J. Fluid Mech. (2006), vol. 560, pp. 103 122. c 2006 Cambidge Uivesiy Pess doi:10.1017/s0022112006000590 Pied i he Uied Kigdom 103 O implodig cylidical ad spheical shock waves i a pefec gas By N. F. PONCHAUT, H. G. HORNUNG, D. I. PULLIN AND C. A. MOUTON Califoia Isiue of Techology, Pasadea CA 91125, USA (Received 4 July 2005 ad i evised fom 6 Jauay 2006) The poblem of a cylidically o spheically implodig ad eflecig shock wave i a flow iiially a es is sudied wihou he use of he sog-shock appoximaio. Dimesioal agumes ae fis used o show ha his flow admis a geeal soluio whee a ifiiesimally weak shock fom ifiiy seghes as i coveges owads he oigi. Fo a pefec-gas equaio of sae, his soluio depeds oly o he dimesioaliy of he flow ad o he aio of specific heas. The Gudeley powe-law esul ca he be iepeed as he leadig-ode, sog-shock appoximaio, valid ea he oigi a he implosio cee. We impove he Gudeley soluio by addig wo fuhe ems i he seies expasio soluio fo boh he icomig ad he efleced shock waves. A seies expasio, valid whee he shock is sill weak ad vey fa fom he oigi, is also cosuced. Wih a appopiae chage of vaiables ad usig he exac shock-jump codiios, a umeical, chaaceisics-based soluio is obaied descibig he geeal shock moio fom almos ifiiy o vey close o he eflecio poi. Compaisos ae made bewee he seies expasios, he chaaceisics soluio, ad he esuls obaied usig a Eule solve. These show ha he addiio of wo ems o he Gudeley soluio sigificaly exeds he age of validiy of he sog-shock seies expasio. 1. Ioducio The poblem of a implodig shock wave is ieesig fom a fudameal gasdyamical poi of view, ad has impoa applicaios agig fom deoaio ad fusio iiiaio o he desucio of kidey soes. Gudeley fis ivesigaed he poblem by cosideig a cylidical o spheical shock wave, iiially a a vey lage adius, popagaig iwad hough a pefec gas a es ad he eflecig fom he axis o cee, see Gudeley (1942). Gudeley cosideed oly he case whee he shock is aleady so iese ha he sog fom of he shock jump elaios apply. Fo his case, he foud a similaiy soluio, i which he shock adius is give by he ime elaive o he ime a which i eaches he cee aised o some powe smalle ha uiy, so ha he shock segh becomes ifiie a he cee. The value of his powe is he same fo he icomig ad efleced shock waves. Sice Gudeley s wok, his value has bee ecalculaed wih geae accuacy by seveal eseaches. Hafe (1988) deived he equaios i Lagagia coodiaes ad used powe seies o solve hese. By doig so, he was able o fid he expoe value wih a vey high umbe of sigifica digis. The poblem was also sudied by Chese (1954), Chisell (1955), ad Whiham (1958) wih appoximae mehods, specifically geomeical shock dyamics. I hei soluios, he expoe i he expessio fo he Mach umbe as a fucio of
104 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo shock adius fo he spheical case is exacly wice ha fo he cylidical case. This appoximae esul diffes fom he exac soluio by less ha oe pece. The geomeical shock dyamics mehod is boh simple ad iuiive while povidig faily accuae esuls. Moe ecely, Chisell (1998) descibed he implodig shock poblem aalyically, alog wih he flow geeaed behid i, by makig a few aalyical assumpios. The expoe values ha he foud, usig appoximae equaios, ae faily close o hei exac values, which idicaes ha his descipios ae valid. Chisell also ivesigaed he covegig shock behaviou whe he specific hea aio, γ,edso1oo ifiiy. Fially, Lee (1967) used a quasi-simila appoximaio ad was able o fid he appoximae flow behaviou eve fo fiie Mach umbes. His soluio agees vey well wih he exac similaiy soluio. Gudeley suggesed ha he powe-law soluio should be exeded as a powe seies. This would he pemi elaxaio of he sog-shock assumpio, allowig soluios fo a iceased age of accepable Mach umbes, valid fahe fom he poi of eflecio. The aim of he pese wok is o ideify a geealized implodig-shock poblem usig he full Rakie Hugoio shock-jump elaios, ad o obai a powe seies soluio boh fo he covegig ad efleced shocks i which he Gudeley soluio is he fis em i he ie, sog-shock expasio. Such a soluio would apply i ad ea he sog-shock limi. Also, we seek a seies soluio fo he oue flow descibed by a iiially ifiiesimally weak shock a ifiie adius as i popagaes iwad. Fuhemoe, we also aim o compue he full flow field fo he geealized implodig shock usig he mehod of chaaceisics. Fially, we wa o compae he esuls wih umeical simulaios. I he followig secios we fis caefully defie he geealized implodig eflecig shock flow. The poblem will he be posed, dimesioal aalysis will be used o guide he soluio saegy, ad fially, he geeal equaios will be give ( 2). The he Gudeley soluio will be iepeed ad wo addiioal ems will be added. A seies expasio will also be fomed fo a vey weak shock locaed vey fa fom he eflecio poi ( 3). A algoihm based o he mehod of chaaceisics, ad desiged o fid he complee implodig shock soluio fom ifiiy o he oigi, will be descibed ( 4). Fially, some compaisos bewee expasios, chaaceisic soluios, ad esuls fom a Eule solve will be peseed ( 5). Noe ha he seies expasio calculaios ivolve vey leghy expessios. Fo his easo, oly he mehod o solve he poblem, ad o he expessios hemselves will be peseed i his aicle. Fuhe deails ae povided i Pochau (2005). 2. Poblem defiiio 2.1. Geeal oaio Coside he oe-dimesioal poblem of a shock popagaig, fom ifiiy, hough a saioay iviscid pefec gas, ad eflecig a he oigi. This shock ca have eihe cylidical (ν = 2) o spheical symmey (ν = 3). The poblem, as defied by Gudeley (1942), has o chaaceisic legh; he shock comes fom ifiiy ad eflecs back o ifiiy. Fo he puposes of his pape, eal gas effecs ad shock isabiliies ae o cosideed. The idepede vaiables i his poblem ae he adius,, ad he ime,. The shock posiio is give by R s () ad is velociy is U s (). The oigi of he idepede
