Tutorial lectures on hydrodynamics instabilities

Transkript

1 Tuoril leures o hdrodmis isiliies Leure oes preseed he Isiue of Lser Eieeri. Os Uiversi (4--6-(7--6 Jvier Sz Reio ETSI Aeroáuios. Uiversidd Poliéi de Mdrid

2 Coes Leure: The lssil Rleih-Tlor (RT isili. 3 Pheomeolo of he isili. Iompressile fluids: Heurisi derivio of he lier isili rowh re (Seod Newo Lw; lsis mehod (dispersio relio; Awood umer effes fiie hiess effes. Exmples. Compressile fluids. Dispersio relio. Iroduio o olier lssil RT isili. Pheomeolo. Lzer models pproh. Leure : Lier live RT isili 4 D lio fro sruure. Isori model d orol models.silizio mehisms. Heurisi derivio of he dispersio relio i shrp lio fros (Seod Newo Lw. Lier lsis mehod. Froude umer depedee. The riil surfe proximi effes (lo wveleh perurio d Ldu isili. Leure 3: No lier live RT isili 6 Shrp lio fro model: Therml equio momeum equio d ime evoluio equio of he ierfe. Sile mode perurios: surio mpliude iversio of spie ule smmer o lier uoff wve umer smpoi ule veloi. Referees: S. Boder Phs. Rev. Le (974. H. Te K. Mim L. Mohierh d R. Morse Phs Fluids (985. H. J. Kull Phs. Fluids B 7 (989. J. Sz Phs. Rev. Le (994. R. Bei e. l Phs. Plsms 3844 (995. A. Piriz J. Sz d L. Iñez Phs. Plsms 4 7 (997. J. Sz J. Rmirez R. Rmis R. Bei d R. P. J. Two Phs. Rev. Le (89 95 (. P. Clvi d L. Msse Phs. Plsms 69 (4.

3 THE CLASSICAL RT INSTABILITY 3

4 Pheomeolo of he isili (* η ( << λ = π η η γ η η e γ γ Visosi Surfe esio η η 3 loη.λ lo h α ( α.65 * - Lord Rleih Pro. Lodo Mh. So. 4 (883 7; G. Tlor Pro. R. So. A (95 9; D. J. Lewis Pro. R. So. A (95 8. Aelered fluid lers : P P > P 4

5 Rleih-Tlor isili Geophsis Asrophsis Teholoil ppliios: Ieril Cofieme Fusio (ICF x Iompressile fluids d uiform desi. ν = ( v v v + = p + 5

6 Seod Newo Lw: m = F λ = π ξ ξ = ξ + ξ + ξ 3 3 ξ = Ce γ ξ ξ Awood umer γ A T = AT = + > γ=± A T Usle. RT modes < γ=± i T A Sle. Grvi wves Mehod: ν = d ( v v v + = p + Equilirium soluio: v = p= p ( p + = Ierfes: Perured soluio: p v v v ix+ γ ix+ γ ( e = ( x e v + iv = x γ v = ip x γ v = p ix + j ξ e γ 6

7 Mehod: Soluio for eh fluid ler: p p = j+ j j d p = Ae + Be i vx = ( Ae + Be γ v = ( Ae Be γ Boudr odiios: = p + ξ p p + ξ oious j j j v oious d v = γξ j =± vriles mus eouded Compii odiio ->Dispersio relio : A B A D( γ = B ξ j How he sperum is?: For eh vlue we hve differe vlues of γ : γ γ. The umer of vlues m e ifiie u i se he sperum is disree. ( D de ( γ = Adved omme: Lple rsform i : Poles of K (disree sperum: e γ (*Brhe pois of K (oiuum sperum: α γ e (* E. O PRL 98 K( se s ds 7

8 Exmple : A+ ξ = B+ ξ A= B= γξ γ γ p = Ae p = Be ( A B = ξ γ γ A T = AT = + Exmple : d ξ ξ p Ae Be = + d d A+ B+ ξ = Ae + Be + ξ = d d ( A B = γξ ( Ae Be = γξ γ γ d d e e A γ B = ξ ξ d d γ e e 4 γ = γ =± γ =± i γ = ξ = ξ d e RT γ = ξ = ξ d e GW 8

9 Compressile fluids: p = d ξ p ξ Equilirium soluio: Perured soluio: = p + p / = os p = p ( + / d = ( + / d /( /( d = p/ = ( + / d = p ix+ γ ix+ γ = + e p = p + pe ix+ γ v = ( v v e x Equios: γ + ( v + iv = x γ v = p + γ v = ip x p = Boudr odiios: p + ξ = v = γξ ; = p + ξ = v = γξ ; = d Dispersio relio: γ ( + + ( = p p p γ ( + p p = = d p = A Hper( γ + B Lue( γ D( γ d = 9

