ANALYSIS OF IMPLODING SHOCK WAVES BY THE CHESTER-CHISNELL-WHITHAM METHOD

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ANALYSIS OF IMPLODING SHOCK WAVES BY THE CHESTER-CHISNELL-WHITHAM METHOD

J. Tyl, E. Wlodarczyk

To cite this version:

J. Tyl, E. Wlodarczyk. ANALYSIS OF IMPLODING SHOCK WAVES BY THE CHESTER- CHISNELL-WHITHAM METHOD. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-267-C8-272.

�10.1051/jphyscol:1984850�. �jpa-00224351�

JOURNAL DE PHYSIQUE

Colloque C8, supplément au n ° l l , Tome 45, novembre 1984 page C8-267

ANALYSIS OF IMPLODING SHOCK WAVES BY THE CHESTER-CHISNELL-WHITHAM METHOD

J . Tyl and E. WTodarczyk*

S. Kaliski Institute of Plasma Physics and Laser Microfusion, 00-908 Warsaw, Poland

Military Academy of Technology, 00-908 Warsaw, Poland

Résume - On a présente la possibilité d'utiliser la méthode CCW pour n'importe quelle équation d'état du milieu. Pour des

équations d'état choisies on a obtenu des solutions analytiques.

En prenant comme exemple des matériaux choisis, on a montré une bonne concordance des solutions obtenues avec les résultats des calculs numériques.

Abstract - The paper presents the possibility of the CCW method application to any equation of state of medium. Analytical solu- tions were obtained for the selected equations of state. Good agreement of numerical and analytical solutions was shown (con-

sidering the selected materials: Cu, Mg) .

Using the Chester-Chisnell-Whitham (CCW) method /1 - 3/ for analysis of concentric shock waves it should be borne in mind that it describes the amplification of the parameters at the wave front, which is con- nected with the phenomenon of cumulation only. It being borne in mind that a plane stationary shock wave is produced as a result of compres- sion of the medium by a piston moving at a constant velocity /A/, it may be expected that the CCW method will describe well the propagation process of a concentric shock wave, if the velocity of motion of the boundary (a cylinder or a sphere ) is also constant.

I - FOHBUJLATIOK OF THE PROBIEM

According to the CCW method the relations to be satisfied at the wave front are the laws of conservation of mass, momentum and energy, the equation of state for the medium considered and the differential rela- tion along a characteristic moving in the direction of the wave front.

Thus, the relations describing the problem can be expressed as follows p (D - u ) =p D (1.1 )

P = P0 + P0D u (1.2)

e = eo +| (P + P o)(p-L-|) (1.3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984850

JOURNAL DE PHYSIQUE

where D, u, c, p , ~ a n d p a r e the v e l o c i t y of the wave f r o n t , the velocity of motion of the medium and t h a t of sound, the pressure, the i n t e r n a l energy and the density ; ^II i s t h e c o e f f i c i e n t of symmetry and r i s the coordinate of the wave f r o n t .

I n the subsequent solutions the symbols having no indices w i l l denote parameters of the wave f r o n t f o r t > 0, the index 1 being reserved f o r the wave f r o n t parameters a t the i n i t i a l i n s t a n t of time ( t = 0 ) and 0

-

f o r t h e parameters of undisturbed medium.

The motion i n the (r, t )

-

plane i s i l l u s t r a t e d i n Fig. 1 . The curves 1 and 2 represent the t r a j e c t o r y of t h e shock wave f r o n t and the motion o f the boundary, respectively, the l i n e s 3 representing negative charac- t e r i s t i c s . Positive c h a r a c t e r i s t i c s i s s u i n g from t h e f r o n t l i n e a r e not shown. I n agreement with the assum- p t i o n of t h e CCW method the d i s t u r - bances moving along such characte- r i s t i c s do not influence the motion of the wave front. The i n i t i a l va- l u e s of t h e parameters of the shock wave a r e d e t e d n e d by prescribing

e

r l r an value f o r the velocity ^V of the

boundary ( ul = V )

.

Big. 1

TI

-

^{THE MOTION OF A} STROEJG CONCEEJTRIC SHOCK WAVE I N GAS The equation of s t a t e of gas has the form

I n the case of a strong shock wave i t can be assumed t h a t po = E o = 0.

Then, from (1.1 )

-

( 1 . 3 ) and (2.1)

,

^{we have}

On s u b s t i t u t i n g the r e l a t i o n s ( 2.2 ) i n t o ( 1.5 ) ( i t being borne i n mind t h a t u < 0 , D < 0 ) we f i n d

where

Now, on i n t e g r a t i n g (2.3 )

,

with the i n i t i a l condition, and making use of the r e l a t i o n s ( 2.2)

,

^{we find}

where

The values of ^c~ obtained f r o m (2.7) a r e compared i n Table 1 with those obtained by solving s e l f -similar problems / 5 / .

