evolution of discontinuity in self-similar flows
TRANSCRIPT
IL NUOVO CIMENTO VoL. 11 B, N. 1 11 Settembre 1972
Evolution of Discontinuity in Self-Similar Flows.
S. G. TAGAR~
Depa~tment o] Applied Mathematics, Indian Institute o] Science - Bangalore
(ricevuto il 30 Dicembre 1971)
S u m m a r y . - - The self-similar solution of the point explosion problem without a secondary shock is valid only for a very short time, when the radius of the shock front is sufficiently small. In this paper we have studied the propagation of small disturbances on the self-similar flow due to the point explosion problem in a medium whose density varies according to the [(7 - - y)/(y § 1)]-th power of the distance from the centre and we have determined the situation under which a shock wave will appear at the leading or trailing front of an initial continuous disturbance. The evolution of a discontinuity in a self-similar flow may give some answer about the origin and existence of secondary shocks in blast wave problems.
1 . - I n t r o d u c t i o n .
F o l l o w i n g ~ v e r y s t r o n g exp los ion , t h e r e is, gene ra l ly , a v e r y sma l l r e g i o n
filled for a v e r y sho r t t i m e w i t h h o t m a t t e r a t h igh p r e s su r e , w h i c h t h e n s t a r t s
to e x p a n d w i t h i t s f r o n t h e a d e d b y a s t r o n g wave . W h e n we cons ide r th i s
p h e n o m e n o n as a r e s u l t of p o i n t exp los ion , t he flow is se l f - s imi la r a n d we ge t
on ly one sphe r i ca l shock wave enc los ing t he ~ h o l e d i s t u r b a n c e i n i t s i n t e r i o r .
The se l f - s imi la r so lu t i on of the p o i n t exp los ion p r o b l e m w i t h o u t a s e c o n d a r y
shock is v a l i d o n l y for a v e r y sho r t t i m e , w h e n t he r a d i u s of t he shock f r o n t
is su f f ic ien t ly smal l . A p h y s i c a l l y r ea l i s t i c b l a s t w a v e is n o n s e l f - s i m i l a r a n d for
a c e r t a i n r a n g e of t i m e i n t e r v a l we can r e g a r d the b l a s t w a v e p r o b l e m to be
a sma l l d e v i a t i o n f r o m the se l f - s imi la r flow. T h u s i t seems t h a t s t u d y of the
p r o p a g a t i o n of smal l d i s t u r b a n c e s on a se l f - s imi la r flow m a y p r o v i d e some
i n f o r m a t i o n a b o u t t he a p p e a r a n c e of t h e s e c o n d a r y shocks. I n th i s p a p e r we
h a v e t a k e n the bas ic se l f - s imi la r flow to be a p a r t i c u l a r p o i n t exp los ion p r o b l e m
(*) Present address: Molecular Biophysics Unit , Indian Ins t i tu te of Science, Bangalore.
73
74 S . G . TAGARE
in a medium whose dens i ty varies according to t h e - - [(7 --7)/()~ + 1)]-th power
of the distance f rom the centre (~). We create at some given t ime t > 0, a small continuous spherically symmet r i c d is turbance over a small por t ion of the self-similar flow and t ry to find out the s i tuat ions under which a shock wave will appear at the leading or trai l ing f ront of the wave. A d iscont inu i ty
ma y appear even in the inter ior of the d is turbance , bu t we ~ssume tha t this s i tuat ion is absent or, even, if it appears, i t does not influence the flow in the
neighbourhood of the leading and trail ing f ront of the dis turbance. To discuss the appearance of the d iscont inui ty we make use of the me thod described by JEFFREY and r]~ANIUTY (~) and corrected by PRASAD and TAGARE (a).
2. - B a s i c e q u a t i o n s .
