Turbulent mixing and beyond

1. Turbulent mixing and beyond Snezhana I. Abarzhi Many thanks to: K.R. Sreenivasan (ICTP), L. Kadanoff (U-Chicago) R. Rosner (ANL), S.I. Anisimov (Landau Institute) The work was partially supported by NRL (Steve Obenschain) and DOE/NNSA

2. Turbulence • is considered the last unresolved problem of classical physics. • Complexity and universality of turbulence fascinate scientists and mathematicians and nourish the inspiration of philosophers. • Similarity, isotropy and locality are the fundamental hypotheses advanced our understanding of the turbulent processes. • The problem still sustains the efforts applied. • Turbulent motions of real fluids are often characterized by • non-equilibrium heat transport • strong gradients of density and pressure • subjected to spatially varying and time-dependent acceleration • Turbulent mixing induced by the Rayleigh-Taylor instability is • a generic problem in fluid dynamics. • Its comprehension can extend our knowledge beyond the limits of idealized consideration of isotropic homogeneous flows.

3. Rayleigh-Taylor instability Fluids of different densities are accelerated against the density gradient. A turbulent mixing of the fluids ensues with time. • RT turbulent mixing controls • inertial confinement fusion, magnetic fusion, plasmas, laser-matter interaction • supernovae explosions, thermonuclear flashes, photo-evaporated clouds • premixed and non-premixed combustion (flames and fires) • mantle-lithosphere tectonics in geophysics • impact dynamics of liquids, oil reservoir, formation of sprays… RT flow is non-local, inhomogeneous, anisotropic and accelerated. Its properties differ from those of the Kolmogorov turbulence. Grasping essentials of the mixing process is a fundamental problem in fluid dynamics. “How to quantify these flows reliably?” Is a primary concern for observations.

4. rh rl Rayleigh-Taylor instability Water flows out from an overturned cup Lord Rayleigh, 1883, Sir G.I. Taylor 1950 P0 = 105Pa,P = r g h rh ~ 103kg/m3, g ~ 10 m/s2 h ~ 10 m

5. The Rayleigh-Taylor turbulent mixing Why is it important to study?

6. Photo-evaporated molecular clouds Stalactites? Stalagmites? Eagle Nebula. The fingers protrude from the wall of a vast could of molecular hydrogen. The gaseous tower are light-years long. Inside the tower the interstellar gas is dense enough to collapse under its own weight, forming young stars Hester and Cowen, NASA, Hubble pictures, 1995 Ryutov, Remington et al, Astrophysics and Space Sciences, 2004. Two models of magnetic support for photo-evaporated molecular clouds.

7. Supernovae Supernovae and remnants type II: RMI and RTI produce extensive mixing of the outer and inner layers of the progenitor star type Ia: RTI turbulent mixing dominates the propagation of the flame front and may provide proper conditions for generation of heavy mass element Pair of rings of glowing gas, caused perhaps by a high energy radiation beam of radiation, encircle the site of the stellar explosion. Burrows, ESA, NASA,1994

8. Inertial confinement fusion • For the nuclear fusion • reaction, the DT fuel should • be hot and dense plasma • For the plasma compression • in the laboratory it is used • magnetic implosion • laser implosion of DT targets • RMI/RTI inherently occur • during the implosion process • RT turbulent mixing • prevents the formation • of hot spot Nishihara, ILE, Osaka, Japan, 1994

9. BACKLIGHTER LASER BEAMS Imprint MAIN LASER BEAMS QUARTZ CRYSTAL 1.86 keV imaging 2D IMAGE RIPPLED CH TARGET Feedout Richtmyer-Meshkov BACKLIGHTER TARGET Si Magnification x20 Time STREAK CAMERA Rayleigh-Taylor Inertial Confinement Fusion Nike, 4 ns pulses, ~ 50 TW/cm3 target: 1 x 2 mm; perturbation: ~30mm, ~0.5 mm Aglitskii, Schmitt, Obenschain, et al, DPP/APS,2004

10. Impact dynamics in liquids and solids MD simulations of the Richtmyer-Meshkov instability: a shock refracts though the liquid-liquid (up) and solid-solid (down) interfaces; nano-scales Zhakhovskii, Zybin, Abarzhi, Nishihara, Remington, DPP, DFD/APS 2005

11. Solar and Stellar Convection Solar surface, LMSAL, 2002 Simulations of Solar convection Cattaneo et al, U Chicago, 2002 Observations indicate: dynamics at Solar surface is governed by convection in the interior. Simulations show: Solar non-Boussinesq convection is dominated by downdrafts; which are either large-scale vortices (wind) or smaller-scale plumes (“RT-spikes”).

