_numerical study of an explosion in a non-homogeneous medium with and without magnetic fields

8/2/2019 _numerical Study of an Explosion in a Non-homogeneous Medium With and Without Magnetic Fields

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C ~ .,,,,.aPadd,tVa. 7.PP. 97-101 OM.~"/9"JO/~I-O0~'//~O0/OO tNqlmmWeu Ltd., ~ . Wired l OmUBltm

N U M E R I C A L S T U D Y O F A N E X P L O S I O N I N AN O N - H O M O G E N E O U S M E D I U M W I T H A N D W I T H O U T

M A G N E T I C H E L D SS. M . ~ and S. T. Wu:~

Mechanical l::.,n~neeringDepartment, 'The University of Alabama in H uatsvt]le, Huntsville, AL 35807,U.S.A.an d

Y . NAro~^w^High A ltitude Observatory, National Center for Atmospheric Research, Boulder, CO 80303, U.S.A.

(R~e/ved 27 February 1978;/n r e v i s e d ] o t t o 1 1u/y 1978)A h m m - - A v e r s i o n o f t h e t w o s t e p I . a x - W e n d r o f f d i ~ e r e n c e m e t h o d w i t h s e c o n d o r d e r a cc u r a c y i s u s e dt o s e e k mlutiomt of the u n s te n d y ma lp ~ to b y d ro d y e ~u n /c (M I ID ) e q u a t io n s fo r t h e s tu d y o f a n e x p lo s io n ina no n-h om ojeu ous reed/urn with and w i t h o u t m a p e t i c 6 e ld s . T h e ex p lo s io n i s i n / ~ t e d b y / n t r n d u c i n g a

a m o u n t o f e n e r g y i n a s m a l l v o l u m e o f g a s w h i c h l e a d s t o a n i m t a n t a n e o u s i n c r e a s e o f g a s~ ' m l ~ ' m n e e . Two types o f ma g n e t i c ~ ~ t m s ( i .e . open and c losed) a re cons idered m illustratethe dependence a nd differences of magnetic and a non-magnetic flow motion. Numerical results sh ow thestrong dependence of induced Idl.ID flow Reid on the magnetic Se id configuration and st~ngth. Thedevelopment aad decay of the fast and s low MHD shock waves, as w el l as the ordinary psdyna mic shockwaves, are also represented well b y the present computer simulations, in conclusion, we have demon-sU'ated hat the num erical scheme we have used is a reliable one for studying 2-dimonsionalM ]~ p ro b lem sie non-homogeneousmedium.

I N T R O D U C T I O NI n r e c e n t y e a r s t h e r e h a s b e e n a n i n c r e a s i n g i n t e r e s t i n t h e t r a n s i e n t m a g n e t o h y d r o d y u a m i c( M H D ) f lo w s a n d t h e i r re s u l ti n g w a v e p h e n o m e n a , d u e p a r ti a ll y t o t h e g r o w t h o f t h e f u s i o nre sea rch t ry ing to unde r s t and the i n t e r ac t ions be tween the p l a sma and magne t i c f i e ld i nTo kam ak fus ion dev ices and the o the r due to t he a s t rogeophy s i ca l f lu id p rob lem s , i n pa rt i cu la r ,t he s tudy o f t r ave l ing in t e rp l ane t a ry d i s tu rbanc es , r e la t ed to t he so l a r- t e rr e s tr i a l env i ronm en t .B e c a u s e o f t h e m a t h e m a t i c a l c o m p l e x i t y o f t h e n o n l i n e a r M H D f l o w e q u a t i o n s , a n a l y t i c a ls o l u ti o n s a r e a l m o s t i m p o s s ~ l e t o f in d . T h e r e f o r e , it b e c o m e s n e c e s s a r y to u s e t h e c o m p u t e rf o r s e e k i n g t h e n u m e r i c al s o l u ti o n s f o r t h e s e c a t e g o r ie s o f p r o b l e m s .N u m e r i c a l s t u d i e s o f t r a n s i e n t M H D f lo w s h a v e b e e n p u r s u e d b y m a n y a u t h o r s , s u c h a s i nt h e f u s i o n r e s e a r c h f ie ld , v a r io u s l - d im e n s i o n a l M H D c o d e s b e c o m e r o u t i n e [ l ] , a n d s o m e2-d imens ioua l codes become ava i l ab l e [2 ,3 ] r ecen t ly . Howeve r , a l l t he se s tud i e s were con-c e r n e d w i th a h o m o g e n e o u s l a b o r at o r y p l a sm a . I n c o n t r a s t, i n a s tr o g e o p h y s i c al M I I D p r o b l e m sthe n on-u n i fo rm p la sm a m us t be cons ide red . S t e ino l f son , .Drye r and Nakagawa .[ .4 ] u sed a 1 -d i m e n s i o n a l M H D c o d e t o s t u d y i n t e r p la n e t a r y s h o c k p a ir s a n d N a k a g a w a a n d W e I Ic k[ Y Je x a m i n e d t h e m o d u l a t i o n s o f s o l a r w i n d u s i n g a 2 - d im e n s i o n a l M H D c o d e .