O implodig shock waves i a pefec gas 105 IV III 0 II I Shock posiio Fis chaaceisic family (paicle ajecoy): d/d = u Secod chaaceisic family: d/d = u + c Thid chaaceisic family: d/d = u c Las chaaceisic eachig he icomig shock Figue 1. Skech of he diagam of he poblem. The shock posiio is epeseed by he hick cuves. The hee diffee chaaceisic families ae also show. u ad c ae he local velociy ad speed of soud of he flow, especively. vaiables is such ha he shock eflecs a =0 whe =0, i.e. R s (0) =0. (2.1) I his cofiguaio, he icomig pa of he shock is chaaceized by <0 ad he efleced pa, by >0. The medium is a pefec gas wih a aio of specific heas γ, ad he flow upseam of he icomig shock is a es wih pessue P I ad desiy ρ I. Noe ha he oaio i his wok, which is suiable fo he seies aalysis, is slighly diffee fom ha chose by Gudeley. The eie poblem ca be divided io fou sepaae egios i he (, )-plae. Regio I is he udisubed flow ( <0ad<R s ()). I his egio, he desiy ad he pessue ae cosa ad he flow is a es. Regio II coespods o he flow behid he icomig shock ( <0ad>R s ()). Regio III coespods o he flow upseam of he efleced shock ( >0ad>R s ()). Ad, fially, egio IV coespods o he flow dowseam of he efleced shock ( >0ad<R s ()). I he cue oaio, f I efes o he value of f i egio I, f II o he value of f i egio II, ec. Figue 1 shows hese fou egios, as well as epeseaive shapes of he diffee families of chaaceisics. Fis, a paicle ajecoy is show as he hi solid cuves. I egio I, he flow is a es ad heefoe he paicle ajecoies ae jus lies of cosa. Afe he icomig shock, he gas flows owads he cee bu slows dow i ime due o he accumulaio of mass. Afe he efleced shock, he flow is
106 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo dieced away fom he cee ad slows dow o eveually come o es whe he pessue becomes cosa i he whole domai. The secod family of chaaceisics is defied by d/d = u + c ad is epeseed by doed cuves. I egio I, hese ae saigh lies epeseig waves movig a he speed of soud. Afe he icomig shock, hey ae defleced owad he cee ad hey sop a he shock. I fac, he secod family of chaaceisics i egio IV epeses waves ha avel fase ha he shock ad sop whe hey each i. Fially, he hid family of chaaceisics is defied by d/d = u c ad is epeseed by dashed cuves. I egio I, hese chaaceisics ae also saigh lies ha descibe waves movig a he speed of soud. They sop a he icomig shock sice, i egio II, hey avel fase ha he shock. The hid family of chaaceisics i egio II o III eachig he efleced shock ae jus defleced by he efleced shock. Noe ha he paicula chaaceisic ha eaches he shock a = 0 is of special impoace sice i is he bouday of he domai ha iflueces he icomig shock. This chaaceisic leads o sigulaiies i he equaios as explaied i lae secios. Regio I is aleady kow ad egios II, III ad IV saisfy he Eule equaios. Shock jump codiios mus be saisfied alog he boudaies bewee egios I ad II ad bewee egios III ad IV. The vaiables mus be coiuous bewee egios II ad III ad fially, fom physical agumes, he flow mus have o velociy a he oigi. I is impoa o oe ha he shock Mach umbe eds o ifiiy a he oigi i he icomig case, bu is fiie i he efleced case. The easo fo his is ha, alhough he shock velociy is ifiie a he oigi, he speed of soud a he oigi, i egio III, is ifiie as well. 2.2. Dimesioal aalysis Accodig o Buckigham s pi heoem, sice we have seve vaiables, we ca fom fou idepede o-dimesioal umbes. We will ake hese fou o-dimesioal paamees o be ν, γ, θ = c I,η= R s R s, whee c I is he speed of soud i he udisubed egio (c I = γp I /ρ I ). These fou o-dimesioal vaiables ae sufficie o descibe he complee soluio o he poblem. This meas ha i (θ, η) coodiaes, hee exiss a uivesal soluio fo a give γ ad a give ν. The desiy, pessue, ad velociy ca be expessed as ρ = ρ I ρ(ν, γ, θ, η), (2.2) P = P I P (ν, γ, θ, η), (2.3) u = c I u(ν, γ, θ, η). (2.4) The shock moio ca also be ivesigaed by cosideig he velociy of he shock, U s =dr s /d. Sice he shock is locaed a η = 1, we have U s = c I K ( 1) (ν, γ, θ), (2.5) whee K ( 1) is he ivese fucio of K wih espec o U s /c I. K is a ukow fucio defied as ( θ = K ν, γ, U ) s. (2.6) c I
O implodig shock waves i a pefec gas 107 This equaio is useful sice i is valid fo all shock moios, egadless of he paicula scalig of he poblem. Howeve, i is o coveie whe diec pedicios of he shock posiio ae desied. I fac, (2.6) ivolves hee vaiables (, R s () ad U s ()), ad is a oliea diffeeial equaio fo R s (). This ca be see if we wie (2.6) as ( c I R s () = K ν, γ, 1 ) dr s (). (2.7) c I d To fid moe suiable equaios, we ca dop oe of he vaiables ( o U s )ad defie a chaaceisic ime, τ, such ha c I τ is he adius a which he icomig shock has a Mach umbe of 2 (abiay choice). I his case, we ca show ha we ge he followig expessios: ( R s c I τ = F ν, γ, U ) s, (2.8) c I R s c I τ = G (ν, γ, τ ), (2.9) whee F (ν, γ, 2) =1, (2.10) G(ν, γ, 0) =0. (2.11) The seies expasios will be wie i he same fom as (2.9). The fucios F, G ad K will be used o make compaisos bewee he soluios fom he seies expasios, he mehod of chaaceisics ad Eule compuaios ( 5.3). 3. Seies expasio soluios To obai a simple soluio, i is useful o fid he limiig behavious of he shock i he fom of a seies. Gudeley obaied he fis em of he expasio seies soluio close o he oigi, fo c I τ (o equivalely, fo M s 1). I he ex subsecios, we will fis asfom he equaios o make hem moe suiable fo solvig he seies expasio poblems ( 3.1). Gudeley s soluio will he be examied ( 3.2) ad exeded wih wo addiioal ems ( 3.3). Fially, seies expasios will also be fomed fo he weak shock case, i.e. fo c I τ,ofom s 1 1( 3.4). These ca be vey useful as saig codiios whe usig a umeical mehod o fid he soluio i he eie domai. 3.1. Iiial chage of vaiables Lookig a figue 1, we see ha he fou egios have complicaed shapes ha ae o kow apioi. I is much easie o asfom hese egios io oes wih fixed shapes o fid ou exacly whee he shock jump codiios have o be applied. This ca be doe by makig he chage of vaiables (, ) (η, ), whee η = R s(). (3.1) The shock jump codiios ow occu a η = 1 ad ude his chage of vaiables, figue 1 is asfomed io figue 2.
