

-

+ vk

da'' da* duT du&

3LW

_g«Pg°Tginh

<9u«

da*' dai Qak dun du

T

duP

6W da*_da?_ da' da1' dai \ r Vj T + du du? du du^dvF HuP ) da* dai Qak

•£(VVW, dun

duT du? '

Shock wave propagation

49 49

through a gas

,baia ^ T\ rA ^xraa,^,, 2ff gaH .. V\a _2H== g°f>b gaaP naP , Q^?CT - baxT baX -6 b 6 ?A iX a ==^^2 1 _ €T OU"

Here H # is the mean curvature of the shock front, K is its Gaussian curvature. This is the geometrical compatibility condition of the third order. TVT „.~ \A ±i~~ u : ~ ~ ~ ~ „ + ; ~~ i : u ; i u „ „ i:*.: T?: ± „ r „ n i_x Now„ we comnatibilitv consider the kinematicall compatibility conditions. First of all let the external unit normal to us deduce the expression for the time-derivative of the the wave front. As is denned, we have 9< ^ ^W = fti"V = 1. We differentiate this identity with respect to t: -^ ■ !^ii /i V / V+ 2+i 2i/,~ 2i/,(48) (48) / , - ^— - = ==00. . v ; dt 3 dt dt dt 3 A. Let us write a scalar product of two vectors: the unit normal to the wave rfront and the tangent vector to that one j,

da' dai

n

Trhe h e derivative of this expression pression along a ray is equal to da;; <%.'" dt

d ( ha1 \

•

a

J

du

Da* diyj

a

•"•' dua

-" dt\du J

(49) " '

dt

Multiplying Eq. (49) by the gaP f ^ r and taking into account Eq. (48), we have dvk

1 dg dgtj tj

vvv vvv ~dT == 2^rvvv -dT=2^tT

{•

•■kk

~dT 2^T

dgij dg tj

-^r ~1T999

ik ik

-^f

dakk d d (da (da%1 \\ da

• ■

v v v

Ui9al3 Vi9 d^Jt{d^)-Vi9ai3 d^Tt\d^)-

- ^du^JtVd^)-

(50) { ((5 50 0)) ™>

The derivative ^ -$%*p is equal to dan

dv? OxT

^ r__^-M ^^ -lf

+ +

dv? dxT „ ,{gjrT , „+9i . T «. A ^ ^ + G+ zGl// {9jrTi l*T^ >« + '9iJ• ^l) ■

(51) (51) (51)

Making use of Eq. (33), we have da I/ 0UU> a * " \\

dG it UKjr

=U G 9 a dt V r)it I d^- f)n Jt\-d^) \

„(I ^y

a

\

_ rLi or

6

da{

U

m UU> , l,da rii

\\

- r)n ^ + i / 9f}ii. ^ ar " "rnJ i' T

,co\

,52)

,5.

Substituting u n g Eq. JC/q. (51) (5i.) and and Eq. r>q. (52) (52) into into Eq. t/q. (50), (50), we we have have k dv dv*

=

9

-df ~dT = -°

„„ „ a (( ntt

dG dG 8G

1 kk dvf\ dv" da dv™\ dv? \ da daK

+nm nm +

{d^ ^)w{d^ ^)d^dvm ik dx™ dv? dxf 9 kii-^r9' -~dV ~w

„ „ ,,,

M , fc. i/¥W

GVV TV Gvv

»i> (53)

(53)

Propagation and reflection of shock waves

50

It is the expression for the time derivative of the contravariant component of the unit external normal. The time-derivative for the covariant component of the normal is equal to dUrn

v k dgmk

. +9mkdv

^r

dt or

k

^r

k

dv[ dv-n ap 3G da (54) + GI/.I/T' + ni = -9mk9 dt *""** dua duP ' "' dam In an analogous manner, the time-derivative for the component of the normal in the frame xlx2x3 is equal to dr>P___ dt ~

af3 g

( dG_ &4 \du<*+Uidu<*

dx{

(55)

dup'

In contrast to Thomas (1961) formula, Eqs. (53), (54) and (55) contain derivatives for the gas velocity ahead of the shock front. The reason is as follows. Thomas uses the velocity of the wave displacement, and we use the velocity of the wave propagation relative to the gas ahead of the wave front. Let us write the time-derivative for a flow parameter on both sides of the wave front #i ctyi . „ idi d2 d<j>2 , „ id2 dt = — dt h Gl/ -£—r, da* ' —77dt = -^7dt + Gv da1 ' Subtracting the former expression from the latter one, we get

30 = -GL^ dt

+

(56)

dt

Eq. (56) is the kinematical compatibility condition of the first order. It differs from the analogous expression written in the Eulerian variables in that it contains the velocity of the wave propagation instead of that of the wave displacement. The kinematical compatibility condition of a higher order can be obtained by substituting <j) —>• -^. If the medium is uniform and at rest ahead of the wave front then it is convenient to make use of the following expression

dLW

L\(*)

dt

- GI<*+1) +

dL{k-m)

1

duP '(k)

Jfe!

■9

(fe - m)\

(XmPr,

h

<*it

b« n - l « n

QG

duP

_

dtdai-.da*

m=

1,2,-

I ™M

(57)

Shock wave propagation

through a gas

51

This expression can be easily deduced by induction. Let us obtain the kinematical compatibility conditions of the second order. Substituting —> ^ into Eq. (36), we obtain

d2 k

= L^k+9jk9«eJL

dL dt

is given by

da dt The derivative

dt or

dL(l)

r(D Lt

-L&G-

~~dT

d dt

da,j

d da*

dvl dt

d<j>

dv%'

da*

~dT

\

(58)

(59)

Taking into account Eqs. (36) and (53), we obtain the following expression for the third term on the right-hand side of Eq. (59) d<j> 9

dt ~

da*'

du<* \8uP "*" k

m

du?

r

dv x dxk

da

i ■k jl

-L&

dv

nm + GviV** T $ , ) ■

(60)

In view of Eqs. (56), (59), and (60), Eq. (58) is reduced to d2$ dakdt

H - i ( 2 ) G +dt? + i ( 1 ) fdv? » ™ + G ^ .aafid[4>] dua

(6G \duP

, „ „ap 9 +9 9

dv? du?

(d[]

i^\ir+L

*

(1)

8V? dx? dv duP

G

\ da>

) d&'

(61)

Substituting -¥ §£■ into Eq. (56), we have

~d2f\ dt

2

= -GLP +

'df\ dt

dt

(62)

Propagation and reflection of shock waves

52

The second term on the right-hand side of Eq. (62) is equal to d_ 8$ H dt

H(-^f)=4(

i(1>

°)+

d2[]

dt2

(63)

In consequence of Eq. (63), Eq. (62) yields:

av

L^G2

2

dt

- 2G

dLW dt

■^(f + Gg^OWVli) r

a 0 d[] i zdG _z_

dv\ dv,P '

dv,P d^

s-i

k

zw

+

dv\ dxi dv duP

d2[<j>] dt2 '

(64)

Eqs. (61), (64) are the kinematical compatibility conditions of the second order. They contain the derivatives for the gas velocity ahead of the wave front just as Eqs. (53)-(55) do. Now we consider the kinematical compatibility conditions of the third order. We assume that [<£] — 0. In addition, we hold x\ = a% (dx\/dt ^ 0) while deriving the first two equations. We substitute d into Eq. (65). The right-hand side contains the quantities Lj. ', Lrt \ The derivative for L^ along the ray is as follows d2

dL^ dt

dt

dtfdai 2

d2

dt

.(\)di/

+ Ll(2)) . + 2L{\

GL^

tJdi/_

+ 2 dtfdai

(65)

dt

The quantities LrJ are defined by Eq. (37). The quantities Ly fromEq. (57). Taking into account Eqs. (51) and (65), we have dz<j> l

da dasdt

■{-

L^G+dJ^

, . a0dLW +2g

+a

° iM-

at f dG

+

2 L i ^ an dv[ dv[ + ni+

dxi\\

can be found

d^{d^ d^ ^d^)rii/s

L(2)G+

dL&\

dL^ dua

dv? dn

Shock wave propagation

through a gas

dua 9

9

53

- ^

V dn \\

^

O^c

a ..* adu ^n du%

( 1 ) G

da1

>

du"

W' + W*

L

S~^ du"

£a

w

da1 das du7* uP

^(-*4^ S)} dLM

d_ dn

das

.•A

(66)

By making the substitution —> d

d3<s>

^-2GdJ^-L^ddt4+d2L{1)

dakdt2

+

dt2

dt

(-—^yi dvl_

+ L&1

dn

d (dvl

+ dt

dn

2

+ L&G

d v? -9 dndn

a/3

dG dvj du? + duf*ni+ ,<*/?

i

3 G

aw

weG

dua

du<*

dv\ dxj \ \ Vk dn duPJ)

da" d (2rdLW du<*duP V dt

(1)dG

dt (67)

dn )

In order to obtain the last kinematical compatibility condition, we make use of Eq. (62), by substituting d<j>/dt instead of <j>. We suppose as in the previous cases that [
d3 dt

3

at

Ln£2.. 2 dt

dLW

dv\

\duP

du?

aB

+

dt (dG

^ ( f +» )

f dG ^

dG

2

dG f dG dua \duP

dv1? dxk dv du@ ^dv\ du?

„ fit;*

dt

(SL^G-L^h)

+ 3 G8LM

^

da1 uu, dxk dv duP

k

,_.• \

e

„ .da1

_

m

_k l

ji

Propagation and reflection of shock waves

54

^*^^lt)+«?!»*-I<»0{j^,

^"(i^ + ^ ^ l ^

?7 r»

A = ^ + GvVi*r*,.

(68)

Eqs. (66)-(68) are the kinematical compatibility conditions of the third order in Lagrangian variables in the general case when there is velocity distribution in the medium ahead of the shock wave. The compatibility conditions enable us to find the derivatives for the flow parameters behind the wave front. But that is not all. We have deduced them from the fact that there is no break-up of the wave front. As will be shown below, the compatibility conditions yield the connection between the distribution for the flow parameters in the neighbourhood of the wave front. If there is no connection between the distributions of different flow parameters then the wave front must transform into two fronts. In conclusion, let us evaluate the vorticity of a gas flow behind a curvilinear unsteady shock wave. Generally, there is a jump of the vorticity across the shock front. In other words, a potential flow ahead of a shock wave becomes a rotational one behind the shock wave. We shall compute the values of the components of vorticity behind the shock wave. Such an evaluation is of interest if one investigates the interaction of a vortex with a shock wave. We make use of Eq. (61). Let the axis x3 be normal to the shock front at the instant in question. Denote as /i and 12 the arcs of the curves defined by the intersections of the shock front with the planes xxx3 and x2x3, respectively. Using the equations of motion and Eq. (53), we get r L

i1 J

=

(i-g)2flQ , (i-g) 2e 8l2 1e , (1-g) 2piG \s do?

(H \8a2

da2)

dv\\ da3)

Shock wave propagation

r 1

2] J

through a gas {1-efdG 2s

_ (l-£) 2PlG

55 (1-e)

dh

2e

(}_dPi__C2 \e da1

( H , H \ V^1

#«3/

d

Pi\ da1]'

[u, 3 ] = 0, x\ = a\

^ # 0, of = \rotlf, e = ^ . (69) at i p2 It is interesting to note that the j u m p of the normal component of the vorticity is equal to zero. The j u m p of the component ut1 and that of OJ2 depend not only on the Mach number of the shock wave and the distribution of Mach number along the shock front but also on the flow parameters ahead of the shock front. The expressions for [a;1] and [u;2] contain the components of the tensor Sij — | (-£fc -f -^ J and, in addition, the derivatives of the pressure and density ahead of the shock wave. Eqs. (69) are valid for any gas which is in thermodynamical equilibrium. They coincide with the analogous expressions obtained earlier ( R a m et al. 1966) if the gas is uniform ahead of the shock wave. 6. R a y m e t h o d for c a l c u l a t i o n o f u n s t e a d y s h o c k w a v e s The notion of rays is used in optics and acoustics for constructing a wave front. If the position of the wave front is known at an initial instant of time then its positions at subsequent times may be found from geometrical considerations. The matter is more complicated in the case of shocks. For the motion of a shock is determined by the distribution of gas parameters not only ahead of the wave front but also behind the wave front. Nevertheless, geometrical considerations allow to construct various methods of the calculation of shock propagation. A well-known ray method for the calculation of shocks was proposed by Chester, Chisnell and W h i t h a m ( W h i t h a m 1974). It gives excellent results in some cases. This method takes into consideration the shock curvature and neg­ lects the dependence of the shock velocity on the distribution of flow parameters behind the shock front. There are modifications of that method (see e.g. Best 1991). The method proposed below takes into account the influence of the flow behind the shock on its propagation at the expense of loss of simplicity. Let us introduce new variables u 1 , w 2 , u3. Here u1, u2 are curvilinear co­ ordinates of the shock front, u3 — s, s being the distance along the ray between the point in question and the position of the shock front at a given instant. Thus the coordinate u3 is orthogonal to the shock front. The kinematical compatib-

56 56

Propagation reflection oj of SHOCK shock waves rropagauon and ana reflection waves

[ $ ] - G [§£]. [|f]. ndition is as follows f f l = W\ [f£]. ility ■ condition follo\ s while using new variables: [[f£] i of the metric tensor ^ have the values ahead of the The covariant of the metric tensor gjk have the values ahead of the variant components The( fc component = 1,2; 1,2; #
J = 1,2; 1 2;

A =1,2;

3a g = /3 g3a=gP

3

= = 0,

3 /g33 = 1.

We have behind the front: fro:t: gjk = #<*,?, gal}, Jj = = 1,2; k = 1,2; 5gaa33 = = 0, 0, gg3333 = = 22 k a 3 33 a3 33 jk al3 0,733 = gi = =< < P, = 1,2; 1,2; * = = 1,2; 1,2;
[^|=G(l-e)n<. Thus

(70) (70)

i _ !l = - =((-l)n r[ _—^— (C={(-l)n\ £ r- _ l )lnW * \\

/rite Eq. (35) in in the he rectangular We write ctangular reference frame

.-

_

dij-ninj_g

g a a0 ae

dx* dx* BX' dx' dx< dx{

_____.

Multiplring iplring it by th< the quantity ity [ ^^-11 , we get _.

-

_

I ^^

J

ravi = / r a v i i\ i ,.++ af" /afr a v i a*A a_« nVjnnn + ** *U"aP"J "([l£\w [1*1 ^5_?J J a^' [1*1 UI a a^ ^ JJ ""V ("aP"J So­ [d^\= = U 2 oting [*V1 °& B£\ .ting [^] nn,j 1as ££< „ B£\ have Denoting [ ^ ] ] n,< 2 \\ [fe]l * £ ] |°& £ as _ i 2 \ we have I I

) = 4 2 ) « 8 '+^Bi 2ouP |4-

com] In ourLr case the compatibility compatibilityy conditions can be be written as follows

[ml =vy{~ [f]=^(^-5(1))' =1 B B [w]= [w\ \d2<j>] {1 f{o(2)

„,^ = W ra<*i 5d) |"a_l) R(l) BBCD = r ^ l . =[ds-\>

2 ds(1-+ + + B + d d2 "rfT +2^p-) ^p-J 2^V o^r [}\ 2-di{idy { „ m^r)> [<£A 5
)==[ a[ ?0J1' ' y,-rJ y==G'° «(2)=[-^l B

'

dj t_dJ dT =Ll_ ^G-

57

Shock wave propagation through a gas

It is easy to show that the kinematical compatibility conditions can be presented in the symbolic form \dm<j>]

^d

(

nd

(

/ ^dN\\

d

\

[m^l - ^dl {^dl \yyjl ■ •' V^'oT))
-AT

- t s

^Mud) '

-t *M-idBli~l) v ' dl

'")'

, ^.--a *'(»"-2) d 2 ^ ' " 2 ) ' 1-2 d/2

y

As one can see, the kinematical compatibility conditions contain the quantities £('). Let us compute the quantities B£> = r ^ 1 l n5. These quantities will be usethe theequation equation of of continuity continuity which which may may be be used in the sequel. To do this, we use written in the form //

x 2 \ 2

»=(*)'...

(7!)

g being the determinant of of the metric tensor. We differentiate this relation with respect to s and use the identity (Kotchine 1965) dg ^-=99

as

kidgik

-z— ■

as

We get the following expressions for the quantities 2 #

2 22 2 e ( l - 2)(2#2 2)(2 / / 2 - tf) tf) + + I (5i (BW) 2HeBW -2e(l )) + 2HeBW 5CT = -°n — —4c(l. - £c )\Afi - i\. ) + ~ I J-ffi J T 4ii c±J^ '

4yz

dua duP

dua "

222 g 3Vi 0 022PlPl ^± ^ ^ ee (( 22(d (d(dp£\ 2V fd (d (dPl \Ay\ \ 2\2 2 _ (d£l_Y\ fd PP2V PA PPi ^, g + + £ £ -dsT 7i 71 \V£ \~dTJ ~\d7J )J'' + pP~ -1-ds^ 1~ds^ + + ^~dsT rf \ \~dT -{^7 ' 2) 3) 33 22 B£> = -B^B£> 4eH(l e*)(W 3K) ~ \B%>) 2#> J 4 4 4eff(l )(4ff -3K)-1 (Bl -4eff(l ~* )(4tf ~m-y [B^ £~ /\ ~n —n —n \ / 2 -22 \ J ,b

2 2 W °l> 2eHB& *!±ZjW + 6e U222B^ (2H* 2eHB& _{1~J ^LflK p-^%L jj$?^ pj B^(2H - -K)K) ++ 2eHB^ -_ * 9g9ap9^

3*

„( ) a? ap °y dy °y dy

2 D(2)

~4etf~nn" ~4etf~ "

3(1 --£)g) „c<0 _a0 dy°y R( 3(3> ) *v

9 d^duJ~~^y~~9

Ba d^Ba

58

Propagation and reflection of shock waves T

y

y

" 7 dua

n 5

e

"

5«« "

y

1y

+

n 9

ey

Se g g baaB?a +

2y2

y

^ dua du°

du° du? 9 9 <W duP

^

+ la"?— 5 (3) + May«/» ^La _^_ _ A a^_^La W_ *

+

du<* "

4y 2 g

5

BV

dkxfl Osk

du duP

du<* 9w^ +

a0

e d3p2 pi ds3

V

\Pi ds3

Pl

2_(dPi\3 p\ \ ds )

9* du<*

Ti—,

,

9

a

2y

du du?

8uP

ds Os2

+

p\ ds Os2

2e3 (dp2 p\ \ d, n

k > 2,

-

'

B<«> = sHBM - (1 - * ) y « - HflW(BW + y W) + 3 (j#> + y(3)f - 3e(y n ( 3 )) 2 + 6e(A" - 2H2)(B^)2 - 96e(l - e4){7H2{K

+ 12e{8# 2 ( A - # 2 ) - K2}

- H2) - 2K2} + ^ ( 4 2 ) ) 4

+ 96e3HBW(4H2 - 3K) - p(fl( 2 )) 2 (£( 3 ) + y]')) - 1 4 e ( 2 # 2 - A')y n ( 3 ) - -e9af}(BW + yW)(flW + y « ) + 4e<^YJ 2 )Yf - e « ^ <

+^ ' " W

+ Y^iBf

+ ^(3)) +

^ r W

- f ^y«2) ( ^ + y i 4 ) + 3 ^ ( 2 ) n ( 2 - ^ v i 3 ) )

Shock wave propagation through a gas + 4SsK(s2g^B^

59 e2)ga^2})

- (1 -

(2)

4 8 £ 3 / ^ ^ L ^ L _ ^ ^n r (-1 ) J 3 )

+

+ 3 {l(5(3) + Y n (3) ) _i! (j B( 2 ) )2+ ^J

x ^ ^ + y ^ j ^ + yj2)) + 12 £ /^YJ5 + ^{ 5 ^(5i 2 ) + ^i2))}2 +

- 6eHg^g^bp
lSe{g^Y^Y^}2

+yf)

7

+ 6 £ 5 ( W ^ 7 ( B S + YJ5) - ^ - V ^ g + y W)(^3 + y,(3) - &,« V ' y $ y $ + 24(4£jff - | l t f V V ' t o r W r W - 18 £ ^^/"6«yi 2 )y^) - 7s2gal3g^ba<7qav + Ueg^g^b^Y^ + 96e(l - e2)Hg^g^baaY^ + 24eg^g^^r^

-

- m3Hga^g^ba

- ^(B™

{2 %s3Hg^g^baaBi,P } d£

d£

du@ dui

+ Y^)

x ^ + ^V'r}1^ -

24eHg^g^baaY^

As (dpi d3pi _ 2dp2 d3p2 p\ \ ds ds3 ds ds3

3e J /'aVi V £2 ( f ^ V l _ 6£ / ( 2 a V _ e4 / ^ Pi 1 V ^s2 / V ®s2 / J Pi 1 V ^s J ^ \ ds +

12£ f fdpA2 5V P\

oua

\ \ ds)

ds2

3

m

/a^V

5^2

yds ) ds2

60

Propagation and reflection of shock waves r<2) = fl(4> + y(4) + 3S<2'

" +e

r = B « + yj3> + e f ^ + 25<2) ^

^ = 5S + <

+

" +

"

- e / " ^ « ) + Y^),

yi3 + yg

+ 2 ( & ^ - B£))baP - 2e>Kgae + 2 ^ * , „

_

B(3)

, o(3)

,

- Zeg^{baaB^ Ay2y

y(3)

y(3)

+ bopBW) + QeB^(2Hba/} -

KgaP),

duadu0'

8y3yy du^duP 4yiy dua du? y(5) _ 1 1 ^ _,<*/? _%__%_ _ 9 y , „ « < ? ^ dV Ay2 9 dua du? Ay39 dua duP

Ay29

9

' du<* du^

_ 329 «/»W_ J?£ _ 29±a«0Wdy_ a a Ay

du duP

16*/4 V

y(2) =

y

du du?

^y2\Ay

du«duP)

_±J^_ 2y du<*' 2y \2ydu<* y\

Sy2

du<* du°

9

ai

du") '

)

9

9

°in du° fat

Shock wave propagation through a gas iy' dy' iy ay

1 i dy" ay

Zy' iy

61 61 dy i 1f ay (

(

(

dy' ay

-T^^Sr"^"^^' dd22V y a

dy dy _ _

ou aui

o^(M

*(*) B

^

^ =

au° "■•' o(*W

~d^-B°

r

^'

,

/ y

dy

.•>

„

y =

=Tv


(721 (72)

~d¥■

B{k) will be given later. The expression for B£' Now we consider the mean curvature of the shock front. fron The time-deriva­ time-derivafor nnantitv is tive this quantity ive I rtft

2dH__ dg^_b dt dt ITT

g0lp

dba[S_ dt

^ (73)

11

In view of Eq. (45),), we write dba0aP _dm db dm i,. ^, x +ni

-^ = ^ ^

djapap _ dx> =

aV \ d ( d'x* ni

^r

d fdx'\ (dx'\ rra a ni T

Jt\a^b^)- Jt\d-^) ^

Taking into account Eqs. (33), (40), we get dbaP d2G dG ra _„ ^ ^ r +nk -dT -dT == d^d^-d^ d^d^~d^ ^ T^+nk

(( d d22v\ v\

dvl dv\ ra \\ ^ n h T G{2HKp R9a ) T \fr^-a^ {d^d^-d^ "P)«p)-G{2Hbap--K9ap) e -(741 (74)

We used me the laenmiy identity ^±nomas (Thomas 1966) ve usea iyoo; g^b^bp,,

=

2Hbal3-Kga0.

The time-derivative for gaaaP is

^ = =g-g* W -_*££*_ M |f*V „« V (( W ** ^ -duM |f* V T y y u duT duT dudu°) dt V d° °J

(75) (75) y '

t.ino- Ens nto E n (731. Substituting Eqs. (74V (74), 1751 (75) iinto Eq. (73), we we have ap iV G 6 ^ -j0—0-) 2—— = 2g £ ^I(Gbp^ 2 ^ = 2 J g, -baV

+ fp ( d2 d2° G __^dG_ ( dH\ _M H r>\ + ++ nk nk (-^ rO Q°P ( r % + gaP K K \du«duP

du° e

-G(2Hbaa0 - a0 Kg aP)\. -G{2Hb p-Kg )]. ap-Kg a0)\

{du^u?

du° ?)

Propagation

62

and reflection

of shock

waves

Or

2

f= 2 G < 2 * 2 - *> -2rt^ dS fe+AG+nkAv*'

As

(76)

^(s^?-^)-

Formulae given above enable us to deduce a system of equations that de­ scribe the propagation of a curvilinear shock wave in a non-uniform gas. First of all, we compute the equation that contains the first time-derivative of the Mach number. T h e equation of motion has the form 5V dt2

_ _ 1 jkdxi dp p9 dui duk'

(77)

Multiplying this equation by n,- and subtracting the quantities ahead of and behind the shock, we have

[0V dp

1_ fdp2 d]h_ pi \ dv dv dp _ dp t dn dx{

Tli =

dp

(78)

We took into account that v\ — v[/e. The left-hand side of Eq. (78) can be written with the aid of the kinematical compatibility condition of the second order in the form B(2)G2

>U) _2QdBn _ at

fl(i)GVVl/fcr?

J

_

B

^ G ^ n ov

m

= -

pi

dp2 _ dlh dv dv

Substituting the values of B^\ B£] from Eq. (72), we obtain

Pi

(dpi__ \dv

E

1 dG — = dt

de dp2\ .2He{l-e)G*-2G-+(\-e) dt dv)

dp2 dv

dpi dv

(79) The derivative of the density is dp: dv

" cl\dv

\ds)i

ds_ dv

(80)

Here S is entropy. If there is no dissipation ( ^ = 0) ahead of and behind the shock then dS2 _ 1 d[S] + dth (81) dv G dt dv '

Shock wave propagation through a gas

63

We assume the gas to be perfect. From Eqs. (79), (80) and (81), we get i dM _ c\ at

i dP2 pi Of

I dPl p\ ov

+ A5-—±ni C\ ov -{M2 (M*-\)± Ai =

I dPl pi ov

I

p\c\

dPl at

+ A6H,

kl

= Jl±^M*

k2 =

27 7-1 k3 = 1 M 9 Ar4 = —qM, k6 = 2qk5 = -Mq, ,.-, - ..0-.,(^+1), 2 4 2 2 F = 2(2 7 - 1) 4- M + (7 + 5)M - 7 + 1, q = 2 7 M - 7 + 1, 27

2

w = ( 7 - l ) M 2 + 2.

(82)

Eq. (82) has a clear physical meaning. It is seen from it that a plane shock which propagates through a uniform gas decreases if dp2/dn > 0 and increases

ifdp2/dn<0. A converging (H > 0) shock intensifies if dpildn = 0, and a diverging (H < 0) shock decays. We can take a derivative along the normal for any quantity behind the shock as the first term on the right-hand side of Eq. (82). Eq. (82) coincides with that one obtained by Sedov (Sedov 1959) on the assumption that the flow is one-dimensional, the gas ahead of the wave is uni­ form and at rest. It was derived by Shugaev (Shugaev 1976) for the case of a three-dimensional flow ahead of the shock wave. An analogous formula was obtained by Wright (Wright 1976) for the gas with an arbitrary equation of state. We can continue the procedure. Differentiating Eq. (77) with respect to time and taking jumps, we obtain r^3 <9V

dt3

l(igjkdxidp_ dt \p

dui duk

(83)

Taking into account the kinematical compatibility conditions of the first and third orders as well as Eq. (76), making use of Eq. (83) and differentiating Eq. (82) along the ray, we get a system of two equations with two unknowns

*Jlf/itt» and £ ( £ ) ( « £ ( £ ) ) . Those equations are given below (Shugaev 1983). The gas is assumed to be uniform ahead of the shock wave, for the sake of simplicity.

Propagation and reflection of shock waves

64

p\C\ at \ ov J pi dvdv

* \p\

4\VM)>

+

4^ £H,

+

d2M duaduP

a

— gn P AM =

ov J

d

dM dM

dM dvF

(v^) 2 = ^ ^ E j -

ap

(M2-I)N^/E,

AP =

E = 2(7 - 1)(2 - 7 ) M 6 - (7 - 1)(13 + 37)M 4 + 2( 7 2 - 7 - 4)M 2 -

m

(1

N? N.( i

K

(i

AT)( i

N-(2

jvi :

N? N? MW r(2

2 7

- 2 7 - 5, 7+ 1 4

WJ M(M2-

_ _

^ / „ 4M V 2w

(M 2 - 1) F22

p

2

1),

+

(F 7 (M 2 - 1) + F 5 F 8 ) (2F 2 )

^_2F6^lzJfc

Fu ((7 + l ) F | ) ' ((7+1)^2)

+ (7+l)M 2w 2 F 12 ((7+l)M)'

+

3

-2

m

(7+1)'

...