In t h i s pape r , a 2 -d imens iona l t r ans i en t MH D code in t he sph e r i ca l coo rd ina t e s i s de sc r ibed .T h e c o d e i s u s e d t o e x a m i n e t h e r e s p o n s e o f a n e x p l o s i o n in a n o n - h o m o g e n e o u s m e d i u m w i t hand wi thou t magne t i c f i e ld s . The spec i f i c de t a i l s o f numer i ca l r e su l t s and the i r phys i ca ls lm, f l cance a r e , h ow eve r , r epo r t ed e l s ewh ere [6 , 7 ] .

F O R M U L A T I O NBasic equations

Th e d e r iva t ion o f t h e M H D equa t ions a r e descn ' bed in t he f i te r a tu re [8 , 9 ] . The bas i cequa t ions o f an inv i sc id , ad i aba t i c and inf in it e con duc t iv i ty f lu id fo r t he 2 -d imens iona l num er i -tRm estch A ssociate.SProfessor.|Research Scientist and Ad jtmct Professor of UAH . The National Center for A tmospheric Research is sponsored bythe National ~Jence Foundation.

97


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98 S.M . I-hN a aLc a l c o m p u t a t i o n a r e w r i t te n i n t h e f o U o w i a g q u a s i - c o n s e r v a ti o n f o r m i n t h e s p h e r i c a l c o o r -d ina tes ( r , O, O) o n the equ a to r ia l p lan e (O = 90 ) [10] .

O W o _ 1 o _ _- ~ - + ~ + ~ - - ~ , (1 )w h e r e

W =

~ V ,rB ,

+ ~ plVl + 2+o)(2)

F =

r2pV,r~/_ + _ . , . ~ B ~ - B ,hv V v , ' ~ 2 ~ to ' ]

0r ( V , B , - V , E )

(3 )

G =

v 2 p ~

r2 (p + p V * 2 + 2 ~,0 /r (V ~ , - V ,B ,)0

(4 )

a n d

S =

0 ~ + ~ ? +2rp + r2p ~r ~ rpV ,:r B ~ B ~ + r o

o O00

(5 )

w h e r e p d e n o t e s t h e m a s s d e n s i t y , V = ( V , , 0 , V ~ ,) d e n o t e s f l o w v e l o c i t y v e c t o r , T d e n o t e st e m p e r a t u r e , B = ( B ,, 0 , B , ) d e n o t e s t h e m % v n e ti c f ie ld , a n d t h e o t h e r s y m b o l s h a v e t h e i r u s u a lmt ' J n i n e : p = p RT b e i l ~ t h e g a s p r e s s u r e , h t h e g r a v i t y p o t e n t i a l , y t h e s p e c i f i c h e a t r a ti o , a n df i n a l l y ~ e t h e p e r m e a b i l i t y i n v a c u u m . I n d e p e n d e n t v a r i a b l e s a r e t i m e ( 0 , r a d i a l ( r ) a n da ~ i m u t h a l ( ~ ) c o o r d i n a t e s .


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Num erical stud y of an explosion ~a a non-homogeneousmedium 99I n i t i a l c o n d i t i o n s

T h e i n i ti al c o n d i t i o n a d o p t e d f o r th e p r e s e n t s t u d y i s a n i s o t h e r m a l a n d h y d r o s t a t i c s t a te , i .e .T o ( r , 4,, t = 0 ) = T ~ ( c o n s t a n 0 , V o ( r , ~ , t = 0 ) = 0 , ( 6 )

w h e r e s u b s c r i p t 0 d e n o t e s t h e ~ c o n d i t io n . S u b s e q u e n t l y , t h e ~ d e n s i t y d i s t ri b u t io n i so b t a i n e d f r o m b a s i c e q n s ( 1 ) - ( 5 ) a s

1 1po(r, 4), t = 0) = p , e x p L R T ~w h e r e s u b s c r i p t s d e n o t e s a r e f e r e n c e v a l u e a t r = R , a n d ~ = ( & 2 ) . S i n c e t h e ~ s t a t e i sc h o s e n t o b e h y d r o s t a t i c e q u ih ' br iu m , t h e ~ m a g n e t i c f ie l d c o n f ig u r a t io n i s d e t e r m i n e d f r o mt h e f o r c e f r e e c o n d i t io n 0 7 x B = a B ) a n d t h e s o le n o id a l c o n d it io n 0 7 . B = 0 ) . F o r c o n -v e n i e n c e , t w o t y p i c a l c o n f i g u r a t i o n s w e r e c h o s e n f o r t h e p r e s e n t s t u d y ; n a m e l y

( a ) C l o s e d - f i e l d c o n f i g u r a t i o nB r - ~ c l r - ' l c o s ~ ,

B r = c 2 r - ' l s i n ~ ,

B ~ f - c 2 ( ~ - ~ ) r - ~ c os ~ ,

Co) Open- f i e ld conf igu ra t ion

(S )

(9 )

w i t h ~ = ( % /( 5) + 3 ) /2 , c l a n d c 2 b e i n g c o n s t a n t s d e t e r m i n e d b y t h e i n i ti a l m a g n e t i c f i e ld s t re n g t ho n t h e b o u n d a r y . T h e i n it ia l m a g n e t i c f ie l d c o n f i g u ra t io n s g i v e n b y e q n s ( 8 ) a n d ( 9) a r e s h o w n i nF ig . I .