108 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo η = R s () 1 I < 0 II 0 η = R s () IV 1 > 0 III 0 Shock posiio Fis chaaceisic family (paicle ajecoy): d/d = u Secod chaaceisic family: d/d = u + c Thid chaaceisic family: d/d = u c Las chaaceisic eachig he icomig shock Figue 2. Skech of he η diagam fo he poblem. The shock posiio is epeseed by he hick lies. The hee diffee kids of chaaceisics ae show as well. The cosses coespod o he same poi i he (, ) domai. Noe he shape of he las chaaceisic eachig he icomig shock: i he η diagam, his paicula chaaceisic eaches =0 a a fiie η<1. As will be show i he ex secios, he flow close o he oigi is self-simila ad he ajecoy of his chaaceisic coespods o a cosa η ha is diffee fom uiy. This paicula chaaceisic leads o a sigulaiy i he domai. 3.2. Gudeley s soluio Gudeley s wok focused o he egio close o =0 ( c I τ). Thee, i ca be assumed ha bewee egios I ad II, he sog-shock jump codiios ae valid. Noe ha his assumpio cao be made fo he efleced shock bewee III ad IV. Alhough he efleced shock velociy eds o ifiiy whe eds o 0, is Mach umbe is fiie sice he speed of soud eds o ifiiy as well. Fuhemoe, Gudeley hypohesized ha ude he sog-shock assumpio, he shock posiio ca be wie as popoioal o ime aised o he powe ±. Usig he oaio ioduced i (2.9), his meas ha ( ) ± R s () = c I τβ ±, (3.2) τ whee he supescip ± efes o a cosa ha has diffee values i he icomig ad i he efleced cases, wih ad + deoig he value i he icomig ad i he efleced cases, especively. To simplify he expessios i lae secios, i is moe
O implodig shock waves i a pefec gas 109 coveie o wie (3.2) as ( ) ± R s () =, (3.3) α ± τ s whee α ± is a cosa ha is chose o be 1 fo he icomig shock, ad ha has a ukow cosa posiive value α + fo he efleced shock, ad whee τ s =(β ) 1 1 τ. (3.4) The expoe ± is a ukow cosa ad, based o he wok so fa, is value is o ecessaily he same i he icomig ad i he efleced cases. Is value lies bewee 0 ad 1 sice R s (0) = 0 ad he speed of he shock eds o ifiiy as eds o 0. Fom (3.3), we ca fid he shock velociy, which is give by ( ) ± (Rs ) ± 1 ( ) ± ± 1 Rs U s (R s ) = c I = c ± α ± α ± I. (3.5) τ s α ± Sice he shock posiio will oly be accuae fo R s, he soluio ha we obai will oly be valid fo small. I addiio, sice he chaaceisics comig fom he shock ae oly coec fo small (o small R s ), he soluio is also oly coec fo τ s. The mehod o solve his simplified poblem will be discussed i he followig subsecios. New shock jump codiios will fis be wie ad ew vaiables will be ioduced ( 3.2.1). Fom he esulig equaios, he poblem will be solved fo ± adhefoα ± ( 3.2.2). 3.2.1. Self-simila poblem We fis wie he sog-shock jump codiios fo he icomig case, which ae γ +1 ρ II (1,) = ρ I γ 1, (3.6) ( ) 2γ Us (R s ) 2 η=1 2γ ( ±) 2 ( P II (1,) = P I = P I γ +1 c I γ +1 u II (1,) = 2 ( γ +1 U 2 ± s(r s ) η=1 = c I γ +1 ) 2( ± 1) ±, (3.7) ) ± 1 ±, (3.8) whee we used he fac ha (R s ) η=1 = ad ha fo he icomig case α = 1. These equaios sugges ha we y a soluio of he fom ρ(η, ) = ρ I ρ 1 (η), (3.9) ( ) ± 2 ( ) 2( ± 1) P (η, ) = P I P 1 (η) ±, (3.10) α ± ( ) u(η, ) = c I u 1 (η) ± ± 1 ±, (3.11) α ± whee ρ 1 (η), P 1 (η), adu 1 (η) ae ukow fucios of η oly. The subscips 1, 2, ad 3 deoe he fis, secod, ad hid ems i he seies expasios. Sice, whe his soluio fom is subsiued io he Eule equaios, hee is o loge a depedece o, he assumed fom of he vaiables is accepable ad he poblem becomes self-simila.
