Generation of Nonuniform Vorticity at Interface and Its Linear and Nonlinear Growth (Richtmyer-Meshkov and RM-like Instabilities) K. Nishihara, S. Abarzhi, R. Ishizaki, C. Matsuoka, G. Wouchuk and V. Zhakhovskii Institute of Laser Engineering, Osaka University Incident laser light ablation surfaceshock front ablation surface shock front mass density vorticity International Conference on Turbulent Mixing and Beyond, Trieste, Italy, Aug.18-26, 2007 Introduction of Richtmyer-Meshkov instability and Laser Implosion introduction (shocked interface) After an incident shock hits a corrugated interface, ripples on reflected and transmitted shocks are induced and RM instability is driven by velocity shear left by the rippled shocks at the interface. IS I dv0a shocked interface dv0b vortex sheet from linearized relation of the shock Rankin-Hugoniot dvoa ko 1 where ust usi u vi , dvob ko 1 sr usi vi v1 Matsuoka, Nishihara Fukuda (PRE(03)) A=0.376, ξ0/λ=0.02 0 , k ; amplitude of the initial interface corrugation and its wave number, usi , ust , usr ; incident, transmitted and reflected shock speeds, and vi , v1; interface speed after the interaction and fluid velocity behind the incident shock. introduction (accelerated interface) Acceleration of different mass fluids also drives velocity shear at the interface. after Jacobs & Sheeley, PF (96) http://infocenter.ccit.arizona.edu/~fluidlab/papers/paper4.p df two fluids with surface perturbation spring 0 K M after the contact http://scitation.aip.org/getpdf/servlet During the contact of a container with a spring, phase inversion of the corrugated interface occurs and velocity shear is induced due to the acceleration. uy ux before the contact outline of talk We first show that the RMI is driven essentially by nonuniform velocity shear induced at an interface, instead of impulsive acceleration. In early stage of the growth, we show the importance of the interaction between the corrugated interface and rippled shocks through sound wave and entropy wave. There exist similar instabilities caused by the interaction, such as rippled shock interaction with uniform interface, and instability of the laser ablation surface. Nonlinear evolution of the instability is analyzed, treating the interface as a vortex sheet with finite density ratio for incompressible fluids. Nonlinear evolution of the instability in cylindrical geometry is investigated both analytically and with the use of molecular dynamic simulations. introduction Better understanding of hydrodynamic instabilities is essential for laser fusion. laser Instability of ablation surface effective gravity ablation surface ablator main fuel shock contact surface vapor fuel RMI In this talk, we will mainly discuss instabilities associated with nonuniform vorticity deposited at the interface. RTI; Rayleigh-Taylor Instability RMI; Richtmyer-Meshkov Instability Richtmyer-Meshkov instability (initial perturbation and wave equation) rippled corrugated rippled transmitted shock interface reflected shock dv0a y x dv0b Perturbation of shocked interface, and ripples on reflected and transmitted shock surfaces linear RMI IS I IS I IS TS t=t1 RS t=0 t 0 time transmitted shocks TS t=t2 0 r 0 reflected shocks TS t2 I space t1 incident shock trajectory I t=t2=0+ RS RS I ; interface IS; incident shock RS; reflected shock TS; transmitted shock initial amplitude of rippled shocks (t=t2=0+) ust kt 0 k0 (1 ) usi usr kr 0 k0 (1 ) usi linear RMI Consider interaction between corrugated interface and rippled shocks through sound wave and entropy wave between them. Initial velocity shear (t=0+) dvoa ktovi dvob kro vi v1 Solve wave equations in the regions between interface and shock fronts for sound wave and entropy wave with proper boundary conditions dv0a y 2 2 2 2 dp (kcs ) dp cs 2 dp 0 2 t x dp exp(iky) Boundary condition at shock front normal velocity; from linearized shock jump condition with respect to ripple amplitude x x ust as (t )eiky tangential velocity; continuous x dv0b Propagation of a rippled shock driven by a corrugated piston (a) P S (b) P S y x Consider interaction between corrugated piston (interface) and rippled shocks through sound wave and entropy wave between them. ripple shock Solve wave equation for pressure perturbation between shock and contact surface with proper boundary conditions pressure perturbation dp1 dp1 ( x, t ) exp(iky) wave equation 2 2 2 2 2 d p c d p c k dp1 0, 1 1 1 1 2 2 t x change variables where ( x, t ) (r , ) , where 2 1 1 2 dp1 dp1 