(o )

j.~2"" , ,

4 " )~ " 0 " "~ ~,._s Axis of" ~ . ~ , : ~ / - - s F n m e tr y

(c )

~( , , o ,~ )

(b ) Id }F~. I . ~ ~ ~ co n f~ am ~ om solar oq,~_,~ial O(me (0 - ~2) , coordmte sy ,umm ~ I~m p, t ad o~ mesh m~MemeaL (a ) Open f ie ld c o n f ~ o u . C o )C~ned ~Id c o n f ~ n ~ . (c) Grid l~a ta r r a n g e m e n t . ( d) C o o r d i n a t e s y s t e m .


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100 S.M . HAN a aLBoundary and perturbed conditions

T h e r e a r e f o u r b o u n d a r y c o n d i t i o n s t o b e s p e c i f i e d i n t h e p r e s e n t p r o b l e m . T h e l o w e rb o u n d a r y ( r = R , ) i s a s s u m e d t o b e a s o li d b o u n d a r y ( i.e . t o t a l r e fl e c ti v e b o u n d a r y ) , A n d t h ea x i s ~ = O r /2 ), i s t a k e n a s t h e a x i s o f s y m m e t ry , w h i l e t h e o th e r r e ma in in g tw o b o u n d a r i e s ( i nt h e r a n d 0 d i r e c t i o n s ) a r e a s s u m e d o p e n b o u n d a r i e s ( i . e . f r e e b o u n d a r y ) . T h e d e t a i l e dn u m e r i c a l p r o c e d u r e i s d i s c u s s e d i n t h e n e x t s e c t io n .P e r tu rb a t io n s a r e i n t ro d u c e d in a fi n i te r e g io n ( a s s h o w n in F ig . 1 ) i n t e rms o f i n s t a n t a n e o u st e m p e r a t u r e e n h a n c e m e n t t o s i m u l a t e s u d d e n t h e r m a l e n e r g y r e l e a s e d u e t o a n e x p l o s i o n .S u b s e q u e n t f l o w mo t io n in t e r a c t in g w i th s p e c i f i e d ma g n e t i c f i e ld is t h e n in v e s t i g a t e d b y u s in g aM I - ID c o d e d i s c u s s e d in t h e fo l l o w in g s e c t io n s .

N U M E R I C A L M E T H O DF i g u r e 2 s h o w s a d i s c r e t e m e s h s y s t e m u s e d i n t h e p r e s e n t s t u d y . A p p l y i n g a f i ni te

d i f f e r e n c e m e t h o d d u e t o R u b i n a n d l k r s t e i n [ l l ] t o e q n ( 1) r e s u l t s i n t h e f o l lo w i n g f in i ted i f f e r e n c e e q u a t io n s :Intermediate step

+ 1 , - G " " ' I + t (1o)

a n dy r . . I ~ e . . l 1 W " A t . F " ' F " F " ~1i,~+ m = = ~ (W i ~ ,j l + J - 4 - ~ [ ( F ~ + L ~ + t - i - L i l )+ ~ i+ L j-- i - L m

A t I , . G " ~ A t .A 4 ,r , ~ " ~ ' J : ~ j j+ -~ - (S , .~ l+S , : i ) , (11)

w h e re t h e s u b s c r ip t s i a n d j r e f e r t o s p a t i a l g r id c o o rd in a t e s s h o w n in F ig s . l ( c ) a n d 2 ; t h es u p e r s c r ip t r e f e r s t o t h e t ime s t e p . S imi l a r i ly t h e i n t e rm e d ia t e v a r i a b l e s W c 1 a n d VCD ~ c a n b ef o u n d a t C ( i - (1 /2 ) , j ) a n d D(i , j - (1 /2 ) ). Us in g the in te rm edi a te var iab les . . ~.j and th e o ldvar iab les W ~"j, the f ina l va lue o f W ~ 'j j is fou nd f ro m the fo l low ing equa t ion :

Final step, , , o - T O r ' ' , j ' - ' . ) + A t _ G

+ + + + + . g : + ' + (12)

W i th t h e s e e q u a t i o n s a ll f lo w v a r ia b l e s a t t h e i n t e r i o r m e s h p o i n t s a r e a d v a n c e d t o t h e ( n + l ) A tt i m e s t e p e x c e p t a t f o u r b o u n d a r i e s .A t t h e a x i s o f t h e s y m m e t r y , t h e r e f l e c t io n r u l e i s a p p l ie d a n d a t t h e f r e e b o u n d a r yn o n - r e f l e x iv e c o n d i t i o n s a r e a p p l i e d , i . e . (a2W /Or ) = (1/F)(O2W /O4,2) = 0 . A t t h e s o l id b o u n -d a r y , a s i m i l a r t r e a t m e n t d u e t o L a p i d u s [ 1 2 ] i s a p p li e d . T h i s m e t h o d f o l l o w s t h e c o n c e p t o f t h eB A i n t o ao n s e r v a t i o n a l p r i n c ip l e . N e w g r id s a r e c o n s t r u c t e d a s s h o w n in F i g . 2 . T h e f l u x " ' "r e c t a n g l e w h i c h c o n t a in s p o i n t A d u r i n g t im e Atl2 isB ~ = - ~ - - ~ [ ~ ( F ~ + F ~ j - I ) 1A t -~(F~-1+F]~-~[~(GJr+G~r~(G~dr2+GT~rz]" A t 1 . . 1

+- -~ [~ (S ~j '+ "S2j + S,j-I" + S2j- ,)] ." (13)

U s i n g t h e s e f l ux e s B ~ , t h e i n t e r m e d i a t e c o n s e r v a t i v e v a r i a b l e v e c t o r ~V" +u2 a t t h e p o in t A a n d


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Numericalstudyof ,m mq=losiou a non-homogeneousmedium

s + ' ; i I( , , / i f I I

J ' J m x / + l ./ - I / , 2 1 -1 i i ' t ' l / - I m o xFig . 2. D iscretizedm esh systems or in te r ior don ~i , and sol id boundary.