110 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo Oce agai, we have o made ay assumpios abou he value of he expoe ±. Usig he chage of vaiables defied ealie, he coiuiy codiios bewee egios II ad III ca be wie as ρ 1,III (0) = ρ 1,II (0), (3.12) ( ) + 2 ( ) 2( + 1) + P 1,III (0) = P α + 1,II (0) ( ) ( ) 2( 2 1), (3.13) ( ) u 1,III (0) + + 1 ( ) + 1 = u α + 1,II (0). (3.14) These codiios mus be saisfied fo all. This is oly possible if we impose he codiio ha ± = + = =. Fially, as oed befoe, he full Rakie Hugoio shock jump codiios ae used bewee egios III ad IV sice he efleced shock Mach umbe is fiie. Usig he defiiios ioduced i his subsecio, he esulig equaios fom a o-liea sysem of diffeeial equaios ha ca be fuhe simplified by applyig a fial chage of vaiables: φ 1 (η) = η 1/ u 1 (η), (3.15) η π 2/ P 1 (η) 1 (η) = γρ 1 (η)[1 φ 1 (η)], (3.16) which leads o a sysem of diffeeial equaios of he fom dφ 1 (η) dlogη = f 1(φ 1 (η), π 1 (η)), (3.17) dπ 1 (η) dlogη = g 1(φ 1 (η), π 1 (η)), (3.18) dρ 1 (η) dlogη = h 1(φ 1 (η), π 1 (η)). (3.19) I is easie o ea φ 1 as he idepede vaiable ahe ha η. Weheobaihe followig wo diffeeial equaios: dπ 1 (φ 1 ) = g 1(φ 1, π 1 (φ 1 )) dφ 1 f 1 (φ 1, π 1 (φ 1 )), (3.20) dlogη(φ 1 ) dφ 1 = 1 f 1 (φ 1, π 1 (φ 1 )). (3.21) Alog wih hese equaios, a fis iegal ca be foud: [ π1 (φ 1 ) ] ν ρ 1 (φ 1 ) = C η(φ 1 ) (1 φ 1) 2(1 )+ν ν(γ 1) 2(1 ) 2 ν, (3.22) whee C is a cosa ha akes a diffee value i each egio. The poblem is ow elaively easy o solve sice (3.20) ca fis be cosideed aloe o solve fo π 1 (φ 1 ). The, (3.21) ca be iegaed o fid η(φ 1 ). ρ 1 (φ 1 ) is obaied usig he fis iegal (3.22). 3.2.2. Discussio ad soluio Fis, coside he limiig values of he poblem. I egio II, jus dowseam of he shock, φ 1,II ad π 1,II ae give exacly by he shock jump codiios. A he
O implodig shock waves i a pefec gas 111 2.0 1.5 IV Soluio Zeo locaio Pole locaio Paicula pois Bouday pois 1.0 π 1 0.5 0 III II 0.5 0.50 0.25 0 0.25 0.50 0.75 1.00 1.25 Figue 3. Zeo ad pole locaios of he igh-had side of he π 1 (φ 1) equaio (3.20), fo ν =3, γ =5/3, ad =0.68838. The cicles ae he paicula pois ha ae iesecios of he zeo ad he pole cuves. The soluio is epeseed by he solid cuve ad he cosses show he boudaies of he egios. φ 1 jucio bewee II ad III, ρ 1, P 1,adu 1 ae coiuous ad fiie, meaig ha, sice η eds o 0, φ 1 ad π 1 also ed o 0 a he jucio (see (3.15) ad (3.16)). Fially i egio IV, whe η eds o ifiiy, P 1,IV mus be fiie, meaig ha π 1,IV (1 φ 1,IV ) mus also ed o ifiiy. I addiio, φ 1 cao chage sig wihi a egio, ad π 1 (1 φ 1 ) mus always emai posiive. Fo ay give case, ν ad γ ae fixed. Fo each value of, we ca plo, i he (φ 1, π 1 )-plae, he zeos ad poles of he igh-had side of (3.20) ad (3.21). Figue 3 shows hese fo π 1 (φ 1) (3.20). I egio II, he soluio ajecoy i he (φ 1, π 1 ) domai mus go fom he iiial poi o (0, 0). Alog his ajecoy, he vaiables mus be coiuous. This is oly possible if he ajecoy cosses a pole while simulaeously cossig a zeo ad vice-vesa. I fac, i ca be show ha if his is o he case, he soluio will o be smooh wihi he egio. I is cocepually easy o fid sice i coespods o he value a which he ajecoy cosses oe of he cicled pois i figue 3. I is impoa o oe ha he sigula poi coespods o a poi o he limiig chaaceisic, which is he chaaceisic ha eaches =0 a = 0 ad is epeseed i he chaaceisic skeches by a dash-do-do cuve. Coside ow he ajecoy i egio IV. By examiig (3.20), we see ha he oly way φ 1,IV (η) /η 1/ ca ed o ifiiy while keepig π 1,IV (η) [1 φ 1,IV (η)]/η 2/ fiie, is fo he ajecoy o become age o he pole cuve aoud is sigulaiy. This codiio leads o a specific value of α + (α beig 1). As a example, figue 3 epeses he soluio ajecoy fo ν =3 ad γ =5/3, i he (φ 1, π 1 )-plae, alog wih he coespodig pole ad zeo cuves of he π 1 (φ 1), equaio (3.20). Noe ha i his case, π 1 (φ 1) becomes ifiie ad chages sig wihi egio III. Fially, as discussed i 2.2, τ is a chaaceisic ime. The aio τ s /τ ad he cosa β ±, i (3.4), cao be evaluaed usig his expasio seies mehod. The eie poblem has o be compued usig a diffee mehod ad he cosas ae deemied so ha he shock ajecoy fis he expasio. This pocess will be explaied i moe deail i 4.