Let us consider the spherically symmetr ic mot ion of a poly t ropic gas,
neglecting any dissipative mechanism. The equat ions of mot ion are
( 2 . 1 )
(2.2)
and
(2.3)
e~ + Uer + eu~ + 2Qu __ 0 , r
e(u, + uu~) + p~ = 0
(p~ + ups) _TP ( ~ + uQ,) = o ,
where ~ is the mass density, u the part icle velocity, p the gas pressure~ t the t ime, r the distance f rom the centre of s y m m e t r y and Y the rat io of the specific heats. We assume tha t the init ial dens i ty d is t r ibut ion in the medium at rest
is propor t ional to 1/r ~, where (o is cons tan t given by
7 - - 7 ( 2 . 4 ) ~ = y + l
I f a s t rong explosion takes place at the origin, then the mot ion is self-similar (1)
and the d is turbance at any t ime is confined wi th in a sphere hounded by a shock
wave with radius
(2.5)
where A is a cons tant and
(2.6)
i~(t) = A t ~ ,
2 y + l ($-- 5 --o) 3 y - - 1
(1) L . I . SEDOV: Similarity and Dimensional Methods in Mechanics (New York, 1959). (3) A. JEFFREY and T. TA~IUTI: Nonlinear wave Propagation (New York, 1964). (3) P. PRASM) and S. G. TAGARE: Zeits. Aug. Mech. Phys., 22, 359 (1971).
EVOLUTION OF DISCONTINUITY IN SELF-SIMILAR FLOWS 75
To discuss t he p r o p a g a t i o n of a smal l p e r t u r b a t i o n on t he a b o v e u n s t e a d y self-s imilar so lu t ion , we i n t r o d u c e a ne w se t of d e p e n d e n t va r i ab les g, g a n d v and a new se t of i n d e p e n d e n t va r i ab les $ a n d ~ (4) t h r o u g h the equa t i ons
(2.7)
where
p(r , t) = qo(t) P ~ ( $ , v) ,
u(r, t) = / ~v ($ , v) ,
q(r, t) = qo(t)g($, ~) ,
r 1 - - R( t ) ' v = -~ ln R( t ) ,
(2.8) qo(t) = q*t - ~
w i t h ~* a c o n s t a n t . W e shall f ind i t m o r e c o n v e n i e n t to w o r k w i t h the va r i ab l e z i n s t e a d of z ,
whe re
~Yg (2.9) z - -
g
E q u a t i o n s (2.1)-(2.3) t r a n s f o r m to
(2JO) U ~ + A U e + B = 0 ,
whe re
g
U = v l, A = I
Z _ J
m
(v - - ~) ~ g5 o
z - - ( v - - ~) ~ - Y g
0 (~ - - 1 ) z ~ (v - - ~)
and
( 2 . 1 1 ) B =
7
r + ] J _
(4) P. L. BHATNAGAR and P. I~ Proc. Roy. Soc. Lond., A315, 569 (1970).
7 6 $ . G. TAGA_R:E
The sy s t em (2.10) governs an a r b i t r a r y u n s t e a d y spher ica l ly s y m m e t r i c flow. A self-similar flow is the z d n d e p e n d e n t solut ion of (2,10). The solut ion of the po in t explosion p r o b l e m is (~)
m
go(~)
(2.12) Uo(~)= Vo($)
Zo(~)
7 + 1
2
7 + 1
2 ( 7 - - 1 ) ~ ~ T ~ _
o < $ < 1
The sy s t em (2.10) is an h y p e r b o l a a n d the th ree charac te r i s t i c curves are given b y
d~ d~ (2.]3) d~ : (v--~)~ and d~ ~ (v--~:)~ :/: V ~ .
F u r t h e r the coefficients A and B are i ndependen t of the new t ime va r i ab le 3. l~ow it is a s imple m a t t e r to discuss the evolut ion of a d i scon t inu i ty in v
a t the wave f ron t of an a r b i t r a r y cont inuous p e r t u r b a t i o n p resc r ibed a t a g iven (~ t ime >~ v~ on the basic self-similar solut ion (2.12). We jus t need to follow the simple m e t h o d described in ref . (2,3) keep ing in mind t h a t the self-similar flow here corresponds to the s t eady solut ion of the re fe rence (~) or if). v plays the role of t and ~ t h a t of x. We follow the ref . (~) for all o the r no ta t ions .
The e igenvMues of the m a t r i x A are
(2.14) ) .o ) - - (v - -~e )6 , 2(2)=(v--~)(~+(~,V/Z , 2 ( a ) = ( v - - ~ ) 8 - - ( ~ v / z
and the corresponding le f t e igenvec tors are
(2.15) = = , y~/~, 1 ,
= , - - y ~ / z , 1 .