!01

B a re g iven by~ / A . i n _ 1 . . .- ~ ( W ~ j + W I . ~ - , + W ~ j - , + W ~ j ) + B ~ ( 14)

a n d

- ~ ( W , . I + W ~ j - , ) + B A . ( 15)Similar express ions for points C and D can be w ri tten. The f inal variables a t the point ( l , j ) a rethen fo und b y applying the conservat ional pr inciple to rec tangle A BC D

( ~ . + l / ~ F , ~ l# 1 \ -I A ._ 1 rl+~--'~-A-rr- x l c X I D r i l l - i - ' ~ A l li i " ' a " + i t l + i s " i r i + i c " + i r l iD " + Ira ) ( 1 6 )2 T # jIn qn (16), the intermedia te flux vectors ~+ trz , ~ , .+ira and source v ector ~+lra are evaluated byusing th e in term ediate con serva tive flow variables ~V +lrz. Com putation s of c ons erva tivevariables a t the solid boun dary are perfo rm ed by us ing eqns (13)-(1 0. One cr i t ica l assumptionneeded for th is method is tha t the rec . tangle ABCD is so smal l tha t the var ia t ion of f lowquanti fies a t the bou ndar y point (1 , j ) i s equal to th e var ia t ion of the f low quant it ies a t the p ointE. (see F ig. 2 .) Th e radia l m om entum at point E fou nd by eq n (16) m ay not be zero. S ince theradial m om entum at the sol id bo und ary should be zero, th is radial m om entum at point E is ar ti ficial ly conv erted to thermal e nerg y of the gas p 'e , i .e .

There fore , the p res sure and the rad ial ve loc i ty a t the sol id boundary becom esp 'F ~ i ffi p 2 + l + p ' E ,

a n d

(17)

V n + lj ffi O. (18 )


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102 $. M. HAN a a/.Numerical stability criteria for non-linear partial differential equations such as eqns (1)-(5)is not ye t comp letely established. T he cau se of the nu me rical instabili ty c an be sum marized asfollows: (1) gradients in the physical flow quantifies are to o large, (2) tim e step is too large, and(3) improper handl ing of the boun dary condit ions . The f ini te di fference metho d em ployed herecontains dissipative term s such tha t sho rt wav e length oscillations ate dam ped ou t effectively.

H ow eve r, whe n there are strong gradients in the flow field, additional artificial viscosity term sbecomes neces sa ry .In the presen t s tudy, Lapidus diffusion, which has b een successful ly applied to m any M HDcalculations [2, 3] is em plo yed . Th e Lapidus sch em e essentially add s an artificial viscosity t erminto all partial differential equations to be solved, so that eqn (1) becomes0 W e D + D , + D e = 0 (1 9)Ot

w h e r eOF 10G

anda C l _ . v iDe = - b2(rA~)2~ ~ '~ k l r a I r o e / ' ( 2 o )

w here b, and be are co nstants of ord er of unity. Equation (19) is then solved b y a fractional t im estep m etho d. Ef fects of artificial viscosity on the o verall flows field hav e bee n c arefully studiedto me et the requ irements specif ied by Richtmy er and Morton[13].The maxim um allowable t ime s tep At is de term ined by an analogous m ethod ofRich tmyer [14]. Also A t is restricted b y th e artificial diffusion velocity. Th eref ore , the max imu mallowable t ime step, At, chosen at every t ime step satisfies the following three inequalit iessimultaneously

At At 1Ar ' rAO - v 2 ( I v l + C A + c , )

a n d

I L v I < 12 b * ~ A t I rO c b I (21)

w h e r eI v l = = v ? + v ~ = , c A = = c ~ = + c ? . c ? = P

andI B , . , IC , + = V L o ~ o ) "

The d is turbed mag net ic f ie ld poten t ia l A is foun d by us ing the induc t ion e qua t ion, i .e .(22)


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N ~ , m e r i c a ] s t u d y o f a n e x p l o s i o n i n a n o n - h o m o g e n e o u s m e d i u ma A - v B = ~ -a d .O t

1 0 3

( 2 3 )

For the present problem eqn (23 ) reduces toO A- ~ - = r ( v . B , - V , B . ) = L . ( 2 4 )

Therefore, the perturbed t ime dependent magnet ic potent ial A is calculated by

A ' + ' = A~ + 2 ( L ~ + L ~ f ' ) ( 2 5 )

A com plete set o f dependent f low variables and the m agnetic potent ia l at a new t ime l evel ,(n + l )At , i s thus provided by these equations .

Fq~ure 3 sh ow s a f low chart used in the present s tudy.

