Acceleration and Deceleration Phase Nonlinear Rayleigh-Taylor Growth at Spherical Interfaces

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1 Aeleion nd Deeleion Pse Nonline Ryleig-Tylo Gow Speil Inefes Pesened o: HEDP Summe Sool Dniel S. Clk nd Mx Tbk Lwene Livemoe Nionl Lbooy Augus 5 Tis wok ws pefomed unde e uspies of e U.S. Depmen of Enegy by e Univesiy of Clifoni Lwene Livemoe Nionl Lbooy unde Con No. W-745-Eng-48. Lwene Livemoe Nionl Lbooy, P.O. Box 88, Livemoe, CA UCRL-POST-434

2 Abs Te lssi Lyze model fo e nonline evoluion of bubbles in e Ryleig-Tylo insbiliy s eenly been genelized o e se of eleing, speilly imploding inefes [D.S.Clk nd M.Tbk, Pys.Rev.E 7, 553R 5.], e onfiguion mos elevn o ineil onfinemen fusion. Hee, e model is fue exended o e se of bubble gow duing e deeleion sgnion pse of speil sell implosion nd o e gow of spikes duing bo e eleion nd deeleion pses. Diffeenes in e nonline gow es fo bo bubbles nd spikes e found wen omped wi e pln esuls. Te model pediions e veified by ompison wi numeil ydodynmis simulions. UCRL-POST-434

3 Speil onvegene modifies nonline RT gow fom Lyze model expeions pln speil g Lyze s [955] model is e lssi model fo desibing nonline RT gow bu is sily vlid only fo pln inefes We ve ombined Lyze-like RT model nd Kidde-ype self-simil implosion o develop model fo speil inefes moe elevn o insbiliies on ICF psules Model evels iniil gow like e Lyze model, followed by le ime modifiions Compison wi -D simulions onfims pedied bubble spike gow UCRL-POST-434 3

4 Moivions Undesnding e nonline pse of RT gow is impon fo IFE wee opimized psule designs my pus e limis of ydodynmi insbiliy. Lyze s model simply quie uely desibes nonline RT gow bu is sily vlid only in pln geomey. Is ee n nlogous model fo speilly onveging sysem moe elevn o ICF ges? Finding n nlogous model mig evel ineesing sling popeies of nonline bubble gow. Su model ould povide igoous elevn nonline es poblem o vlide ydo. odes. Te speil poblem s no been solved, bu ppes solvble UCRL-POST-434 4

5 Ouline Review of Lyze model Modifiions neessy fo speilly onveging inefes Conneion o self-simil implosions of [Kidde`74] RT bubble model fo onveging geomey: eleion pse gow sgnion deeleion pse gow Lyze-ype models of RT spike gow RT spike model fo onveging geomey: eleion pse gow sgnion deeleion pse gow Appliion o non-self-simil implosions Summy/Conlusions UCRL-POST-434 5

6 Lyze model is e lssi model fo nonline pse of RT gow unsble inefe g evy fluid line RT gow nonline gow Lyze model z lig fluid spike bubble g u S Appoxime nonline RT pse by bubble ising wiin ylindil boundy UCRL-POST-434 6

7 Model is nonline bu lol soluion of e ydo. eqns. oninuiy eqn. inompessible/ ioionl Benoulli eqn. momenum inefe moion eqn. φ d d F φ S lnρ Lyze nsz dominn mode: φ φ S ; v φ p gz ρ ; S S Z z nonline in mpliude g u z S k z / φ φ A e J k / / sisfies ydo. eqns. o lowes ode in bu lol UCRL-POST-434 7

8 Bubbles ise wi onsn veloiies pln inefes inompessible keep only s mode k z / φ φ A e J k / subsiue Lyze model F φ S p φ gz ρ φ S ; S S Z z bubble eig line pse -D simulion Tylo expnd in ububble g / k wee J k ime bu model is sily vlid only fo pln inefes wile ICF ges e speil nd ompessible UCRL-POST-434 8