12. cooling instabilities liquid instabilities heating Non-Boussinesq turbulent convection Thermal Plumes and Thermal Wind Sparrow 1970 Libchaber et al 1990s Kadanoff et al 1990s Sreenivasan et al 2001 helium T~4K Re ~ 109, Ra ~ 1017 The non-Boussinesq convection and RT mixing may differ as thermal and mechanical equilibriums, or as entropy and density jumps

13. FIRES • shear-driven KH • buoyancy-driven RT 1 m base Tieszen et al, 2004 hydrogen and methane FLAMES • Landau-Darrieus (LD) + • RT in Hele-Shaw cells 1.0 mm x 200 mm, 0.17 mm/s, Atwood ~ 10-3 Ronney, 2000 linear nonlinear Non-premixed and premixed combustion • The distribution of vorticity is the key difference between the LD and RT

14. Turbulent mixing induced by the Rayleigh-Taylor instability What is known and unknown?

15. l g rh ~ h rl Rayleigh-Taylor evolution • linear regime • nonlinear regime • light (heavy) fluid penetrates • heavy (light) fluid in bubbles (spikes) • turbulent mixing RT flow is characterized by: • large-scale structure • small-scale structures • energy transfers to large and small scales

16. Nonlinear Rayleigh-Taylor / Richtmyer-Meshkov Krivets & Jacobs Phys. Fluids, 2005 • large-scale dynamics • is sensitive to the • initial conditions • small-scale dynamics • is driven by shear

17. Rayleigh-Taylor turbulent mixing Dimonte, Remington, 1998 3D perspective view (top) and along the interface (bottom) • internal structure of • bubbles and spikes

18. Rayleigh-Taylor turbulent mixing FLASH 2004 3D flow density plots broad-band initial perturbation small-amplitude initial perturbation The flow is sensitive to the horizontal boundaries of the fluid tank, is much less sensitive to the vertical boundaries, and retains the memory of the initial conditions.

19. Unsteady turbulent processes Our phenomenological model identifies the new invariant, scaling and spectral properties of the accelerated turbulent mixing accounts for the multi-scale and anisotropic character of the flow dynamics randomness of the mixing process discusses how to generalize this approach for rotating and reactive flows and for other applications

20. How to model turbulent processes in unsteady (multiphase) flows? Any physical process is governed by a set of conservation laws: conservation of mass, momentum, angular momentum and energy Kolmogorov turbulence transport of kinetic energy isotropic, homogeneous … Accelerating flows transports of momentum (mass), anisotropic, inhomogeneous… potential and kinetic energy Rotating flows transports of angular momentum momentum, mass, potential and kinetic energy Unsteady turbulent mixing induced by the Rayleigh-Taylor is driven by the momentum transport

21. 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

22. Asymptotic dynamics • characteristic length-scale is horizontal L ~ l nonlinear • characteristic length-scale is vertical L ~ hturbulent a ~ 0.1

23. In the turbulent mixing flow: • length scale and velocity are time-dependent • kinetic and potential energy both change • changes in potential energy are due to buoyancy • changes in kinetic energy are due to dissipation • momentum gains and losses Accelerated turbulent mixing • The turbulent mixing develops: • horizontal scale grow with time l ~ gt2 • vertical scale h dominates the flow and is regarded as • the integral, cumulative scale for energy dissipation. • the dissipation occurs in small-scale structures produced by shear • at the fluid interface.

24. rates of momentum gain and momentum loss are scale and time invariant • rates of gain of potential energy gain and dissipation of kinetic energy • are time-dependent • ratio between the rates is the characteristic value of the flow Unsteady turbulent flow P remains time- and scale-invariant for time-dependent and spatially-varying acceleration, as long as potential energy is a similarity function on coordinate and time (by analogy with virial theorem)

25. Kolmogorov turbulence is inertial (Galilean-invariant), isotropic, local and homogeneous. • Energy dissipation rate is the basic invariant, • determines the scaling properties of the turbulent flow. RT turbulent mixing is non-inertial (accelerated), anisotropic, non-local and inhomogeneous. • The flow invariant is the rate of momentum loss • We consider some consequences of time and scale • invariance of the rate of momentum loss in the • direction of gravity. Basic concept for the RT turbulent mixing • The dynamics of momentum and energy depends on directions. • There may be transports between the planar to vertical components. • It may have a meaning to write the equations in 4D for momentum-energy tensor and study their covariant and invariant properties in non-inertial frame of reference.

26. energy dissipation rate ~g, time- and scale-inv time- and scale-invariant time- and scale-invariant not-Galilean invariant time-dependent time-dependent energy transport and inertial interval • rate of momentum loss time- and scale-invariant not a diagnostic parameter transport of momentum • enstrophy • helicity Invariant properties of the RT turbulent mixing RT turbulent mixing Kolmogorov turbulence

27. local scaling more ordered • scaling with Reynolds • viscous scale mode of fastest growth Scaling properties of the RT turbulent mixing RT turbulent mixing Kolmogorov turbulence transport of momentum transport of energy • similarly: dissipative scale, surface tension

28. Kolmogorov turbulence: • spectrum of kinetic energy kinetic energy = • spectrum of momentum RT turbulent mixing: momentum = • spectrum of kinetic energy (velocity) kinetic energy = Spectral properties of RT mixing flow What is the set of orthogonal functions? These properties have not been diagnosed. The difference with Kolmogorov and/or Obukhov-Bolgiano is substantial.