I n t e r m e d i a t eS t e p

F i n ~ ]

i ] n l c l a l i z a t l o n I

l t = OI I n t r ~ u c e P e r t u r b a t i o n s [L N i t ,I

l -I O e t e ~ I n e C o m p t a t l o n o ! D . m a i n ]II C i ] u | a t e O l d T im e S t e p F l u x e ~ a n d |a l e o Io u r c e ~ ; ~ I , i ' ~ i . ] a n d ~ l ; J

I C a l c u l a t e I n t e r m e d i a t e ~ b e n d e n t IV a r i a b l e s ; ~ A I ~ I C I D!I C l l c u | a t e t h e I n t e r m e d i a t e F l u x e s II _ - - r r + l - - n ~ l - - ' ~ + 1 ll a n d S o u r c e s : ~ . ~ ^ . ~ h , e t { . I

tI

$ o u n d a r v C o n d i t i o n s i s g e l ] a sP h y s i c a l B o u n d a r y C o n d l t l o n ~tS m o o th o u t ~ t , J b y

A r t i f i c i a l D i f f u s i o nl t ' t + ~ tI p , : : i : : ; . I

.-i I' . .

N o' S T ~ 3 e ~

S t { ,D

F i g . 3 . F l o w c h a r t

N U M E R I C A L R E S U L T SThe in/t /a/vaJues and other pert inent constant values u sed in the actual computat ion are

l i s t ed in Table I . One of the most dominant ef fect s on f low mot ion i s due to the value of /3 ,which is the rat io of the gas pressure to the magnet ic pressure, i .e .


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104 S. M. H ~ ~ aLTable 1. Initial values and constants

Variables Value Unitsp j 10 :s gm/cm~g, 2.75 x 104 cm/sec2

1.2Rs 6.95 x 101 cmL 1.5 x 10~ K~ r 2 x 109 cm 0 3 .6 degree~, 0.7b , 0.5b, 2.0R 8 . 2 x 1 0 e c m ~ / K -s e c ~I=. 35J ~ 1 2B,, 2.10 GaussB , , 2.10 Gauss

Figure 4 shows the in i t i a l cond i t ions a long the ax i s o f symmetry . As ,8 decreases rap id lywi th inc reas ing r , dominan t e f fec t s o f the magne t ic f i e ld a re expec ted a t l a rger rad ia l d i s tance .

B e f o r e i n t ro d u c i n g a n y p e r t u r b a t i o n , t h e s t e a d y s t a te c o n d i t i o n s a r e c h e c k e d b y r u n n i n g t h en u m e r i c a l c o d e t o e n s u r e t h a t t h e n u m e r i c a l s o l u t i o n s a g r e e w i t h t h e p r e s c r i b e d s t e a d y s t a t econdi t ions . A f te r ob ta in ing s teady s ta te so lu t ions , the t em pera tu re inc rease o f T = 10 To a t n inem e s h p o i nt s ( i = 2 , 3 , 4 ; j = 1 , 2 , 3 ) is i n t ro d u c e d (s e e F ig . 1) a t t im e t = 0 . T h e t o ta l e n e r g yre lease in t rod uce d by th i s pe r tu rba t ion i s o rd er o f ~ 6 x 103 e rgs , which i s wi th in the range o f atyp ica l so la r f l a re energy re lease .

The resu l t s o f computa t ion a re shown in F ig . 5 a t the t ime t ~ 600 sec a f te r the in t roduc t iono f t h e p u l s e f o r t h e v e l o c i t y v e c t o r s a n d c o r r e s p o n d i n g m a g n e t i c fi el d fi n es in a n o n - m a g n e t i ccase (Case a ) , an open f ie ld case (Case b ) and a c losed f ie ld case (Case c ) . I t can be seen f romFig . 5 tha t (1 ) d i s tu rbance p ropaga tes fas te r ac ross the magne t ic f i e ld f ines (as seen in Case ba l o n g a z im u t h a l d i r e c t i o n in c o m p a r i s o n w i t h C a s e c , a l o n g r a d ia l d i r e c ti o n ) b y t h e f a s t M H Dw a v e s ( o r s h o c k s ) , ( 2 ) t h e s l o w M H D w a v e ( o r s h o c k ) d e v e l o p e d i n C a s e s b a n d c p r o p a g a t en e a r l y a t t h e s a m e s p e e d a s t h e a c o u s t i c w a v e ( o r s h o c k ) a s s e e n i n C a s e a , a n d ( 3) th e m a g n e t i cf ie ld f ines channe l the f low mot ions (as seen in Cases b and c ) . F igure 6 shows the resu l tan te n h a n c e m e n t s o f d e n s i t y a n d t e m p e r a t u r e f o r C a s e s b a n d c a t t h e t i m e t ~ 6 0 0 s e c . T h e d e n s i t ye n h a n c e m e n t d e n o t e d b y A p /p 0 f o r C a s e h s h o w s a h i g h e r p e a k t h a n C a s e c . T h i s i s d u e t o t h e

I I I ! ! I ! I l ! I I |

~ ,oo F r/r~

~ io ~ _g1 0 - 3 I O

1 t~8 1 6 2 4 3 2 4 0 4 8 56

R o d i o l d i s T o n c e o l o n g o x i s o f s y m m e t r y , x I 0 4 k mFig. 4. How propertiesalong he ax is of symm etry ~ = ~r12) or ,8 = 0.7.


