1. 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 shock frontablation surface shock front mass density vorticity International Conference on Turbulent Mixing and Beyond, Trieste, Italy, Aug.18-26, 2007

2. Introduction of Richtmyer-Meshkov instability and Laser Implosion

3. 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 shocked interface vortex sheet v0av0a v0bv0b from linearized relation of the shock Rankin-Hugoniot incident, transmitted and reflected shock speeds, and amplitude of the initial interface corrugation and its wave number,where interface speed after the interaction and fluid velocity behind the incident shock. introduction (shocked interface) Matsuoka, Nishihara Fukuda (PRE(03)) A=0.376, ξ 0 /λ=0.02

4. introduction (accelerated interface) Acceleration of different mass fluids also drives velocity shear at the interface. after Jacobs & Sheeley, PF (96) http://scitation.aip.org/getpdf/servlet spring two fluids with surface perturbation before the contact after the contact http://info- center.ccit.arizona.edu/~fluidlab/papers/paper4.p df http://info- center.ccit.arizona.edu/~fluidlab/papers/paper4.p df 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.

5. 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.

6. ablation surface shock contact surface ablator main fuel vapor fuel laser effective gravity RTI; Rayleigh-Taylor Instability RMI; Richtmyer-Meshkov Instability Instability of ablation surface RMI In this talk, we will mainly discuss instabilities associated with nonuniform vorticity deposited at the interface. Better understanding of hydrodynamic instabilities is essential for laser fusion. introduction

7. Richtmyer-Meshkov instability (initial perturbation and wave equation) v0av0a v0bv0b x y rippled reflected shock corrugated interface rippled transmitted shock

8. t2 t1 time space incident shock trajectory reflected shockstransmitted shocks Perturbation of shocked interface, and ripples on reflected and transmitted shock surfaces ISI t=t2 TSRSI t=t1 ISI t=0 RSTS IS linear RMI initial amplitude of rippled shocks (t=t2=0+) t=t2=0+ TSRSI I ; interface IS; incident shock RS; reflected shock TS; transmitted shock

9. Consider interaction between corrugated interface and rippled shocks through sound wave and entropy wave between them. linear RMI Initial velocity shear (t=0+) Solve wave equations in the regions between interface and shock fronts for sound wave and entropy wave with proper boundary conditions v0av0a v0bv0b x y Boundary condition at shock front normal velocity; from linearized shock jump condition with respect to ripple amplitude tangential velocity; continuous

10. Propagation of a rippled shock driven by a corrugated piston Consider interaction between corrugated piston (interface) and rippled shocks through sound wave and entropy wave between them.

11. Solve wave equation for pressure perturbation between shock and contact surface with proper boundary conditions pressure perturbation wave equation,where,,change variableswhere,, by introducing solution whereare Bessel functionsare coefficients, ripple shock

12. Amplitude of shock ripple decays with t -1/2 solid line; analytical solution circles; simulation result dotted line; CCW approximation solution shock front ripple, where, are Bessel function, shock Mach number ahead and behind the shock ripple shock

13. Richtmyer-Meshkov instability (linear theory) asymptotic growth rate effects of compressibility Solve wave equations in the regions between interface and shock fronts for sound wave and entropy wave with proper boundary conditions v0av0a v0bv0b x y

14. Both tangential velocity and normal velocity reach asymptotic values linear RMI 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). time evolution of tangential velocitytime evolution of normal velocity

15. Asymptotic growth rates depend on the whole compressible evolution: linear RMI Integrate equation of motion from 0+ to From pressure continuity at the interface, we have for tangential velocity 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. : density at t=0+ By defining the difference between normal and tangential velocities at each side of the interface, we get an exact expression for the asymptotic linear growth rate: In a weak shock limit. the F-terms can be neglected.

16. Efects of the compressibility: Freez-out of the growth asympotically occurs due to the compressibility linear RMI As the shocks separate away, their ripples will change in time, generating at the same time sound waves and vorticity/entropy. A typical spatial vorticity/entropy profile: K. O. Mikaelian, Phys. Fluids 6, 356 (1994), Wouchuk and Nishihara, Phys. Rev. E 70, 026305 (2004)

17. -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 00.20.40.60.81 shock intensity CO 2 - Air Xe - Ar SF 6 - Air normal velocity Efects of the compressibility: At high incident shock intensity the asymptotic growth rate decreases, which agrees well with simulations by Yang et al. linear RMI a rarefaction is reflected back different pairs of gases asymptotic velocity a shock is reflected back Y. Yang et al, Phys. Fluids, 6, 1856 (1994), J. Wouchuk, Phys. Rev. E63, 056303 (2001), and Phys. Plasmas, 8, 2890 (2003).

18. Exact linear formula also agrees well with laser experiments with solid target at high Mach number of 10 and 15 (rarefaction was reflected) linear RMI 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).

19. solve wave equations in regions 1, 2 and 3 with proper boundary conditions. RMI-like Instability (1) Instability induced when a ripple shock hits uniform interface

20. phase 1phase 2phase 3 time derivative of ripple shock front ripple Since shock front ripple oscillates, phase of oscillation at the interaction changes dynamics of interface after instability due to rippled shock

21. Growth rate of contact surface ripple depends on the phase of the incident ripple shock at the incident growth rate of contact surface phase 1 phase 2 phase 3 dotted line; instantaneous value circles; simulation solid line; time integrated value instability due to rippled shock R. Ishizaki et al., Phys. Rev E53, R5592 (1996). Analytical solutions agree with simulations

22. Incident laser light shock frontablation surface shock front (a) nonuniform target surface(b) nonuniform laser irradiation RMI-like Instability (2) Instabilities associated with laser ablation (nonuniform target or nonuniform laser)

23. trajectory of shock, ablation surface, and sonic point Chapman-Jouguet condition at sonic point density profile Energy deposited at heat wave front induces ablation pressure, and laser ablation drives a shock wave ahead (like a piston) Energy deposition at heat wave front corresponds to combustion in rocket engine heat wave temperature heat flux divergence of heat flux ablation surface instability distance time flow diagram ablation surface shock front

24. dash-dot line; ablation surface deformation solid line; ripple shock driven laser ablation dotted line; ripple shock driven rigid piston Ablation deformation monotonically increases, and amplitude of shock ripple is small compared with a case of a rigid piston ablation surface instability 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).

