1 / 40

Generation of Nonuniform Vorticity at Interface and Its Linear and Nonlinear Growth

Incident laser light. ablation surface. shock front. ablation surface. shock front. vorticity. mass density. Generation of Nonuniform Vorticity at Interface and Its Linear and Nonlinear Growth (Richtmyer-Meshkov and RM-like Instabilities)

skip
Download Presentation

Generation of Nonuniform Vorticity at Interface and Its Linear and Nonlinear Growth

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Introductionof Richtmyer-Meshkov instability andLaser Implosion

  3. shocked interface I S I dv0a dv0b from linearized relation of the shockRankin-Hugoniot vortex sheet where amplitude of the initial interface corrugation and its wave number, incident, transmitted and reflected shock speeds, and interface speed after the interaction and fluid velocity behind the incident shock. 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. Matsuoka, Nishihara Fukuda (PRE(03)) A=0.376, ξ0/λ=0.02

  4. Acceleration of different mass fluids also drives velocity shear at the interface. after Jacobs & Sheeley, PF (96) http://info-center.ccit.arizona.edu/~fluidlab/papers/paper4.pdf two fluids with surface perturbation after the contact before the contact spring http://scitation.aip.org/getpdf/servlet introduction (accelerated interface) 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. 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. outline of talk We first show that the RMI is driven essentially by nonuniform velocity shear induced at an interface, instead of impulsive acceleration.

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

  7. dv0a dv0b y x Richtmyer-Meshkov instability (initial perturbation and wave equation) rippled transmitted shock corrugated interface rippled reflected shock

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

  9. 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 dv0a dv0b y x Boundary condition at shock front normal velocity; from linearized shock jump condition with respect to ripple amplitude tangential velocity; continuous Consider interaction between corrugated interface and rippled shocks through sound wave and entropy wave between them. linear RMI

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

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

  12. ripple shock 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

  13. dv0a dv0b y x Richtmyer-Meshkov instability (linear theory) asymptotic growth rateeffects of compressibility Solve wave equations in the regions between interface and shock fronts for sound wave and entropy wave with proper boundary conditions

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

  15. Asymptotic growth rates depend on the whole compressible evolution: Integrate equation of motion from 0+ to 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. 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 From pressure continuity at the interface, we have for tangential velocity : density at t=0+

  16. 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. 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 -0.05 CO - Air 2 -0.1 asymptotic velocity -0.15 normal velocity Xe - Ar -0.2 -0.25 -0.3 SF - Air 6 -0.35 0 0.2 0.4 0.6 0.8 1 shock intensity 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 Y. Yang et al, Phys. Fluids, 6, 1856 (1994), J. Wouchuk, Phys. Rev. E63, 056303 (2001), and Phys. Plasmas, 8, 2890 (2003).

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

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

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

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

  24. Ablation deformation monotonicallyincreases, 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). dash-dot line; ablation surface deformation solid line; ripple shock driven laser ablation dotted line; ripple shock driven rigid piston This instabilitty now called ablative RMI after V. N. Goncharov, Phys. Rev. Lett., 82, 2091 (1999).

  25. uniform laser irradiation target surface deformation 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 areal mass density R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997). T. Endo et al., Phys. Rev. Lett., 74, 3608 (1995).

  26. Fairly good agreements were obtained between experiments and theory, by assuming the ablative Rayleigh-Taylor growth after rarefaction wave returns the ablation surface ablative RTI RMI-like nonuniform laser irradiation ablation surface instability after shock reach rare surface, exponential growth is assumed due to ablative RTI square and solid line: l=100mm, I0=0.4 circle and dotted line: l=75mm, I0=0.1 M. Nakai et al., Phys. Plasmas, 9, 1734 (2002). H. Azechi et al., Phys. Plasmas, 5, 1945 (1998).

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

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

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

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

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

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

  33. Density jump at the interface introduces generation of vortex and thus opposite sign of vortex appears, which causes double spiral structure of spike 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 C. Matsuoka et al., Phys. Rev. E67, 036301 (2003).

  34. Fully nonlinear evolution: Double spiral structure is observed as Jacobs & Sheeley experiment. nonlinear RMI Parameters A = 0.155 kx0 = 0.2 kvlint = 0, 1, 2,,,,12 Color shows the vorticity Jacobs

  35. Cylindrical vortex sheet in incompressible RMI. spike bubble bubble spike Features of cylindrical geometry, ・　two independent spatial scale, radius and wavelengthnonlinear 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 nonlinear RMI A=0.2, n=4 (inner: lighter fluid) A=-0.2, n=4 (inner: heavier fluid) Details by Matsuoka On Aug. 21 C. Matsuoka and K. Nishihara, Phys. Rev. E73, 055304 (2006), Phys. Rev. E74, 066303 (2006).

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

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

  38. 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 1st 2nd 3rd 2nd 3rd 1st

  39. velocity (radial) density initial shock reaches the center reflected shock reaches shell 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) 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, andnonuniform vorticitymay provides a paradigm in vortex dynamics.

More Related