The equat ions govern ing the p ropaga t i on of the d iscont inui t ies in the first der iva t ives across the wave f ront ~----0 moving v i th a charac ter i s t ic speed 2(il (i----2 or 3) are(3)
(2.16)
(2.17)
and
(2.18)
- - (~ ) Uo~) x + lo (j) g = O,
~:~ ~ , + [{w(~'~))}o~] ' ~;o~, + [v~b~')]o~ + (Zo~) ~> ~;o~, + ~o~,~')~ = o
x,, = (Vu~t(*')o~,
i :/: j ,
EVOLUTION OF DISCONTINUITY I!~! SELF-SIMILAR FLOWS 7 7
where the vec to r ~ r ep resen t s the j ump in the va lue of the no rma l de r iva t ive Ur and the p r i m e in the second t e r m of eq. (2.17) denotes the t r anspos i t ion
opera t ion. The wave f ron t ~(~, v ) = 0 is defined b y
(2.19) ~%-~ {(v--~)(~ + s5 V ' z } ~ : 0
with the in i t ia l condi t ion
(2.20) ~(~=, O) : s ( ~ - - ~:~),
where e = 1 when the p e r t u r b a t i o n of the self-s imilar solution forms a for- ward- fac ing and e = - - 1 when i t fo rms u backward - f ac ing wave , so t h a t ~ > 0 corresponds to the u n p e r t u r b e d self-similar region (see F igures in ref . (a))
ahead of the wave f ront .
3. - D e t e r m i n a t i o n o f pos i t ion o f a s econdary shock .
On the w~ve f ron t ~ = O, we h a v e
(3.1)
where
d ~ _ _ Z ~ ) : ( v 0 _ ~ ) 8 + e ~ / z 0 _ fl~ dT' 3 7 - - 1 '
fl=--(7--1)-~s%/27(7--1).
F r o m the eqs (2.16)-(2.18), we ge t
(3.2) 2 7 ( Y - - 1)(7 - - 3 ) ~x + za (7 + 1)~
-67(7-1) [ ~v%7(7-1)~ (3.3)
27( 7 - - 1 p ~ : o (~ + 1) 3
27(7 __ ].)2 ~ n l - - (7 + 1) 3
7+I ----0,
(3.4)
~nd
(3.5)
~[27(~-_l)'~a.l+~TV27(7-1) ~, ~q 8p(7-n [ (7-~-1) 3 - d~ Y J - 1 ~ - ~ - } - ~ (7-~-1) 2 x
27(7 - 1 ) ~ ~ 2s7/~ VsT(7 - ] ) :~3
dx s ( y + 1) 2 7~ 3 /~ ~ = ( 7 + 1 ) ~ . - - .
2 ~/~-~T~- i i
78
(3.6)
where
(3.7a)
and
(3.7b)
S. G. T A G A ~ E
Eliminat ing z~, :~, z3 and its der ivat ives f rom (3.2) to (3.5), we get
~ d~x dx + B ~ ~ + B~x = O,
1 B~ ---- ~ [8eyfi.--/~(y - - 1 ) V 2 y ( r - - 1) + 4s~(r~-- 3y + 1)] ,
1
Equat ion (3.6) is to be solved wi th the conditions
(3.s)
and
x(~) = o
(dx) = l im (y A- 1)2~ s(y A- 1)2 ( 2 )
At (~c, T~), where ~ d iscont inui ty appears, we have
e~c (3.10) (~(vc))~=0_ = 0 , i.e. x ( ~ ) = - - ~ .
YVhen the per turba t ion forms a forward-facing compression wave or a backward-facing expansion wave, the jump [u t~ -uu , ] in the fluid accelera- t ion u t § uu~ should be negative. �9 gives us at
y + l < 0 .
= ~1 the condit ion
A similar consideration for the per turbat ions leading to a backward- facing compression wave or a forward-facing expansion wave gives us
2
Now we discuss the following different eases:
EVOLUTION OF DISCONTINUITY IN SELF-SIMILAR FLOWS 7~
Case 1. ( B ~ - - I ) ~ - - 4 B ~ < O .
F r o m eqs. (3.6)-(3.10), we get
1 I / (3.12) [ ~ J sin k~ In ~ -~- (7 -t- 1)~(v~ - 2/(7 § 1)) ~ 0 ~
where ]~1 :~ ik2 are the roots of the quadrat ic equat ion
(3.13) m 2 + ( B ~ - - l ) m + B~ 0 .
Case 2. (B~ - - 1)2 _ 4172 - - 0 .
F r o m eqs. (3.6)-(3.10), we get
(3.14)
where
(3.15)
ln~ +
2
()~
Case 3. ( B 1 - - 1 ) 2 - - 4B2 ~ 0.