Fl4 Fl5 ilvF^*-6 1) (7 + l ) M V 2 + F2 2F 2 7+ 1 M(M2 - 1)F2, 2 w (F1F2 + ( 7 + 1 ) ( M 2 - 1 ) 2 ^ - ( M 2M 4(M2-1) F21 + 2W(M2-1)^ + ^ r2 r2 (7 + l ) M f 23 (2) _ 2F,24 2 4(M - 1) ( ( 7 + l ) M ) ' iVs - (7+l):

1 f F 2 Fi4

M V(7+l)

^-^ (( +l)MF 7

F1 = 2 ( 7 - l ) M

2

F3 = ( 3 7 - l ) M

- 1 ) ^ )

(2) 6

=

2F2F13 (7 + l)Mw 2 '

1 _MJ11)' F

2)

2

+ 3-7,

F 2 = 3(7 - 1)M 4 + (3 2

2

13

((7+l)M)'

(M2-l) F22

. „

+WF220 0

F

N,( i )

+ 3-

7)M 7 )

2

+ 2(7 + 2),

F4 = jM2 + l,

F5 = (3 - 7 ) M 2 - 7 - 5,

Shock wave propagation through a gas

65

F 6 = 3™2F3 - 4 7 (7 2 - 1)(M 2 - l ) 3 + 4(7 - 1)(M 2 - 1)F 4 F 5 , F7 = (7 - 1)(7 7 - 3)M 4 - 4(7 - 1)(4 7 + 3)M 2 + 4(2 7 2 - 3 7 - 8), F 8 = (7 - 1)M 4 + 2,

F 9 = (7 + 3)M 4 - (7 6

7)M

2

+ 6,

4

F 1 0 = (7 - 1)(37 + 1)M - 3(7 - 1)(5 - ~f)M + 12(37-4)M2+4(10-7), F u = 2wF 5 F 8 - (7 + 1)(M 2 - 1)F 10 ,

Fl2 = 7 M 4 + 1,

F 1 3 = 2(7 - 1)(7 2 + 7 - 6)M 6 + 2 ( - 2 7 3 + 3 7 2 + 67 - 19)M 4 + (7 3 ~ 7 2 + 117 + 2

2

1)M2

+ 4(7 + 3),

4

F 1 4 = ( - 7 + IO7 - 5)M + 2(9 - 7 7 ) M 2 - 2(7 - 7), Fis = 4wF 5 F 8 + (M 2 F 1 6 = - ( 7 - 1)(3 F17 = M 2 + 5,

- (7 + l)Fio),

1){WF7

7)M

4

+ 4(7 - 1)(7 + 3)M 2 + 3(7 + 2)(3 - 7),

F 1 8 = - ( 7 i ) ( 7 + 3)M 4 + 4(7 - 1)( 7 + 2)M 2 + 4(7 + 2),

F 1 9 = 2F 2 F 1 6 - wFl7F18,

F20 = F2F7 + 2(7 +

6

2

F 2 i = (7 - 1)(17 - 7)M + ( - 5 7 - 26 7 + 39)M

l)wF8F17,

4

+ 4 ( 2 7 - l ) M 2 - 1 2 ( 7 + 2), F22 = F2F10 — 2w F5F17,

F 2 3 = - ( 7 + 1)(7 + 4)M 6 + ( 3 7 2 + 4 7 - 9)M 4 - ( 7 + 2 ) ( 7 - l ) M 2 + 2( 7 + 2), F2A = 2(7 - 1)(2 7 - 1)M 6 - 5(7 - 1) 2 M 4 + 2(7 - 1)(3 7 + 2)M 2 -72 + 67+H.

(84)

Eqs. (84) describe the variation of density distribution behind the shock (2)

(2)

(2)

(1)

wave. The coefficients A\ , A 3 , A\ , and A\ } are equal to zero at M — 1. It is interesting to note that the coefficients AJ ,A\ ' change the sign at M = M*, M* being a root of an algebraic equation of the third degree 2(7 - 1)(2 - 1 and 7 —>• 2, the value of M* increases unboundedly M* ** V2(7 ~ 1)"*, 7 - 1 ;

M* ^ V^5(2 - 7 ) " * , 7 - 2 .

Thus we obtain a connection between the first and second derivatives along the normal for the density behind an unsteady shock wave at the point of the front where M = M*.

Propagation and reflection of shock waves

66

Let us consider one-dimensional motion. We suppose the gas ahead of the shock to be uniform and at rest. The second relation of Eqs. (84) takes the form 1 x 2 d2P2 r ( x / 1 dp2 a

2

=il(7) i

*

For l < 7 < 2 , L i i s always positive. Therefore, so is the second derivative along the normal for the density behind the wave. The value of the density in the neighborhood of the front (x = 0) can be expressed as —P2(x,t) = — f>2(Q,t) + £ + \LX? Pi Pi 2

+ ...,

pi OX

Now we proceed to the general case. We differentiate Eq. (77) m times with respect to t and then multiply it by n,- (or by dx*/dua, in order to determine Bla). We use the kinematical compatibility conditions up to the (ra + 2)th order. The quantities Bn ,« = 2,..., m + 2, are determined from the continuity equation while differentiating it with respect to s (see Eqs. (71)). We get the following unclosed system dxi

Ci eft pi 1 d PlCi (ft \ as J LCI dt

5s cte2

pi

\ p i as J

ci 1 d /5>2_\ Pici ctt V dsm J

1 pi

ci (m+l)^ x

ci

as

W+1

>P2 5«( m + 1 ) '

(m+l) / 1 9^2^ f 1 s V A

f 1 <9'p2

t = 2,3,---,m + 2, j + Ar + ..- + Z = m + l , j < m, & < ra, • •, / <

ra.

(85)

Shock wave propagation

through a gas

67

Dots on the right-hand sides mean the derivatives for the pressure along the shock front as well as for other flow parameters along the ray ahead of the front and geometrical properties of the front. The system contains the unknown functions xls, nit G, ^ , ^ • • • d£m#. The number of equations is one unit less than the number of unknown func­ tions. The value of derivative of (m + l ) t h order m a y be determined from the boundary condition of the problem. It is interesting to note that the coefficients Ay, i = 1, 2 • ♦ •, tend to zero as M -> 1. Therefore Eqs. (85) are convenient for studying problems connected with the propagation of weak shocks. For the first time, an analogous system of equations was written by Maslov. He considered a set of differential equations in the case of weak shocks and found that this set m a y be continued to infinity (Maslov 1977). Piskareva and Shugaev discussed the set of equations for one-dimensional propagation of a shock wave which involved the functions M , -^ I ^ ) , ^ ( %ffi) etc. (Piskar­ eva et al. 1978). Later on analogous systems were the object of study by several authors (Grinfeld 1978; Prasad 1982; Anile et al. 1986 a and b, 1988; Sharma et al. 1995). Grinfeld investigated the wave propagation in nonlinear elastic materials. Other authors considered shock waves in gas. Numerical results were obtained by Shugaev (Shugaev 1983). Let a curvilinear shock wave propagate through a non-uniform gas which is at rest (pi = const). Then the unclosed system is as follows (m = 3) dxi _

2-

y n

ly

*

dnl _

dl^

1

2

ap

dl

dxl

dy

e'

d

1 l d d W _ 2v + vil- - JJA ++ ±.?El Zy ^L +V *n Bm+ + JJL »n BW- - -l ±

vBn yti

dl

+9 yBn

B

2y

+9

«

dp

8uP dl

+V

**a du?

-2Q«f>
e d s

Ay9 dP Aydl9

£pi

ds

2 dl

du«duP+Pl9 "

e ds* a

+

2 dl

du«["*> n

+

du?^2y

2 dl n

9

- , « * ^ ^ 9 + a « 9V ^m B 9 dl du?^

_ V ^ W > + ^(2)) + yf^T

n

+

^ 2 dl

du« du? ( 2 ) a

— du*

+ ^

dl

h

dl

68

Propagation and reflection of shock waves

yBW-2y^-

dBP d2B^ +y dl "dP

18 3 Q s ds3

-f { (4 2) - ~Pj B^ + A£HBP + y„(»))}

= M2,

$ = ( 7 + ly^u-tfgVyu /(u) = ( 7 - l ) « + 2, (1)

_ J_dp2

(2)

dM _ dM ds ~ dr

p(«) = 2 7 t i - 7 + l , la^nW _ I ^

1 fPi\lh+ ^L^flW 9p2 ^ ^ -^/»^2x ^ W + yP)),

_ ]_dp2 de_ 1f " ~ Pl ds du° + Pl9 (1)

^

5p2 _ 9 _ / l . / ' p i du» ds\pi \p

^2.

= 2 ( 5 ^ 6 ^ ) - ^ ^ ) j_5p2 f ^

3 Pl

dvf V n'a

,

,( 5 ( 2 > + y( 2 ))l

W dW> du«

2yA"ay)

Eqs. (86) can be simplified for M ~ 1, px = const

dZi

1/7

ai

1 „,

'

(86)

Shock wave propagation through a gas

69

dZ2 = -\yZz + e 2 , dl dZ3 = e3, dl 2 y= M -l, ei =■-^Zf

+ HZlt

- 2eim2 - jHBW - -HZ\ - Ig^A^Zt 7

.^ z

+ m3,

I

+ ZZ\) zzl)^zlz + ^Z?Z2 - f2U-^zt^ > 4 + ^-(Z2el +

e 3 = -^(AZ.Zs Zl 3 - i^Zle, jz

+ \m2 - ^(Zim3 Z 7

+ e\) - HZ,B^ -

Of*

- jHB^ - 1 2 7 m i ^ 3 ) - -zHZ\ + AHZ^UK 7 9HZ!Z2 - 2He2 + 4e : I — HZX - ZH2 + K 7 7 +

j

du^duJ+2Zig

- ^Aaf3Z2

A H 2g

^ ~

-

^ m 7

2

- 1SH2) + Hm3

oua ouP

2 9

+ Zxe2)

9

h

«*d^

7g^g^b^AaaZu

9

ACi

mi = 2H — K,

Z\, m^ — eim 2 + Z\m\, 7 m 4 = 3—r-^iei - ——(Zie2 + Z2e1) + (^i + ^ m 3 + - m 5 - e1m1 2j2 2j 7 2 4 8 3) - 7rai£(, + 2He1m2 + # e 2 Z?mi EZxex + m 3 # 7 7 + 2HZ1{3H2

m2 — E

- 2K) + im 2 flf a/? A tt/ jZi +

^Zl9aPAapH

g du«duP g du«duP' 1 m 5 = - —e? + m 2 m 3 + 2eimi + 2HZ1{AH2 - 3K), 7

2J(3> = 1 ^ - 2 / 0 - ^ 2 , 7

7

7

Propagation and reflection of shock waves

70

BW

= --Z3

+ ^Z,Z2

-

K

-\Z\ - -m1Z1

+ 2HB&

+ l-g^AapZl + « M , 72

7

Kl = 7 + 1, «fc+i = «fe(7(fc + l) + 1),

(87)

k> 1.

The quantities £ « ^ in Eqs. (72) can be determined from Eq. (77). That equation may be written as d2xi dt2

dxi dua

1 dp p dua

(88)

We assume that the gas is at rest ahead of the shock wave. Using the identity [AB] = [A][B] + [A]B1 + [B]AX, we have

d21„i x 2

dt

dxl du"

(89)

1 dp p dua

(90)

Taking into account the kinematical compatibility condition, we obtain from Eq. (90) 1 - e dy e dp2 B& = y dua ypi dua Differentiating Eq. (88) with respect to s, we have with the aid of Eqs. (64), (89)

R(3) _ _ i A ( l

dp

A + y1 n<2) i ^dB"] + l d £

dy

yds \p2 du<*J 2y ^ dl "*" y dl &ua l-edy' _ {2)(l_dy_ +9e_]+2 ds de n \ydu<* du<* ) dldua 2y dua

+ (2 + e^'b^B™ -

a

(1

~£^1

+ 2£)

/"^

(1 - e)y' de

2y

dua

dy dip'

*(*) , k > 3 can be found similarly. The quantities B& Eqs. (87) may be useful in studying the decay of weak shock waves: M — 1 - f J M , S M
Shock wave propagation through a gas

71

obt ahead of the shock wave to be at rest. We obtain third and fourth obtain from the thirc equations

dy = _7+1 dl ~

47

dl

2

yV

~

y ,

1

4VPl1'

2 2255 7

V ,/22

4 7 + 1) )

*

4(1+i)

■ 16(7TT) i o i 7 n y mm , ~ „7

i

d p2 11 &P2

' - 1^El

Pl Pl

Pl Pl

~ ~ Plpi ds' ds' 22 y=M y= M -..-..

„-

1_^l

2 ~~ Plpi ds ds1 ''

(91)

The wave decays if Zi > (7/(7 + l))p[. The two equations (91) include three Z2. In In order order to to obtam obtain aa closed closed set, set, we we rnree unknown unicnown functions: funct,ions: y, y, Z Au\ , and and ZJ2. assume Z to have the same form as in the limiting case y = 0. y=0. assume Z222 to have the same form as in the limiting case y = 0. The equation (91) (91) is is as as follows follows under under the the above-mentioned above-mentioned The solution solution of of the the first first equation assumption assuniption assumption

-.«

„ . . _ / . . 7+1..,.,}

^Y^-ZiHo) f{l/A(s))ds\ Zi(l) = Z!(l Zi(io)A(J) ^Y^Zi{l 1Y±ZI(1 f(l/A(s))ds\ f(l/A(s))ds 1(1) I( 1 + ^±lz 00)) 0)A(l) l(lo)J(l/A{s))dsj

M(l) = - A ) - 1 + (Mo (M 0 - 1) ((f$±-\

\Pi(h)J

4

* exp [ _- 2 ± 1i /j ZlZ(s)ds l{s)ds\

\

47 J

^=eXP(-5(l-^)). \

\

r J- V

u

/ /

X1 j ,

J

,,

'

/

(92 («)

tn a similar manner, we have for spherical shock waves In 1

Z1(0=(l

+

1

/oZ1(/o))/iA(^)'1Z1(/„)^>

. f i \1/4 / -v + 1 r' .. \ M(J) = M{l) = 1+ + (Mo-l)A(l)lj-j (Mo - l)A(l) (p\ expf-^-^ exp(-2±i / S i f oZ^dj,), )*,), A(/) = (pi(/)/ (pi(/)/pPl1{lo) {lo)11/\/\

Mo = M(/0).

(93)

Let Let the the shock shock wave wave have have an an arbitrary arbitrary shape. shape. In In the the case case of of aa weak weak shock shock

Propagation

72

and reflection

of shock

waves

we can put (Thomas 1966) H0 - K0l 1 - 2H0l + K0l2 '

H(l) Consequently, I

,. _ , . .,. „ „ . ,2l / u dl = - 1 , / H dl = - ^ log |1 - 2H0{1 - /o) + KQ(l - l0) lo

The quantities Z i ( / ) , y ( / ) are as follows Z 1 (/) = Z i ( 0 ) ( l + ^ i ( 0 ) ^ y = y0exp(

—

/

F2(S)^)~1F1

/

Z^dsjFf,

F i = |1 - 2 F 0 ( / - /o) + # o ( / - Z 0 ) 2 |, F2(s) = - 7 = log VA0

, V-Ho

-

OL

A 0 > 0, H0 < 0,

+ V-#o - ®

F2{s) = - 7 = log #o \Affo + a - A 0 s + y/H0 - a HQa Ko>0,Ho>0,s< ,

K0s'

AO

^2(«) = — ^ - a r c s m , A 0 < 0 , a = Wflg - A 0 . -A0 a v Eqs. (92) and (93) describe the decay of a weak shock wave in a nonuniform gas and the variation of the pressure derivative (Z\ > 0) behind its front. In order to obtain the pressure distribution behind the shock wave we need more equations. Let us consider the set of three equations. T h e gas is assumed to be uniform and at rest ahead of the shock wave. Those are as follows d

y dl ~

dZy

~~dT dZ2 dl

7 + 1 rr

4 7 •< / ^ i ,

-—2z2 7

2 472

7 + 1 72 27 3(7 + 1 ) ^ , 27

We can rewrite the last two of Eqs. (94) in the form dZi _ 2 7 dy 7+

Z2

2Zi

izl + — , y

(94)

Shock wave propagation through a gas

f.

=

73

.nl?l + ^i.

dy

7

Differentiating the second of Eqs. the first one, we have

y

(95)

y

(95) with respect to y and taking into account

2

29/ d w 1 0 y dw+ ( 2 4 + / ? y ) w = 0

V~ ^

w = Zl

'

/ ? = ( 7 - l ) / ( 7 + l).

(96)

Eq. (96) has the solution Z21=yn/2{C1Mrj)+C2Y5(V)}, rj = A{(3y)1'2.

(97)

J5, Y5 are Bessel functions. Arbitrary constants C\, C2 can be defined from initial values of the functions Z\, Z2. Eq. (97) has physical meaning only if y <^ 1. Thus we can simplify the expression for Z\. We have Z^y) ^ ZM

( ^)

.

(98)

,2/0/

Substituting Eq. (98) into Eq. (94), we obtain Z1(Z) = Z 1 (Z 0 )ft- 1 , y(l) = t/(/ 0 )/>- 2/3 , A = l

?lrtI)z1(/o)(/-/o). 87 These expressions seem to be more accurate than the previous ones as we used three equations instead of two while seeking the solution. If we take four equations then we obtain for Z\ +

+ I / Z ? ^ ( 1 5 6 + 18aiy) - Z?(140 + 42a l2 / + 2a 22 / 2 ) = 0, dy 7+ 1 (7-l)(37-l) 7-1 (7 + l ) 2 We seek the solution in the form Z1=aky°+k.

Propagation and reflection of shock waves

74

The exponents a have the values o-i = 1.2583,
Fig. 22. Pressure distribution behind a weak shock wave at different instants.

7. Path of a particle behind a shock wave Let the shock wave intersect a particle at the instant t, ul,u2 being the curvilinear coordinates of the particle on the shock front. At the instant t + At the position of the particle is as follows x\{v}, u2, s,t + At) = x[{ul, u2, s, t + At) + A*

dxi ~dt~

Shock wave propagation

through a gas

75

At2

d2i 2x„ «

+ The value \^-

At3

d3)3„i X Ot3

+

dt2

(99)

is defined by Eq. (70). The j u m p s of the derivatives of a

higher order can be found from Eq. (77) and the compatibility conditions 82i dt2

lgjkdp^dxi p dui

du

1

1 dp2 2 Pi en ds

dx2 ds

d22„i X

i a0 dx = 9 duP

%

dP2 dx

1*«(> = — 9 P2

k

du<* du~P dx% 1 dpi duP ^ pi ds

1 a/3 dpi pi dua

dx\ ds

Or

dt

2

pi dua

]_dpi

+ n,-

Pi

1 Pi

ds

dua

p\ dp2 ds

(100)

Similarly, [d3xi dt

3

=

1 p9

-y/y

dx{ duk

dp dui

jk

d_ 1 ds IP

jk

dp

dx{

dui

duk

Or ra 3 x ! ' dt3

= Vv

P\

1

d2p

Pi

ds2

+—

dp

J _ djh_

ds~

y/y 2

+

9P2

pi

dpi_

l_d_

ds

pi dl

dt ffh

£

p^ 4.

os

dp Js dP2

a"?

9P2 a

ypi

ou

OuP

^°"°t^)»-H7>>"°°"^ _1_

dv$ dxk du° duT dp

i

ITs 9

lypi

9aa9l(3bai

— Pi 1

p\

d

d

a/3

dv^ dxk\\ duT du°JJ

dp2 dua

dpi dua

dpi \

.2

g ' dp2 p\ ds

duP j_dpi VV

P\

dx1

1

du?

piyfy

aaP

dt

dua !hft

dx

du«

dxi

dxl duP

dx1

dy dvki n a + V £ du^a r * du

Pi ad Pi ds * du« duP

dp2 dua

( dp2_ \€ du<*

+

aP

dp2 a

du

dpi ds

dp2 dua

dpi dua

dv{ duP

dx1 duP

Propagation and reflection of shock waves

76

_ 1 9 «/»fe JL(!!Pi\-JL(!!zi\\ ^ + ±a9°p PI \ dva\ds) dua\ds))dup ^Pl

X9

X

du« V dl ) + 2y dl dua) du? py p ( d ( 1 dpi , dpi\ , 1 [ I dpi a a + + \8u \^/y 8t ds ) 2y \y/y dt dxi dvfi

+

1 ds pi dl9

a0

t +

dpi\ ds J

dy dua

dp2 dx{ dua du? Jo

Eqs. (70), (99)-(101) enable us to determine the path of a particle behind an unsteady curvilinear shock wave up to the third order of accuracy. It is clear that the path is fully determined if we know the distribution of the flow parameters ahead of the shock, the shape of the shock wave and the distribution of the pressure behind the shock. The terms which are proportional to At2 determine the curvature of the path and those proportional to At 3 determine its torsion. 8. Distribution of flow parameters behind an unsteady curvilinear shock wave Let an unsteady shock wave propagate through gas. The gas is assumed to be uniform ahead of the shock wave, for the sake of simplicity. Let the gas particle intersect the shock front at the instant t. We shall determine the gas pressure in the neighborhood of the shock front at the instant t + At. We can write p2(u\

u2, s, t + At) = p2(u\u2,

0, t + At) + s^-{u\u2,

+ \s2^(u\u2,0,t

0, t + A*)

+ At)

+ ^ 3 ~ ( u \ t / 2 , 0 , t + A*) + . . . , pAt

s= -

G(r) dr.

(102)

Here p2(u1,u2, s,t + At) refers to the particle in question, p2(u1, u2, Q,t + At) is the pressure directly behind the shock wave. We use the expansion of

Shock wave propagation through a gas

77

the flow properties in power series with respect to A*. We obtain up to the terms of the fourth order p2{u\ u2, 8, t + At) = p2(u1, u2, 0, t + A*) + Ax At + + A3At3 + _ dp2 ndp2 at os 1 fd2P2 dGdp2 2 V dt2 dt ds

A

M

1

A

= 3

(d3P2 6 V dt3

d2

G9p2 dt1 ds

+G*L f^El) 2 2 dt

+

I ds J

A4At4,

, ^2d2P2\ ° ds2 ) 3

r3d

P2\ ds* )

„dp2 °~ds ' 1 fdG d fdP2 2 1 dt dt V ds

_ G^^EI 2 _ G2± f92P2 dt ds

dt V ds

4

M

= — (£&■ -^±?P 1 + G ^) 4 3 4 24 V dt dt ds i (fGd_ fdpA 6{dt2 dt\ds J

+G

dt\ds 2dGd*p2\

J J

)}■

+ - (—)2

d2p2

ds J 8 \ dt J ds2 £_ fdpA d2Gd2P2 3 dt \ds J dt2 ds2

,3_d {Pp£\ l _ i r do f_ fdpA _ 3

A2At2

2

4 1 dt dt \ds J 1 dGd (d2p2\

1t-d?r] + 2GltJt \J^)

2f_

2

dt \

■

fd2P2 Os2

(103)

All the quantities on the right-hand side of Eq. (103) refer to instant t. Similar formulae can be written for the gas density. Eqs. (99)- (103) allow us to determine the distribution of flow properties behind an unsteady shock wave at the instant t + At if we know these parameters at the instant t. Unfortunately, power series in gas dynamics have poor convergence. In order to improve the convergence, we used continued fractions (Jones and Thron 1980; Ditto et al. 1988) aQ axAt a2At2 a3At3 aAAtA

T + ^~~ + ~ ^ +

1 +

1 +""•

While solving the problem of the shock reflection from a smooth body, we know the coefficients of the power series at the point TV on the body surface, namely, at the point of the intersection of the incident shock with the body surface. We must determine the distribution of the flow properties at the instant t. The time is counted from the instant when the shock wave hits the body. Let the gas particle intersect the reflected shock wave at the instant t±, 0
78

Propagation

and reflection

of shock

waves

introduce the following nomenclature At = it<--t^f^/r /GG Af = Goo~oo.,,

Ah CAt, A A*
0 < CC < 1.

Here -${xl,xl) lere zrg x% = zg = -ip(xl,x%) - # i ? £ , a : g ) is the body equation, the axis ix% g coincides with the velocity elocity of the incident shock wave and has the opposite direction. We have (cf. with Eq. (102)) fAt pAt /.At

s = -

G(T) JAUl JAt

dr.

Tn h i s rcase asp W u s t fulfill thfi following followino- substitutions s u b s t i t u t i o n s into i n t o Eq. E n . (10 (10.^ In tthis weP m must fill the (103)

2 c^ , f->(i-C) ^ . . f^(i-0'f. ■ + U 0 C f d t f ff. (w)G c (1 oc d2

°

/i,.

2 .^d2G G ^d

d2G

/.. n

22 ^.*d 3d G

C H A P T E R 3. I N T E R A C T I O N OF A P L A N E SHOCK WAVE W I T H D I S T U R B A N C E S A N D STABILITY OF S H O C K WAVES

9. L i n e a r i n t e r a c t i o n o f s h o c k w a v e s w i t h d i s t u r b a n c e s The interaction of a shock wave with disturbances having the form of plane waves was studied by many authors (Blokhintsev 1945; Kontorovich 1959). The results were surveyed and some new data were given by McKenzie and Westphal (McKenzie et al. 1968). Here we present a solution in fuller form than McKenzie et al. did. Any weak disturbance may be presented as superposition of three disturb­ ances: acoustic wave, entropy disturbance and vortical disturbance. It is of interest to solve the problem of the interaction of a shock wave with these dis­ turbances. The problem is of practical value due to the fact that there are usually disturbances in the gas through which a shock wave propagates. When a shock wave interacts with disturbances of various kinds, its front and strength (Mach number) alter, the disturbances being amplified on passage through the front. An additional disturbance arises in the gas due to the change of Mach number, namely, an entropy disturbance. We introduce the Eulerian coordinates x1, x2. It is assumed that the shock front is parallel to x2 axis, while the velocity of the shock wave is directed along the x1 axis. We have for the undisturbed gas x\{a},

a 2 , i ) = a1,

x\{a},

a2,t)

= 6Q a2,

6Q =

. P20

Here a* are the Lagrangian coordinates, subscript 0 refers to the undisturbed value. We get for the gas behind the shock wave x\ — (1 - so) Got -f So a 1 ,

79

x\ = e0 a 2 .

Propagation

80

and reflection

of shock

waves

T h e linear wave equation has the following form in the Lagrangian variables

d2p2

2 40 (/ d£)2 p2

£i2 d2p

,

\

2

2

dt

P20 = const. Let an acoustic wave fall on the shock front from the side of a compressed gas. There are no disturbances ahead of the shock as the velocity of shock propagation is supersonic. A reflected wave appears as a result of interaction. We take the equations for the incident and reflected acoustic waves in the form S

JS1 = v U%* - k0Got) ,

Pi

\

n\1] = cos (9,

^

'

Pi

n£ 1} = sin0 > 0,

= i, (nj2V - k0Got) , \

n[2) < 0,

/ k0 = J—.

Here 0 is the angle of incidence of the disturbance on the shock front, n\ is the projection of unit normal t o the front of the acoustic wave in the xlx2 coordinate system, superscripts 1 and 2 relate to the incident wave and reflected one, respectively. We locate the origin of the reference frame a t the point of intersection between the incident acoustic wave and the shock wave. T h e flow is station­ ary in the new coordinate system. I n the previous system, this corresponds to disturbances along the shock front from the incident and reflected waves propagating with the same velocity. This implies proportionality between the arguments of the functions (p and \j) a t the shock front, i.e., {n^ -kQ)al —7-:

+

n[l)a2 w.— = const.

(n^-JfcoK+n^a2 Or (2)

n\

;

j

— ko

~

(2) '

n\}

that is, n2l\n^ - k0) = nPinP - k0). Eq. (104) has two roots: n£ 2) =
(104) - 2Ar0n(11)), on =

1 + k2, and n[ } = n\* but only the first root has physical significance. For I > n[} > 2k0/a1, we have n\

there are incident and reflected waves. For n p <

2k0/ki,

> 0. T h e solution in the case can be constructed, if we take two

incident waves which are not independent. For n^ = l/k0, these waves are

Interaction of a plane shock wave...

81

such that the overall disturbance is zero. In that case, the velocity of gas behind the shock wave is equal to the velocity of sound in the coordinate system related to the point of intersection of the wave fronts. To find ip, we use the expression for d2M/dt2. By substituting into the right-hand side of Eq. (84) the expressions

l - S P 2 = (p + A

SM^l^-iv + i,) ai^MoCit

we get

(Mi - \Ywl(F^

- F6)(F5n^

- F7)

™h

Fx = (2 7 - 1)M04 + 2M02 + 1, "2'Z

_ FSy/wET -

F45

=

(7 + 1)(7 ~ 1)^M 0 (M 0 2 - l ) 2 2Fl^qE

M»(7M2 + i± M0y/(y-l)wQ), w0 F 6 , 7 = ( 7 - 1)(Mf 22 - I K " ± y/j^lMoau F8 = (57 - 1)M 4 + 4M02 - 7 + 1, _ (QI - 2fc0n(11))(nf } a' - G0k0t)

s —

i a

(2)

2

;2

i

(ax - 2*0n(i1)) There is no reflected wave for cos 6 — F 2 and cos 0 = F3. At those angles of incidence, the shock front distorts and an entropy disturbance arises in the gas but an acoustic wave does not propagate through the compressed gas. At normal incidence of an acoustic wave on the shock front from the side of the compressed gas, the resulting disturbance has the form — Sp2(a,t) = (p(a - Gok0t) + k

jfao

io = l + M 2 ( l + l As one can see, the frequency of the reflected wave changes due to the Doppler effect. The reflection coefficient k is negative for 7 < 5/3, i.e., a rarefaction

82

Propagation and reflection of shock waves

wave reflects as a compression wave, and vice versa. For the case 2 > 7 > 5/3, the reflection coefficient is zero at M 0 = M* = ^ ( 7 — l)/(2(2 — 7)) (no reflected wave), k > 0 at M 0 < M*, & < 0 at M 0 > M*. For 7 > 2, the reflection coefficient is always positive. Consider a three-dimensional problem of the interaction of a plane shock wave with a disturbance propagating from the side of the compressed gas. Let the shock wave propagate along the axis x3. Let the disturbance take the form ^Sp(2l)

= j dX j (Zo,X,Q)dO, 0

0

£0 = a1 sin 0 cos x + a 2 sin 0 cos \ + a3 cos 0 — G^k^t xl = e0a2,

}=e0a\

x\ = a3,

x2 = (1 — €Q) Got + e0a ,

xl^eoa1, 2k0 /? = arccos

xl = e0a2,

We assume <j) to be a known function. Then the pressure distribution in the reflected acoustic wave is 1

Jp(2)

_

4

9 0

/?

