10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy was eleased. Such explosions occu fo example in asophysics in he fom of supenova explosions. Bu how fas will he shock wave avel and wha is lef behind? The poblem of he poin explosion is also known as Sedov-Taylo explosion, afe he wo scieniss ha fis solved i by analyic (and in pa numeical) means in he conex of aomic bomb explosions. Today, he poblem can povide a useful es o validae a hydodynamical numeical scheme, because an analyic soluion fo i can be compued which can hen be compaed o numeical esuls. Also, he poblem seves as a good example o demonsae he powe of dimensional analysis and scale-fee soluions. 10.1 A ough esimae Le s begin by deiving an ode of magniude esimae fo he adius R() of he shock as a funcion of ime. The mass of he swep up maeial is of ode M() R 3 (). The fluid velociy behind he shock will be of ode he mean adial velociy of he shock, v() R()/. We fuhe expec E kin 1 2 Mv2 R 3 R2 = ρ R 5 2 1 (10.1) 2 Wha abou he hemal enegy in he bubble ceaed by he explosion? This should be of ode E hem 3 2 P V (10.2) 1
10 Poin explosion whee P is he posshock pessue. To find his pessue, we need o ecall he jump condiions acoss a shock. If he shock moves o he igh wih velociy v 1 = v(), hen in he es-fame of he shock he backgound gas seams wih velociy v 1 o he lef, and comes ou of he shock wih a highe densiy ρ 2, highe pessue P 2, and wih a lowe velociy v 2. The Rankine-Hugonoi elaions fo he shock ell us ρ 2 = v 2 v 1 = γ 1 + 2 ()M 2 (10.3) whee M = v 1 (10.4) c 1 is he Mach numbe of he shock. Fo a song explosion, he sound-speed of he backgound medium is negligibly small, so ha he Mach numbe will end o infiniy in his limi. Fo he pessue, he Rankine-Hugonoi elaion is P 2 = 2γM2 P 1 γ 1 (10.5) As he backgound pessue is P 1 = c 2 1/γ, we hen obain in he limi of a song shock: P 2 2v1 2 (10.6) Wih his posshock pessue, we can now esimae he hemal enegy in he shocked bubble: E hem P 2 R 3 v1r 2 3 R 5 (10.7) 2 This suggess ha he hemal enegy is of he same ode as he kineic enegy, and scales in he same fashion wih ime. Hence also fo he oal enegy E, which is a conseved quaniy, we expec R 5 E = E kin + E hem (10.8) 2 Solving fo he adius R(), we ge he expeced dependence ( ) 1 E 2 5 R() (10.9) 2
10.1 A ough esimae 10.1.1 Dimensional analysis Anohe poweful appoach o make a simila kind of esimae is hough dimensionless analysis. If we assume he posshock pessue is always much lage han he peshock pessue, P 2 P 1, hen he value of P 1 plays no ole. The only paamees of he poblem ae hen E and, and all quaniies appeaing in he soluion can only be a combinaion of E,, and. We can hence use dimensional analysis o idenify some of he expeced dependencies. The dimensions of he pincipal quaniies ae: [ ] = M L 3 (10.10) [E] = M L2 T 2 (10.11) [] = T (10.12) The only quaniy of dimension lengh we can consuc fom his is [ (E ) 1 ] 2 5 = L (10.13) Hence any adius elevan fo he poblem, in paicula he shock adius, mus depend on hese vaiables hough his combinaion. This also moivaes us o define a similaiy vaiable η as η (E 2 / ) 1/5 (10.14) A fixed ime, his is simply a scaled adial coodinae. The shock will be a his ime a some posiion η s in his vaiable. Now, a a diffeen ime, he shock will be a he same value of η s. The self-simila soluion becomes saionay in he appopiaely scaled vaiables. Fo he shock posiion we can hence wie ( ) E 2 1/5 R() = η s 1/5, (10.15) whee η s is a consan of ode uniy. The shock velociy follows via ime diffeeniaion as v s () = dr d = 2 R() 3/5 (10.16) 5 10.1.2 Obaining he exac soluion In ode o obain he numeical value of η s, and he deailed sucue of he ineio soluion in he explosion bubble, we can y o fuhe exploi he scale-similaiy of he soluion (apa fom assuming spheical symmey, of couse). We fis ecall 3