9 Subsnil modifiions e equied o oun fo speil onvegene z Reple ylindil boundies wi onil boundies in speil geomey g u S u β Reple unifom gviy g wi speilly oning oodines / g UCRL-POST-434 9

10 UCRL-POST-434 Hydo. eqns. mus be solved in oning fme fo onveging se Lyze nsz inompessible fo in oning fme, uomilly leds o ompessing flow fixed fme! S S p F S Φ φ φ ρ γ γ φ φ φ φ / exp os 3 ρ ρ P A - φ φ φ oninuiy eqn. in fixed fme oninuiy eqn. in oning fme Lyze nsz fo mode φ effeive g oning fme eqns.

11 UCRL-POST-434 Inefe eqn. is solved fo mpliude nd uvue subsiue expnd inefe spe in oning fme sepe odes in, ;, b R R S { } 4 3 φ O A b A b A b A S S 4 4 : : b A b A b O A A O mpliude uvue inegion ons. O

12 UCRL-POST-434 Two nul soluions fo inefe eqn. bubbles spikes 4, 4 O R bubble-ype inefe: inefe spe, 4, >> R,, >> R pidly inesing uvue ses iniil uvue self-simil /ons. uvue bubble, R Flow sould sympoe o one of ese spes depending on iniil vlue of spike-ype inefe: > b < b, R

13 UCRL-POST Fo bubbles, Benoulli eqn. yields oupled ODE s fo mpliude nd sle fo os φ φ P A subsiue inompessible in oning fme R S p F S 4, φ ρ γ γ φ φ Benoulli eqn. on inefe Tylo expnd in 3 : ons. : O O γ onion sle fo keep only s mode mpliude oupled ODE s

14 In limi >>, O eqn. beomes Kidde s eqn. fo self-simil implosion >> Fo ypil se, eqns. deouple nd s ODE edues o eqn. fo Kidde s self-simil implosion Kidde s eqn. 3γ ons. / / Kidde implosion ime ; γ 5/3 implosion digm self-simil isenopi pofiles fee fll /.7 /.5 /. /.7 /.5 /. UCRL-POST-434 4

15 Wi speified, nd eqn. n be solved by WKB meod Kidde / Assuming, WKB ppoximion yields bubble eig ~ exp ± d / ~ / / / / / 3/ in good geemen wi numeil soluion of oupled ODEs / ; 3/ 8 numeil soln. WKB soln. u Lyze d u Lyze / Iniilly line gow wi ime s in Lyze model followed by enned bubble gow le in implosion UCRL-POST-434 5

16 Enned bubble gow onfimed by -D simulions ig pessue lig fluid spike one wlls 8 -D simulion numeil soln. R, ompessing dense fluid u Lyze WKB soln. u Lyze d bubble /. /.5 / onvegene/implosion /.5 Simil esuls found fo nge of mode numbes ; equied subsnil ALE /.75 UCRL-POST-434 6

17 Kidde s eqn. lso desibes self-simil sgnion By evesing sign of ons. in s ODE flipping dieion of pessue gdien, se desibes self-simil sgnion Kidde s eqn. 3γ >> ons. / oose sgning implosion veloiy / sgnion ; γ 5/3 implosion digm self-simil isenopi pofiles fee fll /.7 /.7 /.5 /.7 /.7 /.5 /. /. /.5 /.5 /. /. UCRL-POST-434 7

18 Enned bubble gow lso ous duing sgnion Fo sgnion pse, eigenmodes i.e.,, so WKB soln. wi is φ A P sgnion os e ppopie sgnion ~ ~ 5 5 exp ± exp n n d / ; / implosion 8 WKB soln. u Lyze explosion u Lyze d Lyze uve uses pek deeleion / u Lyze Lyze UCRL-POST-434 8