29. We assume Rate of temperature change is Landau & Lifshits dynamical system asymptotic solution Time-dependent acceleration, turbulent diffusion The transport of scalars (temperature or molecular diffusion) decreases the buoyant force and changes the mixing properties with A. Gorobets, K.R. Sreenivasan, Phys Fluids 2005

30. vs time t Asymptotic solutions and invariants buoyancyg dr/rvs time t dimensionless units • Buoyancy gdr/rvanishes asymptotically with time. • Parameter P is time- and scale-invariant value, and the flow characteristics

31. Randomness of the mixing process Some qualitative features of RT mixing are repeatable from one observation to another. As any turbulent process, Rayleigh-Taylor turbulent mixing has essentially noisy character. Kolmogorov turbulence RT turbulent mixing Noisiness reflects the random character of the dissipation process. velocity fluctuates velocity and length scales fluctuate energy dissipation rate is invariant energy dissipation rate grows with time • We account for the random character of the dissipation process in RT flow, • incorporating the fact that the rate of momentum loss is • time- and scale-invariant value, and fluctuates about its mean. • Observations focus on the diagnostics of integral scale and • do not provide necessary information on dissipation statistics. with M. Cadjan, S. Fedotov, Phys Letters A, 2007

32. is stochastic process, characterized by time-scale and stationary distribution is non-symmetric: mean mode std Stochastic model of RT mixing 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

33. 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

34. 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

35. Statistical properties of RT mixing < P > time-dependent acceleration turbulent diffusion uniform and log-normal distribution < a > t/t • The value of a = h /g(dr/r)t2 is very sensitive parameter • Asymptotically, its statistical properties are very sensitive to noise and retain a time-dependence. • The length-scale is not well-defined

36. Statistical properties of RT mixing 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 for any type of acceleration • a robust parameter to diagnose

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

38. Kolmogorov turbulence unsteady turbulent flow RT mixing: between order and disorder ? Turbulent mixing is “disordered.” However, it is more ordered compared to isotropic turbulence Is a “solid body acceleration” being the asymptotic state of RT unsteady flow? Group theory approach, Abarzhi et al 1990s: In RT flow, the coherent structures with hexagonal symmetry are the most stable and isotropic. Self-organization may potentially occur. How to impose proper initial perturbation? Faraday waves (Faraday, Levinsen, Gollub) can be a solution This imposes very high requirements on the precision and accuracy in the experiments.

39. Diagnostics of unsteady turbulent processes Basic invariant, scaling and spectral properties of accelerated mixing differ from those in the classical Kolmogorov turbulence. • In Kolmogorov turbulence, energy dissipation rate is statistic invariant, • rate of momentum loss is not a diagnostic parameter. • In unsteady turbulent flow, the rate of momentum loss is the basic invariant, whereas energy dissipation rate is time-dependent. • Energy is complimentary to time, momentum is complimentary to space. • In classical turbulence, the signal is one (few) point measurement with detailed temporal statistics • Spatial distributions of the turbulent flow quantities should be diagnosed for capturing the transports of momentum and energy in non-Kolmogorov turbulence

40. Verification and Validation • Metrological tools available currently for fluid dynamics community • do not allow experimentalists to perform a • detailed quantitative comparison with simulations and theory: • qualitative observations, indirect measurements, short dynamic range … • The situation is not totally hopeless. • Recent advances in high-tech industry unable the principal opportunities • to perform the high accuracy measurements of turbulent flow quantities, • with high spatial and temporal resolution, over a large dynamic range, • with high data rate acquisition. • a new research platform is attempting to launch for studies of turbulence • in unsteady turbulent flows • leverage existing technology, the unique experimental facility, holographic data storage http://en.wikipedia.org/wiki/HDSS

41. Conclusions The model • We suggested a phenomenological model to describe the • unsteady turbulent mixing induced by Rayleigh-Taylor instability. • The model describes the invariant, scaling and spectral properties of the flow. • The model considers the effects of randomness, turbulent diffusion, … The results • Unsteady turbulent flow is driven by the transport of momentum, • whereas isotropic turbulence is driven by energy transport. • The invariant, scaling, spectral properties and statistical properties of the • accelerating mixing flow differ from those in Kolmogorov turbulence. • The rate of momentum loss is the basic invariant of the accelerated flow, • the energy dissipation rate is time-dependent. • The ratio between the rates of momentum loss and gain is time and scale- • invariant, for sustained and/or time-dependent acceleration Works in progress • The model can be applied for rotating, compressible and reactive flows • The results of the model can be applied for a design of experiments and • for numerical modeling (sub-grid-scale models) • Rigorous theory is on the way. New experiments are attempting to launch.