9/12

N m e x i c l study of u explosion in a nou.homoseneous m ediumA x i s o f e / m m e w y . .: t ~ ' i i N o t e

...x \ i ~ ~ i i u ~ ' ~ : v H " ~ " ~ ' ~ ' ' ~ ' ~ ~" . " . . - ~ i : : i i l - l O m l n- ' : " -. . . . . . . . . - . . . . . .L ;i ~ ~ , , , < , , : ~ , , , , , e ~ , , , , , , , =.- .1 7 .i" -...:- . ' ~ " " " " ~ , . , O m , , '~ I d l i n e s )

, ~ ' . . . - . . " . . / ' - - ~ i i ~" - ' " . . . ' . - . . 'A \ \ \ ~ I \

. . - . . - . . - ~i . ~ . . . , '. ..

l i ~ i ' "~ 'I I c l l ~ i l i i l " 0 - 7-F i l l . 5. V e l o c i t y v e c t o r s a n d m a l p l e t l c f i e l d li n e s a t t i m e t , = 6 0 0 s e c f o r C a s e a ~ = = ) , C a s e b ( ~ > = . = 0 . 7 ), a n dCase c ~ = ~ d = 0.7).

,8== ,. - 0 .7 ~


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1 0 6 S. l ~ I - l ~ a eLmagnetic f ie ld channel ing effects of the f low. In Case c due to the density decrease along theradial direction and the perpendicular field lines, temperature changes (AT/To) and densityc ha ng e s ( A P / P o ) o c c u r o v e r wide domain, yet perturbed ( A p / p o ) and A T ~ T o remains s t o a t ] i nc o m pa r i s o n w i t h t ho s e i n C a s e b . N o t e t h a t a peak of temperature at larger radial distancewh ich i s due to a fa s t MI lD shoc k w ave front i s order o f 25% . Figure 7 show s the flowproperties a long the a x i s o f s y m m e t r y a t t , - 4 5 3 s e c for Cases a , b and c . We observe tha t thewave front propagates faster in Case c and almost the same in Cases a and b. Contact surfacein Case c , how ever , propagates s lower than that o f Ca ses a and b du e to the s trong re tardingeffects of the Lorentz force . Complex variat ions of f low properties (particularly V ,) near thesofd boundary are due to ref lect ion at soEd boundary and interactions of f luid with magneticf ie lds and other non-l inear effects .

Computat iona l resul t s shown in Figs . 5 -7 agree genera l ly wed wi th phys ica l expec ta t ions .To check the accuracy o f the numer ica l re sul t s fur ther , the jump condi t ions for d is turbancespropagat ing fas ter than charac ter i s t ic wave speeds a t the MHD shock front a long the ax is o fsymmetry are examined. Der iva t ion o f the MIlD jump condi t ions appropr ia te to the presenta n a l y s i s i s g i v e n i n t h e A p p p e n d i x I . M e a s u r i n g t h e s pe e d o f d i s t u r b a n c e a t p o s i t i o n r =4 .4 x I0~ Km ('FIB. 7 ) for Case c an d wi th f low cond i t ions sho wn in Fig . 4 , we hav e the A l fvenM a t h n u m b e r M A ~ 10 .50 . Theoret ica l jum p cond i t ions g iven b y ILankine -Hougonio t jum pcondi t ions [eqns (A 9) - (A 11) in the App endix 1 are pz/p~ = 1.322,B ~,2/B f~ ~ I. 19, Tz/T~ --- 1.17 and%: = 314 km/sec . From F ig . 7, we m ay obtain approximate jum p condi t ions g iven by the num er ica lresults; p2/p] ~-1 .19 , T2 /T~ ~ 1.1 7 and V,~ = 320 kin/see. W e can see a sat isfactory agreementbetw een th ese results. T he re lative error in com pu ted gas pressure (ep /po) for th is examp le case i sthen ~ 0 .11 , which is a re latively small error considering the small values of /~ at this region. (SeeAppendix 2 for de ta i l s . ) The same procedure for Cases a and b a l so showed that the jumpcon dit ion s are s imilarly sat isf ied b y present num erical results .

T

2="

J~

4 O 03 2 02 4 01608 0

0- 4 0

i0 e

io '

,o '2x I07

I I I I I I I I I [ I I I i I 1Nonrno(}netichock MHD shockN 5 3 s e c ) ~ ,~ _ ~ ( Pa ro l le / ) L I / \ / ( 4 5 3 s e c ), " , ; 1 ~ / Y iH D s ~c k/ .4,11 r~ ,~ .I / ~.. , (Perpendiculor)

~ ' ~ ' ' ~ \ . ' V ~ M H D S h oC k ". y , : X \ , ' / ~ . / (Perpendicular) -

~ . , P ~ ~ f ~ "~ \\ ~ _In i t io l $ to te J

B "=o (Co==a)

i ' ~ . " ~ , j Nonmognet ic shock (4 .53 $ec) !" ~ -',~ [~ ~ M H D s h o c k ( p a r o l l e l lc o n t a c t ~ 1 ~ ( 45 3 s ec l~ o,n ~u ~= w " ~ t ~ ' - ~ , M H D shock (perpendicular~ . s u r r o c e ~ , ~ " . , , , , - ~

I n i t i

. ~ , . ~ ~ ContoctI O ' / j ~ J J ~ s u rf c e

' ' , ~ . / M H D s h o c k ( p o r a l l e ( ) -~" " - , ' \ / / ( 4 ~ , . ~ _' ~ , . ~ ' / M H D r ,hock (perpendiculo }. . . . ~ . . . . . . " ~ . _ ~ ,5 3 s e c ) -I I I I I I I I I 1 I |

I Oe 0 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4R a d i a l d i s t a n c e a l o n g t h e i x i $ o f s y m m e t r y , x 10 4 k mF i g . 7 . D i s t u rb e d f l o w v a rm b l e s a l o e s t h e a x is o f s y m m e t ry a t t i m e t , - 453 sec for Cases a , b ~ n d c .