Table 2 presents the normalized time i n which the c e n t r e i s reached by t h e front. I n the paper /6/ time t c = t,

-

^I

v

lr;' was calculated numerically. In this paper i t i s expressed by the equation

Table 1 Table 2

(28-270 JOURNAL DE PHYSIQUE

111

-

CONCENTRIC SHOCK WAVES I N A SOLID

Below we present an a n a l y t i c solution describing the propagation of a concentric shock wave i n a medium characterized by t h e following r e - l a t i o n

The c o e f f i c i e n t s co and s a r e t o be determined from experimental data. Because, i n a considerable range of parameters, shock adiabates and isentropes p r a c t i c a l l y coincide f o r s o l i d s /7/, speea of sound was determined s t a r t i n g out from the shock adiabate.

Assuming po = so ⁼ 0 and introducing t h e notation U = -u ,parameters of t h e shock wave f r o n t can be represented i n the form

i p = p 0 u ( c o

+ s u )

ⁱ

( 3 . 2 )

On s u b s t i t u t i n g ( 3.2 ) i n t o ( 1.5 ) we f i n d d i f f e r e n t i a l r e l a t i o n between q = U/cO and r

.

On i n t e g r a t i n g this equation with t h e i n i t i a l con- d i t i o n , we o b t a i n

0

- e

[ a r c t g

fw -

a r c t g

/- 1)

⁼

)

^(3.3)

Figs. 2 and 3 show examplary r e s u l t s . They present that cumulative e f f e c t s i n the s o l i d a r e more intense f o r a more intense shock wave.

It i s i n f e r r e d that the r a t e of increase of the parameters o f the wave f r o n t is, f o r a given u, higer f o r a material of lower c o

.

If c, i s the same f o r both materials, the cumulation e f f e c t s a r e more pronounced f o r a higher s

.

I n order t o appraise the accuraty of t h e s o l u t i o n ( 3 .3 ) the propagation of a c y l i n d r i c a l shock wave i n copper was analysed by numericalmeans. The p r o p e r t i e s of copper were described b

8

the equation of s t a t e (paper /8/)and i t was assumed t h a t V =

-

^{2.10 m/s}

.

Pig. 4 presents comparision of the r e s u l t s (broken l i n e

-

numerical analysis, s o l i d l i n e

-

^{b y t h e CCW method)}

.

Big. 2 Fig. 3

The p r o p e r t i e s of s o l i d s a r e o f t e n described by t h e equation o f s t a t e i n t h e &lie-Gruneisen form

where p E

,

^r a r e f u n c t i o n s of density.

Making o? Eqs (1.1)

-

( 1 . 3 )

,

^(1.5) and ( 3 . 4 ) and assuming t h a t p o . = c o = 0, we f i n d

where P, - rp E x

P = i

1

-.(" ^{* P o}

^{- 1 )}

30URNAL DE PHYSIQUE

Fig, 4 Fig. 5

The s o l u t i o n ( 3.5 )

,

^{( 3.6 )} was used t o make a diagram p ( r 1 f o r t h e c y l i n d r i c a l shock wave i n magnesium ( V =

-

2 . 1 0 ~ 4 s f

.

^{It i s con-}

f r o n t e d ( ~ i ~ . 5, s o l i d l i n e ) w i t h t h e numerical a n a l y s i s (broken l i n e ) .

~ c t i o n s p,

,

^{E x}

,

^r f o r magnesium were assumed according t o t h e paper /9/.

The method submitted here i s a e f f e c t i v e means o f a n a l y s i s o f cumu- l a t i o n processes. The s o l u t i o n procedure i s reduced t o t h a t o f s o l v i n g o r d i n a r y d i f f e r e n t i a l equations. The a n a l y t i c form of t h e s o l u t i o n e n a b l e s u s t o a p p r a i s e t h e i n f l u e n c e o f t h e parameters c h a r a c t e r i z i n g t h e p r o p e r t i e s of t h e m a t e r i a l on t h e cumulation phenomena.

REFERENCES

[ I

1

CHESTER W., P h i l . Mag.

fi

(1954) 1293.

[ e l

CHIS= R.F., J. F l u i d Nech,, 2 ⁽¹⁹⁵⁷⁾ 286.

[ 3 ] WHITHAM G.B., J . F l u i d Mech.,

4

(1958) 337.

[ 41 ZELDOVICH Ya. B., RAIZEB Yu. P., P h y s i c s of Shock Waves and High Temperature Hydrodynamics Phenomena (Eoscou, 1963 )

.

[ 5 ] LAZARUS R.B., RICHTNYER R.D., Rep. LA-6823 (1977)

.

[ 61 DYKE Y.V,, GUTTLLLANN A.I., J. F l u i d Mech,, 120 (1982) 451.

[ 71 STANYUKEVICH K.P., P h y s i c s of Explosion ( ~ o s c o u , 1975

.

[ 81 TILLOTSON I.%, Rep. GA-3216 (1962 1

.

[ 91 ZHARKOV ^V.N., KALININ V.A., Equation of S t a t e f o r S o l i d s a t High P r e s s u r e s and Temperatures ( ~ o s c o u , 1968

.