27T

X

^

/" ,

/*

(cos0-F2)(cos<9-F3)

,/A

^

f/l

0

5 = — ( a i — 2&OCOS0)

o-ia3 + cr 2 (a 1 cosx + a 2 sinx) + G o M

.

«2

The resulting disturbance is the sum Sp2 + Sp2 '. Let us consider a one-dimensional head-on interaction of a plane shock wave with an acoustic wave. The equation of the incident acoustic wave has the form Spi -
A transmitted acoustic wave arises as a result of the interaction: Sp2 = ip(a + koM0Coot),

kQ = \/q0/w0.

Poo

Substituting Eqs. (105) and (106) into Eq. (82), we get — t o - 7—TTY~ ( 2 M o + 2 ( ^ + ^ ^ o + (37 " 1)M02 - 7 + 1) Poo

( 7 + 1)J0

(106)

Interaction of a plane shock wave...

83

(M0 + l)(a + koMoCoot) M 0 (l + fc0) z 2 = e0a + (1 - €o)M0c00t.

xi = a,

In conclusion, we consider the incidence of a plane shock wave on a nonuniform region of a gas at rest. We represent the gas density in the non-uniform region as pi(x1,x2)

— />0(1 + h(a)),

a = nf'x\

h= —-, Po A reflected wave arises behind the shock front = $ (n^a1

-

h < 1,

po — const.

MClk0t)

Pi

The condition for proportionality between the arguments of h and $ implies that „?> = - ( r f ) ) ' - (a3ni1])2)

( ( n ^ ) 2 + („£>)) " '

(konP+l)

x (n^+ito/)-1, 1 /«

/ = ((r4 1} ) 2 - a 2 ( e 4 1 } ) 2 )

,

<*3 - ek0.

(107)

For 1 > n[* > fa = a 3 (l + af) , we have ny < 0; for ny = fii, we have n\ } — 0, i.e., the front of the reflected acoustic wave becomes perpendicular to the shock wave front. For fa > n\ ' > /?i = e ( a 2 / ( 1 + <*2£2)) , the quantity n\ ' is positive. For n\ < pi, the expression in the root in Eq. (107) becomes negative, and no solution exists in the linear treatment (this means that the flow in an arbitrary small region around the point of intersection between the wave fronts cannot be considered as a locally uniform for such values of n[ , see e.g. Pekurovsky et al. 1974; Dyakov 1957). In that case, the gas velocity behind the shock wave is subsonic in the coordinate system bound up with the point of wave-front intersection. The function ip is

_ 21W(Mj -1) M 2 ) + * 0 y f

(7+l)4M04F9 ^ 3

n (D

6

)

x(((7+l) M0 + ^)(n(l))2_U)3^(7?)! (2)

F9 = 2Fi(n?))2

+

2F8^-Fio, k0

Propagation

8 4 84

and reflection

of shock

waves

4 22 2 F = 3(7 3(7 + + 1)M 1)M 004 + + 2(7 2 (7+ + 3)M 3)M ++ 11, Fww10 = 1,, 2(7 3)M -77 + 00 0- 7 a _ _ a(1) -M )) (nfW ( fn ,< j V -M -Moc.kpt) (2) >> 0Q l fV 00C llk 00t) C1M) (2) 0C {2) n^;+&o nn^ 00 ni 2 ) + + k&o

l

"~

'

For For

//

(2) >> Q

''

v MO. 1l 22 \\\ !l// 2

3 3 3

n »P = ((7 + l)3M06 + u; 3J "i^

P = U + ipij + uJ /

and and and ^

/1 1 /

1

M < ((± ^ --2( (777722222 Mo M oo<< (j-(7 Y

v>2 \\\ l/» 1/2 ^ 6 7 3 1)((7 1)(7 9))) )j + 67 67 3 + (7 (7 + 1) 1) ((7 ((7 + 1)(7 + 9))) 9 ) ) ) ) ) )(7 + 9))) )J ,,,, 1/ + + 6677 -- 3 + (7 + 1) ((7 + 1)(7 + 9))) ) J 1/ 11 22 1/

Mo Mo < < (^ (± (( 7 + 67 - 3 + (7 + 1) ((7 + 1)(7 + 9))) ) J

,

there no reflected id wave. there is is n no wave. o reflected there is no reflected wave. s h o c k w a v e t htrough rough a region of non10. P r o p a g a t i o>n n o f a pplai l a n e sho* u n i f o r m d e n s i t yf (((nnnooonnnllliiinnneeeaaarr ccaassee)) gh an inhomogeneous region he motion »n of of a shock wave through inhomogeneous inhomogene We consider the Suppose pi(a) fn(a) = po = = const onst for a< and Pl , (a) = in one-dimensionali\ case. Suppose a < 0, an Supp here SPl < 0,5 0,SPl {0) = ag Po + + Pl Sfn{a) i{a) > 0, where = 0. 0. Here, a is a LLagrangian Po+S (a) for a > where Pl{0) P6pi{a) e subscript pt 1 refers to the state ini front of the wave. \ shall coordinate and the wave We thomogeneity, neity, \Spi\/ |\S/n\/p{0) f y i |PM < 1. We represent den (0)0 ) « the the density p2 assume a weak inhomogeneity, inhomogeneity We p22 behind id the M of the and the pressure p number M t h wave, as well as the the Mach number wave in the form 22 + + ..., .... + + 6S62oo(a.t) p2(a,t) (a,t) + ..., P2

= p2(a,t)=p poo(a.t) = p20+S por,n+8oo(a.ti 2(a,t) 2{a,t) 20+Sp 20+S 2(a,t) P2P(a,t)

2 pP2(aJ) (a,t)=P20+Sp (a,t)+S p2(a,t)++S62p2p2(a,t) = = P20 P202+ +Sp Sp2(a,t) 2(a,t) 2(a,t) 2(aJ)

M(a0)

2

= = M0 M0 + + SM{a SM(ao) + S M{a0) 0)

++..., ..., ..., + ...,

p2(0,0),p P20 = = p22(0,0),Mo ( 0 , 0 ) , Mo = M ( 0 ) . P20 = /> 20 2 0 = 2 (0,0),p

(108) (108

Here oordinate of i coordinate 0 is J.J.CH3 a U.Q l O the U11C ^UU1U.1±1CLLC U l the W.1C shock O l l U ^ A . 1front. 1U1IL. Using Eq. (108), we obtain ob from Eq. (1 08), we (77) nonlinear wave equation (77) Using Eq. (108), we obtain from Eq. (77) the the n nonlinear wave equation

dd222SS8p d SP2 P22

fp,(P*\ 5p2d15p2 2 2\d2 8p2_ 22fp2\d 22

2

+ ^ fdSp p 2 \2\22 _ 7 -7 + + l11 /(Q

( \d 8P2_i +l fdSP2\ 22 P2 -_ {7o)~da^~^P~7 _°C2 2 \7o)~da^~~ ~4PV\\~dT)

P2 2

~M -W-_° {7o)~da2-^7

\~dT)

WP2 _ cC (109) 4C 2 {i m da HS/ Po^~^~ \Po. L / Po~da~~da" KS/ Po^r~ir-

Here, cc2 ^locity of sound behind id the wave wav i.e in the1 region a < an 2 is the velocity T h e solution must satis st satisfy boundary conditi, The the boundary conditions on the contact •ontact surface. suri Inn addition, it is necessary Eq. (84). We make the substiessary to take into aaccount c:count c Eq. lake sub tution t * a o ution t-Kioo 0

Interaction of a plane shock wave...

85

Up to the quadratic terms, f3 = (^2(^5 «o) — p2o)/po is given by /3{a, a0) = A!h{a) + A2h2{a) + A3h{X) + A 4 /i(a)M0 + A 5 /i 2 (£) a

+A6

a

t,

I h{a)ti(T))da + A7 f h{a)ti(£)da 0

+ A8 f

0

Sp2 ao)

0

2

^

h{a)ti{()da,

= Nihix) + N2h {Q + N3 P

h(a)h'(9)da

+ N4 [ h(a)h'(C)d<*, Jo

x2(a, a0) = e0a + bia0 + 62 /

A(a)da + 63 /

/i(a)da,

X = ( + m i ( a 0 - a) A(0 + ™2 0/ ft(a)da + m 3 0/ ft(a)da, _ a + apfco _ 2a + a(k0 — 1) _ 2a + a0Ar0 - a ?7 ^ ~ 1 + &0 ' ~ FT&o" ' " 1 + *^ ' . _ 2(l + Aro)a + (a + afeo)(Ar 0 -l) (l + *o) 2

^"

f,.KT-W-ifW V wogojo JI 2(72-l)(M2-l)3

_ A2 —

2

V

:

Jo

V

V2

iVqowoJ

JO y

+ ^ T ^ ( ( 3 M 0 2 + l)w0q0 - 2M02(M02 - 1)(( 7 - l)q0 + 21Wo))) , 2wo?oJo / 2 _2(7+l)M0^(M -l) _ A3 —

,

:

A4

— 60^1^3,

qojowo A5 =

, 4 3 f l + 3M 2 f\ jo

^ 2-

Jo

2(7-l)M02(M02-l) g0Jo

qoy/mqo ) ) •

\ 2

^6 =

3 2

4(7-l)M (M2-l)3 . ,., +, ni)ni : Ai, ' qoWojo(l

£0^3 A7 = 2(1 + Ar0)'

86

Propagation and reflection of shock waves .

2(M02 - 1) Mh-1) & , . M „2 I 2 = 7—TTT77?> ( 7 +l)M 0 ' " (7+l)M 0 2 ' (7 + l ) M 2 ( M 2 - l ) ;

.

&

°l = 7 mi =

2Wo(M02 - l)fc2 (7 + l)Jo?o '

JOA/90Wo(l + fco)2

1 / " " " 2(1 + to) V

io =

3=

l + M 2 (l

+

Ma) = ^ ,

+

2(7-l)(M2-l)3\ qowjo ) '

_ jo + 1 - M 2 "" 2jo(l + to) '

A

*'(,) = £ ,

po

dr) 4

F 1 = (3-7)M 0 + 2 ( 7 - l ) M 0 2 + 7 + l , F2 - 2( 7 - 1)M04 + ( 7 2 - 2 7 + 5)M02 + 2( 7 - 1), 2(M02 - 1 ) ^o , 2(l + fc0)2^o(M2-l) ( 7 + l)M 2 U T ( 7 + 1 ) M 0 2 T (7+l)gojo a0

7h{a)da _ 2 (/- 1 )( M °-;) 3 Jh{a)da.

X

(7 + l)goJoM02

y

(110) v

7

;

If MQ ~ 1 then we have

vi (7+I)2 7(7 ~ 1) ^ = -::;+1y2(M0 -1)3, 2(7 (M, A3 S A, 2 7+ 1 2 - L947 +169 297 + ^ J7^ HF (Mo -l) 2 } 4( 7 + 1) ^ (MQ-1) f 2(2- 7), „ , 27 7 2 -507+19, a , 2\ A5 1+ (Mo 1)+ 2 (Mo 1} --%ny\ "7Tr " 4 (7+ i) ~ }> (Mo_1)3

n^wfr*-*

^--(7TF

^ MQ-1f

'

5-37,,,

..

3572-947 + 5 5 . ^

^1

Interaction of a plane shock wave...

87

2

^ - ^ ( M o - l ^ l + ^ M o - l H ^ ^ M o - l )

} ,

^2 = - ^ T T T ( M o - 1) ( l + (Mo - 1) - -(M 0 - l) 2 = 4 ( ^ ) ^ 0 - 1) { l + (Mo - 1) - i(M 0 - I) 2 } ,

^

»r ^

£o =

61

7(5-37) , , ,

n

s

! _ _ l _ ( M o _l) + _ l _ ( M o _ 1)2 _ Jl_{Mo _ 1)8>

= 7^1 I 1 - I( M ° - !) + 3 ( M ° - ^ =

4(7-1)(MQ-1)3

62

(7 + I ) 2 2 /w ,x l\ 3,,, ,. 357 - 1387 - 45, „, l X ,l 7 (Mo - I) 2 } , 63 = — ( M 0 - 1) ( l - -(M 0 - 1) + 4 ( 7 + 1]2 4

- = !<*• -') I1 - $ £ > 1

™2 =

4

f,

1,„

(M

°"1} + ^ T I ) ^ ( M o - 1 ) 2 } '

^ . 5-3

,„,

„

- ( l - 2 ( M o - l ) + 4 T ^7( M o - l ) 2

+"

37

l! 5 8 r 6 7 (M 0 -i)3},

8(7+1) 2

m3 = 7 \ 1 - v(Mo - 1) + J,1 ~ I,v(Mo - l) 2 4[ ' 4(7 +1)

As one can see, there are two waves behind the shock, one of them propagating downstream, the other being a reflected wave. The change in the Mach number is as follows M ( a )

_Mo=(M02-l)M0

M

(M02 1)MQ

l-A (1 + H

V

Fl

)+

2y/qQW0J

M

l±l

jo

Propagation and reflection of shock waves

88

(r + iMMzi) (I - Mill) _ I) *<„) k0(k0 + l)jg

J o

d£

_ 2a + a(A?o-l) l + *o *

, ^

< _

> ^

The first two terms in Eq. ( I l l ) correspond to the break-up of the wave at the weak inhomogeneity Sp(a) = const and are dependent only on the difference p(a) — po. The third term gives the dependence of the change in M on the profile of the non-uniformity. If the density change is fixed, it follows from Eq. ( I l l ) for a quadratic distribution of the density within the inhomogeneity Spi/po = k\a + /?2«2 that Mo — M(ao) has its minimum for k\ = 0 in our approximation. It follows from Eq. (110) that the ratio of the density gradients over a shock front is equal to

2

ff/lh^M^+^^W!))- }-

(112) Eq. (112) was verified experimentally (Shugaev 1983). A single diaphragm shock tube of square cross-section of 28.5 x 28.5 mm 2 area was used. Density gradient was created with the aid of a heater. Density distribution was close to a two-dimensional one. Gas density was measured with the aid of Fabry-Perot interferometer. Typical interferograms are shown in Fig. 23. The heater is on the left. The first interferogram shows the pattern of the inhomogeneity in the absence of a shock wave. The subsequent photographs show the incident shock. The interval of time between the second and third photographs is 24 ps, and between the third and the fourth 9 ps. Outside the inhomogeneous region, the shock wave is planar, the density behind it being constant. Near the heater, a strong density gradient arises behind the shock front, this gradient being several times greater than the density gradient ahead of the shock wave. The initial density gradient ahead of the shock wave was about 3 x 10~ 4 g/cm 4 . In Fig. 24, we have plotted the experimental values (dots) of the ratio a — {dp2/dn)/(dpi/dn) of the density gradients over the shock front. Each of the dots is the mean value of several experiments. Solid line is the result of calculation in accordance with Eq. (112). It can be seen that experiment and calculation agree well.

Interaction of a plane shock wave...

Fig. 23. Interferograms of the shock wave propagating through a non-uniform region.

89

Propagation and reflection of shock waves

90

Fig. 24. Ratio a = (d p2 / dn) / {d p\ / dn) against Mach number.

11. Nonlinear one-dimensional interaction of a weak disturbance with a shock wave We assume that small disturbances are imposed on the uniform flow. We start from Eq. (109). We make the substitution t -> ao, ao being the coordinate of the shock front. Let us introduce the characteristic variables £, rj a+ = /

j/(a+(a,f),a)da,

y{a ao)

C2(a,a 0 )p2(a,a 0 ) c1(a0)p0(a)M(aoy

' -

/•a 0

a" = /

y(a~(a,£),a)da,

Eq. (109) may be written in the form 1 d2P2 Pi d£dri K

*'V)

*(t,v), 2\da0da0 1 dM pi \M dao

da daj 1 dci c\ dao

{

p\ d£

®El®L+ d£ da0

p 2 dn dp2 drj

drj dao,

Interaction of a plane shock wave...

91

y2 dp0 fdp2 d£ pipo da \ d£ da

dp2 dr?" drj da

7 + 1 fdp2_d£_ \v\Pi V d£ 9a0

dp^drj^ 2 drj da0,

c

d2 i oaQ

2 ^ oaz

2

2d

AT

r) d2r] da1 oa^

Let us consider the reflection of a weak disturbance from a shock front. We take the solution of the linear problem as the first approximation. Then the second approximation may be written as

= {£) +Hn) + I ^ I *(«,/?) da.

Pi

(113)

Jo Jo The function $ on the right-hand side of Eq. (113) is

*(£, n) = cW(£(0V(Ci) + Qit'iCiWfa) +
(y-l){T+l)*{M§-l)h 8 7 M 0 2 ( g o ^o) 3 / 2 '

V 2

~

(7+l)(M02-l)fc w0k\

27^oA?2VW^o TJ(0) = (a + k0aQ)/k2,

£(°) = a 0 _ a /& 0 )

2

* 3 = 1 - l/* 0>


g0 = ?(M 0 ), 2

w0 = w(Mo), 2

™ = (7 - 1)M 2 + 2,

&0 = v W ™ o ,

2

X: = (M - 1) {2(2 - 7 ) M -

h = 1 +fc,fc2= l + &o,

7

+ l}/(g 0 j 0 2 ),

io = l + M02(l + 2/* 0 ).

(114)

Substituting Eq. (114) into Eq. (113) and taking into account Eq. (82), we have

-Sp2(a, a0) = HO + W ) + A1(f>2(C2) + A2^(Ci)^(C2) Pi

+ A 3 f\(a)^(Cs)da Jo

+ A4 f %(<*)<£%) da, Jo C (o)

z 2 (a, ao) = <*£o + «o(l - £o) + 6 1 / K2

+ b2

(a) da

rCi

<j)(a)da + b3 /


92

Propagation and reflection of shock waves SM(a0) = £ ^ M & ) + W(A.) + {At + A2- | ± i ( l + k)2)<j,2(k3a0) (a)(A3f((3) +

+ / Jo

JM'(C4))

da},

(°)) + m 2 / 2 0(a) da JCe yd Ks + m 3 / 0(a) d a + 7724 / 0(a) da, JO JCr

£ = ^(o) +

mia^(f

A = M + W 0 ) <£(C 2 ) + L2 (* 0(a) da Jo + L3 / Jo

<^(a)c?a + L 4 /

0(a) da,

JO

?7 = ?7(0) + cn(a 5 - (*)(&) + ^2 /

0(a) da

JC2

/•Ci

/*Cs

+ <x3 / 0(a) da + a o ) |a=a 0 5

£s = f (a 5 Oo) |a=a 0 5

Ci = * 3 a,

Cs = ^(* 2 a - *o*!i/ 0 ) ),

C2 = W ° \

C4 = 5 { * 2 a + ( l - * o K ( 0 ) L Cs = * 3 ao,

Ce = k0k3^/k2, (0)

C8 = *o*|»? /*2, , , ,,„. x ^ - ^3/(2*0), A4 = -A3/k0,

(7 = * 3 £(°), 0)

C9 = *l»/ , ^ = Ar4/(27iogo), . _ (72-l)(M02-l)2P^2 4, S^M!^ ' k4 = M02k2h/k0

- k 6 - k7/(M^ - 1),

f2 2 !x *1*2

* 5 = (7-l)(M0 -l) M^k 0 - ( 7 + l ) { ( 7 + l)(l

+

* / 2 ^

2 k7 = *i{(7 +

2k%q0y/q0wo 1)(2M02

- l)V«5u^(l - * + *i/*o) + i 2 w 0 /(2M 0 2 )}

- ( 7 + l)M 2 u, 2 fc 2 (l + kk2/2)/q0,

mi = ( ^ t ^ ,

Interaction of a plane shock wave...

m2

m

_ (j+l)kk22w0 -4iqo(Mi-iy

m z

93 (7+l)(M02-l)M2 2-fMZqo

-

kik2^/qoWo (7+I)2 LT 4 = -T-7757T75 l ~ 4 7 M 0 2 ( M 0 2 - 177' )' 27Mo ' _ k2 ((j+l)kk2w0 ( 7 - 1)(M 0 2 - l)A?i fciy/gowp 2 2 + 2 + " 2 7 I 2g 0 (M - 1) M g0 2 M 2 ( M 2 - 1) _

(7 + l)fefc|w 0

_

fclfoygOWp

3 _

""2

4 47<7o(M 0 2 -l)' -47M2(M2-l)' *o!i (7-l)(M02-l)fc1 — ^1
_

1

,

—

}■

(7 + l) 2 fc

^ ~ 2 7 fc 2 V?oW hqo —" 47M2(M2-1)'

AJW0&2

ft

l =

, „ ■2 ) , 02 = — 7(7 + 1)M 2 (M 2 - 1)*„' 7 feoM ' _ (7-l)(M2-l)fc1fe2 6 3 ( U 5 ) " 7(7 + l ) M 0 % • In this case, the shock Mach number depends on the prehistory of the process. At Mo ~ 1 we have

2

^

^(7-l)(5-37)r 27(7+1)2

4 {Mo 1)

~ '

~ (7~l)(5-37) UA M - J ( l)2 ( ° ) ' 7 7 + 7+1 k3^M01, mi £ i f - , 27 ~ 2(7-1), . m

A * A ^ 4 - ~A3' (5-37)(M0-l) m2 £ - i ^ ^ i, 87 7+ 1 , ^ 7 + 1

A

3 _

327(7+l)v

47(M0-1)'

2

""1 = —

^(M 0 -l) ,

-^-STT^I)' 1

64 £

4

47(M0-1)'

.3.^(Mo-l),

^4T<*>-I>.

~ 5 " 3 7 -(Mo - 1), "5-27(7+l)V "

C2 S i/(°)(M0 - 1),

**£,

"

S 7 ~ ^ ( M o - 1), 7(7 + 1)

>24 Ci = «(M 0 - 1),

Cs = {1 + i ( M 0 - l)}a - i { l + 3(M 0 - 1 ) } ^ ° \

94

Propagation and reflection of shock waves U = {1 + | ( M 0 - l)}a - \{M0 - 1)^°),

Cs S a0(M0 - 1), Cs S ^(Mo - 1 ) V 0 ) ,

C6 2 | ( M 0 - 1K<°\


C9-(M0-1)V0),

*<°> = «o - «/Mo,

,(0

f +^i.

=

( n 6 )

Mo -r 1

One can see from Eqs. (115), (116) that the reflection coefficient is propor­ tional to (Mo — l ) 2 in linear case and to (Mo — 1) in nonlinear case. Let us consider the nonlinear interaction of a shock wave and an acous­ tic wave in the case of head-on collision of waves that propagate in opposite directions. The pressure distribution behind the wave front which propagates toward the shock is as follows 2y

£L = l + if,\a + tCoo ( M

I

0

< L

Here subscript oo refers to the undisturbed flow. The solution of Eq. (109) up to the second order is given by

^ = V>fo) + p dp j *(a,/3)da. Jo

Poo

Jo

Using the solution of the linear problem, we have

POO

Jo i+m-gfcwm + A3<j>2{el)+A4(0i){e2)+As + Ae f1

[ ' {a)'{0A)da

4>'(a){65) da + A7 f * {a)<j>'(07) da,

JO

J»6

( = v + r,(l + (1+l)(r])/(2y))/M0

~ 2 7 (A^+1) {7 + l ~ (Mo " ^Fl^ J ' *(<*) <*"> e, = mVW,

92 = ^m{,( i + l/fc0),(«») - £(°)/* 0 },

63 = rn(l - *o)i7 (0) /(l + *o), (?5 = | ( 1 + l/*o)a,

£ (0) = «o -

#4 = | { ( 1 + l/*o)a - m(l/k0

06=mtW/(l

+ k0),

a/k0,

- 1)*7(0)},

Interaction of a plane shock wave...

95

07 = | { ( 1 + l/ko)a - m^/k0},

Ax = F0/(j0(l

+ 7)),

A2 = -2M 0 (M 0 2 - l)/(io(l + 7)), F0Fs F0 (7-1)(MQ + 1)2(MQ-1)3 = 2 (7+l)?oio (7+l)io 2 7 (7 + l) g o 2 J ,> 0 (7+l)F4 f F32 / 2g0

f n

ilJho l(7+l) Jo 3 4 0 - l) + ( 7 + l8)7JMo 02 V(M ^(l + lAo)3 + A

=

Afi =

(

r

2

1 3

2

V ( 7 + 1)5

ui%q%{q*F2m2°

~1)+2Mo(M°2"1)F5h

(7-l)(M0 + l)2FiF3(Mo-l)3 27(7 + l)g0MWo

Af02(l + l / * o ) { (Mo + l ) 2 2 22 2 F 1 F3 (M (7 - l)(Mp + l))2 Fif 3 (M 0 - 1)F!F 3 f F 3 ( M - l )

■fqowojo 2

2 M„ M 0

I (7 + i)io?o

M„0 22 - 1n M

+•jott>(l + l/*o) A

7+1 (7 - l)(Mp + 1) 2 (1 + l/feo)FiF 3 (M 0 - l ) 3 47(7 + l)tfoWoj"o '

m =

1 + M0 M o

F 0 = 2(2 7 - 1)M04 + (7 + 5)M02 - 7 + 1 , Pi = 2(7 - 1)M0 + (7 + 1 ) ^ ° ^ + 7 - 3 , JF2

= 2(7-l)Mo + 7 - 3 ,

F3 = 2M04 + 2(7 + 1)M03 + ( 3 7 - 1)M02 - 7 + 1 , F 4 = 4 ( 2 7 - l ) M 0 2 + 7 + 5, F5 = 67(7 - 1)M02 - 2 7 (3 - 7 )Mo - (7 - I) 2 At M 0 ~ 1 we have 5-37,M ^ (9 +l)(7-7) A 2 S — - f (Mo - n1), AA 3 S - 7 7+1 ~32(7-l)a(„ <„ (7-I)2 A 1)3 AA A ~ ^ +1 ) 2 ^ ° - ^ ' - 47(7 + 1 ) 2 '

AiSl,

(H7)

Propagation and reflection of shock waves

98

„2(-,-l)2

wiofwuMfcMMwiinnnm

lIlliiniiL..-—JmMB£B8L^t*MiSI!S^

% - l )

2 3

The expression for the pressure in Eq. (117) can be separated into two parts. . One of them refers to the break-up of an arbitrary discontinuity, the other depending on the pressure distribution behind the front of the acoustic wave. 12. Instability of shock waves Experiments fulfilled in shock tubes in the seventies showed interesting features of shock waves at high Mach numbers. A distortion of the shock front was observed. It was found that density distribution as well as electron number density became non-monotonous within the relaxation zone (Griffiths et al. 1978; Glass et al. 1978; Ryazin 1980). Those phenomena reveal a threshold character, the threshold depending on the shock velocity and the initial pressure. Small impurities of hydrogen influence the flow in a stabilizing manner. Fig. 25 shows interferograms of unstable shock waves (Ryazin 1980). Thus the problem of shock stability arises.

Fig. 25. Interferometric streak-camera records of unstable shock waves in xenon, a - po = 1.3kPa, M — 15.8; b - po = 1.3kPa, M = 1 7 . 8 (Ryazin 1980). Theoretically the stability of a plane shock wave was studied in the fifties (Dyakov 1954; Kontorovich 1959; Yordansky 1960). The aforementioned au­ thors investigated the interaction of a plane shock wave with small perturbations

Interaction

of a plane shock wave... wave.

97

inn the flow behind the shock Lock front. is as ffollows. There The problem under consideration consi re is an initially plan plane and down shock hock wave. The flow upstream upstre downstream is uniform. Let the small niform. ipstream sma shnrV WRVP b p disturbances isturbances behind the shock shod wave be P2 = = P20 P20 + Sp2,

P20 = const,

Sp2 = Sp> p(jfer-arf). 2020exp{kr-u;t).

(118)

If the amplitude of the disturbances increases es as the ttme [me grows then an instability occurs. If the amplitude diminishes as the time>grows the grows then then the the shock shock wave the amplitude does not grow and ai does not ave is stable. In the case when amplitude ude ot hen diminish nl C U L l d l OtClUllltJ'. stability. iminish we have IdVC the t l l C lneutral It was found two phenomena (1) corrugat instabilke place: .) corrugation riomena m a y take instabilm d that fr p ity of shock waves; ofsound by by the shock ffront. In the Leous emission of sound aves; (2) sspontaneous so latter coefficient wave sr case, the reflec ient of an acoustic aco oustic wave avei from >m the shock wave ficient le reflection acoust takes an infinite large value but the perturbations do not grow in 1 time. Those s •tions »t ow perturbat TThose ut i t e larg« perturbation authors lors stated1 the co ty as well spontanell as ffor spoi he conditions con iinst :or corrugation spontanthe conditions for for corrugation instability sou ilue of the le reflection is eouss emissioni of>f sound. An infinite large value reflecti ction ccoefficient coeffick soun value finite vala value if we use aa defect of linear theory. The reflection coefficient has a finite defect 01 linear theory. The renection coemcient has a fniie value 11 we use nonlinear nonlinear equations. equations. Let us us consider consider the the reflection reflection of of small small disturbances disturbances from from an a initially Let an initially plane plane shock front in a gas with arbitrary equation of state. We r make use of the shock front in a gas with arbitrary equation of state. We make use of the f n l l n w i n c r P m n ttion inn following equation l E

..IdG , _ , 1 dp .„ I +^ 2 *(. i( 1-., , i~^f, ( 1 + 2n*,r2 Mi-a¥=-(i-Mj)-^% M

—m,

2

2

0 rr

+

M

G

(119)

>ng the Hugoniot Here: subscript H denotes that the derivative is taken along Hugoniot adiato the gas downst: :, M2 is the Mach number of the shock wave referred to downstream. batic, Eq. (119) is similar to Eq. (82). Differentiating Eq. (119)I) along the ray, ttaking into account Eq. (118), and retaining the members onlyf of of the the first first order, orde we first order, we have 1 _^

1 A^Xn

1

A / £>An„\

AAtf

(120) From the Hugoniot adiabatic we obtain Sp2 == dp? = F

F

—

2Pl G(l-g) rfG 2piG(l~g) :

1+j

1+j

-0U.