dp1 2 2 dp1 0 2 r r r r x x v x1t , t t r cosh kc1t, r sinh kx f 1 0 by introducing Us dp1 f (r ) g ( ) d2 1 d 2 f f (1 2 )f0 0, 2 dr r dr r d2 2 g g 0 2 d piston solution shock c d dp1 ( Da e Db ecx)(cD J ( r ) D N (r )) cos u 1 0 s where J (r ), N (r ) are Bessel functions , Da d are coefficients ripple shock Amplitude of shock ripple decays with t-1/2 solid line; analytical solution circles; simulation result dotted line; CCW approximation solution 0.01 Shock front ripple as / 0.005 0 -0.005 -0.01 0 1 2 3 4 Normalized time v 5 s 6 7 t/ shock front ripple as (rs ) 1 (2 22 1) M s2 J 0 (rs ) 2 J 2 (rs ) , where rs kc1t 1 s2 , 2 2 a0 1 (2 s 1) M s J n , M s , s are Bessel function, shock Mach number ahead and behind the shock Richtmyer-Meshkov instability (linear theory) asymptotic growth rate effects of compressibility dv0a dv0b Solve wave equations in the regions between interface and shock fronts for sound wave and entropy wave with proper boundary conditions 2 2 2 2 dp (kcs ) dp cs 2 dp 0 2 t x y dp exp(iky) x linear RMI Both tangential velocity and normal velocity reach asymptotic values time evolution of tangential velocity time evolution of normal velocity 0.1 0 dimensionless velocity -0.02 dimensionless velocity tangential velocity at the contact surface in fluid "a" a =1.8, a =1.1, R 0 = 3 -0.01 -0.03 -0.04 -0.05 -0.06 0.08 0.06 normal perturbation velocity at the contact surface a = 1.8, b = 1.1, R 0 = 3 0.04 0.02 -0.07 -0.08 0 0.5 1 1.5 xt / 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 xt / J. G. Wouchuk and K. Nishihara, Phys. Plasmas 4, 1028 (1997), J. G. Wouchuk, Phys. Rev. E 63, 056303 (2001), Phys. Plasmas 8, 2890 (2001). linear RMI Asymptotic growth rates depend on the whole compressible evolution: Integrate equation of motion from 0+ to ddv y ddv x dpi dt x dt From pressure continuity at the interface, we have for tangential velocity kdpi af (dvya dv0ya ) bf (dvyb dv0yb ) a,bf : density at t=0+ By defining the difference between normal and tangential velocities at each side of the interface, dvi dv ya Fa dv yb dvi Fb we get an exact expression for the asymptotic linear growth rate: 0 0 d v d v bf Fb af Fa bf yb af ya dvi bf af bf af In a weak shock limit. the F-terms can be neglected. which is valid for any value of the initial parameters: shock intensity, fluid density and fluid compressibility. It should be noted that the F terms are proportional to a spatial average of the vorticity field left by the rippled shock fronts. linear RMI Efects of the compressibility: Freez-out of the growth asympotically occurs due to the compressibility As the shocks separate away, their ripples will change in time, generating at the same time sound waves and vorticity/entropy. dimensionless velocity A typical spatial vorticity/entropy profile: 0,02 vorticity (a.u.) 0,01 0 normal perturbation velocity at the contact surface a = 1.8, b = 1.1, M i = 5 0.02 R 0 = 1.1579271... 0.01 0 -0.01 -0.02 -0,01 -0,02 -0.03 -0,03 0 0.5 1 1.5 2 2.5 3 -0,04 -2 -1,5 -1 -0,5 0 x/ 0,5 1 1,5 2 xt / K. O. Mikaelian, Phys. Fluids 6, 356 (1994), Wouchuk and Nishihara, Phys. Rev. E 70, 026305 (2004) linear RMI Efects of the compressibility: At high incident shock intensity the asymptotic growth rate decreases, which agrees well with simulations by Yang et al. a shock is reflected back a rarefaction is reflected back different pairs of gases 0.3 0 VMG -0.05 CO 2- Air Air - SF 6 R-M 0.2 0.15 our model and Yang et al simulations 0.1 normal velocity asymptotic velocity asymptotic velocity 0.25 -0.1 -0.15 Xe - Ar -0.2 -0.25 0.05 -0.3 0 SF 6- Air -0.35 0 0.2 0.4 0.6 shock intensity 0.8 1 0 0.2 0.4 0.6 0.8 shock intensity Y. Yang et al, Phys. Fluids, 6, 1856 (1994), J. Wouchuk, Phys. Rev. E63, 056303 (2001), and Phys. Plasmas, 8, 2890 (2003). 1 Exact linear formula also agrees well with laser experiments with solid target at high Mach number of 10 and 15 (rarefaction was reflected) 0 LP100/14 -2 LF100/4 HP100/14 mo del prediction linear RMI -4 HF100/4 LF150/10 LF100/10 -6 HF150/10 -8 LF100/14 HF100/10 -10 LF150/10 -12 -12 -10 -8 -6 -4 -2 0 Nova experiment ( m/ns) J. Wouchuk, Phys. Plasmas, 8, 2890 (2001). G. Dimonte et al., Phys. Plasmas 3, 614 (1996); R. L. Holmes et al., J. Fluid Mech. 389, 55 (1999). RMI-like Instability (1) Instability induced when a ripple shock hits uniform interface solve wave equations in regions 1, 2 and 3 with proper boundary conditions. instability due to rippled shock Since shock front ripple oscillates, phase of oscillation