10 Poin explosion ha ouside of he shock, he soluion is v = 0, ρ =, and P = P 1 0. Inside he shock, fo 0 < < R(), we can make he ansaz ρ(, ) = ρ 2 A(η) (10.17) because all physical quaniies inside he shock can only depend on η. In his ansaz, we have aleady incopoaed he posshock densiy ρ 2 = γ 1, (10.18) of a song shock, so ha we know A(η s ) = 1. Similaly, we can make he ansaz P (, ) = P 2 ( η η s ) 2 B(η), (10.19) so ha a he shock we ge B(η s ) = 1. The exa faco (η/η s ) 2 is hee inoduced fo convenience lae on, bu i could also be absobed ino a edefiniion of B. Finally, we can adop ( ) η v(, ) = v2 lab C(η) (10.20) η s o descibe he un of he adial velociy, wih C(η s ) = 1. We need o be a bi caeful abou he meaning of v2 lab, which is no he posshock velociy v 2 in he shock s esfame ha we consideed ealie. Rahe, v2 lab is elaive o he esfame of he backgound medium (he lab fame). If he shock moves wih v s o he igh, we have v1 lab = v s v 1 (10.21) v lab 2 = v s v 2 (10.22) whee posiive v 1 and v 2 descibe moion o he lef, as in ou skech. Since v lab 1 = 0, i follows v 1 = v s, and fom he jump condiions we have v lab 2 = v 1 v 1 γ 1 = 2 v s (10.23) We have now wien ρ(, ), P (, ) and v(, ) in self-simila fom in ems of hee funcions A(η), B(η) and C(η). To deemine hese funcions, we use he Eule equaions in spheical symmey: [ ρ(u + 1 2 v2 ) + 1 2 (2 vρ) = 0 (10.24) v + v v = 1 ρ ] + 1 2 P [ 2 ρv(u + P ρ + 1 2 v2 ) (10.25) ] = 0 (10.26) 4
10.1 A ough esimae This is augmened wih he ideal gas equaion of sae, P = (γ 1)ρu. (10.27) We can subsiue ino hese equaions ou hee self-simila ansaz funcions. Wheneve he hemal enegy appeas, i can be eplaced hough a combinaion of pessue and densiy, u = P/[(γ 1)ρ]. In addiion, we also wan o cay ou a change of vaiables, fom (, ) o (η, ). This can be accomplished by noing ha: ( ) ( ) η = η + = 2η η 5 η + (10.28) η = ( η ) η + ( ) = η η η (10.29) Using hese elaions and ou ansaz subsiuions in he Eule equaions leads o a se of coupled odinay diffeenial equaions in η. Fom he mass consevaion equaion, we obain η da dη + 2 Similaly, one ges fom he momenum equaion C 2 5 η dc ( dη + 4 C 2 + Cη dc ) = 2 γ 1 1 5() dη 5 A and fom he enegy equaion: 2(B+AC 2 ) 2 5 η d 4 dη (B+AC2 )+ 5() d dη (ηac) + 4 AC = 0 (10.30) ( 2B + η db ) dη (10.31) ( 5C(γB + AC 2 ) + η d [ C(γB + AC 2 ) ]) = 0 dη (10.32) The equaions (10.30), (10.31), and (10.32) ae hee 1s ode, non-linea, coupled diffeenial equaions fo he funcions A(η), B(η) and C(η). They can in pinciple be easily numeically inegaed fom he poin A(η s ) = B(η s ) = C(η s ) = 1 o η = 0. Thee is only one cach hee we don acually know he saing value η s ye! If we simply guess a value fo η s, we can cay ou he inegaion of he hee funcions and will ge some kind of soluion. The ouble is, anohe guess will also give us a soluion, bu a diffeen one. Howeve, fo he igh soluion, an addiional consain mus hold. The oal enegy in he soluion mus be equal o he oal enegy E eleased by he explosion. This coesponds o he consain R() 0 ( P γ 1 + 1 2 ρv2 Inseing ou self-simila ansaz, his becomes 32π 25(γ 2 1) ηs The soluion saegy is heefoe as follows: 0 ) 4π 2 d = E. (10.33) (B + AC 2 )η 4 dη = 1 (10.34) 5
10 Poin explosion 1. Guess a value fo η s. 2. Calculae a numeical soluion fo A(η), B(η) and C(η) based on equaions (10.30), (10.31), and (10.32). 3. Check how well equaion (10.34) inegaes o 1. Adjus η s accodingly and epea unil convegence. Sedov acually managed o make consideably fuhe pogess in analyically solving hese equaions, deiving algebaic expessions fo he shapes of he funcions A, B and C. Bu even in his appoach, hee is howeve sill one numeical inegaion lef ha is needed o deemine η s. 10.2 Sucue of he soluion Fo a γ = 5/3 gas, η s is given by η s 1.15. The densiy sucue descibed by A(η) a ime = 0.4 fo an explosion wih enegy E = 1 in a cold backgound medium wih uni densiy ρ = 1 looks as follows: 5 4 3 ρ 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 R We see ha a lage pa of he mass in he bubble is indeed swep up close o he shock, wih he densiy appoaching zeo a he vey cene. Ineesingly, i un ou ha he pessue is almos consan in he bubble; i dops slighly fom he pos-shock value and is hen almos consan. The velociy pofile on he ohe is appoximaely ising linealy fom he explosion sie o he edge of he shock. 6