SHOCK WAVES IN SPARK CHANNELS, PART I

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SHOCK WAVES IN SPARK CHANNELS, PART I

M. Kekez, P. Savic

To cite this version:

M. Kekez, P. Savic. SHOCK WAVES IN SPARK CHANNELS, PART I. Journal de Physique Collo-

ques, 1979, 40 (C7), pp.C7-255-C7-256. �10.1051/jphyscol:19797125�. �jpa-00219097�

JOURNAL DE PHYSIQUE ColZoque C7, supplBment au n07, Tome 40, JuiZZet 1979, page C7- 255

SHOCK WAVES IN SPARK CHANNELS, PART 1

M.M. Kekez and P. Savic.

National Research CowzeiZ, Ottawa, KIA OR6.

INTRODUCTION

This work r e p r e s e n t s an a d d i t i o n a l s t e p i n t h e systematic development of t h e theory of spark chan- n e l formation i n a s e r i e s of p u b l i c a t i o n s [ l ]

-

It i s intended t o f u r t h e r emphasize t h e use of t h e mechanics of compressible media, i n p a r t i c u l a r a s r e l a t e d t o t h e f i e i d s of hypersonic flow and deto- nation, t o e l u c i d a t e t h e problem of gas discharges.

We intend t o show t h a t t h e paraboloidal shock f r o n t which surrounds t h e advancing t i p of a highly con- ducting spark channel w i l l degenerate i n t o a weak d i s c o n t i n u i t y s u r f a c e , i . e . a sound wave, a t some d i s t a n c e t r a n s v e r s e t o t h e a x i s of t h e channel. A formula f o r t h e time of t h e t r a n s i t i o n from hyper- sonic t o sonic motion i s derived.

THEORY

The hypersonic analogy provides 8 r e l a t i o n between t h e constant-energy non-steady s i m i l a r flow behind t h e b l a s t wave a r i s i n g , i . e . from detona- t i o n , and t h e steady hypersonic flow about a blunt- nosed cylinder. I t s value l i e s i n t h e f a c t t h a t it can e s t a b l i s h t h e asymptotic behaviour of t h e flow not j u s t when t h e v e l o c i t y i s i n f i n i t e l y l a r g e but a l s o when t h e p r o p e r t i e s of t h e flow a r e t o be pre- d i c t e d a t moderate supersonic speeds. Even f o r not t o o slender bodies, t h e drag c o e f f i c i e n t , and o t h e r aerodynamic c h a r a c t e r i s t i c s , remain p r a c t i c a l l y un-

We f i r s t b r i e f l y explain t h e relevance of S v i g a r t l s a n a l y s i s t o t h e theory of spark channel formation. Consideration of t h e processes occur- r i n g a t t h e sharp t i p l e a d s us t o assume t h a t a t every i n s t a n t t h e energy absorbed i n t h e t i p a c t s l i k e a powerful chemical heat r e l e a s e which d r i v e s a shock wave i n a l l d i r e c t i o n s . The "chemical"

heat i s replenished by t h e energy of e l e c t r o n s heated by e l e c t r i c f i e l d concentratioh ahead of t h e t i p , and then t r a n s f e r r e d t o t h e heavy charged par- t i c l e s behind t h e shock f r o n t v i a Coulomb,colli- sions. As t h e f i e l d concentration i s s t r o n g e s t a t t h e a x i s of t h e channel (th: r a d i u s of t h e curva- t u r e a t t h e t i p i s s m a l l e s t ) , t h e p r e f e r e n t i a l h e a t i n g of t h e e l e c t r o n s w i l l make t h e spark chan- n e l elongate mainly along t h e a x i s of t h e channel

( i n t h e d i r e c t i o n of t h e s t r o n g e s t shock wave )

.