at the interaction changes dynamics of interface after time derivative of ripple shock front ripple phase 1 phase 2 phase 3 as as instability due to rippled shock Growth rate of contact surface ripple depends on the phase of the incident ripple shock at the incident phase 3 phase 2 phase 1 growth rate of contact surface dotted line; instantaneous value circles; simulation solid line; time integrated value Analytical solutions agree with simulations R. Ishizaki et al., Phys. Rev E53, R5592 (1996). RMI-like Instability (2) Instabilities associated with laser ablation (nonuniform target or nonuniform laser) Incident laser light ablation surface shock front (a) nonuniform target surface ablation surface shock front (b) nonuniform laser irradiation ablation surface instability Energy deposited at heat wave front induces ablation pressure, and laser ablation drives a shock wave ahead (like a piston) trajectory of shock, ablation surface, and sonic point heat wave temperature Te flow diagram Te time heat flux ablation surface divergence of heat flux Te shock front distance Energy deposition at heat wave front corresponds to combustion in rocket engine Chapman-Jouguet condition at sonic point density profile Ablation deformation monotonically increases, and amplitude of shock ripple is small compared with a case of a rigid piston Normalized time r 0 10 20 a 30 40 50 Shock front ripple as / a 0 1 10 8 0.5 6 0 4 -0.5 2 -1 0 0 10 20 30 Normalized time r dash-dot line; ablation surface deformation solid line; ripple shock driven laser ablation dotted line; ripple shock driven rigid piston s Ablation surface deformation aa / a 0 ablation surface instability 40 R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997). This instabilitty now called ablative RMI after V. N. Goncharov, Phys. Rev. Lett., 82, 2091 (1999). ablation surface instability Analytical solutions for both shock front ripple and areal mass density perturbation agree well with laser experiments. comparison with laser experiments (squares) shock front ripple (a) areal mass density (b) Areal mass density perturbation d l / ( d l) 0 Shock front ripple as / a 0 1 0.5 0 -0.5 -1 5 4 3 2 1 0 0 0.5 1 Normalized time u uniform laser irradiation 1.5 s 2 t/ 0 0.2 0.4 0.6 Normalized time u 0.8 s t/ target surface deformation R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997). T. Endo et al., Phys. Rev. Lett., 74, 3608 (1995). 1 ablation surface instability Fairly good agreements were obtained between experiments and theory, by assuming the ablative Rayleigh-Taylor growth after rarefaction wave returns the ablation surface after shock reach rare surface, exponential growth is assumed due to ablative RTI 10 -3 Areal mass density perturbation ( g / cm 2 ) ablative RTI RMI-like 10 -4 10 -5 square and solid line: =100m, I0=0.4 circle and dotted line: =75m, I0=0.1 10 -6 0 nonuniform laser irradiation 0.5 1 1.5 2 2.5 3 3.5 Time ( ns ) M. Nakai et al., Phys. Plasmas, 9, 1734 (2002). H. Azechi et al., Phys. Plasmas, 5, 1945 (1998). Richtmyer-Meshkov instability (nonlinear theory) (incompressible fluid approximation) nonlinear RMI Acceleration of different mass fluids drives velocity shear at the interface. http://infocenter.ccit.arizona.edu/~fluidlab/papers/paper4.p df two fluids with surface perturbation spring 0 http://scitation.aip.org/getpdf/servlet J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996). K M after the contact uy ux before the contact nonlinear RMI We can obtain velocity shear induced at the interface due to the acceleration during the contact between spring and container. coskx Integrate equation for the amplitude of the interface perturbation over the interval of the contact t f ( 2d ) / 0 d2 Ag (t )k 2 dt after the contact v02 02 g (t ) g0 1 1 2 sin 0 t d g0 g K d arcsin 2 0 2 2 0 , g v M 0 0 0 where uy before the contact K ; the spring constant, M ; mass of the container, g0 ; the earth gravity and v0 ; initial velocity of the container d uy , dt u 0 , ux (ux , uy ) ( sin kx, coskx)e ky nonlinear RMI Nonlinear evolution of circulation at the interface with finite density ratio: Bernoulli equation Define interface velocity by mass weighted velocity as By introducing velocity potential and circulation u i i 1u1 2 u 2 d x u 1 2 dt 1 2 which satisfies boundary condition u1 n u 2 n u n Introducing vorticity u1 u 2 q 1 (u1 u 2 ) 2 u becomes u q A 1 2 1 2 A 2 1 1 2 q 2 We obtain from Bernoulli equation d 1 1 A d 2 A q q q dt 2 8 2 dt d u where dt t Circulation does not