F r o m eqs. (3.6)-(3.10), we get
(3.16)
~here
(3.17)
(~C~/r [ 1 - [~r162 (7 + 1 ) . ( ~ - 2/( 7 + 1))
and
B 1 - - 1 k3-- 2
B 1 - - 1
2
~- 1 V(B~- - 1)2-- 4B2
Our ~nalysis is valid for an a rb i t r a ry value of 7. 7 - 7 corresponds to water (5) and for any physical ly realistic gas we have 7 ~ 1. Thus we consider
the range 1 ~ 7 ~ < 7 . For e ~ 1 , we find tha t the roots of (3 .13)are real and unequal and the equat ion giving ~c is {3.16). W h e n s z - - l , i . e . for a back- ward-facing wave~ $c is to be de termined f rom (3.12). I n the c~se of a forward-
f~cing wave $c/$1 must be between 1 and 1/$1(~ 1) and in the case of ~ back- ward4~eing wave it should be be tween 0 and 1. We c~n see tha t eq. (3.12)
(5) R. COURANT and K. 0. FRIEDRICHS: Superson ic F l o w and Shock W a v e s (New York, 1948).
80 s . G . ~ A O A ~ .
(with s : - - 1 ) and eq. (3.16) (with s : 1) give real values ~ only in the case of a compression wave and not in the case of an expansion wave. Thus a
d iscont inui ty appears only when the pe r tu rba t ion is such tha t i t forms a com- pression wuve. The s t rength of the compression wuve can be measured f rom
the magn i tude of ~--2/(y-F 1) and, when ~t--2/(y q - l ) tends to - - 0 % we get f rom (3.12) and (3.16) tha t ~ / ~ tends to 1.
Asymptotic cases:
i) Weak compression wave.
I f ~--2/(y ~-1)<<0 , we get f rom (3.12) and (3.16)
(3.1s)
and
(3.19)
(~ + 1)~(~-2/(y + 1))
(7 § 1)~(~-2/(~ § 1)) �9
ii) y just greater than unity.
Subs t i tu t ing y = 1 + 0 in (3.7) and neglect ing higher powers, we get
. ~ d~x dx (3.90) ~ 4 ~ -F (2 - - s) ~ ~ - - 2sx = 0.
For forward-facing wave s : 1 and we get
(3.21) ~ ] 1 - - ~ ! j + ~ - 1 + 0 / 2 0 .
For backward-facing wave e - - 1 and we get
(3.22) sin l n ~ ----- ~ - - 1 q-0/2 "
F r o m (3.21) and (3.22), we find t h a t if g ~ - - I is finite, then ~o/~-+1
as 0 - + 0 .
Discussion o] numerical results. Tables I and I I give us the values of ~c/$1 for various values of wave s t rength ~--2/( 7 ~-1). The initial por t ion ~1 of the wave f ront satisfies 0 < ~1 < 1, and our calculat ion shows tha t in the case
of a backward-facing compression wave there always exists a value sat isfying 0 < ~c < ~1. However , in the case of a forward-facing compression wave it
may happen tha t ~o > 1, and this resul t is meaningless for us as our flow exists
]~VOLUTION OF DISCONTINUITY 111~ SELF-SIMIL&R FLOWS
TABLE I. -- Determination o] ~/~1 ]or backward-lacing compression wave.
8 1
~ - - 2/(~, + 1) - - 0 . 1 - - 0 . 5 - - 1 - - 1 0
7 0.233 0.582 0.718 0.956
6 0.263 0.581 0.711 0.952
5 0.248 0.564 0.695 0.948
4 0.233 0.547 0.679 0.943
3 0.226 0.537 0.669 0.939
2 0.246 0.554 0.681 0.941
1.8 0.260 0.572 0.692 0.944
1.6 0.296 0.590 0.710 0.948
1.4 0.327 0.622 0.737 0.954
1.2 0.387 0.688 0.789 0.966
1.1 0 .47i 0.753 0.839 0.976
TABLE I I . - Determination o] ~c/$1 ]or /orward-]aeing compression wave.