10 Dispersio relio: γ ( + + ( = p p p We loo for iompressile modes! γ ( + p p = = d v ( v iv γ γ+ + + x = ( + p p = ( + / / ( / d γ p = C + d e d 4 γ = ( γ = γ = 4 ( γ D ( SM γ d = Nolier lssil RT isili p os Hrmois eerio (sri i he wel olier phse. Suhrmoi sde. Bules d spies show differe ime ehvior. Bule ompeiio.

11 v = φ r Δ φ = ( α x ξ r = φ ( φ = ξ φ + os Wel o-lier resuls ξ ( x η os x + η os x + η os3x η ξl ξl 4 η ξl ( ξl e γ γ = 3 η3 ξ 8 3 L mpliude 3...

12 Spie ule smmer s spie mpliude: ule mpliude: ξ + ξ s L L ξ ξ L L lo S lier heor lo Lo wveleh modes eerio = + γ ( γ + γ ξ = ξ ( ξ ( e γ γ ( γ+ γ γ ( +

13 Bule ompeiio. Aelerio of ule fro..8 h h h h = ( α.65 α v = φ x r Δ φ = ( α ξ r = φ ( φ = ξ φ + os 3

14 LINEAR ABLATIVE RT INSTABILITY 4

15 D lio fro sruure: re Lser d L v T L T v D lio fro sruure: Isori model T v v = v = m v v = p+ T T mt ( T = KT T 5 T T L / ( / / ( / L ( L 5 T KT mt = KT L = (..3 μm L 5m v v Fr = (.5 = << L M T / L T T ( + e ( L / L ( e Miimum desi rdie sle leh L = L + m + ( / 5

16 Cold ompressed re: α α P/ = P / = os + ( v = P+ O M ( ( d L L v T BC : P( = d( = P( = = P v ( = d ( = d ( v ( = = v α / = / ( + / α ( p p d p = α OM ( α d ( + ( v = v α d ( α d = d v α Lier live RTI: Silizio mehisms. Sli Lws π lio surfe ξ Δ m < Δ p < Δ m > Δ p > flow he ( << m = >> / V V m Δpd ξ 3 ( + ξ ( ξ Δ p d m ξ Hdrosi pressure Dmil pressure Alio: Fire polishisi. Vorii 6

17 Lier live RTI: silizio mehisms π ξ Δ m < Δ p < flow Δ m > Δ p > he ξ e γ lio surfe Therml pressure Roe effe 4V V γ + γ + AT + r r Fire polishi + Vorii V V γ = + + A VV + r + r T r = L / ( / A T r V = V = + r r uoff L F < /( r T v Lier lsis mehod: Isori model ( M << MF r << T v Equilirium soluio v= v = m v v= p+ T T mt ( T KT T 5 = Perured soluio Perured quiies re expded s ix e γ + 5h ODS γ + ( v + v + iv = γ x v + = p+ γ v + = ip ( T + T = x 5 ( ( vt KT T = = ouded modes = + 3 ouded modes Numeril eievlue prolem for γ γ L F( L Fr = v 7

18 v v Alil model: Isori model ( L << ξ e = ( / L v = v = Ve + φ ( + V v = p p + = x V = iae = Ae p = ( γ + V Ae / Q = p ( + ξ + V m Cold reio γ+ ix Ho reio ix+γ m e ix+γ Q e / Mss lio Momeum V γ + ( + f Vγ + q = ( L / Q ( L / q / V ξ m f V ξ q( γ f ( γ? Alil model: Isori model ( L << / ˆ γ( L / /( V << γ Sli: v T p / Normlized vriles: F = F( η + F ( η γ + ; η ˆ q q q = + ˆ γ + f = f + f γ + ˆ ( << v Eievlue prolem for: ( q f ( q f q f q / Γ (+ / ( < q < ξ e γ+ ix Ho reio ix+γ m e ix+γ Q e Momeum Mss lio re 5h ODS γ + ( v + v + iv = γ x v + = p+ γ v + = ip ( T + T = x 5 ( ( vt KT T = 8