Thus t h e mechanism of spark channel elongation i s q u i t e s i m i l a r t o t h e detonation of explosive mate- r i a l s , o r t o a slender body i n hypersonic f l i g h t . Note t h a t i n t r o d u c t i o n of t h e s e concepts has r e c e n t l y a ( s a t i s f a c t o r y ) d e s c r i p t i o n of a long (10 m) spark s t u d i e d by Les RenardiGrzs Group ( s e e ' [ T I and [ 9 ] ) and i n a d d i t i o n , breakdown

(u-curve, c r i t i c a l voltage vs d i s t a n c e , e t c . ) and r e l a t e d c h a r a c t e r i s t i c s have r e c e n t l y been derived

[81.

changed f o r f r e e stream Mach number g r e a t e r than 3

From Swigart we have Eq. ( 1 ) o r ). When t h e Mach number of a blunt-nosed cylin-

d e r and i t s bow shock shape a r e known, then t h e d

p r o p e r t i e s a t every point on t h e shock t r a j e c t o r y which describes t h e r a d i u s of t h e shock f r o n t , R ,

,

can be determined from c y l i n d r i c a l b l a s t wave about a blunt-nosed s l e n d e r body hemispherical t i p theory. The a n a l y s i s of such a b l a s t wave and i t s of diameter, d , a s a function of a x i a l d i s t a n c e k p p l i c a t i o n t o hypersonic flow p a s t blunt-nosed from t h e nose, ( z / d ) , f l i g h t Mach number, M, and

s l e n ~ . ~ c y l i n d e r s , was developed by Swigart [ l o ] nose drag c o e f f i c i e n t , CD. ( s e e Fig. 1.) to t h i r i order i n inverse Mach number. We n o t e i n

passing

\

later times , f t e r t h e i n i t i a l explosions, while

!

@$

t h e first term describes a s o l u t i o n f o r t h e flow ^d

-\

f i e l d during th&lkeXPlosi~n and, a s such, has been usetf i n our e a r l i b r work [

1 1

Pig. 1. Flow about a blunt-nosed i n steady f l i g h t .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797125

We i n t r o d u c e :R = E/po, E b e i n g t h e energy p e r u n i t l e n g t h of t h e s l e n d e r body, and po and po background p r e s s u r e and d e n s i t y r e s p e c t i v e l y and by d e f i n i t i o n t h e sound v e l o c i t y

c2

Thanks t o t h e hypersonic analogy we can s u b s t i t u t e z/Uo (where Uo i s t h e v e l o c i t y of undis- t u r b e d gas with r e s p e c t t o t h e o b s e r v e r f i x e d i n t h e body) f o r t h e t i m e , t , t h e r e b y r e l a t i n g non- s t e a d y flow behind t h e b l a s t wave t o s t e a d y hyper- s o n i c flow. By d e f i n i t i o n t h e d r a g c o e f f i c i e n t i s

Thus Eq. ( 1 ) becomes

References

[ l ] Kekez M.M. and Savic P . , 1974, J. Phys. D:

~ p p l . PWS. 7, 620

[21

,

Phon. Ion Gases, Eindhoven, 1 6 1

[31

,

I n t . Conf. Gas Disch. Swansea, 129

[41

,

54, 2216

-

[ 5 ] Savic P. and Kekez M.M., 1977, Can. J. Phys.

55,

[6] Kekez M.M., Makomaski A.H., and S a v i c P., 1977, Proc. 1 1 t h I n t . Symp. Shock Tubes-Waves, S e a t t l e , 602

[ 7 ] Kekez M.M. and Savic P., 1978, Proc. IEE 5 t h I n t . Conf. Gas Disch., L i v e r p o o l , 336

[81

,

.

where El = = C ~ R :

"0 Conf. on Plasma S c i . ( t o b e ~ u b l i s h e d )

The v e l o c i t y of t h e r a d i a l expansion i s

dR a t

1 ,

c a l c u l a t e t h e t i m e when t h e hypersonic motion de- Appl. Phys. ( t o b e ~ u b l i s h e d ) cays i n t o a sound w a y , i. e . = C. Taking t h e

a t

9,

='1'47 we find from Eq' ( 2 ) that the time O f the

[11] Les Renardigres Group (1975), E l e c t r a , 53, 31 t r a n s i t i o n , t i s

0'

-21755 El2 t o - c 2

Experimental v e r i f i c a t i o n of t h i s t h e o r y w i l l be attempted i n our companion paper.