conserved for a finite Atwood number A nonlinear RMI Interface dynamics with Lagrangian maker Modified Birkhoff-Rott equation Defining complex z from the interface position (x), y) z x iy : Lagrangian parameter the interface trajectory is obtained from Modified Birkhoff-Rott equation * z d * A z u * q* dt 2 s s2 z z* 1 1 ' q d ' ' s cot z ' z 4i 2 Normalization * Bernoulli equation becomes Finite Atwood number induces locally stretching and shrinking of the interface. kvlint Nonlocal. d A d 2 Re z q * 2 1 A 2 Re z q* dt s dt s kz The similar equations have been obtained by Kotelnikov (PF(00)) but for different u. Solve above coupled equations with initial conditions x y k0 cos( ) 2 sin Weakly nonlinear Theory of a Vortex Sheet : Expansion Comparison with experiments nonlinear RMI accelerated interface shocked interface 5 amplitude (cm) 4 spike spike 3 bubble 2 1 bubble 0 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 time (sec) J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996). G. Dimonte et al.,, Phys. Plasmas, 3, 614 (1996). C. Matsuoka et al., Phys. Rev. E67, 036301 (2003). expansion up to 3rd order X n X (n ) Y n Y (n ) 1 i n n n n (n ) e nky cos nk x nonlinear RMI Dynamics of vortex sheet with density jump nonlinear vortex generation, their self interaction analytical model K Vlin t = 0.80 K Vlin t = 0.05 double spiral shape of spike and vorticity in simulation K Vlin t = 6 K Vlin t = 12 Density jump at the interface introduces generation of vortex and thus opposite sign of vortex appears, which causes double spiral structure of spike C. Matsuoka et al., Phys. Rev. E67, 036301 (2003). nonlinear RMI Fully nonlinear evolution: Double spiral structure is observed as Jacobs & Sheeley experiment. Color shows the vorticity Parameters A = 0.155 k0 = 0.2 kvlint = 0, 1, 2,,,,12 Jacobs nonlinear RMI Cylindrical vortex sheet in incompressible RMI. spike bubble bubble A=0.2, n=4 (inner: lighter fluid) spike A=-0.2, n=4 (inner: heavier fluid) Features of cylindrical geometry, ・ two independent spatial scale, radius and wavelength nonlinear growth depends strongly on mode number ・ ingoing and outgoing of bubble and spike nonlinear growth depends inward and outward motion rather than spike and bubble Details by Matsuoka On Aug. 21 C. Matsuoka and K. Nishihara, Phys. Rev. E73, 055304 (2006), Phys. Rev. E74, 066303 (2006). Richtmyer-Meshkov instability (Molecular Dynamic simulation) (cylindrical geometry) z Fij R Potential barrier as Piston LJ atoms R MD RMI Nonlinear evolution of Richtmyer-Meshkov instability in cylindrical geometry mass density mass density vorticity shock passing interface mass density vorticity reflected shock hits interface bubble Mach stem appears shock pass through interface spike shock reflected anomalous mixing occurs MD RMI Molecular dynamics simulations show RM growth driven by multiple shocks for different mode numbers. Decay of nonlinear growth is mode dependent and higher mode decays slower, which agrees with the model of cylindrical vortex sheet trajectory growth rate 160 0.4 8 1st 140 buble 5 3 80 5 60 spike growth rate 3 100 40 8 0.2 0 shock ~ t 0.55 3rd 2nd 3rd 120 radius 2nd ~ t 0.7 1st 5 3 0 8 -0.2 20 -0.4 0 0 20 40 60 time 80 100 120 0 20 40 60 time 80 100 120 MD RMI Whenever shocks pass through interface from heavy to light, phase inversion occurs, which causes generation of higher harmonics Richtmyer-Meshkov instability at shell surfaces (light-heavy-light) velocity (radial) density initial shock reaches the center reflected shock reaches shell density Conclusion ・ Both exact and asymptotic linear growth rates of the Richtmyer-Meshkov instability and RMI-like instabilities were obtained for compressible and incompressible fluids, which agrees with experiments. ・ By introducing mass weighted interface as a nonuniform vortex sheet between two fluids with finite density ratio, we have developed a fully nonlinear theory of the incompressible RM instability, which also agrees fairly well with experiments. ・ The theory is extended to a cylindrical geometry, in which nonlinear growth is determined from the inward and outward motion rather than bubble and spike, and it depends on mode number. ・ Molecular Dynamic simulation provides a new tool for a study of hydrodynamic instabilities, when CFD fails. We observed enhancement of the growth for sandwiched shell. New features of such a system with density difference across interface, and nonuniform vorticity may provides a paradigm in vortex dynamics.