Multi-scale character of the nonlinear dynamics of the Rayleigh-Taylor and Richtmyer-Meshkov instability Snezhana I. Abarzhi Many thanks to the co-authors: M. Herrmann (Stanford, Arizona State), J. Glimm (SUNY) P. Moin (Stanford), K. Nishihara (ILE), A. Oparin (ICAD), R. Rosner (ANL) International Conference Turbulent Mixing and Beyond, 18-26 August 2007, ICTP, Trieste, Italy
Preamble Ptolemeus, 100 AD new methods of measurements and calculations; Geocentric model with adjustable parameters, describing epicycles. The curve fit was perfect for 1500 years. Copernicus 1543: Heliocentric model provided worse agreement with observations A model, based on a wrong idea and used adjustable parameters, may agree with observations Supernova 1572: Brahe, Kepler, Newton Chandra pictures, 2003
Rayleigh-Taylor / Richtmyer-Meshkov instability Fluids of different densities are accelerated against the density gradient. A turbulent mixing of the fluids ensues with time. • RT/RM turbulent mixing controls • fusion, plasmas, laser-matter interaction • supernovae explosions, thermonuclear flashes, photo-evaporated clouds • flames and fires, geophysics, impact dynamics, spray formation,… RT/RM flows are non-local, inhomogeneous and anisotropic. Grasping essentials of the mixing process is a fundamental problem in fluid dynamics. One of the primary issues is the dynamics of the large-scale coherent structure of bubbles and spikes. This structure appears in the nonlinear regime of RTI/RMI. The nonlinear regime bridges a gap between the initial and turbulent (perhaps, self-similar) stages of RTI/RMI.
l rh g h rl RT mixing,Ramaprabhu & Andrews2004 Rayleigh-Taylor /Richtmyer-Meshkov instability laboratory experiments in fluids and gases nonlinear RMI, Jacobs et al2004
l g rh h rl Rayleigh-Taylor / Richtmyer-Meshkov evolution • linear regime • nonlinear regime • light (heavy) fluid penetrates • heavy (light) fluid in bubbles (spikes) • turbulent mixing • Richtmyer-Meshkov (g=0) RT/RM flow is characterized by: • large-scale structure • small-scale structures • energy transfers to large and small scales
scale separation group theory Abarzhi1990s, 2001 passive active Large-scale coherent dynamics Conservation laws no mass flux momentum no mass sources initial conditions symmetry Singular and non-local aspects of the interface evolution cause significant difficulties for theoretical and numerical of studies of RMI.
Fourier expansion • Local expansion conservation laws local dynamical system ODEs Dynamical system • Layzer-type approach regular asymptotic solutions are absent • non-local dynamics interplay of harmonics singularities • continuous family regular asymptotic solutions • family parameters symmetry, 2D/3D • physically significant is the fastest stable solution
Universal dynamics for all A Nonlinear dynamics of RT bubble RT evolution is characterized by two length scales, periodland positionh time curvature velocity , ,
z A , RT z = - Ak 8 L z = - k 8 D Velocity and curvature vs Atwood, RTI non-local theory empiric model drag model • empirical / drag / Layzer-type solutions violate the conservation of mass • velocity is not a sensitive diagnostic parameter • curvature is sensitive parameter, which tracks the conservation of mass
Nonlinear dynamics of RM bubble RM evolution is characterized by two length scales, periodland positionh time curvature velocity • flat bubbles move faster and experience stronger deceleration and drag • for smaller values of A, bubbles move faster • for velocity, for a finite sequence of data points and short dynamic range • the exponent -1 and coefficient C may be hard to distinguish
Empirical drag models and single-mode solutions • Drag model assumes • RM evolution has a single-scale character, h = h (l) • curvaturezis uniquely determined by period l • Drag model suggests an empirical formula Shvarts et al 1995, 2001 • To calculate the drag model solution in a single-mode Layzer-type approximation, Goncharov 2002 introduced a time-dependent inhomogeneous mass source of the light fluid. • Experiments do not have any mass source yet report a reasonable agreement with the drag model for the position h (velocity v). • The formula is a curve fit with free parameters (exponents, coefficients). • To prove the formula as an empirical law, • huge statistics and large dynamic range are required. • Statistics standards: 30 data points per each parameter, ~3011 (~1.7x1016).
Nonlinear dynamics: power-laws Velocity of RM bubble given by the non-local theory with account for next-order correction in time (solid) and the single-scale drag-model Reliable quantitative statements are very hard to make. Flattening of the front of RM bubble is a qualitative effect.