19 -D simulions onfim enned gow bu sow influene of KH R, ompessing dense fluid ig pessue lig fluid bubble one wlls 8 -D simulion WKB soln. u Lyze KH jeing u Lyze d /. spike /.5 onvegene/implosion KH / Simil esuls found fo nge of mode numbes /.9 /. inefeing voies je ino elee bubble UCRL-POST-434 9

20 Lyze models n lso desibe spike gow [Zng`98] s sown Lyze-ype models n lso desibe sympoi pse of spike gow: bubble z spike z g u φ φ g u z Z, find spikes fee-fll fom inefe nd sympoe o infinie uvue Z, ~ g Z, ; UCRL-POST-434

21 Model fo spike gow wi onvegene n lso be onsued Fom genel soln. of inefe eqn. ose spike-ype inefe eleion pse: b R, O 4 deeleion pse: b R, R, pidly inesing uvue, >>, >> 4 R, R, Flow sould sympoe o spike-ype spe if iniil uvue is lge enoug UCRL-POST-434

22 Benoulli eqn. gin yields oupled ODE s fo mpliude nd onion sle fo spike spe Lyze nsz b φ A P os subsiue ino Benoulli eqn. on inefe nd expnd oupled ODE s O : O : ons. 3γ b 4 b b 4 b / O d d Keeping only leding ode in, eqn. edues o mpliude onion sle fo UCRL-POST-434

23 Spikes fee-sem in fixed fme O Wi deemined s befoe by, spike mpliude is simply eleion pse: deeleion pse: ; > i.e., in fixed fme, spikes oninue fee-seming moion s in [Zng`98] [ ] ; < < b / ~ ± m Cuvue ip lso s found by [Zng`98] Spike moion is independen of mode numbe, i.e.,., boundy ondiions onil wlls e ignoed s sould be. UCRL-POST-434 3

24 Model pediions gee wi -D simulions fo eleion pse ig pessue lig fluid spike KH 8 KH model soln. ompessing dense fluid bubble ½ -D simulion u Lyze d /. /.5 onvegene/implosion /.5 /.75 one wlls / Idenil esuls found fo nge of, i.e., spike gow independen of bounding wlls UCRL-POST-434 4

25 Model pediions gee wi -D simulions fo sgnion pse ompessing dense fluid bubble one wlls 8 model soln. ½ -D simulion KH/wll effe u Lyze d ig pessue lig fluid /. spike /.5 onvegene/implosion /.5 / Gow sues by KH nd ollision wi one wlls /.75 UCRL-POST-434 5

26 Model pediions gee well wi simulions fo nge of mode numbes bubbles spikes deeleion pse eleion pse UCRL-POST-434 6

27 Typil ICF implosions e lose o n self-similiy g ons. fee fll : g ons. O Wiou espeing eqn., model esuls fo bubbles o spikes my sill pply fo neby 's Kidde: self-simil bubble spike ~ ~ 3/ exp [ ] d UCRL-POST-434 7

28 WKB esul my pply o non-self-simil implosions Seing g, WKB esul beomes ~ 3/ ~ exp exp exp ± sin sin g R g R /3/- /3/- in oug geemen wi simulion d u Lyze -D simulion numeil soln. WKB soln. u Lyze d g / R Simil esuls found fo oe mode numbes ; efinemen of mes is on-going Fuue wok: invesige modifiions of Hn suion model due o onvegene. UCRL-POST-434 8

29 Speil onvegene modifies nonline RT gow fom Lyze model expeions pln speil g Lyze s [955] model is e lssi model fo desibing nonline RT gow bu is sily vlid only fo pln inefes We ve ombined Lyze-like RT model nd Kidde-ype self-simil implosion o develop model fo speil inefes moe elevn o insbiliies on ICF psules Model evels iniil gow like e Lyze model, followed by le ime modifiions Compison wi -D simulions onfims pedied bubble spike gow UCRL-POST-434 9

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