(121)

Propagation and reflection of shock waves

98

The variation of the mean curvature of the shock front is equal to 2

dSH _ 1+j d2Sp ~dT2^(1-e)e*da*da*'

2 a

_ ~Xv

2 [

, '

The acoustic disturbance is taken in the form Sp2

4 („«< _ ^ ) + ^ ( n j V -

^ )

Pi

The flow is steady in the reference frame which is fixed at the point of inter­ section of the wave fronts. Consequently, we have r4 2) = {(1 + Af|) cos 0 - 2M 2 } /{Ml - 2M 2 cos $ + 1), 4 2 ) = (1 - M | ) sin0/(M 2 2 - 2M 2 cos 0 + 1). Taking into account Eqs. (120), (121), (122), we define the function V> V> ( n p V -

C

j-t) = Wt),

k =

Ni/N2,

£ = (M22 - 2M 2 cos 9 + 1) (nj2)a*" - —<) / ( l - JW|), JVi = e(l - M 2 cos 0)(1 + 2 M | - 3M 2 cos 0 + j ( l - M 2 cos 0)) -M2(l+j)sin20, N2 = e(l - M 2 cos 0)(j(l - M 2 cos 0) + 2 M | - 1 - M 2 cos 0) + M2(l+j)sin20,

j =

^{^A\.

The reflection coefficient becomes infinite for j in the range e(l - Ml) - Ml < j < l + 2M 2 . e(l - Ml) + Ml

(123)

Eq. (123) is also the condition for neutral stability of the shock wave. The instability occurs if i < - l o r j > l + 2M 2 . (124) The instability cannot arise in an ideal gas with a constant adiabatic index 7. According to Dyakov-Kontorovich-Yordansky criteria, the instability can arise only due to non-monotonous form of shock adiabatic. It is interesting to note that Anile and Russo (Anile et al. 1986 a) obtained just the same criteria in nonlinear case, using the ray method.

Interaction of a plane shock wave...

99

In fact, on assumption the gas ahead of the shock wave is uniform and at rest, Eq. (82) may be written as follows

+ 2Ge(l +

j)(l-e)H.

At the initial instant the shock propagation velocity is assumed to be constant and the first member on the right-hand side to be zero. The last assumption means that the perturbations propagate through the shocked gas from the shock front and not towards it initially. Consequently, the corrugation instability takes place if j < - 1 or j > 1 + 2M 2 . In those cases ^2 > o at H < 0 (convex shock front), ^ < 0 at H > 0 (concave shock front). The inequality j > 1 + 2M2 is a condition for instability against break­ up (Gardner 1963). Namely, if this condition is satisfied for a state 2 behind a shock then the states 1 and 2 can be connected not only by the shock but also by a weaker shock followed by a rarefaction wave which propagates in the direction opposite to the direction of motion of the shock wave. Let the shock wave propagate from left to right. We have: vs — v\ = {(p — P\){ri — r ) } 1 / 2 , r = 1/p. Here v$ is the velocity of gas behind the shock wave. If vi = 0 then vs = {(p-pi)(n-r)}^.

(125)

We have behind a rarefaction wave vr = v2+ I' C(T,S2)

— ,

*2 = { ( P 2 - P i ) ( r i - 7 * ) } 1 / 2 .

(126)

Let us consider the curves vs(p), vr(p) in the plane p, v. The curve vs(p) is obtained by eliminating r from Eq. (125) and the Hugoniot adiabatic A(r,p)-A(Ti,pi) - -^(p-p^ir

+ n) = 0,

where h is enthalpy. The curve vr(p) is obtained by eliminating r from (126) and the equation of state p = p(r, S). It follows from Eqs. (125), (126) dvs _ J_( _ -x dp ~~ 2m

_J2 c2'

d^r _ _J2_ dp ~ c2'

_ n _ V2_ n r2'

100

Propagation and reflection of shock waves

The = another 2-. There The curves curves vvss{p) {p) and and vvrr(p) {p) intersect intersect at at vv = — vvrr,, pp = — pP2±nere ii ii snother snotner ppinn ppinn of intersection. If p decreases, v finally decreases and becomes s of intersection. If p decreases, vs finally decreases and becomes equal equal to to zero zero the contrary, contrary, {p) increases as pp decreases decreases because at . On at pp = = Plpi. pi. On the the contrary, vvvrr(p) (p) increases as as decreases because because r

d2vr/dp2

= (r5/2c5)(52p/5r2)s

> >0

uwing et al. (Houwing et al. 1983) pointed out that the for normal gas. Houwing condition of spontaneous emission of sound can be identified as the condition where a normal shock can split into a stronger oblique shock and an oblique rarefaction wave. The region j < - 1 was identified by Fowles (Fowles 1981) as that one where a shock wave splits into two shocks propagating in the same direction. For shocks in carbon dioxide, it was found that there is a reasonable agreement nent between the observed instability and the lower boundary of the region of spontaneous pontaneous emission of sound (Houwing et al. 1983). However, the instability region egion is predicted to occur at somewhat higher velocities. The instability of a shock wave in monatomic gases was the object of investigations vestigations by many authors. Baryshnikov and Skvortsov (Baryshnikov et al. 980) proposed that intermediate stage of ionization is exothermic and is able 1980) too destabilize the flow. Mishin et al. (Mishin et al. 1981) assumed that the exothermic stage occurs as follows. First of all atom-atom collisions lead to the population of metastable states at excitation temperatures which are higher than the translational temperature of the atoms. Thus the excess of energy is stored stored and and later later on on released released through through superelastic superelastic collisions. Yushchenkova Yushchenkova (Yushchenkova 1980) predicted are achieved achieved behind (Yushchenkova 1980) predicted that that maximum maximum populations populations are behind the shock shock front. front. However, experiments fulfilled by Houwing the However, experiments fulfilled by Houwing et et al. al. (Houwing (Houwing et al. al. 1986) 1986) did confirm this this hypothesis. et did not not confirm hypothesis. It was suggested that the instability appears as a result of precursor radiation from the gas behind the shock wave (Egorushkin et al. 1990). There a r A two t-air, stages c t a f f e c of n f A c process. n m ^ F i r c t r,f all a a neutral nci.fral o i a k i l H , , of ^ f the t k shock o V i ^ U wave ™,a„ are the First of all, stability appears 1992) and and then appears (Egorushkin (Egorushkin and and Uspensky Uspensky 1W2) then the the shock shock wave wave becomes becomes unstable due to precursor radiation. Shock waves in argon were unstable due to precursor radiation. Shock waves in argon were considered. considered. The conditions conditions for for neutral neutral stability stability of of aa plane plane shock shock wave may may be be written written Lock wave The as follows (Teshukov kov 1986) C O

0

0

(-(£)J(I)—((!)-)• dp\

Ws

=

c2

-^

^ ^

(127)

Interaction of a plane shock wave...

101

We can write Eq. (127) in the form

£*('(*),-')/('♦-&, We use the Saha equation a2 _ 2Dj l - a ~ {ni+na)Da

fmekT\3/2 J T e- /* , \ lira2 )

Ni D

J = I > j exp(-4/ArT),

j = a; i,

f

where a is the degree of ionization,
dAi

/

rrida\

+ (! + <*) ( A

; = ^:Eci^4exp(-4/T)'

/3

1

f

i = a ; *>

(128)

where Cj are weight coefficients. If 7 = 5/3 then Eq. (128) transforms into

A = (l-a)Aa(T)+aA,-(T). Fig. 26 shows the function F = T ( | ^ ) a - (/ + AA)(1 + a) against Te (p = 0.1 kg/m 3 ). The neutral stability appears at 1.01 eV< Te < 1.25 eV. Let us find the values of the Mach numbers that correspond to the neutral stability of shock waves. The Hugoniot adiabatic is as follows e(T, r) - ei + \{r - n)(p + Pl) + Q(T) = 0,

e(T,r) = - ^ - + a / + C i A ( T ) . 7-1

p = PRT(1 + a),

102

Propagation

and reflection

of shock

waves

Fig. 26. Function F against T e . The region F > 0 corresponds to the instability of a shock wave (Egorushkin et al. 1992).

Here q accounts for absorption of radiation. The value q = 0 corresponds to the full absorption of radiation. It was shown that the range of neutral stability m a y become a region of instability due to the effect of precursor radiation (Egorushkin et al. 1990). The flow is assumed to be one-dimensional. The mechanism of instability is as follows. A r a n d o m temperature disturbance behind the shock wave causes a local enlargement (or decrease) of the flux of precursor radiation. This fact leads to thermal inhomogeneity in the flow ahead of the shock wave and involves the increase (decrease) of the velocity of an appropriate section of the shock front. Neutral stability occurs in the range of Mach numbers 17 < M < 21 (first ionization) and 30 < M < 35 (second ionization). Initial pressure is equal to 1-5 Torr. Mond et al. (Mond et al. 1994) pointed out the role of the kinetics of elec­ trons in the occurrence of spontaneous emission of sound. In the opinion of the authors, the existence of perturbations that do not decay in time shows that the flow behind the shock wave is unstable. A frequency range exists for which the perturbations in electron temperature differ from that of the heavy particles while ionization equilibrium is maintained. The range of thermal

Interaction of a plane shock wave...

103

non-equilibrium between the electrons and heavy particles is defined by the inequality i/£ < w < 2^i on , where wlon is the characteristic frequency of ionization by electron impact and ve is the inverse of the relaxation time of the temperature of the heavy particles due to their collisions with the electrons. This results in reformulating the stability conditions. The experimental data agree reasonably with the theoretical calculations.

C H A P T E R 4. REFLECTION OF A SHOCK WAVE FROM A CONVEX BODY

13. Reflection of a plane wave from a b o d y of arbitrary shape When a plane shock wave meets a rigid wall, a reflected wave appears. The reflection m a y be regular or irregular (Mach reflection). The irregular reflection of shock waves was discovered by E. Mach who used two simultaneous sparks. A wave pattern which appears when a plane shock meets a rigid wall, depends on the shock strength and the angle of incidence. At small incidence, we have a pair of shocks, an incident wave and a reflected one (regular reflection). At high incidences, a three-shock system arises. It consists of the incid­ ent shock, reflected shock and the Mach stem (irregular, or Mach reflection). Different cases of shock reflection are shown in Fig. 27. At high incident shock strength, additional patterns m a y arise, namely complex Mach reflection and double Mach reflection. These patterns arise only in unsteady flows (Bazhenova et al. 1977). One can imagine the appearance of the Mach configuration in the following way (Bazhenova et al. 1977; see also Korobeinikov 1989). Let the plane shock front be normal to the flat wall having a small projection on its surface. The flow is assumed to be subsonic behind the shock wave relative to the wall. A disturbance arises after the shock wave passes by the projection. The spherical front of the disturbance intersects the shock front. T h a t phenomenon is the limiting case of Mach reflection. The Mach reflection has no analogies in acoustics or optics. A theoretical analysis of shock reflection has been performed by von Neu­ m a n n (Neumann 1963). He used the following assumptions: 1) shocks are discontinuities with finite curvature; 2) two-dimensional steady-flow theory is applicable in the neighborhood of the point of the shock intersection; 3) the flow deflection in regular reflection is zero; 4) for three-shock reflection, downstream pressures and flow angles behind the incident-reflected shock pair and the Mach stem are equal. This implies the occurrence of a contact discontinuity which 105

106

Propagation and reflection of shock waves

Fig. 27. Reflection of a plane shock wave from a rigid wall, a regular reflection; b - Mach reflection; i - incident shock wave; r - reflected shock wave; m - Mach stem; s - slipstream; T - triple point; a - angle of incidence: 3 - angle of reflection. separates flows of different velocities and densities. The two-shock theory yields two solutions for the reflected wave, and the three-shock theory gives multiple solutions for strong incident shocks. Exper­ iment shows that the weaker of the two possible reflected shocks in regular reflection occurs. We choose the solution that gives the weaker shock solving the problem of irregular reflection. When a plane shock wave reflects from a rigid wall a regular reflection takes place at small angles of incidence. The angle of reflection f) is not equal to the angle of incidence a. The simplest case is the reflection of a plane shock wave from a flat wall at zero incidence. In that case, the gas is at rest behind the reflected wave. The quantity v 2n - Vi„ does not depend on the choice of reference frame, n being the external unit normal. On the other hand, the velocity of gas behind the shock wave in the reference frame which is at rest relative to the undisturbed

Reflection from smooth body...

107

gas is

«2 =

J-Ci V 7

^(7 + 1 ) ^ + 7 - 1

Here c is the velocity of sound. We can write V2 __ C2

zi =

z2 7

/ ,

Z\ 1 2

. ,

N /

(l + ^+ ^1 2 ) 1 / 2

(£LzM,

Z2 =

Pi

7[(l + ^ i ) ( l + ^ ^ i ) ] 1 / 2 '

(£3^1.

(129) V

P2

;

Subscript 3 refers to the reflected wave. It follows from Eq. (129) 4(1 + *i)(l + ^ s i ) = *?(1 + ^ * 2 ) . It is a quadratic equation, its roots are 1 + "Vf *1

1 + *1

The physical meaning has only the first root. Thus the pressure behind the reflected shock wave is (37-1)^-(7-1)

(7-l)£+7+l The Mach number M of the reflected shock is given by M =

/27M^-7+l ( 7 - l ) M £ + 2'

MQO being the Mach number of the incident shock. The value of the angle of reflection can be determined from the boundary condition for inviscid flow. The pressure behind the reflected wave decreases as the angle of incidence grows, reaches its minimum and then begins to increase, exceeding its value at normal incidence. The angle of reflection of a shock wave is as follows (Courant and Friedrichs 1948). 2 cos aA\ P =

^ ^ A2 + (A1- 4(7 + 1 ) ^ ) ^ '

A1 = (7 + 1)A3 tan 2 a - (4A23 - (7 + l) 2 ) tan a - (7 + 1)(3 2

A2 = cos a ( 4 ^ 3 ( ( 7 + 1) tan a - A3) + (7 + 1) )> (7+l)(M£-l)tana Az = ( T + l)M 2 ) + ( 2 + ( 7 - l ) M 2 D ) t a n 2 a -

f)A3,

Propagation

108

and reflection

of shock

waves

Here a is the angle of incidence. The Mach number of the reflected shock wave is M = B + (1 + 5 2 ) 1 / 2 ,

B = i(7 + 4

l)t,!-^_. ci cos p

Here the subscript 1 refers to the state behind the incident shock wave. At high angles of incidence, the Mach, or irregular, reflection arises. Besides the reflected wave, the Mach wave and slipstream appear. At a given strength of the shock wave, there is a range of angles of incidence within which both regular and Mach reflection are possible. As a plane shock wave hits a smooth convex body, the angle of incidence increases continuously. In other words, one can observe a transition from regular to Mach reflection. We place the coordinate origin at the point where the incident wave is tangent to the body surface. The axis x3 is directed in the opposite side as compared to the flow behind the incident wave. The equation of the body surface is given by X

0

=

~Y\X0i

X

0J'

We introduce curvilinear coordinates on the reflected shock in the following way. Let a ray intersect the reflected shock and the body surface at points TV' and TV, respectively. We take the coordinates a?J, XQ of the point TV as curvilinear coordinates of the point TV7. The time is measured from the instant when the incident shock wave reaches TV. The coordinates of the reflected shock wave are a?*(a?o,a?o,t) = x{0 + v\At 1 v\=v\

= 0,

v3

A

+ ciMn t -At l 2dMn

1

A

sd

= -Goo(l-£0O) = -V)

2

Mnf A* = ( < - V / G o o ) . ( 1 3 0 )

Here Goo is the velocity of the incident shock wave, G is the propagation velocity of the reflected shock, M 0 is the Mach number of the reflected shock wave at the point N. T h e boundary condition at the body surface is as follows x3 = -iP(x\x2),

a j ^ i ' o + A**',

Ax* = x[(t) + [xi]At + \[x\tW>

x[(t) = -vAt,

[x\] =

[dx'~\

[df\

+ l[4tt]At3

+ ^[x'tm]At4

+ ...

Reflection from smooth body... \d2xi W»] = dt2

109 (131)

At = t - rP/G0

The quantities [dV/d**] are defined by Eqs. (64), (99), (100). In our case these expressions must be simplified because the gas behind the incident shock wave is assumed to be uniform. The expressions for the derivatives [dkx%/dtk] may be written in the form [xit\ =

X^ni,

1 m

Pi

dut dun'

dx{ Out 3W>

indyi V 2 / 39

ey/y

£n 9

Pl

£ dp2 (v9y_dx^_,iVy 2ply/y dl * dut dui +

8 fdp2\ dx{ Out \dl ) dui

Pl

%

[x\ttt] = X < 4 V +

2X^9^

e(l-e)dP2

p\

9

X^

=

v(4)_

P l

9P2 { d s h

V&i,

V(3)

7 +

,

G^

_ ,y/ydp\ 6 ) V ^ ^

to

7 ^

V

Pi os

d

P2 9xl

Zr, o1 A 7

7f9

a-yi

de dxi dvP duv

{ ]_dl>_2 „£„ d fdP2 dx du£ V ds dvV Pi 9s

62

dp2 dp2 duG du-y

\E71

X(2)

{l-e)G,

x(3)_Vli X

- e

.0/1

2 ( 1

3y_d^_

£r?

de dx% dut du*i

r-ru

de dx dut dvP

dxi dui'

en d (dpi » dui \ ds

d

P2

^,p 2 — —

d x i

e

y

£r,dh2

dx

*

1 dp2 Pi ds '

6(1 - e) ^ dp2 dy 2y 9 du^ du^9 V dp2 D(3) ep\ ds

tnBW dP* \ B * du*)

^y dp2 dB{n] ep\ ds dl

VS1, e ^

j/ dp2 2epi ds

(2)

110

Propagation and reflection of shock waves

spi ds dl2 e epi os 2ydP2 f (2)de (l-£)(3-2£) ePlds9 « du"+ 2Pl +

9

ou^ ovP dp^d^ 9 ( °duiduv

|_(l_e)^+_ft7|^____fc^,flU_

+ (l-g)af„^i Pi dut 1 - g /rfp 2 2ypi \dl

dy _ 2y c, (2) SAi. du^ p i y € 5W 1 dp2\ iv dy dy 2e ds J u dut dvfl

■ y a£vB(3)9p2

y

£r,dBW

dP22y(2-e)

f

g5

( 2 ) ^

_ i . - f t f l W ^ , iz£acv^2 ay 1

2p/ « aw _ dp2 dp2 ~ dl ds' dP dl\dsj d ( i\ a /1 cW \ P 2 / as \p2

hA=

dvtdw>'

ds2

dp V^J ~ 2S7 val w J + a^ ^

d / 1 ap 2 \ «5 = *— W \/>2 as / h =*L 6

2pi *

a / I ap2 ds \p2 ds

(L^i\ _ 2 - (- (L^i\\ . i l fl 5 ^

d/2 V/>2 ds )

dl \ds \P2 ds ))

+

ds2 \P2 ds

Let the points TV and N' coincide. Using Eq. ( I l l ) , we obtain the following expressions at that point na = Vv sin(a + /?)/V^, dxi

# -L. f

yi

G

n 3 = cos(a + /?)

A /

Reflection from smooth body... body...

I111 ll

^ = =^ *l + VVI) ii)) (((2 ^■OO

«.

\

- 2 - ^ - s i n ( a + /?)VY>, Goo a

Va = 0d^/dx V W0, ,

VV = yjtl

vcos(a + (3) /?) , 66 = - = Goo ^

-

+ tl,

I1 dG0 (f da 8a ,

da da ,, \

+ + + ^ G ^ fe^ G o o ^ G ^ fe^ 5«H &(*') G , Gt_

_G_G , ,Gt_ Gt_ Q ai Vai + g,2G^°^Var-V G^

(132) 132) (132)

Here subscript 0 refers to the value at the point N. It follows from Eq. (132) that at the point N the mean curvature of the reflected shock is rr

cos2 a ffsm(a

H = -——

I

.

+ f3)\

I cos a (—(V11 (—(Vn ++ i ^22) M ++ 6666))

VV sina J y d cos 3 adM 0 C3+ d t a n 2 „ ^ \ _2 _2 ^ G" s^ h" si T ^ "n "Vddaa""<^33++GG 2i : lt a n * ~5T) a~dT)'' 2 ^ 2 v w—G —^—, 6 = 1-2— + —^——p5—,

G^"sTnV^ Gi:tan a^ri' ^00

^00

- 1^^^11111 + 222^^^111^^^1221^ 21- 2- -^^2222 , 6 =" 6 = ^1^11 2 ^ 2 ^ 1 2 + ^^^2222 , 2^1^2^12 ^ 1 1 + 2^1^^12 • 2ay 2 G^J, i2v\ > \ L - l + sm sin a^/ V - < ^^% ,. ) G^ ^J, Q

,, _d± _d± V'-dxV V'-dxVV^Q

Q

d2j> OH

((133;

^ OXQOXQ *ij ~ ~ dxidxV dx* 0dxi'

From Eq. sui ace we From from t q . (133) {i.66) and ana the tne boundary Dounaary condition condition on on the me body Doay surface sun we get get wojMJ-1) 1 dM ==_ WQ(MJ-I) o s 222 aaa // /f( ,,a m0(M02 - 1) //(c cos WQ{M$-1) / l/_J>_\ / _ vv \\ C 0 008 0O8i H 1a dc^~dT~ o cooJ// ° "^ ci dt V U )1 )V \V CI (ft IL V °°VV \V \U Goo) G 7 L v(7(7(77+++1+l i) v \V Goo/ x~dT~ ( , ( v (v2 xxx (^f --it(yi^n+i i+f+efe) f)e+)++, 626(^(1l--- ,22— ^g --+ + ^ G Cp^ +— —

g) PJ

l J) x ( _ W l l + f e ) + 6 ( l _ 2 i + «^p))

o ci /'rfMo^ d M00\\0 \ ddci /fdM fdM *VV"~da "^^^ "T" /JJ;; \~da~) V " G ^ \~da~J 222 "-~ GGoo ^ GZ ~ 22G~Z

22 5 3 cos^o. ((77 + cofa (7 + 11) cos cccos oo ss3a33^ tt tt pp tt \^ ^ _ (T ll)l)))22^rrrr»»iar^i //ifff2^2222cccccSooooosssssS555QQQa^::: + l _ _ L M 3 1<3 ^llsin V) ^l1& ^^l*^^~^^0^~L- M° c° 0o°C083^ s3/? ^^a~^ ~ "~"~2w cos ^a~ ^€3V )J _"~ °cos3/? iir^^^na""~ V cos3/? ^ V >/

M C ci-»;cos(a c il-vcos(a --« c os(a + 3 (/ _ _ Mnci M M0o CI-?;COS(Q; «cos(a + + + /l3)\ //? ? ??)))2\\\2222 . A\\ 0 0 a. 0CI-?;COS(Q; osp — xlc I cos p p + -f- cosa cos a I ^3 £3 II

MpcA M00cA cA M GGoo) oo/

112

Propagation and reflection of shock waves x (F 2 _( 7 +l)M 0 2 M^F 3 tan 2 /? F1

=

( 3 7 - l ) M 0 2 - 7 + 3,

F2 = 2 ( 2 7 - l ) M 0 4 + ( 7 + 5 ) M 0 2 - 7 + l , F3 = 2(2 7 - 1)M 2 - 7 + 5, 1 dp2 _ JWQ CI / c o s 5 a dM0 pi dn (7 + 1) 2 L Goo \ sin a da

dM 1 . dt Goo

\ J

(v \* \ci)

cos5 a ( „ M0C1 — vcosia + 8)\ ^ x - —3 cos/? + c o s a - ^ —-i ^ 6cos- p \ Goo /

,*^A. 134

It follows from the boundary condition (Eq. 131) Y1 At + Y2At2 + Y3At3 + Y4At4 + . . . = 0. Or putting Y{ (i > 2) equal to zero 1 d

?2 1 ( , *\ , v7 / • / , mi ^—{cos(a + /?) + V^sin(a + /?)} P\ os

£

u - G 0 c o s ( a + /?) ^ ( 1 ) L ai —* g e),l)\ Pi Goo

+ G 2 ( l - £ ) 2 V v 7 ^ 7 = 0> X<3) {cos(a + /?) + V^sin(a + /?)} - f 1 - ^ - V ^ s i n ( a + /?)") y/y

+

(s(sK(s)W'*-^S

^ ^ r ipv-tip(,^gi''ef)ipfl G0 ( l - e ) 3 G 3 s i n 3 ( a + /?) + 7^M3 i>ino^n^a (V)3 G 4 ( l — £^4 ( v

,4

j= 0,

sin 4 (a + P)tl>tozil>T,il>vri
Reflection from smooth body...

113

+2(1 - e ) 2 ^ J s i n 2 ( a + / ? ) ^ C T ^ ^ e ( 9 ) +X( 4 )(cos(a + /3) + Vsin(a + /?)) + 2X (3 )e< ? <7r e ( 2 ^ T

dp2, dl

Pi al / g(l-g)jg3 ,

Pi *„ (5) .

J_5p2

(4)

e2

X

Pi Pi (^p2odada dp2p d2a I
_i^__d_

fdp2_\

Goo C>4 V *

A

2

^Vv d p G?o

^

^ d f dp2 Goo dx% \ dt , dp2 V>C/i

2

*

Goo

(1)

^

0

1 CM

J

- 2 e ( l - e)Vy {As + 2e(l - £ ) i f } / ' / ^ f T e ( ^ , Pi

*

e = 1 - — VV> sm(a + /?) + ^Joo

(i) _ ^ 2 0 ^a dp 2 ° ~ ~dxj ~G^~df'

e

-^

(2) _
e

(3) _ _JL f^El\ _ J^__^_ / d P 2 \ ~ dx° \dl ) Goo dt V dZ y '

6(7

^,

^OO

(4) _ dy^da_ _ tpa dy ~ da <9z£ Goo A '

e<7

Propagation and reflection of shock waves

114 o(5)

_ _9_ f^P£\ _ J0£__^ (^P£\ dxaQ \ds ) Goodt \ds ) '

(6) _ ^}l_ _ ^o dhl ° dx° Goo dt '

(7) _ dVo_ _ J0o_^/ (8) _ 9h^ _ ipg dh2 C<7 ~ dxaQ Goo dt' dx° Goo dt ' g G ^ s i n ( a + /?) g (1) (9) _ _ 1 ^ P 2 , sin(a + /?) _ ^ (1) e 9 - ~ P l ds** V^ p / ^ +PlGoo W * *" a — arctan(V^),

6(7

41)

=

B a ) ™ ( ° + fl + B £> co S (a + /?) + flW +

B<#g*Ti>.MeT

-B$gKTi>Ki;T, G dGp ^ = G 2 ^da- ^^7 ^ ' . n (D

s

Gt . .

G

4 (3) _ GG t _ ^ G^ \ G 1

^ = ^H,

+

- = G ^ - G ^ - (^(WF 1 <*G0 1

1

+ E V G2 \ 1x

+

da

\ \ _

-G^^W;/^7' R(2) = _ J L (1 + W\ D ( 3 ) CT7

^

^

_!i+G^ Q^ ^

(4)__G_dGo _ 2

G , da ^

= 1+ (V^,

^rfGoJD^_GG, + ^ rfa ^ ^ G3^ VW 7 , f G sin(a + /?) , t>£2^ U o o £i(VV) 2 Go

g2=

Gcos(a+ /?)-,

(135)

The expressions for the derivatives of higher orders can be obtained sim­ ilarly. Those derivatives together with Eqs. (132)—(135) have been used for the calculation of the unsteady reflection of a plane shock wave from various smooth blunt bodies (a sphere; an ellipsoid; a blunt cone; a concave body). We took into account the derivatives dM/dt, d2M/dt2, d3M/dt3, d/dt(dp2/ds), d/dt(d2p2/ds2), dp2/ds, d2p2/ds2, d3p2/ds3. Continued fractions were ex-

So ■3, O

o 3

1* 3 3

o o o

1"

Fig. 28. Reflection of a plane shock wave from a smooth blunt body; a - reference frame; 6, c, d- pressure distribution; b - elliptic cylinder, the ratio of semiaxes is 2 : 1, M^ = 10; / - angle of attack f3 = 0; 1 - p2/pOQ = 745; 2 - 761; 3 777; ^ - 791; 5 - 8 0 8 ; (5-810; 7 - 8 1 3 ; 5 - 8 1 5 ; 0 - 8 1 8 ; 10- 820; / / - f3 = 0.5arctan2; 1 - p2/Poo =746; 2 - 7 6 2 ; 3 - 7 7 7 ; ^ - 7 9 3 ; 5 - 8 0 8 ; (5-811; 7 - 8 1 3 ; 5 - 8 1 5 ; 0 - 8 1 8 ; 10-820; //1-/3 = arctan 2; 1 - p2/Poo = 736; 2 - 7 5 4 ; 5 - 772; ^ - 789; 5 - 807; 6 - 810; 7 - 813; 8 - 816; 9 - 819; i0 - 822; c - ellipsoid of revolution, the ratio of semiaxes is 2 : B : 1, M^ = 10, (3 = arctan2; I- B = 4; 1 - p2/Poo = 732; 2 - 7 4 9 ; 3 - 7 6 6 ; ^ - 782; 5 - 799; 6 - 8 0 2 ; 7 - 8 0 5 ; £ - 8 0 8 ; 0 - 8 1 0 ; 1 0 - 8 1 3 ; II-B = 2; 1 - p2/p^ = 720; 2-735; 3-750; ^ - 7 6 5 ; 5 - 7 8 0 ; £ - 7 8 3 ; 7 - 7 8 5 ; £ - 7 8 8 ; 0 - 790; 10 - 792; d - hyperboloid of revolution, semiangle of tapering is 60°, (3 = 0.15, M^ = 2; 1 - p2/Poo = 14.3; 2 - 14.4; 3 - 14.6; ^ - 14.7; 5 - 14.9; 6 - 15.