11/12

Nu m eri~ study of an explosion in a n o n - ~ m e d i u m 107I n t h e n u m e r i c a l r e s u l ts t h e s h o c k t r a n s it i o n o c c u r s o v e r r o u g h l y 6 m e s h p o i n t s a s i t c a n b e

s e e n i n F ig . 7 d u e t o t h e ar ti fi ci a l v i s c o s i t y e ff e c t s . W i t h s ~ n e r v a l u e s o f b , a n d b , t h a n t h o s er e p o r t e d h e r e i n , f u r t h e r c a l c u l a t i o n s w e r e m a d e a n d r e s u l ts s h o w e d a n a r r o w e r s h o c k t r a n s i t io nb u t w i t h s l i g h t n u m e r i c a l o s c i l l a t i o n s .

C O N C L U S I O NA n e x p l o s i o n i n a n e x p o n e n t i a l a t m o s p h e r e w i t h m a g n e t i c f ie ld s is s t u d ie d u s i n g a v e r s i o n o f

t h e m o d i f i e d L a x - W e n d r o f f d i f f e r e n c e m e t h o d . I t i s s h o w n t h a t t h e m a g n e t i c f i e l d s t r e n g t h a n di t s c o n f i g u r a t i o n h a v e v e r y s u b s t a n t i a l i n f l u e n c e o n t h e r e s u l t i n g f l o w f i e l d f r o m a n e x p l o s i o n .T h e p r e s e n c e o f b o t h f a s t a n d s l o w M H D s h o c k w a v e s , a s w e l l a s g a s d y n a m i c s h o c k w a v e s ,a r e d e m o n s t r a t e d i n c o m p u t a t i o n a l r e s ul ts . I t i s s h o w n t h a t t h e M H D s h o c k j u m p c o n d i t i o n so b t a i n e d f r o m t h e n u m e r i c a l r e s u l t s a r e i n s a t i s f a c t o r y a g r e e m e n t w i t h t h e t h e o r e t i c a l l ~ a n k l n e -H u g o n i o t j u m p c o n d it io n s . I n o t h e r w o r d s , t h e n u m e r i c a l s c h e m e s u g g e s te d c a n b e c o n s i d e r e d ar e li a bl e t o o l f o r s t u d y i n g 2- d im e n s i on a l M H D p r o b l e m s i n n o n - h o m o g e n e o u s m e d i u m .A d a ,o w h ~ # ~ - - T w o o f th e ~ t ho r s (S MI-Iand ST W) wish to express their sppre cin t~ to the National Aeronauticsand Spac e .~tmln;ttratinn~/arshall S pace Fight Center for their support of this work thronjh Cm m wt NAS8-28097.W ealso wish W thank the reviewer for his vainabk comments.

R E F E R E N C E S!. K. V . R o b ~ and D. F~ Potter, Magnetohydrody-~mi~dr~dstio~ in M~ todJ m Com p. Phys. Voi. 9, p. 339.Academic Press, New York ( 1 9 7 1 ) .2. I. Lindem uch m d J. Killeen, Alternating direction impi/cit techniques f ~ ~ ms~'tohydmd3nuJm~

c a l c u la l / m n. . C o m p . Phys. D , 1 8 1 ( 1 9 7 3) .3 . P . C ., cc aD le , u m e r i c a l s t u d y o f a c y li n dr i ca l l a s m a e x ] ~ m m % ~ s f i c S e l d . . C , m p . P h y s . 1 4 , 3 7 1 (1 9 74 ) .4 . R . S . S t ei n ol f so n, . D r y e r a n d Y . N a k a g a w a , N u m e r i c a l M H D s i m u l a U o n s o f i n ~ rp l a ne l a ry h o c k I ~ ur s . . C ~ o p h y s .

R ~ . 8 O , 1 2 2 3 ( 1 97 5 ).5. Y. Nakqawa and IL E. WeIlck,Num erical studies of azim uthal moduint/onsof the solar wind with m a ~ fields. So/atPhy s. ~ 42 (1973).6. Y. N akagawa , S. T. W u and S . M. Ha rt. Masnetohydrodymunicsof atmospheric mmsients---l. Basic results o ftwo.dimensional plane analys is. Astrophys. J . 219, 314 (1978).7. S. T. Wu, M. Dryer, Y. N,~%nwa and S. M. Ha n, M qne toh ydr o~ -m ics of almospheric laus/onts--H. Two-d i m e n s i o n a l n u m e r i c a l r e su l t s o r a m o d e l s o l a r o r o n a . A s t m p h y s . . 2 1 9 , 3 2 4 ( 1 9 78 ) .8 . O . W . ~ m m a n d A . S h e r m a n , E a g i ~ . u m g l ~ t o h ~ m d y ~ m i e s . M ~ ,r a w. I -l i ll , e w Y o r k ( 1 96 5 ).9. A . JeBrey , ~d~,m~ohydrodyumics. Oliver and Boy d, Londo n (1966).I0. S. M. Ham,A Numm~ical~ u d y o i T w o - D i m m s io ~ , T I m e . D q x m d ~ 1 ~ a o h y d m d y ~ m m i c F lo w .Ph.D . Thesis. Thel.lu/vm~ ty of A labam a n l.lumXsvme 1976).