19 Alil model: Dispersio relio ( L << V γ + ( + f Vγ + q = ( L / / q q q = + ˆ γ + f = f + V γ + ( + f + q V + = γ / ( L /( q / ( q Γ (+ / m ( V ξ f xq ( V ω r = Blow-off o lio desi rio Q p ( L / V ( ξ ( q q γ / + ˆ / iv L q + qˆ ω ( / ( ξ ( γ uoff: F ( L = ( > / r / q Fr Exesio of he lil model: (for ever F r vlue A T Awood umer effes r = + r ( = ( q / ( = r L Corol model = L Lerl he rspor he lio fro: (sed i SBM f + L ( + L ( + L + r LF r ( + L γ = AT V + r + r r + r Fr < < 9

20 Cuoff wveumer versus Froude umer V = F r Lerl he rsp. Sle leh desi- A umer orreios Roe effe-vor.ov. Fire polishi-vor.ov A umer orreios.. L = F r Cuoff wve umer versus Froude umer.5.. V β =.75.5 γ emp = βv + L m.5. β = F r

21 Ldu-Drrieus isili V Δ p < Δ p > < ( << γ γ LD LDRT V V + q V ( F / >> r >> γ / ( L Ldu-RT isili i ICF : M Criil surfe V T T I II III L Isori pproximio wih / d ever leh muh lrer h L. Reio I iompressile d poeil flow ( V = m. e. / Reio II : T = T ( / = ( / / u = u / ( / where ( / = L/. Reio III : = T = T u = u

22 Ldu-RT isili i ICF : Alio surfe Criil surfe M I II III x ξ e γ ix ξ e + γ ix+ V γ + ( + f Vγ + q = ( L / / i v V m x + ξ = ( + γξ Q ( QL ( / q = ξ ξ / / u( V ( m f V ( ξ qγ (? f( γ? Ldu-RT isili i ICF : Alio surfe Criil surfe M I II III x s ix ix ( ( ξ ξ e / ξe ξ e γ ix ξ e + γ ix+ + ( v = v + v v = p+ e Pv ( KT T = Iδ 5 ( m v ix+ γ ( v = θ + vr + γξe e ix I mv θ v r γξe e δ 5P + + ( + = ( = v r + = + v v v p e / ( θ = os. ( θ T

23 Ldu-RT isili i ICF : M Perured equios ( < s < : γ vr m (( ξ ξ/ θ/ s + ( ssθ θ+ ( s( ξ ξ/ + ξ = s γiv + v iv = rx s rx p (( s ξ ξ / + ξ ( + γu + γ uθ u ( ξ ξ / + u ( uθ s γv + v u + u v = p r r s s r s uθ uθ u ssθ u s ( u su sθ γu sθ γ + s s (( γs+ u su + + γu( ξ ξ / + γ ξ v + iv = s r rx Ldu-RT isili i ICF : M Boudr odiios: s + θ = sθ + = m / m + ( ξ ξ/ i m v r = vrx = ( + γξ p = Q θ = s θ = ( ξ ξ / s ( p ξ u v + = + u + iv r γξ ( ξ rx γ γ Q( γ m ( γ ξ ( γ 3

24 Ldu-RT isili i ICF : M Resoluio mehod: ( / ˆ = O ( F r = O( >> γ /( u >> ˆ/ Q q= q ˆ + qγ + u ( ξ m f = f ˆ + fγ + V ( ξ ˆ/ γ ˆ γ = u d ll perured quiies re perured i he sme w The ssem of ODE is ierivel solved q ( ˆ f ( ˆ q ( ˆ f ( ˆ Ldu-RT isili i ICF : M Dispersio relio: q= q + q ˆ γ + V γ + ( + f Vγ + q = / ( L / f = f + f ˆ γ + qv γ + + = ( + f+ q Vγ / ( L / Vorii: ω v + iγξ + iv rx r ˆ/ + iω ˆ ω( s = = q ˆ qγ + ( ξ u + xp = m ω( s = 4

25 Ldu-RT isili i ICF : qv γ + + = ( + f+ q Vγ / ( L /.5 q 5 L = 5/ = f 5.5 q Ldu isili Silizio F r / ˆ ˆ ˆ e e + + ˆ q ˆ = + ( ( ( / osh ˆ + osh ˆ Γ Γ = ξ f = h( q + h( = ξ osh( Ldu isili i ICF : ( F >> r γ V q ( ( L / / <.7 ( q < Ldu isili >.7 ( q > Osillios / ˆ ˆ ˆ e e + + ˆ q ˆ ( = + ( ( ( / osh ˆ + osh ˆ Γ Γ = q q / >> = = Γ (+ / (.67 = 5/ << q ( / 5