I—1 1—*

116

Propagation and reflection of shock waves

Fig. 29. Density distribution behind the shock wave reflected from a sphere. M^ = 1.2; 1 - p/Poo = 1.669; 2 - 1.672; 3 - 1.676; 4 ~ 1.682; 5- 1.690; 6- 1.712; 7 - 1.753; 8- 1.796.

Fig. 30. Pressure distribution along the axis of symmetry. M^ = 3. a-i= 0.0108; 6 - 0.0215; c - 0.0323; i=t-cOQ/r,p = p/p2.

Reflection

from smooth

body...

117

ploited in order to convert power series into functions that represent the solu­ tion over a larger domain. In the case of a sphere, a comparison was made with numerical results of other authors. The m a x i m u m difference does not exceed 1-2% showing the efficiency of the proposed method. Fig. 28 illustrates the dis­ tribution of flow parameters behind the shock wave reflected from blunt bodies at incidence. Figs. 29 and 30 present some results concerning the distribution of flow properties over a sphere. We note some characteristic features of the reflection of a plane shock wave from a sphere. The density distribution at the initial instant and for t ~ ts — r ( l — c o s a 5 ) / G o o , where as is the angle when the flow behind the reflected shock becomes sonic, has an entirely different character. For t <^ts the minimum value of the density is reached at the point of stagnation. As the time increases further the density minimum moves away from the axis of symmetry and then (for t ~ ts) it is on the shock. The point with the m i n i m u m value of the pressure lies on the body for short times; for long times and for high Mach numbers it is on the wave, and for long times and for small Mach numbers, on the body. For low Mach numbers the pressure distribution is similar to that of the density. In this case the pressure increases in the direction from the body towards the shock.

14. T r a n s i t i o n f r o m r e g u l a r t o M a c h r e f l e c t i o n Three types of flow can be pointed out while studying shock reflection. First of all, it is a steady flow. Secondly, it is an unsteady flow which can be led to a steady one by appropriate transformation of coordinates. The reflection of a plane shock from an infinite wedge refers to this case. As there is no characteristic length, it is possible to introduce self-similar variables. Such a flow is referred to as a pseudostationary one. Finally, there is an unsteady flow which is not self-similar. As an example, one can mention the reflection of a plane shock from a curvilinear wall. The first two cases are investigated extensively. There are regions in a, M , 7 where both two- and three-shock pattern may occur. Therefore transition criteria are needed. There are three alternative criteria which determine the transition from regular reflection to Mach one, namely, the detachment point a p , the sonic point a 5 , and the von Neumann point a^. The first criterion is connected with the m a x i m u m flow deflection in the steady flow. It corresponds to the half-wedge angle at which a bow shock

118

Propagation

and reflection

of shock

waves

wave in supersonic flow becomes detached. The corresponding shock angle is denoted as ap. Von Neumann also proposed another criterion. Namely, transition occurs when an infinitesimal Mach stem normal to the wall can link the point of intersection of incident and reflected shocks to the wall. This condition for transition gives a value for incidence angle ajsf. The criterion a — as was proposed by Hornung, Oertel and Sandeman (Hornung et al. 1979). The flow behind the reflected wave becomes sonic at a = as- There is also the Henderson-Lozzi criterion (Henderson et al. 1975). According to it, the transition occurs at an angle such that the pressure behind the reflected shock wave is equal to the pressure behind a single shock wave normal to the flow direction. As shown by numerous experiments, the angle a^ correctly predicts transition for stationary shock system. There is an agreement between the theory and experimental data for strong and weak regular reflections and for strong Mach reflection. There are large discrepancies for weak Mach reflection. For some values of shock strength and angle of incidence, the theory has no physically acceptable solution. In experiments, a Mach reflection is observed for this case. For the unsteady reflection of weak shock waves, experiments show that regular reflection persists for conditions that should make its appearance im­ possible from the point of view of von Neumann theory. This phenomenon is referred to as the von Neumann paradox (Birkhoff 1950). A hysteresis effect was predicted in the transition from regular to Mach reflection for steady flow, when the reflection angle is changed (Hornung et al. 1979). Later on it was reported that this effect was observed in experiments (Chpoun et al. 1996). Nu­ merical studies also show the existence of the hysteresis. Skews (Skews 1997) calculated the limits which should be imposed on experimental tests in order to avoid three-dimensional flow influences arising from the use of finite width wedge. It is known that for the three-dimensional case of two axisymmetric bodies in close proximity with their axes parallel, the intersection between the bow shocks would change from regular reflection (more precisely, interaction) to Mach reflection as one moves transversely from the plane containing the axes. Thus the finite aspect ratios can influence the flow in the double wedge geometries. If Mach reflection conditions exist on the plane of symmetry, the flow downstream being subsonic, there m a y arise transverse influences. Exper­ imental evidence of these influences leads to the conclusion that the existence of hysteresis in strictly two-dimensional flows has not been found yet (Skews 1997). Hornung, Ortel and Sandeman (Hornung et al. 1979) proposed that sonic point is a criterion for the transition from regular to Mach reflection in unsteady

Reflection

from smooth

body...

119

flows (reflection from a wedge). The reason is as follows. The signal from the leading edge of the wedge can keep pace with the reflection point only if the velocity relative to this point is sonic or subsonic. The Mach reflection is unstable in the range ajy < a < ap. In fact, if due to some disturbance a Mach reflection appears, the subsonic region behind it is limited in extent, so that no signal can be transmitted to the reflection point. As will be shown below, the regular reflection is impossible in unsteady case, at a > as, when a plane shock wave reflects from a curved wall. If a shock wave propagates in a viscous, heat-conducting gas then there is a delay in the onset of the Mach reflection (Virgona et al. 1996). Experimental data lead to a conclusion that existing criteria do not explain the phenomena occurring in the case of unsteady interaction between the shock wave and the curved wall. Galkowski proposed that the transition from regular to Mach reflection is the result of a break-up of the initial discontinuity and is accompanied by the appearance of a rarefaction wave (Galkowski 1989). The ray method can provide some information on the subject. Eq. (134) contains the quantity ^fwhich tends to infinity as a —> a ^ . But in reality the derivative dM/dt and other derivatives as well as the mean curvature of the reflected wave become infinite at a smaller angle of incidence as because the denominator in Eq. (134) becomes equal to zero. The value of as is defined by the equation , t a n

R

_

_1_ /

^2^0

^ - M o V ( 7 + l)(M2-l)F3-

Thus regular reflection is impossible at a > a$. The flow has a singularity at a — as. Density distribution on the surface of the sphere behind the reflected wave confirms the fact that the transition takes place apparently at a = as (Fig. 31). The common theoretical approach is based on shock polar analysis. This approach assumes the flow to be uniform near the point of intersection of the waves. However, numerical and experimental studies suggest that the reflected wave contains a region with high gradients of flow properties. The proposed way of the origin of the Mach reflection is as follows (Tabak et al. 1994; Canic et al. 1996). The incident shock wave interacts with the region where the velo­ city becomes subsonic. The flow at the sonic line has a square-root singularity which gives rise to a weak shock emerging from the singular region. The reflec­ ted shock interacts with the subsonic region, and another weak shock appears. The shock generated by the singularity catches u p with the weak shock gener­ ated by the reflected wave.

Propagation

120

and reflection

of shock

waves

Fig. 3 1 . Density profile along the sphere behind the reflected shock wave (Moo = 2). 1 - a = 37.5°, 2, 3 - a = 46°; 1-3 - experiment, solid line - calculation by the ray method, angle 0 gives the position of the point in question on the surface of the sphere. The observed increase of the density for a = 37.5° supports the assumption that the transition to Mach reflection occurs at a = as. 15. D e v e l o p m e n t o f flow o v e r a b l u n t b o d y b e h i n d a n i n c i d e n t s h o c k wave

15.1.

The interaction

of a plane shock wave with a body

The development of flow past a blunt body was studied both experiment­ ally (Shugaev 1963; Syshchikova et al. 1967) and theoretically (Rusanov 1961; Milthorpe 1995). Below we give the results of experimental investigation car­ ried out by Shugaev. Experiments were fulfilled in a single-diaphragm shock tube of square cross-section of 28.5 x 28.5 m m 2 area. The shock Mach numbers Moo varied in the range 2.2 < Moo < 5, the flow Mach numbers M i were 1.1 < M\ < 1.66. Air was chosen as the test gas. A flat-nosed cylinder (r = 3.5 m m ; 4 m m ) was used as a model. As is known, the flow Mach number behind the shock wave in the laboratory reference frame is given by Mi =

2 ( M ^ - 1)

(woogoo) 1/2 '

Woo = w(Moo),

A streak-camera record is shown in Fig. 32

qoo = tf(Moo). A reflected shock is seen.

A

Reflection from smooth body...

121

rarefaction wave interacts with it. This wave propagates downstream from the edges of the cylinder. The velocity of the reflected shock is at first constant (segment 01 in Fig. 32), then it decreases due to the influence of the rarefaction wave. We denote as i\ the instant when the rarefaction wave catches up with the reflected shock at the axis of symmetry. The time is measured from the instant the incident shock wave reaches the nose of the body. In experiments the velocity of the reflected shock at t < i% coincides with the calculated values for the shock reflected from a rigid wall. The measured values of li agree well with the calculated ones. Fig. 33 presents the path of the reflected shock wave against time for different flow Mach numbers. Fig. 34 shows the velocity of the reflected shock against time.

Fig. 32, Reflection of a shock wave from a blunt-nosed cylinder (streak-camera record). The transition to the quasi-stationary flow may be described with the aid of time interval r during which the velocity of the reflected shock becomes two times less as compared with its initial value. Fig. 35 presents the ratio r/ti against the Mach number Mi. One can see that the quantity rjt\ diminishes linearly as the flow Mach number increases.

to to

I o' 3

1

4 o <*>

sacs

Fig. 33. Path of the reflected shock wave against time for a flat-nosed cylinder. 1 - M M = 1.07; 2- 1.15; 3 - 1.31; 4 - 1 . 3 5 ; 5- 1.46; 6- 1.51.

9?Q 05

Reflection from smooth body...

123

Fig. 34. Velocity of the reflected shock wave against time for a flalnosed cylinder. 15.2. The interaction of a shock wave with a shock layer on a blunt body Let us us consider the interaction of of aaa planar planar shock shock wave wave with with aaa sshock shock layer layer Let interaction of planar shock wave with us consider consider the me interaction over blunt axisymmetric body and the subsequent development of the flow. over aa blunt axisymmetric body and the subsequent development o blunt axisymmetric body and the subsequent development of the flow. Such a problem arises if we study the effect of a blast wave on a body flying Such a problem problem arises arises if if we we study study the the effect effect of of aa blast blast wave wave on on aa bbody flying at a supersonic velocity on condition that the distance from the center of the explosion is much higher than the dimensions of the body. The transient flow in question is very complicated. The incident shock wave interacts with the detached shock and and propagates through the shock layer on the body. Some time later a steady ly supersonic flow arises over a body at a new Mach number. The problem under consideration includes shock interactions, propagation of a transmitted shock wave through a non-uniform region, shock reflection from the body. Hereafter we denote as M\ the primary flow Mach number over the hndv as M2 Mo the Marh body, Mach number of the incident shock shock. Various types >es of facilities are used for studying the interaction of a shock wave with a body )dy in supersonic flow. Bingham et al. (Bingham et al. 1964) ibination of a shock tube and a wind tunnel. Some authors used a proposed a combination toother with a shock tube (Brown et al. 1965; Damkevalaet al. ballistic facility together UX-LXJ.V_- I C t t v l

KM yj Uv^CvVJ. y

O U I-/ V^JL O V / l i l v

JLXVy VV

CvX l O v i J

v / T KsX. KM 1,/V^Vl. y

tit

KM XX Vy TV

J.TX CJtV-'XA XX IXXXJLIL/V^X •

Propagation

124

Fig. 35. Ratio rjt\

and reflection

of shock

waves

against Mach number.

1968). In this case a model is launched into the test section of the shock tube where a shock wave is propagating. Other authors used a double-diaphragm shock tube (Ruetenik and Lemke 1967; Lisin and Shugaev 1969; Shugaev 1983). Apparently the most convenient facility for studying the present problem in a wide range of Mach numbers M\, M2 is the combination of a ballistic set-up and of a shock tube. However, such a facility is too complex and too expensive. Besides, it is difficult to fulfill measurements on the surface of a flying model. The cheapest, simplest and most convenient facility is a double-diaphragm shock tube. It consists of two high-pressure sections. A model is situated in the test section. The double-diaphragm shock tube operates as follows. On the burst of the first diaphragm, a shock wave appears in the low-pressure section. At Moo > 2.07 for 7 = 1.4 (Moo being the Mach number of the first shock), the gas flow is supersonic behind the shock relative to the model. Thus the supersonic flow over the model occurs. In some period of time the second d i a p h r a g m bursts, and the second shock wave arises. This shock interacts with the model which is in supersonic flow. T h e i m p o r t a n t advantage of the double-diaphragm shock tube is the fact that the incident shock is planar. Thus one can easily observe different stages of the interaction during the propagation of the shock through the shock layer

Reflection

from smooth

body...

125

near the body. In addition, the model is at rest relative to the facility. It simplifies the fulfilment of measurements on the surface of the model with the aid of pressure transducers etc. The defects of the facility are as follows. It does not enable one to study oblique interaction of a shock with a body. Besides, the attainable range of flow Mach numbers is comparatively small. Fig. 36 shows a scheme of flow in a double-diaphragm shock tube. Here 1, 2 are shock waves, 3, 4 a r e contact discontinuities in the flow, 5, 6 are rarefaction waves, a and b are the positions of the first and second diaphragms, respectively. Main requirements to be satisfied are: (1) the flows behind the first and second shock waves must be uniform; (2) the flow duration behind the first shock wave must be large enough, so that a steady supersonic flow past a model occurs; the second shock wave must reach the body when a steady supersonic flow over the body does exist. Let us denote as At the time interval between the arrival of the first shock and that of the second one at the test section. The value of At depends on the length of the first high-pressure section, on the velocity of sound in the driver gas, on the time of rupture of the second diaphragm, on the distance between the first d i a p h r a g m and the test section, on the Mach numbers M ^ , M2. The real flow in a double-diaphragm shock tube is very complicated. The second shock wave moves in the gas flow behind the first shock wave. There are disturbances in the flow downstream the contact surface. Flow non-uniformities may arise due to the rarefaction wave reflected from the second diaphragm. The effect of that wave depends on the time rupture of the second diaphragm, too. As the shock-standoff distance depends on the flow Mach number, one can evaluate the flow non-uniformity according to the variation of that distance in time. In our experiments a double-diaphragm of rectangular cross-section of 40 x 60 m m 2 area was used. The interval between the first shock wave and the second one was 50-150 //s. Fig. 37 shows the shock-standoff distance against time at the axis of symmetry. The incident shock wave was planar and moved at a constant velocity. Curves 1 correspond to the single-diaphragm shock tube, curves 2 correspond to the double-diaphragm one. Small oscillations are seen after the flow became quasi-stationary. A finite distance between the walls of the shock tube involves the reflection of the bow wave from the walls. This fact may influence the flow past a body. Our set-up enables one to obtain a steady flow at Mach numbers M\ > 1.15 for cylindrical models of diameter 8 m m . Fig. 38 shows the streak-camera record of the interaction of an incident shock wave with a shock layer on a flat-nosed cylinder. Here 1 is the shock

1—l

to

i 1 OK

o 3

Fig. 36. Scheme of the flow in a double-diaphragm shock tube. 1,26 - rarefaction waves; a, b - positions of diaphragms.

shock waves; 3,4-

contact discontinuities; 5,

1

Reflection

from smooth

body...

127

Fig. 37. Shock-standoff distance against time, a - flat-nosed cylin­ der; b- cylinder with a spherical nose; 1 - single-diaphragm shock tube; a - M^ = 4.35; 6 - M^ = 3.05; 2 - double-diaphragm shock tube; a - M^ = 3.36; b - M^ = 2.91.

wave ahead of the body, 2 is the incident shock, 3 is the bow shock after the interaction, 4 is the transmitted shock, 5 is the incident shock out of the shock layer, 6 is the reflected shock, 7 is the rarefaction wave, 8 is the contact discontinuity. Figs. 39 and 40 present a shadowgraph and an interferogram of the interaction. The process under investigation is as follows. First of all, two new shocks occur that propagate in opposite directions. A contact surface is between them. The transmitted shock reflects from the body surface. It moves in the same direction as the detached shock. T h e reflected shock interacts with the contact surface. A new wave appears. It m a y either be a shock wave (if M i > M 2 ) or a rarefaction wave (if Mx < M 2 ) . T h e refracted wave catches u p with the bow wave. A new shock wave and a rarefaction wave appear as a result. T h e occurrence of waves changes the pressure on the body surface. The velocities of the waves that appear after the head-on collision agree well with calculations fulfilled for plane waves. Let us consider the motion of the transmitted shock along the axis of symmetry (Fig. 41). We use Eq. (82).

128

Propagation and reflection of shock waves

Fig. 38. Interaction of an incident shock wave with a shock layer on a fiat-nosed cylinder (streak-camera record); I - primary bow wave; 2 - incident wave; 8 - bow wave after interaction; 4 ~ transmitted shock wave; 5 - incident shock wave out of the shock layer; 6 reflected shock wave; 7 - rarefaction wave; 8 - contact surface.

Fig. 39.

Reflection from smooth body...

129

Fig. 40.

Fig. 41. Interaction of a plane incident shock wave with a shock layer on a flat-nosed cylinder. 1 - primary bow shock wave; 2 incident shock wave; 8 - transmitted shock wave; 4 ~ bow shock wave after the interaction; 5 - contact surface.

Propagation and reflection of shock waves

130

The pressure and velocity do not change across the contact surface. Thus we get

- q'2M'2{M'l - 1 ) 1 ^ - MW2{M? - 1)^£ 7+ 1 9P2 = P2 gpi 4 7 _ f M^rfM| _ M1M[g2 1 dM{ _ _ ? £ _ , . , , 1, „ 2 1 \ ci rft M ?! c A 7 + r V * 2 2 7 + 1 <9p 2 Miq2w[l 1 SJO'J 0, (7 + 1)M^ pi 5n (7 + l)M[M2qi Pi dn F{M) = 2(2 7 - 1)M 4 + (7 + 5)M 5 - 7 + 1, F/ = F(M/), ff; =

g(M/),

«,{ = «,(M/),

l=.[^-,

kt = M.

V 91^1

(136)

V ^

Here primes denote initial values for quantities after the interaction. Initial values for shock curvature after the interaction can be determined from geo­ metrical considerations r w

] J" 7 (137) Here Hi is the primary curvature of the detached shock wave at the axis. Using Eqs. (136), (137), we get f f

1

1

2

1

f \(

Wl(l

+ Mlh) + 1)M1(M1+M2)

dM[ _ 2{Ml' - l)Fiffi 1 &M'2 _ 2AQ (Mj ; - l)F 3 ffi , 5 eft ~ (7 + l)(M 1 + M 2 )F 1 F 2 ^ (ft (7 + l ) F 2 i ^ ' (9K = 4 7 /?gF 4 giwigiM^ 3n (7 + l) 2 -^2gig2^ 2 (Mi + M 2 ) ' 9P2 _ Q2W[W2 1 3pi dn f3lqiwiw'2p2 dn ' / , = q iW 1^/q2W2/31F5 2qiq2M^2 1 (7+l)Af 2 ^ 2

1 c2 1 P2 1 pi F

w2(l + M[k2) ( 7 + 1)M 2 (M 1 + M 2 ) '

Reflection from smooth body...

Fi — Fi + F3 =

131

fok2F8

M2F6 (7 + DM.M^M, + M2) V1 + ^T*)

^4 = q'2fo [(7 + l)(Af! + M2) - -j^iM?

+ ( 7 + lW

^

+ *'

- 1 ) | + (M : + M 2 )F 9 ,

F 6 = (7 - 1)M^{(7 + l)Mi(Mi + M2) + M ^ V ^ T } - 2fe1{(7 + l)MiM 2 + 2(M2 - 1)},

, 7 . 1 + >!i+iUffi (Jfl ,_ 1)i ^8 - 1 + =

2(7+l)M{2 2 y, (Mi - 1),

gMiM^Mt 2 - l ) ' / ? l 9 ' l g 2 X ^ 2 M22(Mi + M2)F[q\l2w\12

fo = M[ + l/k2,

'

lh = M2- + l/ki,

fo

= ^fi-

(138)

Fig. 42 presents the pressure and density distribution at an initial instant for the case of the interaction of a plane shock wave with a bow wave ahead of a sphere. We used the tables of flow properties for the steady supersonic flow over a sphere (Lyubimov and Rusanov 1970). The equation of motion for the transmitted shock can be solved approximately if one assumes the shock curvature and the pressure gradient behind the shock to be constant in time (see Eq. (138)) . Calculations were carried out for a sphere. The variation of the Mach number for the transmitted shock does not exceed several per cent. Therefore we can put M^ — const. The time during which the wave propagates through a shock layer is given by 1 f6 '=777 /

dx *

,

M£SMJ(0).

Here S is the shock-standoff distance. Fig. 43 shows the positions of the trans­ mitted shock for different instants. Dots are experimental data, solid line is the calculation carried out on the assumption that the Mach number of the transmitted shock does not vary along the ray. A good agreement can be seen

132

Propagation and reflection of shock waves

Fig. 42. Distribution of pressure and density at an initial instant during the interaction of an incident shock wave with a bow wave over a sphere (Mi = 1.5; M? = 1.7).

between experiment and calculation. Let us consider the development of flow. If flow properties are constant behind the incident shock before interaction, a steady flow arises in some period of time. When the transmitted shock reaches the surface of the body, a reflected shock appears. It propagates through a non-uniform flow. There is an essential difference between the flow pattern which corresponds to a sphere and that one for a flat-nosed cylinder. In the former case, the shock reaches the stagnation point of the sphere and then reaches other points on its surface. In the latter case, due to the fact that the shock front is concave, the reflection begins at the sharp edges of the cylinder. The point of intersection of the transmitted wave with the body surface moves to the axis of symmetry. The reflected shock refracts at the contact discontinuity and then it catches up with the bow wave in front of the body. This is the case of the interaction of curvilinear shocks of the same family. The former wave propagates through a non-stationary flow. The above-mentioned interaction results in a new shock, a rarefaction wave and a contact discontinuity. The velocity of the reflected wave diminishes continuously in the reference frame bound up with the body.

Reflection

from smooth

body...

133

Fig. 43. Positions of the transmitted shock wave (Mi = 1.5; Mi — 1.7). Solid lines represent the calculation; dots are experimental data.

It is interesting to note that the Mach number of the shock reflected from a flat-nosed cylinder increases in time. The values of the initial velocity of the reflected shock for the flat-nosed cylinder are shown in Fig. 44 (dots are exper­ imental data; solid line is the calculation). As one can see, the experimental values are 20-50% less as compared with the calculated ones. A similar result was reported by Bazhenova et al. (Bazhenova et al. 1968). It was concluded that there was a loss of energy during reflection. There is some difference in the conditions of experiments under consideration between that case and the present one. In experiments by Bazhenova et al., the body was primarily at rest relative to the gas while we investigated the incidence of a shock on a body plunged in supersonic flow. The discussion of this problem is given below. Streak-camera records show that the bow shock velocity becomes equal to zero relative to the body when it is caught u p with a rarefaction wave which appears during the wave interaction. It agrees with the above-mentioned result (see section 9). Indeed, the intensity of the disturbance reflected from a shock wave is usually small. T h e shock-standoff distance decreases slightly after the interaction of the

134

Propagation and reflection of shock waves

Fig. 44. Velocity of the reflected shock wave against Mach number at the initial instant. Solid line represents the calculation; dots are experimental data.

rarefaction wave with the bow wave. Later on small oscillations of the bow shock wave appear. They may occur due to non-uniformities in the flow upstream. Strictly speaking, flow properties (density, entropy) on the body surface reach steady values after the detached shock wave becomes steady. Let us denote as 13 the instant when the entropy on the body surface reaches its steady value, as t\ the instant when the detached shock wave becomes immovable relative to the body, and as t2 the instant when the contact surface reaches the body. Entropy reaches its steady value at some point, when the gas particle at that point has intersected the steady shock wave. Thus the following inequality is valid: ^3 > ^2 > ^i- The instant £2 can be determined from streak-camera records. The contact surface moves at the velocity of the gas. By measuring the velocity of the contact surface we determine the gas velocity at the final stage of the transition to the steady flow. Fig. 45 demonstrates that the velocity of the contact surface varies linearly at the axis of symmetry near the body. It is interesting to note that the gas velocity varies linearly, too, in the case of steady supersonic flow over a sphere (Lyubimov and Rusanov 1970). Fig. 46 presents the shape of the contact surface for different instants.

Reflection from smooth body...

Fig. 45. Velocity of the contact surface against distance. a - cylinder with a spherical nose;l - Mi = 1.6; M 2 — 1.2; 2 Mi = 1.6; M 2 = 1.4; 3 - Mx = 1.55; M 2 = 1.5; 6 - flat-nosed cylin­ der; 1 - Mi = 1.45; M 2 = 1.1; 2 - Mi = 1.25; M 2 = 1.4; 3 - M x = 1.4; M 2 = 1.4; 4 - Mx = 1.4; M 2 = 1.5; 5 - Mx = 1.45; M 2 = 1.5; 6 - M i = 1.45; M2 = 1.6.

Fig. 46. Shape of the contact surface for different instants.

a

J/

136

Propagation and reflection of shock waves

Fig. 47. Values of r\ against shock Mach number.

Fig. 48. Stagnation pressure against time for the flat-nosed cylinder.

Reflection from smooth body...

137

The transition to a steady flow proceeds in a different manner for different points of the flow. The more distant the point from the axis of symmetry is, the later the above-mentioned transition occurs. Besides, the period of the transition is not the same for different flow properties (e.g. for pressure and for density). The process of the transition may be characterized by the following two quantities: (1) by the period of time Ti during which the detached shock becomes steady; (2) by the period of time T*I during which the value of density becomes steady on the body surface. Fig. 47 shows the values of T\ against shock Mach number.

Fig. 49. Stagnation pressure against time for the cylinder with spherical nose. Fig. 48 gives the variation of the stagnation pressure for the flat-nosed cyl­ inder. The segment 01 refers to the period when there is no flow, the point 1 refers to the arrival of the first shock, the point 2 refers to the arrival of second shock. One can see that the maximum value occurs after the reflection of the in­ cident shock and then the pressure decreases up to the value which corresponds to the steady flow past the body. Experimental data are in fair agreement with the calculated ones. Fig. 49 demonstrates the pressure history in the stagnation point for the cylinder with spherical nose at Mi = 5.16, M2 = 4.45 (McNamara 1967). Curve 1 corres-

138

Propagation and reflection of shock waves

Fig. 50. Density history on the axis of symmetry, a - flat-nosed cylinder; b - spherical nosed cylinder. ponds to the experimental data, curve 2 shows the calculated values (7 = 1.4). The time is counted from the instant the incident shock meets the bow shock. The character of pressure variation in time is just the same as in Fig. 48. If M 2 > Mi then an additional pressure increase occurs. It is due to the fact that in this case an additional shock propagates toward the body after the refraction on the contact discontinuity. Experimental data show that the pressure at the stagnation point begins to decrease on the arrival of the rarefaction wave which appears as a result of the interaction between the bow wave and the reflected shock. The pressure reaches its steady value when the rarefaction wave catches up with the detached shock. A similar result was obtained for the sphere. It is interesting to compare the present results with the incidence of the plane shock on the immovable body. In the latter case the pressure at the stagnation point reaches its steady value before the detached shock becomes at rest relative to the body. The mechanism of the transition to the steady flow is associated with the rarefaction wave that propagates from the edges of the body. Later on that wave catches up with the detached shock, and the velocity of this shock diminishes and becomes equal to zero relative to the body. Now we proceed to the density variation. The density was determined from

Reflection

from smooth

body...

xr—TS'

Fig. 5 1 . Density gradient behind the bow shock wave.

interferograms. The density distribution varies slightly in time in the region between the reflected shock and the body surface. A sudden variation of density occurs after the reflection of the transmitted shock and after its catching up with the bow shock. Fig. (Fig. 51) demonstrates density history on the axis of symmetry for the flat-nosed cylinder and the spherical-nosed one. Curve 1 presents the density variation behind the bow-shock, curve 2 gives that value upstream the contact surface, curve 3 corresponds to the stagnation point. The time t\ refers to the arrival of the incident shock at the stagnation point, ^2 refers to the catching u p of the reflected shock with the bow shock, £3 refers to the instant as the detached shock becomes immovable relative to the body. One can see the decrease of the density at t > ^3, and a slight variation of the density behind the detached shock wave. It is interesting to note that the density gradient increases monotonically behind the bow wave after the reflected shock has caught it u p (Fig. 50). The density gradient was determined with the use of Eq. (82). The shock Mach number was measured from streak-camera records, the shock curvature was found from shadowgraphs. The measured values of the density behind the shock reflected from the flat-nosed cylinder are 1.4 times as high as the calculated ones. For instance, the measured value of the density is equal to 1.7 x 10~ 3 g / c m 3 (Mi = 1.45,

140

Propagation

and reflection

of shock

waves

Mi — 1.4), the calculated value being 1 0 ~ 3 g / c m 3 . The measured values of the velocity for the reflected shock lie below the calculated ones. This fact m a y be due to cooling of the gas near the body surface.