Flow AixCo Marcus Herrmann, Stanford and Arizona State University (JFM) • Solves fully compressible Navier-Stokes equations (two-dimensional) • Operator splitting • Convective terms solved by explicit 2nd order Godunov type scheme • Diffusive terms solved by explicit 2nd order Runge-Kutta scheme • Hybrid tracking-capturing scheme • Interface is tracked by level set approach • 5TH order WENO • Shocks and waves are treated by a standrad capturing scheme • In-cell-reconstruction scheme ensures the interface remains a discontinuity • High Atwood numbers and strong shocks can be modeled
Computational setup • Mach = 1.2 • four cases: A=0.55, 0.663, 0.78, A=0.9 • speed of sound of the light gas cl = 347.2 m/s • viscosity of air for the Navier-Stokes equations • several test runs of the Euler equations • Initial perturbation a(t) = a0 cos(2p x / l) • l = 3.75 cm, a0 = 0.064 l • Computational domain [- 40.667 l, +1.333 l ] x [-0.5 l, 0.5 l] • grid 5376 x 128 (256); equidistant grid cells • outflow conditions in the z-direction • symmetry conditions in the x-direction • Simulations stop as the reflected shock hits the interface. • The run time is longer than in most observations of the nonlinear RMI.
A=0.55 A = 0.663 A=0.78 A=0.9 • RMI develops relatively to a background motion with velocity • at which the interface would move if it would be ideally planar. length scalel time scale • The bubble (spike) dynamics is quantified in the frame of references • moving with velocity Interface evolution induced by RMI
Linear theory of Wouchuk 2001 adequate value of the growth-rate Experiments of Jacobs 1997 A=0.663; Mach = 1.1 A=0.9 Validation • Oscillations are caused by reverberations of sound waves • Oscillations do not result in significant pressure fluctuations • The oscillations induce an error ~10% in theoretical value and • up to ~30-40% in experimental value of RMI growth-rate • Experimental data sampling do not capture the velocity oscillations
The bubble is accelerated “impulsively” and then decelerates. Nonlinear velocity Other earlier observations - set time-scale using the value of the growth-rate v0 - calculated the bubble (spike) position relative to the “middle line” - did not capture the high frequency components Nevertheless - The velocity time-dependence was evaluated quantitatively (pre-factor)
- Time-scale is set by the velocity • Bubble (spike) position calculated in the moving frame of references. • High frequency components are captured. Diagnostics of the velocity Log-log plot of velocity vs time Only asymptotic value of the velocity can be evaluated Accurate quantitative estimate of the velocity time-dependence (exponent) is prevented due to - oscillations caused by reverberations of sound waves - unknown contribution of higher order terms and short dynamic range
curvature is calculated via a least square fit circle, |x/l| < 4/64 Diagnostics of the interface Bubble curvaturezand its rms deviationz‘vs time solid line is z (left scale), dashing lines is z‘/z (right scale), |x/l| < 4/64 • RM bubble flatten asymptotically with time • The flattening process is slower for smaller values of A
Multi-scale character of the dynamics velocityv(t)vs curvaturez(t)with timet white square is our non-local solution, black square is the drag model solution • velocity v=dh/dt and curvature z mutually depend on one another • deceleration d2 h / dt2 and flattening d (zl) /dt are interrelated processes • they indicate a multi-scale character of RM evolution
l g rh h rl Multi-scale dynamics in RTI/RMI The dynamics of RT/RM large-scale coherent structure is characterized by • wavelength – initial conditions • amplitude - small-scale structures • RTI, for all A • RMI, for all A • In RTI / RMI amplitude and wavelength contribute independently • to the nonlinear dynamics. • The multi-scale character of the nonlinear evolution should be • accounted for in a description of the turbulent mixing process. • The nonlinear dynamics is hard to quantify reliably (power-laws).
m m a1 pm pm 2a1 Coherent dynamics in RTI/RMI • Any wave is characterized by wavelength, amplitude and phase. • Coherence, symmetry and order are related to phase. • Is it possible to create and maintain the order in RT flow? • Group theory suggests: it is possible in principleas the flow should • maintain isotropy in the plane normal to the direction of acceleration 2D flow: binary interaction, duplication of the wavelength
2a2 a2 a2 -3a1 p2gm p3m1 p6mm p6mm 3(a1 + a2) a1 a1 a1 Coherent dynamics in RTI/RMI 3D flow: multi-pole interactions, growth of wavelength results in isotropy loss
Kolmogorov turbulence unsteady turbulent flow RT mixing: between order and disorder ? • Group theory suggests that RT/RM coherent structures with hexagonal • symmetry are the most stable and isotropic. Self-organization may occur. • Our phenomenological model, which has found that the unsteady • turbulent mixing is more ordered compared to isotropic turbulence • How to impose the initial perturbation? Faraday waves may be a solution Faraday waves J.P. Gollub Requirements on the precision and accuracy in the experiments are very high
Conclusions • The large-scale coherent dynamics in RTI and RMI • is studied theoretically and numerically. • Theory • obeys the conservation laws • has no adjustable parameters • accounts for the higher-order correlations • identifies the multi-scale character of the nonlinear dynamics • suggests the disordered mixing may have coherence and order • Numerical solution in RMI (M. Herrmann) • models weakly compressible and nearly inviscid fluids • treats the interface as a discontinuity • is applicable for fluids with very high values of the density ratio • The theory and the simulations • validate each other, identify the reliable diagnostics of the interface dynamics • indicate the non-local and multi-scale character of RMI • show that the reliable quantification of RTI/RMI is a complex problem, • still open for a curious mind.