26 NONLINEAR ABLATIVE RAYLEIGH-TAYLOR INSTABILITY 6

27 Lier live RTI: silizio mehisms π ξ Δ m < Δ p < flow Δ m > Δ p > he lio surfe Therml pressure Roe effe d ξ d ξ 4 Vξ V ξ d d uoff L F < /( r Fire polishi + Vorii Nolier model 7

28 Cold reio Ve + φ Δ φ = r ( α x ξ m r = φ + V e m / Mss lio re ( p φ = ξ φ V φ p = q m q / Ho reio x ξ T = P 5 ( Pv KT T = + ( v = ( v v v + p v = + v (/5 K T r ( θ = KT /5 m m = + θ = θ ( r = θ ( l θ vr l θ θ ( x = = Mss lio re : m = m θ 8

29 Ho reio v = m θ + vr ( θ = KT /5 m ξ ( ω e z = vr ω+ ( m θ + vr ω = x v r = φ + Ve τ vr + φ ω( χ ( χ = HC.. of. θ Δ ψ = ω ( χ φ = ( ψ e ψ e x x e θ+ iχ dθ ωdχ = θ Coforml mp ( x ( χ θ d FT o χ φ ( i + τ θ Lier heor : ω = φ ξ = e iχ d ( x x x χ No-lier roe effe: Δ φ = Momeum flux : q= p + m / x ξ h h q flow Δ θ = h q m m m ω dχ = ( + h? 9

30 Resorifore x ξ flow v = m + v θ r ( v + p ω v v / m θ / L θ q p ( dθ + m ωdχ q p m m + m d ( l ω χ IFT o ( l = ( χ q ( m m + m ω Ali surfe equios: Δ φ = Ve + φ ξ x r = φ m / V e ( flow θ = θ = Δ θ = φ = ξ ( φ V φ m = m θ m m V ωdχ θ + iχ e dθ ω d χ = θ ( l φ ( i + τ iχ e d χ θ 3

31 Sile mode resuls Wel o-lier resuls Clssil RTI 3 η ξl ξl 4 η ξl 3 3 η3 ξl 8 ( /( 3 η ξl ξl 4( η ( ξl 3 3 η3 ( 4 ( 4 /3 ξl 8 ( < ξ ( x η os x + η os x + η os3x + 3 ( ( / / ( ξl e γ mpliude 3... / /4 3/4 < / < < < < 3/4 3

32 Iversio of spie ule smmer spie mpliude: ( (/ ξ ξ + s L L ule mpliude: ξ L( (/ ξ L < < < > / ( ( / / lier heor < / Nolier expoeil rowh. Surio mpliude. lssil 5 μm μm 4 μm F = 4.5 λ = μm r.5 λ..5 μm μm S (.5μm..5.5λ γ =.8 3

33 .86 No lier exp. rowh ( ξ. λ. < ξ. λ.. Simulios ART D I V II I V > V.λ. ( s. ( s V III > V II ξ λ...λ. ( s No lier isili for ξ ej >. Bifurio dirm ξe( x = ξej os( jx j= + 3 ξ ξ ( ξ ξ Lier heor shows: ξ ( x ξ os( x e e 33

34 No lier isili for >. Bifurio dirm ξe( x = ξej os( jx j=.. ξ ej π π / Full olier isili ( > =.5 ule 34

35 .86 No lier isili ( >. Simulios ART D ξ λ...λ. ( s ξ λ...λ Asmpoi ule veloi V /3 V super uoff V sv = = s = > 3V 3 35

36 Sili reios:.5 hs / λ F = 5 r / Muli mode resuls 36

37 Lo wveleh modes eerio / γ( γ+ γ ( ξ + = + ξ( ξ( ( γ+ γ γ γ( γ + γ ( e γ + γ γ = Lo wveleh modes eerio ξ = ξ( ξ ( G G = = op op = op / ART SIMUL. MULTI D SIMUL. Full olier heor Wel olier heor.5 Clssil: = / op

38 Bule ompeiio Aelerio of ule fro.4 λ = λ h h h h = α ( α.6 38

39 Aelerio of ule fro λ = λ h h h h = α ( α.6 Aelerio of ule fro λ = λ h.5 h h h = α ( α.3 39

40 Aelerio of ule fro α Fr=5 Fr= Fr=. Fr= h h =α / SC Aelerio of ule fro : ( α = α / SC.7 Fr=3.6 Fr=5 α h h = C V 3 ( h h =α Fr= Fr= C = 3α. / s

41 4