C H A P T E R 5. REFLECTION OF A SHOCK WAVE F R O M A CONCAVE BODY A N D SHOCK FOCUSING

16. R e f l e c t i o n of a s h o c k w a v e f r o m a b o d y w i t h r e c t a n g u l a r c a v i t y Bodies with cavities of various shapes are used widely in aircraft and space­ craft, because the cavity reduces the heat flux to the body nose. Thus the study of shock reflection from such bodies is of practical interest. Many experimental data and theoretical studies indicate that the supersonic flow past bodies with cavities can be steady or unsteady (oscillating bow shock). Shigemi (Shigemi et al. 1976) investigated experimentally the supersonic flow past a hollow cylinder. Pressure oscillations have been found inside the cylinder. Johnson (Johnson 1959) studied a number of cavity shapes at M = 22 in helium. If there was no gas injection into the stagnation region of the cavity then the bow shock was steady. Bastianon (Bastianon 1968) calculated the flow past a concave body. He found that there are undamped oscillations. Bohachevsky (Bohachevsky et al. 1972) found that the oscillations are d a m p e d and the flow eventually reaches a nonoscillatory condition. The supersonic flow past concave bodies was also calculated by Gilinsky (Gilinsky et al. 1976). Two cases were studied: 1) the variation of a body shape during a short interval of time; 2) a sharp variation of the Mach number of the supersonic flow. D a m p e d pressure oscillations are found. Huebner (Huebner et al. 1993) experimentally investigated the flow past a conical-walled cavity with a flat base at M = 10. It was found that there are periodic oscillations of the bow shock. The frequency of the oscillating shock is equal to the fundamental acoustic frequency, in accordance with the theoretical result obtained earlier (Bohachevsky et al. 1972). The oscillating shock amplitude is directly proportional to Reynolds number. Under some conditions there was an aperiodic unstable motion of the bow shock wave. A bulge appeared on the shock. The possible explanation is as follows. The amount of gas gradually increases within the cavity until it is violently expelled and the bow shock is warped.

141

142

Propagation and reflection of shock waves

The flow pattern is given below for the unsteady reflection of a plane shock wave from a body with a rectangular cavity (Serov et al. 1985). Experiments were carried out in a single-diaphragm shock tube of a rec­ tangular 34 x 72 mm 2 or 28.5 x 28.5 mm 2 cross-sectional area. The shock tube was 2.5 m long, the high-pressure section was 0.7 m in length. Helium was used as a driver gas. Air, carbon tetrafluoride (CF4) and dichlorodifluoromethane (CCI2F2) were chosen .as test gases. A model (a body with a cavity) was mounted in the test section which had glass windows.

Fig. 52. Reflection of a plane shock wave from a rectangular cavity; air, M = 1.7, Re = 104, ifh = 0.08, a - i = 3 jus; b - 4Q/is; the time is measured from the instant the incident shock wave reaches the bottom of the cavity (I is its depth); two-dimensional flow. We used models of various shapes. The models of the first kind had the width equal to the distance between the glass windows, so the flow past the model was close to two-dimensional. The models of the second kind had the width equal to 10mm. In this case the flow past the model was threedimensional. The ratio of the depth I of the cavity to its height h was equal to 0.08, 0.4, 2.5 (two-dimensional flow). Besides, we studied the shock reflection from, a thin-walled hollow cylinder with a sharp-edged open end. Its inner dia­ meter was 6 mm, the outer diameter was 8 mm, the depth of the cavity varied

to

3

o o ©

Fig. 53. Eeflection of a plane shock wave from a rectangular cavity; CF 4 , M = 2.6, Re = 5 x 1043 l/h = 0.08, a t = 20/is; 6 - 50 /is; c - 75/is; d - 110/is; the time is measured from the instant the incident shock wave reaches the bottom of the cavity (I is its depth); two-dimensional flow. )_4

CO

144

Propagation

and reflection

of shock

waves

smoothly from 0 to 25 m m . The Mach number for the incident shock wave was equal to 2.1 - 5.5. The initial pressure was 1.3 — 2 0 k P a . A quasi-stationary supersonic flow occurred in the test section after the burst of the diaphragm. Its duration was about 150 - 300 y,s. We took shad­ owgraphs by means of a Q-switched pulse ruby laser which emitted a single spatial mode. The duration of the pulse was 40 ns. We also took streak-camera records. The pressure at the b o t t o m of the cavity was measured with a Kistler transducer. The velocity of the shock wave was measured with pressure sensors. The Reynolds number varied from 10 4 to 10 6 , the characteristic length being the depth of the cavity.

Fig. 54. Values of pressure behind the reflected shock wave against Mach number. 1 - ! / d = 1 . 6 7 ; 2 - 1; 3 - 0.556; 4-0. Figs. 52 and 53 show the flow pattern. A plane incident shock wave (1) is seen. Diffracted shock waves arise near the edges of the cavity, they reflect from the walls inside the cavity. The diffracted shocks interact with the reflected shock, the interaction being the Mach one. Transverse waves and slipstreams appear behind the reflected shock. The slipstreams roll up into vortices due to their instability. The vortices are carried away to the corners of the cavity, and a large vortex arises. Vortices also arise outside the cavity near the edges. As shown by numerous experiments, a reflected shock wave interacts with

Reflection

from concave

body...

145

a boundary layer on walls in a shock tube, which leads to the bifurcation of the reflected shock. The theory (Mark 1957) predicts that such a bifurcation m a y easily occur for gas with a low adiabatic exponent. Due to the bifurcation, a vortex sheet appears behind the reflected shock. A secondary shock wave m a y appear. In some cases a multiple shock system (called pseudo-shock wave) occurs. Owing to it, the pressure behind the reflected shock wave m a y change its value. This pressure was determined by Znamenskaya et al. (Znamenskaya et al. 1990). The model was a hollow cylinder with a flat bottom. The thickness of its wall was 1 m m . A movable b o t t o m enabled to change the depth of the cavity. The diameter of the models was 6 m m , 10 m m and 18 m m . A pressure transducer was mounted in the center of the bottom. The pressure history is as follows. First of all, a sharp rise is observed. It corresponds to the reflection of the shock wave. Then the pressure continues to grow. This effect may be explained by the appearance of a secondary shock. The increase of pressure behind the reflected shock depends on the ratio / / d , where / is the depth of the cavity, d is its diameter. The value of the m a x i m u m pressure p m a x grows as the ratio l/d increases and then becomes constant. The difference between Pmax and pr (where pr is the calculated pressure behind the reflected shock) is noticeable at M > M*, M* = 1.3 - 1.7. Fig. 54 presents the values of pressure on the bottom of the cavity. Dots are experimental data while solid line is the calculated values. For comparison, d a t a are given that correspond to the shock reflection from a flat-nosed cylinder. In experiments, initial pressure was p0 z= 0.4 - 47 k P a . Air was chosen as the test gas. It is worth mentioning the following fact. Sharp edges of the model with a cavity lead to some distinctions as compared with the shock reflection in a shock tube. Shock waves appear near the edges of the model. They reach the bottom of the cavity after multiple reflections. If the ratio l/h is small then the diffracted wave becomes divergent and decays rapidly. Nevertheless, the second wavelet may be observed which occurs as the first one reflects from the opposite wall. In the case of deep cavity (at high values of l/h) there are multiple reflections of the shocks. Successive shock waves appear in the area between the b o t t o m and the reflected shock wave as a consequence. This phenomenon m a y also change the pressure on the b o t t o m of the cavity. Gas pressure and its temperature increase inside the cavity due to the existence of the transverse waves. The value of the pressure is 1.3 times that one which corresponds to shock reflection from a flat wall (M = 4.9, l/h — 0.4, C F 4 as the test gas), in accordance with the data by Znamenskaya et al. (1990). T h e reflected shock wave interacts with the boundary layer and A-configuration appears. It is accompanied by the separation of the b o u n d a r y layer

146

Propagation and reflection of shock waves

and its transition to a turbulent one. One can see it from Figs. 55, 56, 57 and 58. It is worth noting that the appearance of A-configuration in our case differs from what takes place in the shock tube. Namely, the separation of the boundary layer occurs at larger distance r from the part of the shock wave that is normal to the wall. Besides, the triple point is more distant from the wall as compared with the reflection in the shock tube. Fig. 59 presents r and d against s. Here d is the length of the rectilinear part of the shock wave which is normal to the bottom, s is the distance from the bottom, a refers to air, b refers to CF4. For instance, the value of r increases by 25 - 30% as compared with the value defined by the empirical formula (Bazhenova et al. 1977). This discrepancy may be explained by the influence of the diffracted waves which cause the instability of the boundary layer. The part of the reflected shock wave that is normal to the wall disappears completely in CF4 due to the growth of A-configuration (Serov et al. 1985). The reflected shock takes a V-shape. The pressure behind the reflected shock is reduced significantly due to this fact. For instance, the pressure is reduced by 40% as compared with the value behind the normal shock wave (M = 2.1, po = 1-53 x 104 Pa, Re = 10 5 ). The A-configuration arises immediately after the shock wave reflects from the bottom of the cavity in the case of CF4. The flow is all turbulent behind the reflected shock. It is apparently due to a high value of Reynolds number (Re = 10 6 ). The reflected shock interacts with the diffracted waves while leaving the cavity. As a result, slipstreams arise near the edges (see Fig. 53). Figs. 55 and 56 show the disturbed flow ahead of the reflected shock wave and of the bow wave. This phenomenon is due to the above-mentioned interaction of waves with the boundary layer. The disappearance of the part of the reflected shock wave that is normal to the wall was also reported by Kharitonov et al. (1985). They observed the transformation of V-shaped front into an inclined shock (air, nitrogen and carbon dioxide as test gases; l/h = 6.4 or 17.5, M = 2 - 4 ) . This fact is apparently due to the instability of V-shaped front. A secondary shock wave propagates behind the reflected one in CCI2F2. It was observed earlier by several authors (Strelow et al. 1959; Znamenskaya et al. 1980). Its appearance may be explained as follows. If flow separation takes place then the vortices arise behind the reflected shock. They produce a gas flow directed to the bottom of the cavity. On the other hand, flow separation changes the geometry of the flow. Both of these factors may act simultaneously and a secondary shock wave appears as a result.

Reflection from concave body...

Fig. 55. Reflection of a plane shock wave from a rectangular cavity, l/h = 2.5, two-dimensional flow; 1 - incident shock wave; 2 ~~ dif­ fracted shock wave; 3 - reflected shock wave. The time is measured from the instant the incident shock wave reaches the edges of the cavity. Air, M = 2.1, Re = 105, a - t = 40/is; 6 - 60/is; c - 80 jus; d - 110/KS; e - 150 fis; f - 220 ps.

147

148

Propagation and reflection of shock waves

Fig. 56. Reflection of a plane shock wave from a rectangular cavity5 l/h = 2.5, two-dimensional flow. The time is measured from the instant the incident shock wave reaches the edges of the cavity. CF 4 , M = 2.6, Re = 9 x 105, a - t = 30/is; 6 - 50|is; c - 90/is; d - 120/is; e - 140 ^s; / - 200 |is.

Reflection from concave body,..

/;«&"«-N

Fig. 57. Eeflection of a plane shock wave from a rectangular cavity, l/h = 2.5, two-dimensional flow. The time is measured from, the instant the incident shock wave hits the edges of the cavity. OF4, M = 2.9, Re = 2 x 106, a-t = 40|is; b - 60/is; c - 90/is; d 110 fjs; e - 120 fis; f - 190 ps.

150

Propagation and reflection of shock waves

Fig. 58. Reflection of a plane shock wave from a rectangular cavity, l/h = 2,5, two-dimensional flow; 1 - incident shock wave; 2 - dif­ fracted shock wave; 3 - reflected shock wave. The time is measured from the instant the incident shock wave reaches the edges of the cavity. Air5 M = 1.7, Re = 2.8 x 105? a - t = 20 jus; 6 - 50 jus; c 70/is; d ~ 80/is; e - 110 /is.

Reflection

from concave body...

151

Fig. 59. Quantities r and d against s; a - air; b - CF4.

17. O s c i l l a t i o n s o f t h e s h o c k w a v e r e f l e c t e d f r o m a b o d y w i t h c a v i t y The interaction of a shock wave with a concave body can be divided into three stages: (1) the propagation of the incident shock wave inside the cavity; (2) the motion of the reflected wave within the cavity; (3) its motion outside the cavity. We studied the third stage while carrying out the experiments with an axisymmetric model (the hollow cylinder with an open end). The Reynolds number did not exceed 10 5 . The process is as follows. The incident shock wave reaches the b o t t o m of the cavity and reflects from it. Meanwhile a stationary flow m a y arise near the edges. Attached shock waves occur whose parts interact with one another and reflect from the walls of the cavity. It is a primary wave configuration. The reflected wave catches up with it. If the depth of the cavity is small enough

Propagation

152

and reflection

of shock

waves

then the flow near the edges is non-stationary. Afterwards the reflected shock leaves the cavity and transforms into a bow wave. Experiments show that there are d a m p e d oscillations of the bow shock wave during the transition to the stationary flow past the concave body. The time during which the flow becomes steady depends linearly on the depth of the cavity (see Fig. 60). The regularity of the oscillations is violated at high values of the depth of the cav­ ity (l/d > 3). The amplitude of the oscillations depends on the depth of the cavity (see Fig. 61). Solid line corresponds to the calculated values, dots refer to experimental data. An explicit conservative two-layer (with respect to the time) scheme with shock tracking was used. The scheme has the second or­ der of approximation. D a m p e d oscillations also occur if the flow is subsonic. The experimental and calculated values for the damping rate (the ratio of the amplitude of the second period of oscillations of the bow shock wave to the amplitude of the first) increase as the depth of the cavity increases (Fig. 62).

0

tt3

0.6

0.9

1/h

Fig. 60. The value of r against the depth of the cavity, r being the time interval during which the bow shock wave reaches its steadystate position. The frequency of the oscillations corresponds to the fundamental acoustic frequency of the cavity with a wavelength that is four times the distance from the cavity base to the mean shock-standoff distance (Bohachevsky et al. 1972), taking into account the variation of the flow parameters within the shock layer

Here S is the mean shock-standoff distance, v is the mean gas velocity

Reflection from concave body...

Fig. 61. The amplitude of oscillations against the depth of the cavity; dots are experimental data, solid line is the calculation by Grudnitskii et al. (1984); A — (<Jmax — ^min)/(2/i), 6 is the shock standoff-distance from the body.

Fig. 62. Damping rate against the depth of the cavity.

153

154

Propagation

and reflection

of shock

waves

v — ^ , c is the velocity of sound, c = ( Cl + C *). Subscript 1 refers to the values behind the stationary shock wave, 0 refers to the value behind the wave reflected from a flat wall, asterisk corresponds to the value at the stagnation point. Maximum variation of the bow shock velocity is equal to 200 m / s during one oscillation. The corresponding pressure variation is equal to 30%. As mentioned above, a secondary shock wave m a y appear behind the shock wave reflected from the b o t t o m of the cavity. Its velocity is close to that of sound. The secondary shock meets with the reflected shock after it leaves the cavity thus giving rise to a new shock wave. When the reflected shock goes out of the cavity, a rarefaction wave arises and it propagates inside the cavity. The bow shock wave begins to move towards the body after the interaction with the reflected rarefaction wave, and it passes its mean standoff distance. The rarefaction wave transforms into a compression one on its reflection from the bow shock wave. The compression wave reflects from the bottom of the cavity and then meets with the bow shock wave which begins to move away from the body. From the experiments carried out, it is possible to distinguish two basic causes which lead to the appearance of oscillations and which determine the characteristic features of the process: 1) the emergence of a rarefaction wave as the reflected shock leaves the cavity; 2) the appearance of a secondary shock wave which propagates behind the reflected wave, and the subsequent interac­ tion of the waves. In the general case, excitation of flow oscillations occurs when disturb­ ances enter the cavity. The oscillations are d a m p e d due to the weakening of the disturbances as a result of the repeated interactions and reflections. The oscillations of the bow wave and the flow parameters behind it need to be taken into account in the calculation of the effect of shock waves on various bodies with cavities. At the initial instants after the reflection of the shock wave from it, such a body experiences a pulsating load, which can substantially change the effect of the wave on the body.

18. Shock focusing When a plane shock wave reflects from a concave wall, a converging shock arises, resulting in a locally high pressure. This phenomenon is a particu­ lar case of shock focusing. Pioneer investigations in this field were made by Meshkov (Meshkov 1970). Sturtevant (Sturtevant et al. 1976), Nishida (Nishida et al. 1986), Milton (Milton et al. 1987) revealed many features of

Reflection

from concave

body...

155

shock focusing. As shown by Sturtevant, the phenomenon is different in the case of weak shocks and of strong shocks. If the shock wave is weak then the wave fronts are crossed after the focus. The focus is defined as a position where a m a x i m u m pressure p m a x is realized. For stronger shocks, the wave fronts are uncrossed. If the wave is weak then a shear layer arises behind the triple-shock interaction on the trailing wave front. In the case of a strong wave the shear layer arises behind the shock at the focus. The m a x i m u m pressure is limited by nonlinear effects. Transverse shock waves which are accompanied by rarefaction waves pro­ pagate behind the reflected shock wave. Characteristic feature of shock focusing is a sharp pressure peak. The pressure rises near the focus and then decreases due to the rarefaction waves. Babinsky et al. (1996) showed that modification to the geometry of the cavity may change the m a x i m u m pressure as well as the shape of the high-pressure region. Namely, cavities with blunt edges can in­ crease the m a x i m u m pressure and they also increase the area subject to relative large pressures. In the case of vibrational excitation, a pressure peak does not appear (Kishige et al. 1995). The pressure amplification K is the ratio pmax/pi. Here pi is the pressure behind the shock wave reflected from a flat wall at normal incidence. The gasdynamic focus is less than the geometrical one. Flow pattern of unsteady reflection of a plane shock wave from a body with a cavity is given below. The experimental facility was described above. The models had the shape of a parallelepiped whose height was h = 10 m m with a cavity in front. The width of some models was equal to the distance between the glass windows in the test section of the shock tube. Besides, we used a concave model with cross-section a 10 m m square. In the former case the flow past the model was close to two-dimensional, while in the latter it was three-dimensional. The generatrix of the cavity was an arc of circumference. Its radius was equal to 5 m m (deep cavity) or 1 0 m m (shallow cavity). Fig. 63 shows the pressure ratio P2/P1 and pressure amplification K against the dimensionless value a — h/4R (R is the radius of curvature at the b o t t o m of the cavity). The value p^ is the m a x i m u m pressure measured at the b o t t o m of the cavity. The values of K were calculated by Nishida (1990). The shape of the cavity was expressed as y — ax2. There is a good agreement between the experimental values of the ratio pijpi and calculated ones of the pressure amplification though the values of pressure refer to different points in the flow and the shapes of the cavity are not the same in both cases. It follows from Fig. 63 that at M > 2, the ratio P2/P1 does not depend

156

Propagation and reflection of shock waves

Fig. 63. Pressure amplification K and the ratio P2IP1 against geo­ metrical parameter a. a — h/AR, R is the radius of curvature at the apex, h is the height of the model, K = Pmax/Pi> Pmax is the pressure at the focus, p\ is the pressure behind the shock reflected from the flat wall, p^ is the maximum pressure at the bottom of the cavity. 1 - M = 2 - 4 (Nishida 1990), air, K; 2 - M = 1.35 - 3.3, air, p2/pu 3 - M = 4.2-5,CF4>P2/pi; 4 - M = 2.9,CCl2F2,P2M. appreciably on M and 7 = cp/cv and so does the value of K (Nishida 1990). This conclusion is valid for circular and parabolic cavities (Shugaev et al. 1990). Fig. 64 presents the measured and calculated values of pressure against time at the bottom of the cavity. There is a satisfactory agreement between calculation and experiment. The ray method was used. The flow pattern behind the reflected shock wave depends on the fact whether the reflection is regular during the whole period of time or there is a transition from Mach to regular reflection. In the latter, the flow pattern reveals complicated wave interactions. Figs. 65 and 66 show shadowgraphs taken in the experiments. Here 1 is the incident shock wave, 2 is the reflected wave, 3 is the transverse wave, 4 is the slipstream, 5 is the line of intersection of the shock wave with the glass window. First of all, the incident shock wave hits the edges of the cavity and a Prandtl-Meyer flow arises. The incident shock wave reflects from the wall of the cavity. In the case of the deep cavity the Mach reflection exists. The angle of incidence diminishes continuously and the transition to the regular reflection

Reflection

from concave

body...

157

Fig. 64. Pressure at the bottom of the cavity against time, dots are experimental data, solid line represents the calculation by the ray method; t — tvr/R, where vr is the velocity of the shock reflected from a flat wall, R is the radius of curvature at the b o t t o m of the cavity.

occurs. As a result, the reflected shock wave has angular points: one part of the shock front is convex, the other (central) part being concave (see Fig. 66). Slipstreams arise and vortices are generated as a consequence (see Fig. 666, c). Two shock waves arise near the edges of the deep cavity as one can see from Fig. 66 a, 6. One of them is the reflected shock. The origin of the other is due to the fact that the Mach stem becomes curvilinear near a concave wall and its reflection from the wall takes place (Henderson et al. 1975; Syschikova et al. 1976). The second shock has an angular point which probably corresponds to the transition from the Mach reflection of the first shock to the regular one. The interaction of shock waves results in additional slipstreams and transverse waves. The formation of the second shock has influence on the flow pattern of the focusing process. There is no second shock in the case of the shallow cavity when the reflection is regular throughout the time of the process. Figs. 66 c, 67 present a rhombus pattern which consists of contact discontinuities. A jet appears just at once after focusing (see Fig. 66 c and the next). T h e jet carries away the rhombus pattern. It is interesting to note the formation of a specific polygonal structure containing vortices, as seen from Fig. 66 c. This structure consists of contact discontinuities which appear as a result of the interaction between the transverse waves and the second shock. It exists during the time interval At = I.6//C2, c-i being the sound velocity behind the incident shock wave, then it disappears (see Fig. QQ e) and a conventional jet flow is seen, as

oo

;rgf|?;

"8"

a

d

o a.

Fig. 65. Reflection of a plane wave from a concave body (three-dimensional flow). Air, M = 2.1, p 0 =15.3kPa, l/h = 0.4; a-t = 6/JS; b - 9/is; c - 12/is; d - 18/is; e - 27 /is; / - 40 /is; $ - 70/is; 1 - incident shock wave; 2 reflected shock wave; 3 - transverse waves; 4 - slipstream; 5 - line of intersection of the shock wave with the glass window.

O 3

3, S3O

Reflection from concave body...

Fig. 66. Two-dimensional low pattern (air), l/h = 0.5, M = 2.9, p 0 = 5.3 kPa, Re = 3.8 x 104; I is the depth of the cavity, a ~ t = 7fiS]b-10fis;c30 fis; d - 45/JS; e - 5Q|is; / - 85/is; the time is measured after the incident shock reaches the bottom of the caYity; 1 - incident shock wave; 2 - reflected shock wave; 3 slipstream; 4 - transverse wave.

159

\*\ X

Propagation and reflection of shock waves

Fig. 87. A rhombus iow pattern.

Reflection from concave body...

Fig. 88. Three-dimensional flow pattern inside the cavity. M = 2.75 air as the test gas; a - i = 5/is; b - 13/is; c - 15/is; d - 30//s; e 35 /is; / - 90 /is; 1 - incident shock wave; 2 - reflected shock wave; 3 - transverse waves; 4 - slipstream; 5 - line of intersection of the shock wave with the glass window.

161

162

Propagation and reflection of shock waves

shown in Fig. 66 e. Two vortices (or a vortex ring in the three-dimensional flow) are situated at the end of the jet. The jet still exists when the flow becomes quasi-stationary. A rhombus of a size larger than before arises once more when the reflected shock wave leaves the cavity (see Fig. 66 d). Its origin is the result of the interaction of the reflected shock wave with the transverse ones. Fig. 68 shows the three-dimensional flow. One can clearly see the appear­ ance of the rhombus (Fig. 68 6), the polygonal structure (Fig. 68 c) and the jet (Fig. 68 d). Fig. 69 shows the ratio l\/L against the Mach number M of the incident shock, 11 being the maximum length of the jet, while L is equal to the sum of half an arc of the generatrix of the cavity and its depth. The lower and upper data correspond to the three-dimensional flow and to the two-dimensional one, respectively. The data are presented for carbon tetrafluoride (circles and right crosses) and air (triangle and inclined crosses). It is seen that the dimensionless length of the jet increases linearly as the Mach number increases.

Fig. 69. Dimensionless length of the jet against Mach number; /i is the maximum length of the jet; L is the sum of half an arc of the generatrix of the cavity and its depth. 1 , 2 - air, 3 , 4 - CF 4 , 1 , 4 two-dimensional flow, 2 , 3 - three-dimensional flow. Thus the reflection of a plane shock wave from a concave body is accom­ panied by the formation of vortices (Ibrahim et al. 1985; Milton et al. 1987). A jet is found to arise behind the reflected shock wave. The length of the jet depends on the depth of the cavity and the shock wave strength. The jet still exists when the flow becomes quasi-stationary. Two shock waves appear near the edges of a concave body if there is a transition from Mach to regular

So

o

8.

Fig. 70. Unsteady flow past a concave body. CF 4 , M = 2,6, f?o=120kPa? l/h = 0.5; a - t = 30/is; b - 35/is; c 50/JS; d -60/is, e - 100/is. 1 - reflected shock wave; 2 - transverse waves; 3 - line of intersection of the shock wave with the glass window.

C*5

164

Propagation and reflection of shock waves

reflection. A rhombus emerges in the flow behind the reflected shock wave. 19. Resonant excitation of vortices behind the reflected shock wave There are some peculiarities of the flow field past bodies with cavities at high Reynolds numbers. Vortical structures may become unstable. The time during which the flow becomes steady is twice the corresponding value relating to the body without cavity (shallow cavity, ///i=0.15, M=2.6). Let us consider an unsteady flow of CF 4 past a concave body at M < 4 (see Figs. 70 and 71). In this case the vortices are stable. As stated above, the characteristic feature of the flow behind the shock reflected from a con­ cave body is the formation of a jet. It is laminar at low Reynolds numbers. Vortical structures are situated at the end of the jet. Vortices also arise near the walls as a result of the instability of slipstreams. Fig. 72 shows success­ ive shadowgraphs. The vortices move towards the reflected shock. Afterwards they stop and turn into two cylindrical vortices (or into a vortex ring when the flow is three-dimensional). If the Reynolds number is high enough then the vortical structures become unstable. In the case under consideration they become unstable at the end of the jet. Waves propagate along the vortex ring, i.e. azimuthal instability arises. Distortion of the vortex ring takes place, as one can see from Fig. 72. In addition, radial (axisymmetrical) waves propagate along the vortex ring if the Reynolds number is equal t o 2 - 3 x l 0 5 and the Mach number is equal to 4.5 - 5. Instability of vortices may cause a disturb­ ance on the shock front (Figs. 72 e, 73). Fig. 74 shows the oscillations of the vortex ring. The core of the ring becomes elliptic and disturbances emerge on the shock front (Fig. 75). The motion of the disturbance on the shock front is oscillatory. Two oscillations were seen at the streak-camera records. The vor­ tical structures are destroyed periodically, the flow inside the cavity becomes turbulent and the gas flows out of the cavity. Then the vortical structures ap­ pear anew and the process repeats. We indicate some characteristic features of the three-dimensional flow past a body with a rectangular cavity {l/h = 0.4). Fig. 76 shows the quasi-stationary flow pattern inside the rectangular cavity (M=4.9). One can see small vortices which appear during the interaction of the reflected wave with transverse ones. The vortices are carried away to the angles of the cavity and then to the area of flow separation. Thus the vortical structures are situated near horizontal walls and not near the plane of sym­ metry as in the former case. Later on they transform into a single large vortex which oscillates: the diameter of the vortex core diminishes and then increases.

Reflection from concave body,..

Fig. 71. Two-dimensional flow pattern inside the deep cavity; l/h = 0.5; M = 4; pQ = 1.3 kPa; a - t = 20/is: 6 - 30/is; c - 35/is; d 50 /is; e - 80 /is; the time is measured from the instant the incident shock reaches the bottom of the cavity (/ is its depth).

165

168

Propagation and reflection of shock waves

Fig. 72. Three-dimensional flow pattern inside the deep cavity; l/h = 0.5: M = 4.5; p 0 = 6 kPa; a - i = 7 /is; b - 15 /is; c - 30 /is; cf - 80 /is; e - 70/is; / - 90 /is.

Reflection from concave body...

Fig. 73. Disturbance on the shock front, M = 4.5, CF 4 as the test gas.

187

188

Propagation and reflection of shock waves

Fig. 74. Three-dimensional flow pattern inside the shallow caYity; l/h = 0.15; M = 4.9; po = 4kPa; a - i = 7/is; b - llfis; c- 80/is; d - 90/is; c - 100 jus; / - 130/is.

Reflection from concave body...

/

■

Fig. 75. Disturbance on the shock front (shallow cavity).

169

o

I £

s ex

Fig. 76. Three-dimensional flow pattern inside the rectangular cavity; CF 4 ; llh = 0.5; (/ is the-depth of the cavity V M = 4.9;Re = 10 s ; Po = 4kPa; a-t = 50 /«; b - 70/is; c - 90/zs; d - 100 ^s; e - 110/is; / - 120 ps.

i

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55" O

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? O

I

Fig. 77. Disturbance at the shock front (rectangular cavity].

i—4

172

Propagation and reflection of shock waves

The pressure varies non-monotonously at the bottom of the cavity. As a con­ sequence, a disturbance appears against the vortex (Fig. 77). Disturbances appear in the two-dimensional flow at the same values of the Reynolds number and Mach number (Shtemenko et al. 1993; Serov et al. 1995). The amplitude of the pressure oscillations at the bottom of the cavity is equal to 10% of the pressure mean value behind the shock wave in the case of the deep cavity. The period of oscillations coincides with the time during which the vortex moves from the bottom of the cavity to the vortex ring. Fig. 78 shows the pressure variation against time at the bottom of the cavity. Fig. 79 presents the pressure variation against time at the flat-nosed body under the same conditions. Oscillations are not seen in the latter.