I I , E. L. Rubin and S. F. B u r ~ DMerence metheds for the inviscid and viscous equations of a c o m p r e ~ k ~ . J.Com p. Phys., 2, I78 (1967).12. A. L ~ s , A detached shock calcu]al~n by second-order~nite diferences. J. Com p. Phys . 2, 154 (1967).1 3. R . D . R i c h t m y e r a n d K . W . M o r t o n , D i l e r e n c e e t h o d s / o r I ni ti al a l u e r o b l e m s , n d E d n . I nt er sc i en ce , e w Y o r k

( 1 9 S 7 ) .1 4. R . D. R i c h tm y e r , A S u r v e y o f ~ e n m c e M a ~ m ~ / s / o r N o n . S t e a d y ~ D y n a m i c s . N C A R T e c h N o t e 6 3 - 2 0 9 63 ) .1 5. A . J d r e y a n d T . T a n i u , o n . l .K n e a r a v e P v o p a sa t i a n. c a d e m i c P r es s , e w Y m ' k (1 96 4) .

A P P E N D I X l1 " n e ~ I ~ , n ~ , ,e - H u s o ni o t e la ti on s ~ e n b y [ 15 ]

I - ~ W + H . n l = 0 , ( AI )where ! ldeno~ 's that quantities in ! ! are e.onserv~dand ~, is the s hoc k veinc /ty, W is a c~nservat/v~ow variable, H is the~. x v~ctors and n is the unit vocto~ n the dirocl/m~ of shock ~ ~ the axis of symmelry(@= s~/2) in C ase c, = /. !! . a = F (F is the flux in r.diroction in ~ n s (1) and (3)) and thus, ~ (AI) reduces to

V . I W I - l ~ ( A 2)where V, i s t h e a h e ol u t e s h o c k v e l oc i ty l o n g t h e a x is o f s y m m e W / a n d W ~ d F s r ~ l ~ v m b y ~ u s ( 2 ) a n d O ) . U ~components of eqns (2) and 0), we m y Imvealong axis of symm e~'y a Case

p~(V, - Vq) = p , (V , - VO (A~) , , ( V , - V O = , , , ( V , - V O (^4)

, , ( V , - V , f + P , + ~o~o" . f V , - V q)~+ P,+ B--~ ( A 5 ]an d

I B ~ "

w h e r e h = e + ( p /~ ) n d s u b sc ri p t I a n d 2 r e fe r s o ~ a s h o c k a n d p o m h o c k c on di ti on s .


12/12

I ~ S .M . H A ~ a e LR e n ~ a e i ~ e e q n (A S) , w e have

where[ , - (A~)

M . - ( V . - V . ,) l ., , ~ J ~ = ( V . - V . ) l C . , ,C . , = V ( y R T ~ ) ,m e C . , , e ~ ,' V ( p i ~ ) "

l a u ~ d e e l ~ k = C , T , e q n ( A 6 ) c , a b e w r i tt e n a sp , p~ ~ p ~ 2 . ( ' ~ - l )M /. - , . - o [

Comb'min l eq ~ (AT) and (AS) , we have

' ta--~K~,, / ( 2 -. y) ( 2 -~ ,) . W ' - I ~ " ~ , ~ , / t ,: -'v ~ u - ' r ~ where

_~}~ =0,

( A S )

(A9)

(AI0)

O . e I p 2 .PlThe !__,~ ' a t u r e i m p t e d t h e v e lo ci ty j um p ar e I i v e n b y

m - ~ t / 2 C . )an d

(A l l )

V ,~ = V , ( I - ~ , ) + V . , / o , . (AI2)It can easily be checked that w hen there is no ma lnetic field, M A . . , e % eqas (A 9)-(AI2) reduce to well known ordinary gasdynamic shock jump c oud/tions.

A P P E N D I X 2In the governing equations, the gas pressure term p shall be calculated at eve ry time step foe the sixth component Wvector in eqn (2). In the present ca lc ela tm , it shows that kinetic energy is nqlilp'ble com pared with intenud tad mA_neticenergy. Therefore, th e gas pressure is computed from

p r e (y - 1) E - , (A 13 )where f and B are compu ted total en erl y and mallaatic field at a specified space ta d time respectively.Le t F.e and Be be the actual values of the total energy and maInetic field and e~ and e j are the errors introduced byfinite d iference scheme.Introducleg E = Eo + ee and B = Be + ee into eqn (AI3), for small # cases, the error in computation of the p s pressureterm, ~ can be expressed by

Oo # 'where ~, ,~ p - po. Equation (AI4) shows that the relative error in pressure is ~ to ~ I/~. In the present problem,i s -0 .02 a t the ou ter boundary o f the ~ u t a t i o n a l dou u~. I f we assmne tha t I~ ,~ ~0 .01 , then ~ ,/po ~ 0 .2 , imp ly ingthat computed ~ pressure can have ~ 21)9t error at most at the outer rel iou o f the competalioual d o e ~ i . Nemerica/results at the outer reIiou, tbe*efore, should be examined carefully.

_numerical study of an explosion in a non-homogeneous medium with and without magnetic fields

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