112 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo 3.3. Sog-shock seies expasio Gudeley s soluio descibed i 3.2 is oly valid fo vey sog shocks (M 2 1o R s c I τ). I is heefoe useful o exed he soluio i he fom of a seies expasio so ha he appoximae soluio emais valid ove a wide age of Mach umbes as was suggesed by Gudeley (1942). The way o do his is o exed (3.3) ad o assume a shock posiio equaio of he fom ( ) [ ( ) i2 ( ) i3 R s () = 1+a ± α ± 2 + a ± τ s α ± 3 +HOT], (3.23) τ s α ± τ s whee i k <i k+1 ad whee HOT epeses all he highe-ode ems. Esseially, each addiioal em paially coecs fo he fac ha, fo fiie Mach umbes, he sog shock soluio is o exac. Alhough i ivolves leghy calculaios, he mehod o fid he values of i k is saighfowad. Fis, he shock locaio is expaded i ems of a powe seies i. The shock jump codiios bewee egios I ad II, ad he vaiables ρ, P,aduae expaded i a seies as well. A ew sysem of diffeeial equaios is wie fo each em. These sysems of equaios ae sigula a exacly he same pois as hose foud i he Gudeley sog-shock soluio. Each sysem mus possess a coefficie a ± k wih a value such ha he sigulaiy is avoided. If he expoes i k ae o coec, some sysems of equaios will o have a coefficie available o avoid ifiie deivaives a he coespodig sigula poi. Pefomig all he calculaios, i ca be show ha i k =2(k 1)(1 ). Subsiuig he appopiae expoes io (3.23) poduces a shock moio give by R s () = ( α ± τ s ) [ ( ) 2(1 ) ( ) 4(1 ) 1+a ± 2 + a ± α ± 3 +HOT]. (3.24) τ s α ± τ s Usig sadad seies ivesio echiques, we obai ( ) 1/ [ ( ) 2(1 ) (R s ) = α ± Rs τ s 1 a± 2 Rs + (a± 2 )2 (5 3) 2a ± 3 2 2 ( Rs ) 4(1 ) ] +HOT. (3.25) Usig he same echique as used by Gudeley, a seies expasio ca be wie fo he full Rakie Hugoio shock jump codiios bewee egios I ad II ad bewee egios III ad IV, as well as fo he coiuiy codiios bewee egios II ad III. These seies ae i ems of /( ) ad lead us o y seies soluios of he fom [ ( ρ(η, ) = ρ I ρ 1 (η) 1+ρ 2 (η) ( P (η, ) = P I P 1 (η) α ± ) 2 ( [ ( 1+P 2 (η) ) 2(1 ) ) 2( 1) ) 2(1 ) ( ) 4(1 ) ] + ρ 3 (η) +HOT, (3.26) ( ) 4(1 ) ] + P 3 (η) +HOT, (3.27)
O implodig shock waves i a pefec gas 113 u(η, ) = c I u 1 (η) ( α ± [ ( 1+u 2 (η) ) 1 ) 2(1 ) ( ) 4(1 ) ] + u 3 (η) +HOT. (3.28) Whe hese seies ae ioduced io he Eule equaios, he ew expessios lead o a ew seies of equaios. The way o solve his ew poblem is o add oe em a a ime io he seies. Assumig ha he fis (k 1) ems ae solved, a addiioal em ca be added o all he seies (ioducig a ew coefficie a ± k i he pocess). This leads o a seies of sysems of Eule equaios. Give ha he fis (k 1) sysems i he seies have aleady bee solved, oly he kh sysem eeds o be cosideed. Pefomig appopiae chages of vaiables, he followig sysem ca be wie: d φ k(φ 1 ) π k (φ 1 ) = M k φ k(φ 1 ) π k (φ 1 ) + a ± k dφ b k + c k, (3.29) 1 ψ k (φ 1 ) ψ k (φ 1 ) whee M k is a maix, ad whee b k ad c k ae vecos. These ae sigula a he limiig chaaceisic i egio II ad a = 0 i egio IV. To solve he sysem, ak ad a + k ae chose o emove he sigula behaviou i egio II ad i egio IV, especively. Kowig he values of he coefficie a ± k, he fucio G defied i 2.2 ca be obaied: ) R s (ν, c I τ = G γ, τ = τ ( s 1 τ α ± (τ s /τ) τ ) [ ( 1+a ± 2 1 α ± (τ s /τ) τ ) 2(1 ) ( + a ± 3 1 α ± (τ s /τ) τ ) 4(1 ) +HOT]. (3.30) The o-dimesioal fucios F ad K ca be wie i a simila fom. Noe ha i (3.30), he aio τ s /τ is cosa fo a give symmey ad a give γ, ad cao be obaied usig seies expasios. This aio ca oly be obaied by solvig he full poblem (see 4). Noe ha, based o he esuls obaied, he seies peseed i his secio seem o be a leas asympoic. Thei covegece was o sudied howeve. Moe deails abou he equaios ivolved i he calculaios ca be foud i Pochau (2005). 3.4. Weak-shock seies expasio I Gudeley s poblem, he shock is supposed o come fom ifiiy, o avel up o = 0, ad he o bouce back o ifiiy. I is heefoe also ieesig o look fo a expasio soluio whe he shock is sill weak ad fa fom he oigi. This ca oly be doe i he icomig shock case. I fac, i he efleced shock case, he full hisoy of he flow is equied ad o aalyic behaviou ca easily be obaied. If we cosuc a seies expasio simila o wha was doe i 3.2 ad 3.3, bu expad he soluio aoud (, ) edig o (, ) ahe ha (0, 0), we ge R s c I τ = τ w τ [ τ τ w τ + τ τ w τ 19 + 3γ 16(γ +1) log ( τ τ w τ ) + 0 +HOT ], (3.31)
114 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo i he cylidical case, ad ( R s c I τ = τ w τ τ ( τ w τ +log τ ) log τ ) τ w τ 0 + γ +5 γ +1 + 0 + + +HOT τ w τ (3.32) τ τ w τ τ τ w τ i he spheical case. I he weak-shock expasio, boh he aio τ w /τ ad he cosa 0 ae ukow ad cao be foud usig seies expasio. Noe ha 0 acs like a ime shif ha allows he shock o each he oigi a =0. 4. Icomig shock complee soluio I he pevious secio, we ivesigaed he limiig behavious of he poblem ad we obaied soluios ha icluded ukow values. These paicula values ae τ s /τ i he sog-shock expasio, τ w /τ ad 0 i he weak-shock expasio. These ca oly be deemied wih a complee calculaio of he icomig shock poblem. Fo ha pupose, as well as o obai he complee icomig shock soluio, a fis-ode-accuae pogam based o he mehod of chaaceisics was wie. Sice he pogam has o be able o compue he flow saig a R s c I τ ad edig a R s c I τ, compuig he soluio i eal space (, ) would be vey icoveie. To avoid ha, he followig chage of vaiables is pefomed: θ = c I R s (), η = R s(). Wih his chage of vaiables, he ifiie domai becomes a squae-bouded domai wih θ agig fom 1 o0adη agig fom 0 o 1 (see figue 4). The easo fo ioducig θ follows logically fom (2.6). I fac, i solvig he poblem, we will aually fid he ukow fucio θ = K(ν, γ, U s /c I ). I 3.2.1, we saw ha close o he oigi, he shock jump codiios give a fiie desiy behid he shock, bu he pessue, he velociy ad he shock speed ed o ifiiy. Fo a shock vey close o he oigi, we have ha P II (1,θ) R 2( 1) s 1 θ, (4.1) 2 u II (1,θ) R 1 s 1 θ, (4.2) U s (θ) R 1 s 1 θ. (4.3) To avoid ay sigula values i he compuaioal domai, he followig ew vaiables ae used: P (η, θ) = θ 2 P II(η, θ) P I, u(η, θ) = θ u II(η, θ) c I, c(η, θ) = θ c I γp II (η, θ) ρ II (η, θ), U s(θ) = θ U s(θ) c I. Usig his las chage of vaiables, he chaaceisics equaios ca be wie ad he poblem ca be solved fo P (η, θ), u(η, θ), c(η, θ), adu s (θ).