Fig. 78. The pressure at the bottom of the deep cavity against time. A moving vortex is known to emit acoustic waves while changing its velo­ city. The velocity of vortices is not constant inside the jet, namely, they stop near the vortical structures. We calculated the amplitude of pressure oscilla­ tions caused by the deceleration of vortices. It is equal to 12% of the value of the pressure behind the shock. Disturbances caused by pressure oscillations reach the bottom of the cav­ ity, reflect from it, meet with the vortical structures and interact with them. Then they reach the shock front. Disturbances on the shock front arise under appropriate conditions, as was already mentioned. Let us consider the inter­ action of acoustic waves with a vortex ring (Kopyov et al. 1987; Vlasov et al. 1978). The interaction may be resonant or not. If there is no resonance then the vortex ring oscillates as a whole, the frequency being equal to that of the

Reflection from concave body...

173

Fig. 79. The pressure on the flat-nosed body against time. incident acoustic waves. If the frequency of the acoustic waves is a multiple of the natural frequency of the core of the vortex ring then the ring becomes unstable. Radial waves propagate through the vortex ring. The ring emits acoustic waves and their amplitude may be several times as great as the amp­ litude of the incident waves. In our experiments the interaction was resonant. In fact, the amplitude of pressure oscillations increased with time, there were pulsations of vortical cores. One can see an unstable vortex ring from Fig. 74 b (shallow cavity). The fre­ quency of oscillations of the vortex core was measured from the streak-camera records and it was equal to 170 - 220 kHz. We measured the size of the vortex core from shadowgraphs. Its diameter was equal to 0.3 mm. The velocity of the jet was equal to 160 m/s. Consequently the natural frequency of the vortex ring was equal to 250 kHz (Lamb 1932). The frequency of pressure oscillations was equal to 150 - 200 kHz at the bottom of the cavity. The frequency of oscil­ lations was equal to 50 - 70 kHz in the case of deep cavity. In experiments we measured the time between two successive disturbances on the shock front. We calculated the frequency of the acoustic waves that caused these disturbances. We used the following formula (Shugaev 1983). ( 7 + 1) (M 2 - 1) rs

w = (7 + 1) M 2 + 2,

q = 2 7 M 2 - 7 + 1.

Here rp is the period of the pressure oscillations, rs is the period of oscil­ lations of the bow shock. The amplitude and frequency of the acoustic waves

174

Propagation

and reflection

of shock

waves

coincide with those recorded with the aid of the pressure transducer. Thus the following conclusions can be made: 1) disturbances on the bow shock wave are caused by the formation of vortical structures inside the cavity and their instability; 2) they arise due to pressure oscillations within a small region behind the bow shock wave. Fig. 80 demonstrates the natural frequency of the vortex ring core and the frequency of acoustic waves against Mach number. The values of the natural frequency of the vortex ring core were found from its sizes and the velocity of the jet. The frequency of acoustic waves was found from their wavelength. A vortex which moves inside the jet stops at that instant when the successive vortex arrives at the position near the plane of symmetry. In other words, the wavelength is equal to the distance from the b o t t o m of the cavity to the point where vortices stop. Index 1 relates to the shallow cavity, index 2 relates to the deep cavity, a denotes acoustic waves, b denotes vortex ring. The size of the shaded area denotes the uncertainty in frequency determination. One can see that both values of frequency coincide at M = 4 - 5, i.e., in the range where disturbances on the shock front exist.

Fig. 80. Dependence on Mach number of the natural frequency of the vortex core and of the frequency of acoustic waves; 1 corres­ ponds to shallow cavity, 2 to deep cavity, a corresponds to acoustic waves, b to vortex ring. An additional argument, which supports the above-mentioned mechanism, responsible for the origin of disturbances, is as follows. Bright rings are seen at the shadowgraphs near an unstable vortex (see Fig. 74 c, d). T h i s system of bright rings arises as a result of light diffraction, R a m a n - N a t diffraction at

Reflection from concave body...

175

the acoustic waves propagating from a vortex ring (Bergmann 1954). Fig. 81 presents the diffraction pattern which corresponds to the low past the same model when it was turned 90° around the axis of the shock tube. A fringe system is seen near the model. The asymmetry of the diffraction pattern is due to the fact that the vortex ring core emits non-planar waves (Bergmann 1954). Acoustic radiation from a cylindrical vortex is like that from a quadrupole (Dosanjh et al. 1965). Acoustic radiation from a vortex ring is very complex. Probably, it is like the radiation from, octupole. A few narrow beams may arise due to the distortion of a vortex ring.

Fig. 81. Raman-Nat diffraction; the concave body is turned 90° around its axis; a - i = 60|is; b - 100 fis. If the vortex ring emits a few acoustic beams in the direction towards the shock front then a few disturbances may appear on the shock front. In fact, two disturbances are seen on the shock front (see Fig. 73). Let us find the value of acoustic wavelength A. Light beam (Fig. 73) intersects at least two acoustic beams. We use the following formula (Bergmann 1954) d

nK

L = T* Here d is the distance between the maxima of light, n is the number of acoustic waves that lie in the light path; A is light wavelength, L is the distance between the model and the screen. In our case n = 2,A = 6 9 4 3 4 , d = 0.212 m m . L = 150 mm. Therefore A

Propagation and reflection of shock waves

176

is equal to 1 mm. The frequency of the pressure oscillations is equal to J/ =

(£^1 A

=

200 kHz.

Here c is the velocity of sound behind the shock wave, v is the gas velocity. The frequency of pressure oscillations that is found from the pattern of Raman-Nat diffraction coincides with the frequency of pulsations of the vortex ring core and that one of the oscillations at the bottom of the shallow cavity. Analysis of the flow inside the deep cavity confirms the above-mentioned con­ clusions. Fig. 73 / shows the pattern of Raman-Nat diffraction near the shock front. Four acoustic beams arrive at the shock front. Three disturbances are clearly seen on the shock front. The vortex ring emits narrow acoustic beams. Disturbances in the flow past various bodies (CF4 or CCI2F2 as test gases) have been observed earlier in ballistic experiments (Hilton 1957; Bedin et al. 1981; Baryshnikov et al. 1980). It was found that disturbances on the shock front arose at lower Mach number if the body was concave. But there was no adequate explanation for this phenomenon. In our experiments the disturbances arise on the shock front at low Mach numbers of the gas flow (M = 2.3 - 2.6), so that relaxation processes cannot influence the flow. Thus disturbances on the shock wave in front of a concave body may appear under appropriate conditions. The origin of the disturbances is a resonant interaction between acoustic waves and vortical structures inside the cavity. It is worth mentioning that in our case the resonant frequency is five times the fundamental acoustic frequency of the cavity.

CHAPTER 6. P R O P A G A T I O N OF A SHOCK WAVE T H R O U G H A T U R B U L E N T GAS FLOW

Propagation of a shock wave through a random medium is accompanied by nonlinear interaction of the wave with disturbances. The shock wave influences disturbances which in their turn can distort it or change its strength. The passage of a shock wave can enhance turbulent mixing (Alessandri et al. 1995). Experimental studies were fulfilled by several authors. It was found that the energy of turbulent fluctuations increased after the passage of the shock wave (Wintrich and Merzkirch 1995; Briassulis and Andreopoulos 1995). The shock wave is distorted while propagating through a random medium (Hesselink and Sturtevant 1988). Experiments show that the amplification of tur­ bulent fluctuations depends on their amplitude (Azarova et al. 1997). Exper­ iments were performed in a single-diaphragm shock tube. A turbulence grid was used. The incident shock wave reflected from a flat-nosed cylinder in the test section. Then the wave propagated through the turbulent flow. A laserschlieren technique was used. In experiments, the density fluctuations Ap/p were equal to 5% and 9%. Fig. 82 shows the values of the amplification coef­ ficient against the Mach number of the shock wave (a refers to Ap/p = 5%, b refers to Ap/p = 9%). One can see that the amplification coefficient increases as the Mach number increases and decreases as the amplitude of the density fluctuations increases. The turbulent length scale was equal to 0.3 mm ahead of the shock wave and 0.2 mm behind the wave. This fact agrees with the result of numerical simulation by Lee et al. (Lee et al. 1993) who found a decrease of the turbulent length scale caused by the shock wave. A numerical simulation was also performed. The Euler equations were used. The motion was assumed to be one-dimensional. Turbulent fluctuations were modelled as a sequence of ten pulses of gas velocity. Their amplitude was defined as a series of random values, the standard deviation being equal to Mtc\. Here ci is the sound ve­ locity, Mt is a numerical parameter. A scheme with shock tracking was used. The amplification coefficient k is defined as follows k =

(AP2)/(APl), 177

Propagation and reflection of shock waves

178

Fig. 82. Amplification coefficients against Mach number, Ap/p = 5%; b - Ap/p = 9%.

a

Table 4. MM< 1.1 1.2 1.5 2.0 2.5 3.0

0.3

0.05

0.1

1.09

1.05 1.22 1.30

(AP1) = W\)\l\

1.20 1.38 1.62 1.76

(A/*) =

0.5

1.70 1.95 2.23

{p'lfl\

Here p\ and pf2 are the density fluctuations in the points x\ and x2 ahead of and behind the shock wave, respectively. The angular brackets indicate ensemble average and the upper bar refers to a time average. The values of k are given in Table 4. It is seen that the amplification coefficient increases as the Mach number increases, in agreement with the experimental data. The autocorrelation func­ tions of the density fluctuations are presented in Fig. 83 (M — 2,Mt — 0.3).

Propagation of a shock wave... 1.0

Fig. 83. Autocorrelation functions of the density fluctuations.

Fig. 84. Spectral functions ahead of and behind the shock wave.

179

180

Propagation and reflection of shock waves

Fig. 84 shows the spectral functions ahead of and behind the shock wave. The interaction of a shock wave with a turbulent flow occurs if there is a supersonic flow past a spiked body. Separation appears in the flow over such a body. Below we consider an unsteady flow past a flat-nosed cylinder with spikes. The use of spikes is known to reduce drag and heat flux at high speeds

Fig. 85. Positions of spikes and of the pressure transducer (dimen­ sions are given in mm). of flow (Chang 1970). Aerodynamic characteristics depend on the length of a spike / and on the diameter d of the body. Under certain conditions, oscillations may arise in the flow. They are caused by rejunction of the separated flow. The oscillations may be essentially reduced if a small disc is installed at the end of the spike (Belov et al. 1989). The drag is also reduced. Separation arises due to the spike. It is usually assumed that the conical region of separation may be replaced by an equivalent solid cone. The pressure is proposed to be constant within the region of separation, and it is supposed to be equal to the pressure behind the conical shock wave which is caused by the equivalent solid cone. But these assumptions are not valid. The measured value of the stagnation pressure at M = 1.96 is less than that one on the surface of the cone (Chang 1970). In most of the experimental studies only laminar separation of the boundary layer was investigated. Some investigators observed

Propagation of a shock wave...

181

the separation of transitional boundary layer. The drag was 30 - 40% higher as compared with the case of laminar boundary layer. It was found that the drag increases as the Reynolds number increases, and decreases as the Mach number increases. There is a sharp growth of the drag if there is transition from laminar to turbulent flow. Almost all the authors investigated only steady flow. The transition to the steady flow past a body with spikes was studied by Burova et al. (Burova et al. 1993). A flat-nosed cylinder with spikes was chosen as a model. Experiments were fulfilled in a single-diaphragm shock tube of rectangular cross-section of 34 x 72 mm 2 in area. CCI2F2 was chosen as the test gas. The flow Mach number M was equal to 1.8 (initial pressure po = 140 Torr) and 2.6 (po = 30 Torr). The flow duration was 600 fis in the former case and 120 fis in the latter case. The Reynolds number was 5.3 x 105, the diameter of the cylinder was taken as the characteristic length. Shadow pictures were taken and the pressure on the nose of the body was measured. The body diameter d was 12 mm (for experiments in which shadow pictures were taken) or 17.5 mm (for experiments in which the pressure was measured). The quantity N of spikes varied from 1 to 7. Fig. 85 shows the positions of spikes and of the pressure transducer. The length of the spikes varied from d to 3d. A Kistler type transducer was used. a) Flow past models with two spikes. When a plane shock wave reaches the model, weak conical shocks arise at the ends of the spikes. After that the incident shock wave reflects from the flat nose of the cylinder. The separation of the boundary layer occurs at the spikes near the reflected shock wave, the apices of the conical shocks being situated near the points of separation. Shadow pictures are shown in Fig. 86. The time is measured from the instant the plane shock reaches the flat nose. The separated flow is turbulent. The points of separation move upstream along the spikes. The regions of separation join one another, interact, and an X-shaped configuration arises (see Fig. 86, d, e and Fig. 87). Such a flow seems to be unstable. The points of separation begin to oscillate. This effect may also appear due to large distance h between spikes (h/d= 0.6). Fig. 88 shows the flow past a cylinder with two spikes. As compared with Fig. 86, the model is turned 90° around its axis. One can see that the region of separation has two planes of symmetry. The conical region of separation within one of the planes of symmetry includes the bow shock wave while within the other plane (plane B) conical shocks join the bow wave (see Figs. 86 and 88). Such a pattern is a result of the position of the spikes at the nose of the cylinder. The instability of the bow shock wave may be explained as follows. The shock front is not smooth if the wave propagates through a turbulent flow. If some

182

Propagation and reflection of shock waves

Fig. 88. Shadowgraph of the low o¥er the model with two spikes. CCI2F2 as the test gas.

Propagation of a shock wave...

183

Fig. 87. Interaction of conical shock waves. An X-configuration is seen. Mi = 2.8. flow property varies considerably in the direction parallel to the shock front ahead of it, then the splitting of the wave occurs. When the flow becomes quasi-stationary, the separation occurs at a point which is situated at some distance from the end of the spike. b) Flow past a model with three spikes. Such a flow is shown in Figs. 89 (I = $d) and 90 (f = 2d). There is an essential distinction as. compared with the model having two spikes. In the present case conical regions of separation interact with one another in such a way that a single region occurs. That region is bounded by one conical shock. The separation arises at the end of the central spike (I = 2d) or at some distance from the end of that spike (I > 2d). A similar pattern was observed in the flow over a body with one spike having a disc or wedge at its end (Khlebnikov 1995). The flow pattern depends on the dimensions of the model as well as on the flow Mach number and on Reynolds number. It is interesting to note that a single separation region occurs in the flow over a grating of pointed cylinders whose axes are parallel to the undisturbed velocity (Watarai et al. 1997). c) Pressure history in the case of uneven number of spikes. The position of the pressure transducer was 4.5 mm apart from the axis of the model. Thus

184

Propagation and reflection of shock waves

Fig. 88. Shadowgraphs of the flow o¥er the model with two spikes (the same conditions as in Fig. 86). The model is turned 90° as compared with Fig. 86. a - t = 30j*s; 6 - 60 /is; c - 90 /is; d 120/is.

Propagation of a shock wave.

ilfi^v^mmmtMUMmwmM\mimmwwK?»*'» -

Fig. 89. Shadowgraphs of the low of the model with three spikes. CC12F2 as the test gas. Mi = 2.8, Z = 3d. a - < = 10/is; fc - 30 fis; c- 35/is; d - 5 0 p s ; e- 70/is; / - 90/is.

a

Propagation and reflection of shock waves

Fig. 90. Shadowgraphs of the flow with three spikes. CCUF, as the test gas. M, = 2.6, / = 2d. a - t = 80 jis; b - 100 ps; c - "l20 ps.

Propagation of a shock wave...

Fig. 91. Ratio pr/ps against shock Mach number; pr is the pres­ sure behind the reflected shock wave at the initial instant, ps is the stagnation pressure for the steady flow. Solid line represents the calculation, dots are experimental data.

Fig. 92. Pressure history on the surface of the flat-nosed cylinder. Mi = 2.6, / = 2d. Flow duration is 120 ps.

187

188

Propagation

and reflection

of shock

Fig. 93. Pressure history on the surface of the flat-nosed cylinder. M i = 1.8, / = 1.5c/. Flow duration is 600/is.

Fig. 94. Pressure history on the surface of the flat-nosed cylinder. M i = 1.8, / = 2d. Flow duration is 600 /is.

waves

Propagation

of a shock

wave...

189

the measured value of the pressure was 2% less than that one in the center of the nose (Boison et al. 1959). The sensitive part of the transducer has a finite area. This circumstance involves further reduction of the value of the measured pressure by 3 % . The pressure history was obtained in the following cases: a) model without spikes; b) models with one spike, with three spikes and with seven spikes. The ratio l/d was as follows: l/d = 1; 1.5; 2. The flow Mach number was M = 1.8; 2; 3. Fig. 91 shows the ratio pr/ps against the Mach number of the incident shock. Here pr is the measured pressure at the initial instant, ps is the calculated value of the stagnation pressure which refers to the steady flow past a body. Solid line represents the calculated values, dots are experimental values. One can see that there is a good agreement between experimental data and calculated ones. Fig. 92 presents the pressure history on the body surface ( M - 2.6, Re = 7.5 x 10 5 , l/d = 2, p0 = 10 Torr). The time is measured from the instant the shock wave reaches the body. As seen from Fig. 92, when the flow is quasi-stationary (t = 100 - 120//s), the pressure has the values: p = 0.45p 5 (N = 7) and p = Q.3ps (N = 1; 3). A sharp rise of the pressure occurs at t = 50//s. Apparently, it is due to the interaction of the conical shocks. The pressure on the body nose exceeds by 40% that value behind the conical shock wave (JV = 3). One m a y assume that the shock wave does not disappear in the separation region but it is reduced. Pressure oscillations are reduced in the case of N = 7 during quasi-sta­ tionary flow. This effect is clearly noticeable at low Mach numbers. Fig. 93 demonstrates pressure history for Mach number M = 1.8 (l/d — 1.5, po — 140 Torr). One can see that the oscillations are reduced as the number of spikes grows. The relative amplitude of the pressure oscillations Sp/p is equal to 0.11 for N — 7, and it is equal to 0.33 for N = 3. If the model has no spikes then there is no steady supersonic flow under the aforementioned conditions. The stabilizing effect grows as the length of the spikes increases. This fact can be seen from Fig. 94. The Strouhal number is equal to 1.5 and 2, respectively. Probably, there exists an acoustical mechanism which involves the interac­ tion of a mixing layer with the base of the cylinder (Zapryagaev et al. 1991). This circumstance is confirmed by shadowgraphs. One can see small oscilla­ tions of the mixing layer. The increase of the transverse size of the region of separation weakens the aforementioned interaction and reduces the amplitude of oscillations. d) Pressure history for N — 2. In this case the transducer was situated in the center of the nose. Fig. 95 presents the pressure history. The flow Mach number M was 2.6, the initial pressure was p0 — 30 Torr, the flow duration was 120 /is. As one can see, pressure oscillations occur. The pressure increase

190

Propagation and reflection of shock waves

Fig. 95. Pressure history on the surface of the flat-nosed cylinder. Mi = 2.6. Flow duration is 120/is. at t — 80 /is is due to the interaction of the conical shocks. There is no steady flow. The conclusions are as follows. 1) The shock wave weakens while propagating through a turbulent flow, and its front loses smooth shape. 2) The ratio pSl/Ps2i where pSl is the stagnation pressure for the body without spikes and pS2 is that for the spiked body, depends on the flow Mach number. For instance, pSl/pS2 — 5 for M — 1.8, pSl/pS2 = 3 for M = 2.7. The increase in the number of spikes leads to reduction of pressure oscillations. 3) There is no reduction of pressure oscillations for a body with two spikes.

C H A P T E R 7. P R O P A G A T I O N OF A SHOCK WAVE THROUGH A GAS-PARTICLE MIXTURE

The motion of shock waves through particle-laden flows is of great import­ ance for a number of practical problems, e.g. dust explosions and propulsion systems. The process is accompanied by some interesting phenomena. Shock waves decay while propagating in a gas-particle mixture. Relaxation effects arise due to differences in the rate of change for velocity and temperature between the phases. Main features of the process have been studied theoretically by many in­ vestigators. The following assumptions are usually made: 1) there is a dis­ continuity in the gas phase, whereas the particles remain unchanged in their velocity and temperature while crossing the shock front; 2) the particle volume fraction is negligibly small. It is also assumed that the particles have no appre­ ciable partial pressure, and they do not interact with one another. Various forces influence the motion of a particle during shock propagation through the mixture: drag force, added mass force, Basset history force, and the force due to an external pressure gradient. However, only the drag force contributes considerably to the particle acceleration and to the decay of the shock wave at low loading ratio. T h e dependence of the drag coefficient on the particle Reynolds number deviates from the standard drag coefficient for a sphere in a steady flow. Namely, the drag coefficient is smaller for high particle Reynolds numbers (Re > 200) than the standard drag coefficient (Sommerfeld et al. 1993). Apparently this fact is a result of aerodynamic interaction between the particles. Experimental study of shock structure in gas-particle mixture was fulfilled by O u t a et al. (Outa et al. 1976). The facility was a shock tube mounted vertically. The diameter of the particles was equal to several /im. The particle loading ratio 77 was less than two. The quantity rj is the ratio pp/pg, where p is the mean density, subscripts p and g relate to the particles and the gas, respectively. The gas-particle mixture was at some distance from the diaphragm in the shock tube. It was found that the shock wave decayed while propagating

191

Propagation

192

and reflection

of shock

waves

through the mixture. The pressure distribution was measured behind the shock wave. If the loading ratio is small then there is a discontinuous front and behind it the pressure increases gradually. At high values of the loading ratio the discontinuity disappears. The discontinuity pressure rise is well-described by the shock relations when the volume fraction of the particles is small. The propagation of a plane shock wave through a polydispersed dust-gas mixture has been studied numerically by Kutushev and Rodionov (Kutushev et al. 1993). It is assumed that there is a continuous distribution of particles in size. Collisions between particles are also taken into account. The number of particles dn whose radius lies in the range (r,r + dr) is equal to

dn =

N(r,x,t)dr.

Here N is the distribution function of particles in size. The total number of particles is n =

N(r,x,t)

dr.

The quantities rm\n and r m a x are the m i n i m u m radius of particles and the m a x i m u m one, respectively. The equations of continuity take the form

dN d(Nvp) dt ^ dx ~ '

dpg dt

+

d(Pgvg) _ dx

The m o m e n t u m and energy equations are

mp

d(Nvp)

\~^r~

fd(Nep) " H dt

d(Nvl)

+

~dx~ d(Nepvp) dx

+

d(PgV2g)

(dpgVg) +

dt

dx

dp +

12

dx ~

'

Qj(pgEg + Ep) + — (pgEgvg + Epv) + Q^(pagvg + pav) = 0, Pg=Pg0)ai,

^P = ^Pp0)r3, r

EP

Eg = eg + -v2g,

a i + a 2 = l,

Ep=ep

/T'max

nipNEpdr, min

f

+ ±v2p,

max

/

a2 -

E

p v

- \ ^rmin

mpEpvpNdr,

^Ndr,

Propagation

of a shock

J

1

wave...

^

dr.

193

(139)

Here p(°\ a, v, e, E are true density, volumetric extent, velocity, in­ ternal and total energy, respectively; Ep is the total energy of particles per unit volume, Epv is the total energy flux of particles, p is the pressure, f^ is the drag force on a single particle, fc is the collision force on a single particle from the side of other particles, F12 is the force per unit volume on the whole ensemble of particles, qi2 is the heat transfer term. The system (139) was solved numerically (Kutushev et al. 1993). Fig. 96 shows the pressure distribution behind the shock wave propagating through a gas-particle mixture at different values of loading ratio. Solid lines are the experimental data (Outa et al. 1976), broken lines are the computational ones. Fig. 97 presents the decay of a plane shock wave. Dots are the experimental data, solid line is the numerical solution. As one can see, there is a good agreement between experimental data and calculations.

Fig. 96. Pressure distribution behind the shock wave propagating through a gas-particle mixture, a - r] = 1; b - TJ = 1 . 7 . Solid lines are experimental data (Outa et al. 1976), broken lines are the calculated values (Kutushev et al. 1993).

Provaoation waves ProvaQaiion and reflection of shock WQV€S

194

Pip- Q7 TVrav of flip nlanp ^hork Pier qfiork wave in the ffaq-nartirle ffa^nartirle mixture

r^K^e11httaJ 0 r19l3)':1978), soBdline represents

Fig. 98. Concentration of particles behind a plane shock "wave wave.

Se=9189.

Propagation of a shock wave...

Fig. 99. Concentration of particles behind a plane shock wave. Af = 1.6.

Fig. 100. Rise time At for concentration against Mach number.

195

196

Propagation

and reflection

of shock waves

The concentration of particles behind the shock was studied by Shugaev and Shishkina (Shugaev et al. 1996). Experiments were performed in a singled i a p h r a g m shock tube of rectangular cross-section of 40 x 60 m m 2 area. The length of the low-pressure section was equal to 2000 m m . There was a special apparatus which enabled one to fill the test section with smoke. The size of smoke particles was equal to 0.5-1 /im. Air was chosen as the test gas. The shock Mach numbers varied in the range 1.6-2.5. The initial pressure was equal to 100-340 Torr. Light scattering was used in order to detect smoke concentration. A He-Ne laser b e a m was directed along the axis of the shock tube. T h e scattered light passed through a slit of 2 m m width and was detected by a photomultiplier. We assumed the scattering to be single. The intensity of the scattered light was proportional to the particle concentration. The technique enables one to exclude the influence of boundary layer effects. Typical oscillograms are shown in Figs. 98 and 99. In contrast to the pressure distribution, there is no area of constant smoke concentration behind the incident shock wave. The concentration varies non-monotonously in the region between the shock wave and the contact surface. There is a m a x i m u m of concentration at some distance from the shock front. This result was found theoretically for decaying shocks (Korobeinikov 1990). The rise time is equal to 60-500 /is and diminishes as the shock strength increases (Fig. 100). Thus the particles are mainly concentrated within a comparatively thin layer behind the shock front. It is interesting to note that there are oscillations in particle distribution at low Mach numbers (Fig. 99).

C H A P T E R 8. LASER-DRIVEN SHOCK WAVES

A shock wave arises, when a breakdown occurs in a gas. Gas breakdown at optical frequencies can take place in the focused b e a m of a laser. The intensity of a laser beam necessary for it depends on many factors, namely, on the wavelength of laser light, gas pressure, duration of the laser pulse and so on. The formation of laser plasma occurs due to optical breakdown. It arises if the intensity of the laser b e a m reaches a threshold value. This value is proportional to Iu)2/(rpo), where / is the potential of ionization, UJ is a frequency of laser radiation, r is the duration of the laser pulse, po is the initial gas pressure (Ready 1971). The threshold value is equal to ~ 10 1 1 W / c m 2 in the case of air (ruby laser; atmospheric pressure) and 10 9 W / c m 2 in the case of carbon dioxide. For the first time the phenomenon of optical breakdown was studied in 1963 (Maker et al. 1964). It is interesting to note that similar processes were observed near a solid target (a low-threshold breakdown). The intensity of the b e a m may be reduced to 10 7 W / c m 2 in the case of CO2 laser. The phenomenon of a low-threshold breakdown was observed in 1973 (Barchukov et al. 1973). Shock waves that are produced by a focused laser b e a m were studied by Ramsden et al. (Ramsden and Davies 1964; Ramsden and Savic 1964). A surprising result was obtained, namely, after breakdown the spark developed asymmetrically. The luminous front moved towards the focusing lens at an initial velocity of 10 7 c m / s . In order to explain this effect, a mechanism of a radiation-supported shock wave was proposed. After breakdown, a shock wave propagates into the undisturbed gas. Further absorption of the energy of radiation takes place in the plasma behind the shock front moving towards the lens, in the manner of a detonation wave. Light absorption in plasma is due to inverse Bremsstrahlung. A free electron absorbs a photon. T h e electron passes to a state of higher energy of continuous spectrum. The process must take place in the field of either an ion or an atom, or a molecule due to the law of m o m e n t u m conservation. At an initial stage of the process the number of ions is small, and the gas temperature is comparatively low. The absorption

197

198

Propagation

and reflection

of shock

waves

coefficient kw due to inverse Bremsstrahlung has been calculated for the system, which consists of a neutral atom and a free electron, only in particular cases (e.g. in the case of a neutral atom of hydrogen). It has the following value for this system ^ re2(inena{kT)1l2 kw = 14.5 o/9 9

tnna „„uc\ (CGS units).

Here e is the charge of electron, m is its mass, a is the cross-section of elastic collisions, c is light velocity, k is the Boltzmann constant, T is the temperat­ ure, ne is the density of electrons, na is the density of neutral atoms. The numerical value of the absorption coefficient (T — 5000 K, ruby laser) is ku = 1 0 ~ 3 9 n e n a c m _ 1 . If the density of neutral atoms and that one of ions have the same order then the main absorption occurs in the field of ions. The shape of the function ku{ne) changes as the temperature increases. Nevertheless, the absorption coefficient still increases as the electron number density increases. This coefficient is equal to 40 c m - 1 in the case of hydrogen at T ~ 10 5 K if the initial density is 10 1 9 c m - 3 . This means that the laser light is almost fully absorbed by a layer of 0.025 cm in thickness. As the frequency of plasma UJV ~ nj increases up to the value of optical frequency the plasma reflects radiation. So the light does not penetrate into it. This frequency corresponds to the so-called critical density of electrons nc. This quantity is nc = 2.4 x 10 2 1 c m " 3 for ruby laser. Each of the components of plasma (electrons, ions and neutral atoms) has its own temperature. The temperature of ions is usually close to that of neutral atoms. The temperature of electrons m a y be quite different as they absorb the laser radiation. Nevertheless, the period of time during which the distribution of the absorbed energy between electrons and ions takes place is small in the majority of cases. For instance, this value is equal to 1 0 " n s for hydrogen (na = 10 1 8 c m " 3 , T = 10 4 K ) . If the duration of the laser pulse exceeds this period of time then we can put Te = T%. Two stages can be distinguished while considering the expansion of the spark. They correspond to the time before and after the cessation of the laser pulse. During the first stage the plasma front moves approximately as 0.6th power of time, decaying afterwards during the second stage to a value 0.4. The velocity of the shock wave in the direction towards the lens is higher than in other directions. The possible explanation is as follows. A strong shock wave propagates from the point of optical breakdown. The gas behind the shock wave becomes heated and ionized and therefore absorbs laser radiation. The absorption takes place within a thin layer behind the shock front. Thus the motion of the shock wave is supported by radiation within the laser beam.