V~-- 2/ (y+ 1) - - 0 . 1 - - 0 . 1 2 5 - - 0 . 2 - - 0 . 5 ---1 - - 1 0
7 188 98.5 25.3 1.90 1.15 1.01
6 8.07 6.21 3.58 1.45 1.15 1.01
5 9.15 7.04 4.06 1.57 1.18 1.01
4 10.9 8.43 4.88 1.79 1.24 1.01
3 17.9 13.2 7.03 2.13 1.30 1.02
2 10.6 8.33 5.06 2.02 1.34 1.02
1.8 9.72 7.74 4.79 2 1.34 1.02
1.6 8.55 6.89 4.39 1.94 1.34 1.02
1.4 6.45 5.34 3.61 1.80 1.31 1.02
1.2 4.62 3.96 2.88 1.53 1.22 1.02
1.1 2.86 2.53 2 1.39 1.18 1.02
o n l y i n 0 ~ 2 ~ 1 a n d 2 - - 1 c o r r e s p o n d s t o t h e p r i m a r y s h o c k . G i v e n a v a l u e
of y a n d t h e w a v e s t r e n g t h ~ - - 2 / ( y + 1) , t h e r e e x i s t s a c r i t i c u l v a l u e 21 of 21
: such t h u t , w h e n e v e r ~ 1 ~ ~*, t h e v a l u e o f ~ c ~ 1, so t h a t t h e d i s c o n t i n u i t y
a p p e a r s i n t h e f o r w a r d - f a c i n g c o m p r e s s i o n w a v e b e f o r e t h e d i s c o n t i n u i t y
6 - II 5Vuovo Cimento B.
~ 2 s . G . TAGARE
ca t ches t h e p r i m a r y shock. F o r a g i v e n v a l u e of ~ a n d ~ - - 2 / ( 7 + 1)~ t h e
v a l u e of ~* can be o b t a i n e d f r o m t h e T a b l e b y t h e f o r m u l a
(3.23) ~* = ~-~.
TABL~ I I I . - Determinatio~ o] ~* for forward-lacing compression wave.
~ - - 2/(7q- 1) - -0 .1 - - 0.125 - -0 .2 - -0 .5 - - 1 - - 10
7 0.005 0.010 0.040 0.526 0.870 0.99
6 0.124 0.161 0.279 0.690 0.870 0.99
5 0.109 0.142 0.246 0.637 0.847 0.99
4 0.092 0.119 0.200 0.559 0.806 0.99
3 0.056 0.076 0.142 0.469 0.770 0.98
2 0.094 0.120 0.205 0.495 0.746 0.98
1.8 0.103 0.129 0.209 0.500 0.746 0.98
1.6 0.117 0.145 0.228 0.515 0.746 0.98
1.4 0.155 0.187 0.277 0.556 0.763 0.98
1.2 0.216 0.252 0.347 0.654 0.820 0.98
1.1 0.350 0.395 0.500 0.719 0.847 0.98
I n T a b l e I I I t h e v a l u e s of ~* a re t a b u l a t e d for v a r i o u s v a l u e s of 7 a n d
v a r i o u s s t r e n g t h s of t h e w a v e . F r o m T a b l e I I I we can conc lude t h a t for a s t r o n g
c o m p r e s s i o n w a v e ~* a p p r o a c h e s u n i t y , w h i l e for u w e a k c o m p r e s s i o n w a v e ~*
a p p r o a c h e s zero . $ $ $
T h e a u t h o r is g r a t e f u l to D r . P . I~ASAD for h i s c o n s t a n t h e l p a n d g u i d a n c e .
�9 R I A S S U N T O (*)
La situazione autosimile del problema dell'esplosione punt iforme senza urto secondario va]ida per un tempo molto breve, durante il quale il raggio del fronte d 'urto ~ abba-
stanza piccolo. In questo articolo si ~ studiata la propagazione di piccoli disturbi nel flusso autosimile dovute al problema dell'esplosione puntiforme in un mezzo la cui densit~ varia secondo la potenza - - (7 - - y)/(y § 1) della distanza d a l e ntro e si ~ determinata la situazione in cui un 'onda d 'urto apparir~ al fronte di avanzamento o di chiusura di un disturbo continuo iniziale. L'evoluzione di una discontinuit~ in un flusso autosimile pub dare una ri~posta sull 'origine e l 'esistenza 4i ur t i seeondari nei probl~mi dell 'onda di esplosione.
(*) Traduzione a cura della Redazione.
~VOLUTION OF DISCONTINUITY IN SELF-SIMILAR leLOWS 8 3
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(*) IlepeeeOetto pec)atcque~.