O implodig shock waves i a pefec gas 115 0 η* 1 θ = c I /R s () 1 η = R s ()/ Shock posiio Fis chaaceisic family (paicle ajecoy): d/d = u Secod chaaceisic family: d/d = u + c Thid chaaceisic family: d/d = u c Las chaaceisic eachig he icomig shock Figue 4. Skech of he η θ diagam fo he poblem i egio II. The shock posiio is epeseed by he hick lie. Is moio sas a θ = 1 ad eds a θ = 0. The hee diffee kids of chaaceisics ae also show. η is he value of η a which hee is a sigulaiy i he sog-shock seies expasio. Vey close o he oigi he flow is self-simila ad he o-dimesioal ajecoy of he sigula chaaceisic η = R s / appoaches a fiie value ha is diffee fom uiy, whe 0. As see i figue 4, all he hid-family chaaceisics ifluecig he icomig shock sa a (θ, η)=( 1, 1). This meas ha he whole poblem is defied a ha sigula poi. Sice i is impossible o sa hee, he pogam eeds o sa a θ +1 1 ad he weak-shock seies expasio ca be used as he saig flow. Noe ha a fiie bu small θ + 1, he hid-family chaaceisics ha ifluece he shock do o mege o a poi; hey do howeve emai wihi a vey aow age of η. Sice all he chaaceisics goig o he shock sa fom his aow age, we have o be able o add chaaceisics iside he domai. If, duig he compuaio, he spacig of wo successive chaaceisics of he same family becomes excessive, a ew chaaceisic is iiiaed bewee hem. This is doe by ceaig a ode ha is iepolaed alog a chaaceisic of he ohe family. To maiai accuacy duig his pocess, he iepolaio has o be of a ode ha is a leas as high as he ode of he chaaceisic compuaios. As peviously meioed, o sa he compuaio, he weak-shock soluio is used wih a value of θ ha is close o 1. I (η, θ) coodiaes, 0 is he oly ukow cosa appeaig i he saig codiios. If 0 had he coec value, he poblem could heoeically be iegaed up o he eflecio poi ad θ wouldedo0a he same ime as c I /U s. If he guessed value of 0 is oo low, he iegaio leads o a shock adius ha eds o 0 befoe ime eaches 0. This meas ha θ will ed o ifiiy. O he ohe had, if he value of 0 is oo high, he ime eds o 0 befoe
116 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo he shock adius ad θ eds o 0 fo a fiie value of c I /U s. A bisecio mehod is used o obai he coec value of 0. To fid he aios τ w /τ ad τ s /τ, he followig equaio is iegaed saig a a Mach umbe close o 1: 1 dr s c I d = U s(θ) c I = U s(θ) θ = U s(c I /R s ) c I /R s. (4.4) The iiial codiios (R s, ) ca be adom ad he value of τ w is abiaily chose, sice he ukow aios ae idepede of hese values. Oce iegaed, (4.4) gives R s () /c I. Fom hee, i is easy o fid he fucios F ad G used i (2.8) ad (2.9), especively. Fially, τ ca be obaied by usig τ = Rs /c I,wheeRs is he adius a which he icomig shock has a Mach umbe of 2. Fiig he sog-shock expasio esul wih he iegaed soluio, we fially ge τ s. The aios τ w /τ ad τ s /τ ca he be evaluaed. I is impoa o oe ha 0 ad he wo aios wee o foud wih a high degee of accuacy sice he iegaio pefomed was oly fis ode ad ha, fo example, he cosa i he iiial codiios does o appea i ay domia ems. Eve hough he accuacy is o high, he bisecio mehod has o fid he appopiae 0 value wih a high umbe of sigifica digis so ha he chaaceisics ca be compued up o a value of θ ha is close o 0. I ohe wods, we mus fid vey accuaely a value of 0 ha is cosise wih ou iegaio scheme, eve hough he value of 0 iself was o vey accuae. 5. Resuls I his secio, he seies expasio calculaios will be compaed o he chaaceisics esuls. I addiio, some compaisos will iclude compuaioal fluid dyamic calculaios of he Eule equaios made i he AMRITA eviome, see Quik (1998). The sog-shock seies expasio coefficies wee calculaed usig 2000 pois i egio II, 2000 pois i egio III, ad 1000 pois i egio IV. We will compae some aspecs of he esuls obaied by each mehod i he followig secios. Fis, he chaaceisics will be show i he (η, θ)-plae ( 5.1). The desiy, he pessue, ad he velociy disibuios will be compaed fo Eule calculaios ad he seies expasios ( 5.2). Fially, he o-dimesioal fucios F, G, adk ha wee defied i 2.2 will be compaed bewee he mehods ( 5.3). I addiio, some values of he cosas i he expasio seies ae peseed i able 1, alog wih a esimae of he umbe of sigifica digis. 5.1. Chaaceisic esuls Figue 5 shows oe se of he chaaceisic esuls ad illusaes he basic shapes of he secod ad hid chaaceisic family. This calculaio has cylidical symmey ad a specific hea aio, γ, of 1.4. The calculaios wee saed a θ = 0.995 ad wee sopped a θ = 0.0338, which coespods o shock Mach umbes agig fom 1.00257 o 24.6 o o a shock adius aio bewee he fial ad he iiial pois of abou 2 10 12. I he figue, chaaceisics of he secod family ae epeseed wih doed cuves ad chaaceisics of he hid family ae epeseed wih dashed cuves. The chaaceisic ha sepaaes he poio of he flow i egio II ha iflueces he shock fom he es of he egio is clealy disiguishable. The limi of his paicula chaaceisic fo θ edig o 0 is give i he sog-shock expasio