Laser-driven

shock

waves

199

The radiation is fully absorbed by the thin layer of plasma. Consequently, the transverse part of the shock front propagates at a velocity which is less than that of the shock wave within the laser b e a m . This phenomenon is called "lasersupported detonation". The propagation of the shock wave is supported by the energy of laser radiation. Up-to-now we assumed that the shock front coincides with the absorption front. But there are also other types of propagation for the absorption wave. Among them two types are main, namely, subsonic wave (the propagation velocity is less than the local velocity of sound) and supersonic one (the propagation velocity exceeds the local velocity of sound). The radiation from the plasma m a y ionize the ambient gas to such an extent that it begins to absorb the main part of laser radiation. In this case the zone of absorption moves together with the ionization wave. A subsonic radiative wave appears if the gas immediately behind the shock wave is transparent for the laser radiation. The motion of the plasma front behind the shock wave proceeds at a velocity less than that of sound. As a result, the absorption wave remains behind the shock wave, and the pressure is the same for all the volume of the heated gas. A laser supported detonation wave is replaced by a supersonic radiative wave at high intensities of laser radiation, when the radiative mechanism becomes more effective as compared with hydrodynamical one. In this case the velocity of the radiative wave exceeds the local velocity of sound, and the front of the radiative wave is ahead of the shock wave. T h e boundaries for different regimes are as follows (for the radiation of C 0 2 laser) (Danilychev et al. 1983) a) F « 10 6 W / c m 2 — formation of plasma near the target; b) F — 10 6 - 10 7 W / c m 2 — subsonic radiative wave; c) F = 10 7 - 8 • 10 8 W / c m 2 — laser-supported detonation; d) F > 8 • 10 8 W / c m 2 — supersonic radiative wave. Here F is the flux of radiation. If the size of the focus is equal to 1 0 - 1 2 10 ~ cm then laser-supported detonation occurs as a general rule. Below we give fundamentals of classical theory of detonation. This theory was elaborated by C h a p m a n , Jouguet, Crussard, Michelson. During the process of detonation a chemical reaction occurs that propagates from one layer of gas to another. The velocity of detonation wave exceeds the velocity of sound in the primary gas. The endothermic reaction is provoked by shock compression of the gas. The width of the zone of chemical reaction is usually much less than the characteristic length. So the front of a detonation wave is assumed to be a discontinuity which separates the undisturbed gas from that where reaction proceeds. Let the reference frame be bound u p with the detonation

Propagation

200

and reflection

of shock

waves

wave. A particle of the undisturbed gas intersects the shock front, then a chemical reaction proceeds and energy release takes place. Afterwards the particle expands, and its pressure diminishes. The equations of continuity, m o m e n t u m and energy can be written as follows pivi Pl+Pivl ftx + ^

+ Q

=

P2V2,

=

P2+P2V\,

=

/>2 + ^ ,

Vl

= D.

(140)

Here D is the velocity of detonation, Q is the energy which is released on ac­ count of chemical reaction. We can write an analogue of the Hugoniot adiabatic *2(P, r) - fti(pi, r i ) = - ( n + r 2 ) + Q,

r=

1/p.

(141)

The quantity Q is assumed to be constant. We consider the gas that has normal thermodynamical properties. The shock wave compresses the gas, and its state becomes p r , r r . As there is energy release and the gas expands, the shock adiabatic for the products of reaction is situated above the shock adiabatic for the undisturbed gas in the plane r , p. We have from Eq. (140) v2 p-Pi = -(r-ri)-i

(142)

It is a Rayleigh-Michelson line. One can see from Eq. (142) that the density increases as the pressure increases and vice versa. The process due to which the pressure increases as compared with that ahead of the wave front is called detonation. If the pressure decreases then the process is called slow combustion (deflagration). We can deduce from Eq. (140) P2-P1

= PlVi = P2V2-

nA0. (143)

vi-v2 It follows from (143) that the detonation front reduces the velocity of the gas in the reference frame bound u p with the front. On the contrary, the combustion accelerates the gas relative to the front. The Hugoniot adiabatic gives the states behind the detonation front that are consistent with the law of energy conservation. However, there is an additional condition which the process of detonation must satisfy, namely, ^ r -

< 0. n

(144)

Laser-driven shock waves

201

Eq. (144) follows from Eq. (142). The detonation which is described by the point D (see Fig. 101), where the Rayleigh-Michelson line is tangent to the Hugoniot adiabatic, is called the Chapman-Jouguet detonation. It follows from Eq. (141) that

Fig. 101.

TdS + - ( r - n)dp - -(p -

Pl)dr

= 0.

Taking into account Eq. (144), we have at the point D dS = 0. It is valid at the point D

fdp\

fdp\

\dr)H

\dr)s

=(dp\

\dp)sdr

Using Eq. (142), we get

M = 2c. Or

dp= _c2 2

2 2

2 2

p2v2 =

p2c2.

Propagation

202

and reflection

of shock waves

Thus the velocity of the reacted gas relative to the detonation front is equal to the local velocity of sound in the C h a p m a n - J o u g u e t processes. The solution that corresponds to a point S above D (Fig. 101) gives the following value v2 + c^: V2 + c 2 > D. Thus the shock front propagates relative to the reacted gas at a subsonic ve­ locity. This means that any rarefaction wave catches up with the front and reduces it. As a result, the final state moves towards the point D. On the other hand, the solution that corresponds to a point W below D (Fig. 101) gives the supersonic velocity of the front relative to the reacted gas. In this case the energy release is carried away downstream and does not support the shock front. The velocity of the front diminishes, and the final state also tends to the point D. Let us determine flow properties behind a detonation front in the case of the C h a p m a n - J o u g u e t detonation. The velocity of sound is given by c=

y/yp/p.

We assume that the adiabatic exponent 7 does not vary across the detonation front. We have from Eqs. (140) Pi + p\v\ =p2 + P2V\ = p2 + P2{lP2/p2)

= J>2(1 + l) •

Or P1+P1D2 V'2 —

7+1

(145)

If P2/P1 > 1 then P2 =

P1D2 7+1'

From the equation

dp dr

P2-P1 T2 - n

at the point D we have P2 Pi

(7

+ 1)ff-1 ^£2 !

PI

If P2/P1 > 1 then

E l -_ 7 + 1 pi 7

(146)

203 203

Laser-driven shock waves

Let us consider the energy e< equationI (140). (140 Taking into account Eqs. (145), (146), we get Df Q ((77 - \)}D\ 1)}D{ c\ £>? 2{c? + Q(7 Q(-1 \)}D\ + + c? 4 = 0. 0. D\ - 2{c\ Q(-f Hence

+ Q?{j-l)2. D\ = c\ + Q(f Q(72 -1)± - 1) ± ^^2(7-))Qc? /2(7-))Qc? + Q 2 ( 7 - ll)) 2 .. " l — - i ' "evi

*-j -■- y -v /

- v e ^ i ■ "» v /

(147)

*; •

We must choose the upper sign in the case of detonation. If Q/c\ »» 1 then we get instead of Eq. (147) we D = V2(7 v/2(f>2 - 1)QOnly the Chapman-Jouguet processes give a steady value for the velocity of detonation of In this this case of detonation wave. wave. In case no no disturbance disturbance can can catch catch up up with with the the zone zone of chemical chemical reaction. experimental values aa good chemical reaction. There There is is a good agreement agreement between between experimental values of ■ detonation and calculated latter being calculated detonation velocity ones, the quantities detonation velocity and calculated ones, the latter quantities being calculate on on the the basis basis of of the the Chapman-Jouguet Chapman-Jouguet hypothesis. hypothesis. Modern Modern developments developments in treated, e.g. by 1987). detonation are Nettleton (Nettleton detonation Nettleton ii>ettieioii (Nettleton 1987). ueLuiiciuuii are <±rc treated, irecitcu, e.g. e.g. by uy i>eitit;toii iyoi I. laser-supported detonation. We assume assume the the front front of Let us proceed to to laser-supported detonation. Let us proceed to laser-supported detonation. We We assume the front ofof wave absorption as a singular surface (discontinuity) which propagates through absorption wave as a singular surface (discontinuity) which propagates through the gas at the gas at aa velocity velocity G. G. We We choose choose the the reference reference frame frame bound bound up up with with the the front. front. T > i ^ cold r o l r l undisturbed n n r W . i i r V w l gas cr*c transforms + r * n o f n r T n « into i n t o plasma r»l*«rna K ^ i n H the tho ffront. r r m t . The T ^ crao The behind The cold undisturbed gas transforms into plasma behind the front. The gas gas velocity is equal to G ahead of the front. The conservation laws may be written velocity is equal to G ahead of the front. The conservation laws may be written as as follows follows (Raizer (Raizer 1974) 1974) PiG P\G P\G

= p2v2, = p2V2, =

22

= = p2 p2+P2^ p 2 + PP2V P2^ 2 ^ 222, 22,,

Pll+ +PlPlG G

+ + e 2+ ei + .n =~ e2 ++"*" "o"' 6ei1 + + T. + €2 + ~o~ "■^ nG ^T ir2 Pl + 2 2 +"■"^Pl G p77 2

(148) (1 ( 14 48°) )

S Pl First of all, all, we consider a limiting case p2 = px. II II may may take take placc placc en en condition condition thai.t there thPFP PYi«t.s a mechanism m p r h * n i « m of of vPrv ffast W transfer ttransfer r a n c f p r of rvf inni*atirm at at a a velocity v-WJ+.v that re exists there exists a a mechanism of very very fast of ionization ionization at a velocity

much this assumption eater than ch greater greater than that that of of sound sound in in the the heated heated gas. gas. Under Under this assumption the the gas is at rest behind the absorption wave. In this case the radiation is at rest behind the absorption wave. In this case the radiation that that falls falls 2 on 11 cm G. Thus cm 2 of of the the front front is is absorbed absorbed by by the the mass mass of of the the gas gas equal equal to to PlpPlxG. G. Thus :hus PiGe Ge22 Pl

= F. F.

(149) (149)

ex. Now let us return to Eqs. (148). We eliminate the We assume that e2 > » ei. e^ entities G and v2 from the last equation, and we have adiabatic quantities / ,

, \

/

liV((ppi1++1 + ) f l - l )n++I FFw fI -e, = I(p );t; )t(±_ ee22-e, (L-±)+F(, PP)) 22 V'

Pi) '\pi P2j VKPi PPI l P2j P2) V '\pi P2J

\ 1/2

11 /2 P*-n )V / 2 . ^"f _ ^ ^"f \(P2-P1)P1P2J \(P2-Pi)pi

\{P2\{P2-pi)piP2j \{P2-P1)P1P2J \(P2-Pi)pip2j

(150) (150) (150)

Propagation

204

and reflection

of shock

waves

The quantity F is a parameter. Let us consider the supersonic regime. Eq. (150) is valid in this case. We assume that p2 > P i , e 2 > t\ and e — jp/(p{j

- 1)). We have

Here 7 is the effective adiabatic exponent for the plasma behind the front. More accurate equation of energy conservation has the form (cf. with Eq. (149)). P i G e a = F/?,

/ ? = ( l -

( 7

1

" ^

"

£ )

) "

1

-

(151)

Eq. (151) takes into account the motion of the gas behind the front and the change of its density. The quantity f3 is close to unity. The quantity e m a y be eliminated with the aid of the expression

The shock adiabatic in the case of laser-supported detonation is different from the shock adiabatic for explosives. In the former the absorbed energy is pro­ portional to the reciprocal of the wave velocity while in the latter the released energy is constant. The velocity of a laser-supported detonation wave is equal to the local velocity of sound behind the front relative to the gas V2 = C2 = ( 7 P 2 / P 2 ) 1 / 2 .

(152)

As is known, the same condition is valid for the C h a p m a n - J o u g u e t detonation. Making use of Eq. (152), we can find the velocity of a LSD wave and gas properties behind its front

7 + 1 7 + 1 P2 =

P\ ,

7 2 7

/

4

-=M^j)

y/3/.F\2/3

u) •

In this case the quantity (3 in Eq. (151) has the value 0 = 2 7 / ( 7 + 1).

(153)

Laser-driven shock waves

205

Let us consider a spherical LSD wave (Ramsden and Savic 1964). We assume that the intensity of laser radiation does not depend on time. The radiation flux can be written as Wo Wo = a (154) F=

W^j

^"

Here r is the radius of the wave, S is the part of the shock front on which the radiation falls, Q is the solid angle of the converging beam, a is a constant. By substituting the value of F from Eq. (154) into the first of Eqs. (153) we get >,2 _ iW;r/_ 1 ! / 3 dr dt I Pi?2 J Or

'5\3/*f2tf-l)aW0\1/B

3)

X

*

J

'5

t3'\

3 5

Thus r is proportional to t ' . This fact was found experimentally (Ramsden and Savic 1964). Eqs. (153) do not take into account the variation of effective adiabatic exponent due to heating of gas. However, this variation may be considerable. The velocity of an absorption wave may exceed that one of a LSD wave if other mechanism of ionization occurs, namely, radiative mechanism and mech­ anism of electron heat transfer (at temperatures higher than 100 eV). Such a wave propagates at a supersonic velocity relative to the gas behind it, and no hydrodynamical disturbance can catch it up. In this case the compression is less than behind a LSD wave. If the intensity of the laser beam decreases the temperature of the plasma decreases and so does the degree of ionization. The thickness of the absorbing layer increases. Let us determine the threshold value for laser intensity when LSD is still possible. The layer of absorbing plasma expands not only in the direction of the laser beam, but also in the transverse direction. The ratio of energy losses, which are due to the expansion in radial direction and those due to the expansion in transverse direction, is given by 27rr//7rr2 = I jr. A LSD wave can exist only on condition that / -C r. If the radius of a light channel is equal to 10~ 2 -10 - 1 cm then the free mean path of radiation becomes comparable with r at a temperature T ~ 20000 K. The threshold value of radiation flux is F ~ 108 W/cm 2 .

Propagation and reflection of shock waves

206

Now we consider the structure of a LSD wave in one-dimensional case. We assume the intensity of the laser radiation to be constant. The gas ahead of the wave is at rest. The radiation of plasma ionizes the layer of gas ahead of the front of the LSD wave. The initial degree of ionization in the vicinity of the front is aei < 10~ 2 . The free mean path of radiation does not exceed one mm, and the gas is neutral at a distance of several mm from the front. The electron temperature Tei ahead of the front is determined by the absorption of the laser radiation and by losses of energy due to collisions. The temperature Tei has the value T ei = 1 - 2 eV in the case of H2, He, and Ar for the regime of LSD. The period of time from the beginning of photoionization of the gas layer till the passage of the wave front through it is 10~ 8 s. The energy transfer from electrons to atoms is negligibly small due to a large difference in masses of atoms and electrons. Thus we can put ahead of the front that T\
dqk

3

e = —nkT,

p = nkT,

pe — nekTe.

Here k is the Boltzmann constant, e is the internal energy, n is the density of atoms and ions, p, T are the pressure and the temperature of atoms and ions, ne is the electron density, p e , Te are the pressure and the temperature of electrons, E is the electric field arising due to the charge separation, j is the density of electric current, q is the heat flux. The numerical results on the structure of LSD wave in argon were obtained by Fukui et al. (Fukui et al. 1995). The problem is as follows. At the initial instant we have a hot plasma and cold argon. A shock wave is formed by the discontinuity of pressure. The shock wave moves forward, and the hightemperature zone begins to absorb laser radiation. The electron number density increases. The pressure and temperature rise due to absorption of laser radi­ ation, and a blast wave appears. The propagation velocity of the blast wave increases, so that the inverse Bremsstrahlung becomes stronger. The velocity of the blast wave becomes faster than that of the primary shock. The blast wave catches up with the shock wave, then it propagates at a constant velocity,

Laser-driven

shock

207

waves

and a LSD wave is formed. The ionization process is as follows Ar -f e~ <—>Ar+ + e~ + e~. The plasma absorbs the laser radiation through inverse Bremsstrahlung and emits radiation through Bremsstrahlung: Ar + e~ + hv±=±Ar+ Ar + e~«zzLAr

+

+ e~,

+ e~ + /H>.

The variation of radiation intensity is given by dF

where kei is the coefficient of electron-ion inverse Bremsstrahlung absorption, kea is the coefficient of electron-atom inverse Bremsstrahlung absorption. The energy equations are as follows de

m+

d(e+p)v

dx

dee d{ee+pe)v o dt H dx QT = 3peRo{Ta

= $/*-«*■ dpe =

v

~^dx

r QIB

— Te) 2^ ~^r

-QB+QT

+

ee + pe dpe pe dt '

~dt~^Ar'

i

Here QIB and QB are the amounts of energy absorption due to inverse Brems­ strahlung and of energy losses due to Bremsstrahlung, respectively; vej is the effective frequency of collisions between electrons and heavy particles, IAT is the ionization potential. Fig. 102 shows the structure of LSD wave (Fukui, Oshima and Fujiwara 1995). The process of laser-generated shock propagation in a gas can be divided into two stages: (1) the propagation of a LSD wave and (2) the motion of a shock wave after the cessation of a laser pulse. During the first stage a shock wave is a pear-shaped one. The asymmetry of a laser-generated shock wave was studied by several authors (Galkin et al. 1989; Dumas et al. 1995). In the first paper, shock waves were generated by a pulse ruby laser radiation (A = 0.6943//m) which was focused on a target. The targets were of graphite, steel and copper. The duration of the pulse was r = 20 ns, the power of radiation was 108—1011 W / c m 2 , the energy was 0.01 - 1 J. The diameter of the

208

Propagation and reflection of shock waves

Fig. 102. Structure of LSD wave focusing spot varied in the range 100 - 500 /im. The experiments were carried out in air for an initial pressure po = 76 - 760 Torr. We denote as r 0 the quantity (E/po)1^3, as t± the quantity tco/r0, as r\ the quantity r/ro. Here E is the effective explosion energy, CQ and po are the sound velocity and pressure in the undisturbed gas. In the range ^i < 0.02 and F « 1010 - 10 11 W/cm 2 the radius rj increases in time considerably more rapidly than it derives from the point-explosion theory. The shape of the shock wave differs greatly from the spherical one under these conditions (Fig. 103). Evidently, the acceleration of the shock wave in the channel of the laser beam is caused by the appearance of an absorption wave during the pulse. The input of energy to the front of the shock wave propagating outside the channel of the laser beam is practically absent from the instant the absorption wave arises. The velocity of the absorption wave and the shock wave in the channel of the beam during the laser pulse significantly exceeds the velocity of the other sectors of the front. This allows the shock wave outside the beam channel to be treated as cylindrical for t ~ r. The quantity k = (ri — r^jr^ characterizes the deviation of the shape of the shock wave from a spherical one due to the effect of the absorption wave. Here r\ is the path of the shock wave in axial direction, r-i is the path in radial

Laser-driven shock waves

209

Fig. 103. Shadowgraphs of the shock wa¥e near the target (Galkin et al. 1989a).

direction. The dependence of k on the radiation power-flux density F is given in Fig. 104. The measurements correspond to the time i = 0.5/xs (po = 760 Torr). For k = 0 (F < 108 W/cm 2 ) the wave is spherical (the absorption wave is absent). For F > 108 W/cm 2 the shock-wave front begins to stretch out towards the laser beam, which indicates the appearance of the laser-radiation absorption wave. However, a signiicant growth of the quantity k is observed only for F > F* ¥ 2 • 1010 W/cm 2 . For F > F„, the shock wave absorbs radiation just as effectively as for optical breakdown of a gas, whose threshold in air at a pressure p 0 = 760 Torr is equal to 2 • 10 11 W/cm 2 (Eaizer 1974) for A = 0.6943 /*m. Some authors (Kohlberg et al. 1995) observed projections called "aneurisms" at the shock front at low pressures when the laser beam was focused on the target. After cessation of laser pulse, the motion of a shock wave is described well by the theory of point explosion (Sedov 1959; Korobeinikov 1991). If the shock wave is strong then there is an analytic solution obtained by Sedov. The propagation of a spherical shock wave (point explosion) in the case of counterpressure was calculated numerically by Erode (Erode 1959). Experimental data agree well with theoretical ones (Figs. 105 and 106). If the shock wave is

Propagation and reflection of shock waves

210

Fig. 104. Ar against power-flux density, k — (ri — T2)/r2, ri is the path of the shock wave in axial direction, r2 is the path in radial direction.

Fig. 105. Path of the shock wave against time (Galkin et al. 1990). Dots are experimental data, solid line represents the calculated val2.5

t,fis

Laser-driven shock waves

211

not strong then good results may be obtained with the aid of Eq. (82) while substituting on the right-hand side the derivative dp2/dr which is determined from self-similar solution (Sedov 1959).

Fig. 106. Pressure behind the shock wave against time (Galkin et al. 1990). Dots are experimental data, solid line represents the calculated values.

dM _ dr

( 7 + l ) ( M 2 - l ) w 1 dp2 _ 2(M 2 - \)qw 1 27 pi dr 7+ 1 r'

F = M{2(2 7 - 1)M 4 + (7 + 5)M 2 - 7 + 1}, w = (7 - 1)M 2 + 2, q = 27M 2 - 7 + I, 1 dp2 _ S-fE Al Pi dr ~ 25( 7 + l)pic\r*A2' 7 (37-l)(9-27)/? Ai = 0.6 + (7-l)(2-7) (2-7)(7-7) 13 7 2 - 7 7 + 1 2 7 - 0 . 4 + (37 " l)P A2 /? = 5(27+l)(37-l)' 7-7 ' 27+l a(1.4) = 1.175. 'E = Here EQ is the energy of explosion. It is interesting to note that in the case of graphite target, a mass pulse is created which propagates at a velocity considerably in excess of the natural convection velocity (Galkin et al. 1989b). Free convection velocity is equal to 0.16 m/s. The process is as follows. At a time of order of 10 /zs, the expansion of

212

Propagation and reflection of shock waves

the heated region of gas has ended. The graphite evaporates under the influence of the radiation from the hot gas. At first the temperature of the gas above the target is much higher than the temperature at the surface. After that, in 0.2 ms, as a result of convective heat transfer to the cold air, the gas temperature falls below the temperature of the target surface. This results in heat flow from the solid to the adjacent air masses.

CHAPTER 9. SHOCK WAVES IN A LOW-TEMPERATURE PLASMA

A low-temperature plasma may be created by means of a glow discharge, in processes of combustion, by means of radiation. An interesting case is that of nonequilibrium plasma (electron temperature is not equal to that of ions or atoms). We assume the plasma to be neutral. Nevertheless, there arises an electric current in the presence of electric field. The processes that may pass in plasma are as follows: ionization, excitation of atoms and molecules, recombination, dissociation, radiation. We consider an ideal plasma (the Debye length is much greater than the mean distance between atoms and molecules). The propagation of a shock wave through a low-temperature plasma may be of interest in connection with various problems: those of re-entry, processes that occur in plasmachemical reactors and pulse gaseous lasers etc. Nishio (Nishio 1995) proposed a method for visualizing shock waves in three-dimensional flow by means of discharge. The application of a glow discharge for visualizing shock waves in rarefied gas is well known. That phenomenon is also connected with the problem of observation of rockets. When ionized products of combustion flow out of the nozzle, a Mach disc may arise in the jet thus changing the intensity of the radiation of the jet. Furthermore, a shock wave can be a tool for studying kinetics in a low-temperature plasma. It is also worth mentioning that one can obtain a non-contracted discharge at high pressures while using shock waves. It is difficult to create a discharge without arcing under stationary conditions. It is known that the structure of a shock wave in gas can be determined theoretically by solving the Boltzmann equation. In an analogous way, the structure of a shock wave in plasma may be found by solving the appropriate kinetic equation, namely, the Vlasov equation. But it has not been done up-tonow. The structure of a plane shock wave in a fully ionized plasma was invest­ igated theoretically (Shafranov 1957). The case of a partially ionized plasma was considered later (Jaffrin 1965). 213

Propagation

214

and reflection

of shock

waves

The structure of a steady plane shock wave in a partially ionized plasma was investigated using the Navier-Stokes equations for ternary mixture of atoms, electrons, and ions. The basic assumptions are as follows: (1) the characteristic reaction lengths are large compared to the shock thickness, so that no ionization or recombination reactions occur within the shock itself; (2) the Debye length is much less than the mean free path, so that the overall charge separation is negligible; (3) the ion-atom diffusion is small because the ion-atom mean free path is small compared to the shock thickness. The net current is equal to zero. Thus we have mvi — neve = c o n s t . The continuity equation for atoms yields nava

— const.

The equations of m o m e n t u m and energy for monatomic gas are as follows dvj

mjHjVj 5 J / 2 dxx

dpj

d /4

v—s ^

-& ~te ~ Tx U ^ J " ^

n 3 3)

d {, dTA dx \ 3 dx J

- ]P(£jfc + VjPjk), j = i,e,a,

dvj\

+

a = e,

d /4 dx \ 3

3 3

njej x =

dvA dx )

1 2

3

^ ik' 3 3

dv] dx

333^

(j is not summed), ee = - e ,

ea = 0.

(155)

Here Pjk characterizes the m o m e n t u m transfer between species j and k, kj is the thermal conductivity and £jk denotes the energy transfer between the k particles and j - particles. The equations of state are Pj = njkTj.

(156)

Eqs. (155) and (156) represent a closed system. At x —>■ —oo and at x -> + 0 0 the flow is uniform. All the species have the same velocity and temperature. The results of calculations are as follows. There is a thermal layer of elevated electron temperature ahead of the shock front. At low degree of ionization, the atom gas is unaffected by the electrons. The electric field has the direction opposite to the shock propagation. The main results are as follows: (1) the existence of a thermal layer of elevated electron temperature and a precursor ahead of the shock front; (2) rise of an electric field that increases with degree of ionization. The calculated value

Shock waves in a low-temperature

plasma

215

of a j u m p of potential across the shock front agrees well with the measured one (Jaffrin 1965). Experimental investigations that are concerned with the shock waves in a low-temperature plasma are small in number. A special case is the existence of an electric current ahead of and behind the shock front. An energy release takes place due to this circumstance in a large area behind the shock front. Shock waves were used as a tool for diagnostics of a gas discharge (Edels et al. 1957). Miyashiro (1983) investigated the intensity of radiation behind a shock wave propagating through a longitudinal arc discharge. It was found that the shock wave may increase or decrease the intensity of radiation according to the value of electric field. The propagation of shock waves through a glow discharge was investigated by several authors (Klimov et al. 1982; Bystrov et al. 1989, 1990, 1992, 1995). Klimov used a longitudinal glow discharge. Bystrov et al. studied shock propagation through a zone of transverse rf discharge. Results are given below that relate to the shock propagation through a transverse rf discharge. The experiments were fulfilled in a single diaphragm shock tube of rectang­ ular cross-section of an area 40 x 60 m m 2 . The test chamber was dielectric. It had glass windows of 60 m m diameter. Two metal plates were mounted on the upper and lower walls of the test chamber. The transverse discharge was created between the plates by means of a rf generator ( / = 13.6 MHz). The current density was 40 m A / c m 2 . The length of the discharge zone was equal to 80 m m . Initial pressure was equal to 650 Pa. The shock Mach number was equal to 2 - 5. Ar, CO2 and N2 were chosen as test gases. The density of the post-shock flow was measured by a laser-schlieren technique. We used a He - Ne laser with a b e a m diameter of 1 m m . The shock velocity was measured with pressure transducers beyond the discharge zone and from schlieren signals within the discharge zone. The initial translational temperature was measured with the aid of a Fabry-Perot interferometer. The vibrational temperature of CO2 was determined from the infrared radiation by reference intensity tech­ nique. The radiation passed through a slit of 1 m m width and fell on a detector. The initial translational temperature of the test gas was equal to 1200 K, the initial vibrational temperature of carbon dioxide was 2000 K. Fig. 107 shows the shock velocity in plasma against that one outside the zone of discharge. As one can see, the shock velocity in plasma exceeds the value corresponding to the shock that enters into a region of a heated gas. T h a t difference may be due to energy release behind the shock. Fig. 108 illustrates the infrared radiation behind the shock wave (CO2 as

216

Propagation

and reflection

of shock

waves

Fig. 107. The shock velocity v in plasma against that one VQ outside the zone of discharge, dots are the measured v a l u e s ; l , 2 , 3 - com­ puted values of the velocity of the shock that enters into a region of a heated gas; 1 - T = 700K; 2 - T = HOOK; 3 - T = 1500K.

the test gas, vo = 1 6 6 0 m / s ) . We found the vibrational temperature behind the shock. We neglected the interaction between the asymmetric mode and deform­ ation mode of CO2 molecule. We proposed the temperature of the deformation mode to coincide with that of symmetric mode owing to the Fermi resonance and to be equal to the translational temperature. Fig. 109 shows the value of temperature of the asymmetric mode (1) and of the translational one (2) behind the shock. We see that the values of the vibrational temperature exceeds those of the translational. The electron density in CO2 was measured by a microwave interferometer (A = 8 m m ) . In order to investigate a shock reflection a quartz plate was mounted in the test section. The receiver and transmitter of the microwave interferometer were located at the end of the discharge zone or just in front of the plate. The interferometer operated in a linear regime (A