Stochastic model of Rayleigh-Taylor turbulent mixing Is there a true alpha ?

1. Stochastic model ofRayleigh-Taylor turbulent mixingIs there a true alpha? Snezhana I. Abarzhi Meruzhan Cadjan, Sergei Fedotov Illinois Institute of Technology, Chicago, USA FLASH Center, The University of Chicago, USA Center for Turbulence Research, Stanford, USA The University of Manchester, UK Many thanks to: L. Kadanoff (U-Chicago), R. Rosner (ANL) The work is partially supported by NRL and DOE/NNSA published 2007, Phys. Lett. A

2. Rayleigh-Taylor Instability RT turbulent mixing controls a wide variety of physical phenomena ranging from astrophysical to micro-scales, under either high or low energy density conditions. RT turbulent flow is accelerated, anisotropic and inhomogeneous. Its invariant and statistical properties are essentially non-Kolmogorov. We propose a stochastic model to describe some of the mixing flow quantities.

3. linear regime l • turbulent mixing rh h rl g ~ h Rayleigh-Taylor evolution • Some qualitative features of RT flow are repeatable in observations. • As any turbulent process, the mixing dynamics has essentially noisy • character, with non-deterministic details and all scales contributing.

4. The growth of the horizontal scales was considered as a primary mixing mechanism with “unique” constant a • Several folds scatter in the observed values of • indicates a need in new probabilistic approaches • for a understanding the mixing process Significant efforts and large resources were involved. Quantification of Rayleigh-Taylor flows • For nearly two decades, the observations were focused on • the diagnostics of the vertical scale, readily available for measurements • the ascertainment of the “universality” law

5. Random character of Rayleigh-Taylor mixing Noisiness reflects the random character of the dissipation process. Kolmogorov turbulence RT turbulent mixing velocity fluctuates velocity and length scales fluctuate energy dissipation rate is invariant energy dissipation rate grows with time • Our model • accounts for the random character of the dissipation in RT flow • shows that the mixing growth-rate parameter a is not a universal constant • and is very sensitive to the statistical properties of the dissipation • identifies the rate of momentum as the statistic invariant of the flow • and a robust parameter to diagnose • The dependence of the growth-rate parameter on the Atwood number, • the initial conditions and the bubble interaction is not considered.

6. Modeling of RT turbulent mixing Dynamics: balance per unit mass of the rate of momentum gain and the rate of momentum loss These rates are the absolute values of vectors pointed in opposite directions and parallel to gravity. buoyant force rate of momentum gain rate of potential energy gain dissipation force rate of momentum loss energy dissipation ratee dimensional & Kolmogorov Lis the flow characteristic length-scale, either horizontallor verticalh

7. rates of momentum gain and momentum loss are scale and time invariant • characteristic length scale for energy dissipation L ~ h • rates of energy gain and energy dissipation are time-dependent a ~ 0.1-0.2 Asymptotic dynamics a = 2a • ratio between the rates is the characteristic constant of the flow

8. is stochastic process, characterized by time-scale and stationary distribution is non-symmetric: mean mode std Stochastic model Dissipation process is random. Rate of momentum loss fluctuates If Fluctuations • do not change the time-dependence, h ~gt2 • influence the pre-factor (h /gt^2) • long tails re-scale the mean significantly

9. Statistical properties of RT mixing < P > sustained acceleration < a > t/t uniform distribution log-normal distributions • The value of a = h /g(dr/r)t2 is a very sensitive parameter

10. Statistical properties of RT mixing probability density function at distinct moments of time P(P) ~ p(a) sustained acceleration log-normal distribution a P The rate of momentum loss is statistically steady

11. Mixing with turbulent diffusion Transport of scalars (temperature or diffusion) decreases buoyancy. dynamical system asymptotic solution • dissipation and diffusion are random processes

12. Statistical properties of RT mixing with diffusion < P > time-dependent acceleration turbulent diffusion uniform and log-normal distribution < a > t/t • Asymptotically, the statistical properties of a = h /g(dr/r)t2are very sensitive to noise and retain a time-dependency. • The length-scale is not well-defined

13. Statistical properties of RT mixing with diffusion probability density function at distinct moments of time ~ p(a) P(P) time-dependent acceleration turbulent diffusion log-normal distribution a P The ratio between the momentum rates is • statistically steady with or without turbulent diffusion accounted for • a robust parameter to diagnose

14. Is there a true universal alpha? • The growth-rate parameter alpha is significant • not because it is “deterministic” or “universal” • but because the value of this parameter is rather small. • The small value of alpha implies that in RT flows • almost all energy induced by the buoyant force dissipates • slight misbalance between the rates of momentum loss and gain is • sufficient for the mixing development. • To understand the mixing process, • momentum transport should be monitored. • To characterize this transport, one can choose • the rate of momentum loss m (sustained acceleration) or • the parameter P (time-dependent acceleration) • To monitor the momentum transport, • spatial distributions of the flow quantities should be diagnosed.

15. Observations of Rayleigh-Taylor turbulent mixing Our model reconciles with one another the controversial results of early observations. • Existing observations • do not provide necessary information on the dissipation statistics • are often not designed to address the delicate issues on • the momentum transport and the value of alpha LEM by Dimonte et al, 1998 - 2000 • time-dependency of the acceleration • uncontrolled transversal acceleration • meniscus and boundary layer effects • surface tension effects • mixing in stable configuration • uncontrolled initial conditions • short dynamic range in time What is required: - large dynamic range in the observation time - spatial distributions of the turbulent flow quantities - tight control of the experimental parameters

16. Conclusions The model: • Our stochastic model describes • the random characters of the dissipation and diffusion processes in • the turbulent mixing induced by Rayleigh-Taylor instability. • The fluctuations of the rate of momentum loss and gain are accounted for through • multiplicative noise with uniform and log-normal distributions The results: • The mixing growth-rate parameter alpha, conventionally used to characterize the mixing flow, is not a universal constant. • It is extremely sensitive to statistical properties of the dissipation. • The ratio between the rates of momentum loss and momentum gain • is the statistic invariant and the robust parameter to diagnose. • The approach can be extended for the cases of spatially varying and time- dependent accelerations and for studies the effect of the initial conditions • The results obtained can be used for a development of sub-grid scale numerical models and LES for unsteady turbulent processes