25. Analytical solutions for both shock front ripple and areal mass density perturbation agree well with laser experiments. shock front rippleareal mass density uniform laser irradiation target surface deformation comparison with laser experiments (squares) ablation surface instability R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997). T. Endo et al., Phys. Rev. Lett., 74, 3608 (1995).

26. nonuniform laser irradiation Fairly good agreements were obtained between experiments and theory, by assuming the ablative Rayleigh-Taylor growth after rarefaction wave returns the ablation surface ablation surface instability M. Nakai et al., Phys. Plasmas, 9, 1734 (2002). H. Azechi et al., Phys. Plasmas, 5, 1945 (1998). square and solid line: =100  m, I 0 =0.4 circle and dotted line: =75  m, I 0 =0.1 after shock reach rare surface, exponential growth is assumed due to ablative RTI RMI-like ablative RTI

27. Richtmyer-Meshkov instability （ nonlinear theory ） (incompressible fluid approximation)

28. Acceleration of different mass fluids drives velocity shear at the interface. http://scitation.aip.org/getpdf/servlet spring two fluids with surface perturbation before the contact after the contact http://info- center.ccit.arizona.edu/~fluidlab/papers/paper4.p df http://info- center.ccit.arizona.edu/~fluidlab/papers/paper4.p df nonlinear RMI J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996).

29. We can obtain velocity shear induced at the interface due to the acceleration during the contact between spring and container. where ; the spring constant, ; mass of the container, ; the earth gravity and ; initial velocity of the container,, Integrate equation for the amplitude of the interface perturbation over the interval of the contact before the contact after the contact nonlinear RMI

30. Define interface velocity by mass weighted velocity as which satisfies boundary condition Introducing vorticity  u becomes  By introducing velocity potential and circulation We obtain from Bernoulli equation where Circulation does not conserved for a finite Atwood number A Nonlinear evolution of circulation at the interface with finite density ratio: Bernoulli equation nonlinear RMI

31. Defining complex z from the interface position (x  )  y  )   : Lagrangian parameter Bernoulli equation becomesNonlocal. The similar equations have been obtained by Kotelnikov (PF(00)) but for different u. Interface dynamics with Lagrangian maker Modified Birkhoff-Rott equation nonlinear RMI Solve above coupled equations with initial conditions the interface trajectory is obtained from Modified Birkhoff-Rott equation Finite Atwood number induces locally stretching and shrinking of the interface. Normalization

32. Weakly nonlinear Theory of a Vortex Sheet : Expansion Comparison with experiments nonlinear RMI spike bubble accelerated interface J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996). spike bubble shocked interface G. Dimonte et al.,, Phys. Plasmas, 3, 614 (1996). C. Matsuoka et al., Phys. Rev. E67, 036301 (2003). expansion up to 3 rd order

33. analytical model K Vlin t = 0.80 K Vlin t = 0.05 double spiral shape of spike and vorticity in simulation Dynamics of vortex sheet with density jump nonlinear vortex generation, their self interaction Density jump at the interface introduces generation of vortex and thus opposite sign of vortex appears, which causes double spiral structure of spike K Vlin t = 6 K Vlin t = 12 nonlinear RMI C. Matsuoka et al., Phys. Rev. E67, 036301 (2003).

34. Fully nonlinear evolution: Double spiral structure is observed as Jacobs & Sheeley experiment. Color shows the vorticity Parameters A = 0.155 k  0 = 0.2 kv lin t = 0, 1, 2,,,,12 Jacobs nonlinear RMI

35. A=0.2, n=4 (inner: lighter fluid) A=-0.2, n=4 (inner: heavier fluid) Cylindrical vortex sheet in incompressible RMI. nonlinear RMI 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 C. Matsuoka and K. Nishihara, Phys. Rev. E73, 055304 (2006), Phys. Rev. E74, 066303 (2006). spike bubblespike bubble Details by Matsuoka On Aug. 21

36. Richtmyer-Meshkov instability （ Molecular Dynamic simulation ） (cylindrical geometry) Potential barrier as Piston LJ atoms F ij z RR

37. Nonlinear evolution of Richtmyer-Meshkov instability in cylindrical geometry MD RMI mass density shock passing interface Mach stem appears shock reflected reflected shock hits interface shock pass through interface vorticitymass densityvorticitymass density anomalous mixing occurs bubble spike

38. Molecular dynamics simulations show RM growth driven by multiple shocks for different mode numbers. trajectorygrowth rate Decay of nonlinear growth is mode dependent and higher mode decays slower, which agrees with the model of cylindrical vortex sheet 1st 2nd3rd 1st 2nd 3rd MD RMI

39. Whenever shocks pass through interface from heavy to light, phase inversion occurs, which causes generation of higher harmonics MD RMI Richtmyer-Meshkov instability at shell surfaces (light-heavy-light) density velocity (radial) initial shock reaches the center reflected shock reaches shell density

40. 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.