γ α + a 2 a + 2 a 3 a + 3 η 0 τ s τ Cylidical symmey 1.2 0.86116302390 6.099957294 0.56495331 21.4953 5.5264 106.97 0.8797191058 0.7212 195.6 30.66 1.4 0.835323191953 2.8156109349 0.2335140 8.92588 1.0608 52.646 0.85129137696 0.6470 49.26 23.39 5/3 0.8156229691667 1.6947147269 0.5004547 4.391459 0.1756 22.670 0.830529251753 0.5678 26.34 17.87 Spheical symmey 1.2 0.75714181478 6.431231558 0.6267725 50.4650 9.13 613.7 0.91041281789 0.7612 20.93 3.119 1.4 0.717174501488 2.688492680 0.275784 16.614985 2.59 200.2 0.88628918681 0.7052 9.933 2.727 5/3 0.6883740859496 1.5478188813 0.621289 7.06248 4.774 16.5 0.867641792579 0.6426 7.042 2.387 Table 1. Values of he diffee cosas. The umbes ae displayed wih a esimaed umbe of sigifica digis. These esimaes wee obaied by compuig each value wih vayig iegaio sep sizes. τ w τ O implodig shock waves i a pefec gas 117
118 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo 0 η* 0.1 0.2 0.3 0.4 θ 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 0.5 η 0.6 0.7 0.8 0.9 1.0 Figue 5. Secod ad hid chaaceisic families i he cylidical shock case fo γ =1.4. Doed ad dashed cuves epese chaaceisics of he secod ad he hid families, especively. The coss epeses he limi of he las chaaceisic eachig he icomig shock. This limi coespods o he value of η a he sigula poi i egio II i he sog-shock seies expasio. calculaio sice i coespods o he value of η a he sigulaiy (η ). This limi is epeseed by a coss i he figue. Due o he sigula behaviou of he chaaceisics close o η =0 ad o θ = 0, he chaaceisics could o be compued i he eie domai ad wee heefoe copped. 5.2. ρ, P, ad u disibuio Compaisos of he flow popeies wee made bewee he sog-shock seies expasio ad Eule calculaios. The vaiables ρ, P, u, ad ae omalized by ρ I, ρ I U s () 2, U s (),ad s (), especively. I hese expessios, U s () is he speed of he icomig shock whe i cosses he adius, ad s () is he ime a which ha occus. Compaisos ae made i he spheical case, fo γ =1.4. The shock was iiiaed wih a Mach umbe of 5 ad he vaiables wee ake a a adius 10 imes smalle ha he iiial adius whee he Mach umbe was 12.4. Figue 6 shows he vaiaio ove ime of he omalized desiy, pessue ad velociy. Gudeley s soluio is accuae fo vey high Mach umbes, bu a he cue Mach umbes, i shows some discepacies wih he Eule esuls. Mos of hese discepacies appea i he desiy ad he pessue disibuios. The discepacies ae gealy educed whe wo ems ae added o he expasio. As explaied peviously, he shock seies expasios ae accuae fo a low shock adius (R s c I τ). Bu his also meas ha he flow popeies a a give small adius will oly be valid fo small imes ( τ). 5.3. The o-dimesioal fucios F, G, ad K The fucios F, G, adk wee ioduced i 2.2. Give he symmey of he poblem, ν, ad specific hea aio, γ, hese fucios ae fully defied. They ae paiculaly useful sice hey descibe all possible shock ajecoies. The weak ad
O implodig shock waves i a pefec gas 119 120 20 100 16 ρ/ρ I 80 60 40 I II III IV P/(ρ I U s () 2 ) 12 8 I II III IV 20 Gudeley 4 Gudeley 3-em expasio 3-em expasio Eule solve Eule solve 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 / / s () s () 0 u/ U s () 0.2 0.4 I II III IV 0.6 Gudeley 0.8 3-em expasio Eule solve 2 0 2 4 6 8 10 / s () Figue 6. Nomalized desiy, pessue, ad velociy disibuios vesus omalized ime fo a spheical implodig shock (γ =1.4). The calculaios wee saed wih a shock Mach umbe of 5 ad he measuemes wee ake a a adius 10 imes smalle ha he iiial adius (whee he icomig shock Mach umbe is 12.4). Noe ha, alhough i is someimes idisiguishable fom he Eule compuaios, he 3-em expasio was ploed i each gaph i all he fou egios. sog seies expasios of hese fucios will be compaed o he chaaceisics esuls ad o he Eule soluios. The esuls will be peseed fo he axisymmeic case (ν = 2), wih a γ of 5/3. The fucios F, G ad K ae epeseed i figues 7, 8, ad 9, especively. Noe ha i he efleced shock case, hee is o weak-shock seies expasio. Also, he chaaceisics pogam is oly used o compue he icomig shock soluio. I all of he figues, he weak-shock seies expasio shows vey good ageeme wih he chaaceisics compuaio fo lage adii (o fo Mach umbes close o 1). The Gudeley soluio is excelle fo vey sog shocks. The addiio of wo ems o he Gudeley soluio clealy impoves he esuls sice i emais accuae ove a wide age of adii. Discepacies ca be see i he Eule compuaios fo lage. This is due o simplified iiial codiios ad he use of a o-ifiie domai. I addiio, discepacies i he Eule soluio exis i he egio close o he oigi because he gadies become vey high ad umeical eo becomes impoa. Noe ha eve hough he iiial codiios of he Eule simulaio wee o pefecly chose o avoid he ioducio of a legh scale, he compuaio quickly coveges o he seies soluio. This suggess ha, like he oe-em Gudeley soluio, he seies soluio is a aaco.
120 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo R s /c I τ = F(v, γ, U s /c I ) 10 5 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Weak expasio Gudeley 3-em expasio Eule solve Chaaceisics 10 3 10 4 10 5 1 0.75 0.5 0.25 0 0.25 0.5 0.75 1 (u s /c I ) 1 Figue 7. Fucio R s /(c I τ)=f (ν, γ, U s /c I ) i he axisymmeic case fo γ =5/3. R s /c I τ = G(v, γ, /τ) 10 5 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Weak expasio Gudeley 3-em expasio Eule solve Chaaceisics 10 3 10 4 10 5 10 6 10 3 10 0 /τ 10 3 10 6 10 6 10 3 10 0 10 3 10 6 /τ Figue 8. Fucio R s /(c I τ)=g(ν, γ, /τ) i he axisymmeic case fo γ =5/3. 6. Coclusios ad fuue wok I his wok, he implodig eflecig shock poblem was ivesigaed wih cylidical ad spheical symmey. The icide shock oigiaed fom ifiiy, avelled hough a iiially uifom pefec gas a es, was efleced a he oigi, ad avelled back o ifiiy. Gudeley s sog-shock soluio was expaded usig a hee-em powe seies o epese he behaviou of he shock close o he eflecio poi. Aohe seies expasio was cosuced o epese he behaviou of he icomig shock while i is vey fa fom he oigi. Fially, he mehod of chaaceisics was used o solve he icomig shock poblem houghou he eie domai. I ode o hadle he vey lage age of adii, a appopiae chage of vaiables had o be made.
θ = c I /R s = K(v, γ, U s /c I ) O implodig shock waves i a pefec gas 121 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 Weak expasio 0.6 Gudeley 3-em expasio 0.8 Eule solve Chaaceisics 1.0 1.00 0.75 0.50 0.25 0 0.25 0.50 0.75 1.00 (u s /c I ) 1 Figue 9. Fucio θ = K(ν, γ, U s /c I ) i he axisymmeic case fo γ =5/3. Seveal compaisos bewee he powe seies, he mehod of chaaceisics, ad Eule compuaio soluios wee peseed. The esuls show ha he weak-shock expasio seies is vey accuae fo lage adii. Also, he wo addiioal ems i he sog-shock seies expasio moe accuaely epese he acual soluio aoud he oigi. The ex sep o complee his wok would be o use he chaaceisics mehod o compue he efleced shock moio. This poblem is moe complicaed sice boh egios aoud he shock have o be calculaed (egios III ad IV). I addiio, he iiial codiios ae o based o he sog-shock seies expasio sice his seies is oly valid fo small. If he same ype of chage of vaiables as he oe used i he icomig chaaceisics pogam is used fo he eflecig pa, he followig pocedue could be used o obai a iiial codiio. Fis, he icomig shock case should be evaluaed up o jus befoe he eflecio ime. The, he soluio a ha ime has o be asfomed back io vaiables ad usual chaaceisics have o be used o fid he evoluio of he flow i egio II, o coss he bouday bewee egios II ad III, ad o coiue he calculaio up o a small fiie posiive ime. The, he esuls should be asfomed back io he coodiaes used i he chaaceisics pogam o fom he iiial codiios i egio III whee he adius is sufficiely lage ha he sog-shock seies expasio is o loge accuae. N.P. ad D.P. wee paially suppoed by he Academic Saegic Alliaces Pogam of he Acceleaed Saegic Compuig Iiiaive (ASCI/ASAP) ude subcoac o. B341492 of DOE coac W-7405-ENG-48. REFERENCES Chese, W. 1954 The quasi-cylidical shock ube. Phil. Mag. 45, 1293 1301. Chisell, R. F. 1955 The omal moio of a shock wave hough a ouifom oe-dimesioal medium. Poc. R. Soc. Lod. 232, 350 370. Chisell, R. F. 1998 A aalyic descipio of covegig shock waves. J. Fluid Mech. 354, 357 375. Gudeley, G. 1942 Sake kugelige ud zylidische Vedichugssöße i de Nähe des Kugelmielpukes bzw. de Zylideachse. Luffahfoschug 19, 302 312.
122 N. F. Pochau, H. G. Houg, D. I. Pulli ad C. A. Mouo Hafe, P. 1988 Sog covege shock waves ea he cee of covegece: a powe seies soluio. SIAM J. Appl. Mahs 48 (6), 1244 1261. Lee, B. H. K. 1967 Nouifom popagaio of implodig shocks ad deoaios. AIAA J. 5 (11), 1997 2003. Pochau, N. F. 2005 Pa I: 3DPTV: Advaces ad eo aalysis. Pa II: Exesio of Gudeley s soluio fo covegig shock waves. PhD hesis, Califoia Isiue of Techology. Quik, J. J. 1998 Amia a compuaioal faciliy (fo CFD modelig). VKI 29h CFD Lecue Seies ISSN 0377-8312. Whiham, G. B. 1958 O he popagaio of shock waves hough egios of o-uifom aea o flow. J. Fluid Mech. 4, 337 360.