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Presented to: Newton Institute, Cambridge University Ye Zhou and Alfred Buckingham

Scaling criteria for high Reynolds and Peclet number turbulent flow, scalar transport, mixing, and heat transfer. Presented to: Newton Institute, Cambridge University Ye Zhou and Alfred Buckingham Lawrence Livermore National Lab. Introduction.

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Presented to: Newton Institute, Cambridge University Ye Zhou and Alfred Buckingham

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  1. Scaling criteria for high Reynolds and Peclet number turbulent flow, scalar transport, mixing, and heat transfer Presented to: Newton Institute, Cambridge University Ye Zhou and Alfred Buckingham Lawrence Livermore National Lab

  2. Introduction For many turbulent problems of scientific and engineering interest, resolving all interacting scales will remain a challenge in the foreseeable future ● In comprehensive flow experiments or corresponding direct numerical simulations of high Re and Pe number turbulent flow, scalar transport, mixing, and heat transfer, one must consider • the energetic excitation influences of the entire range of dynamic spatial scales combining • both velocity fluctuations and passive scalar variances • However, direct computational simulations or experiments directed to the very high Re and Pe flows of practical interest commonly exceed • the resolution possible using current or even foreseeable future super computer capability (Sreenivasan, this workshop) • or spatial, temporal and diagnostic technique limitations of current laboratory facilities.

  3. Introduction Kane et al., Astrophys. J. 564, 896 (2002) • Problem of practical interest: • Very High Reynolds number:Re~1010 • High energy density physics, • Supernovae and other astrophysical applications • Turbulent mixing of materials • 2.Currently available facility: Moderate to high Reynolds number Re~105 -- 106 • Laboratory experiments • Laser facilities • Simulations Practical needs promote use of statistical flow data bases developed from DNS or experiments at the highest Re and Pe levels achievable within the currently available facility limitations. Supernovae 1987 Muller, Fryxell, and Arnett A&A (1991) Omega Laser

  4. Question: Is it enough to understand the physics of the turbulent flows of interest using methodologies available? • At what turbulent flow condition can investigators be sure that their numerical simulations or physical experiments have reproduced all of the most influential physics of the flows and scalar fields of practical interest? • Can one define a metric to indicate whether the necessary physics of the flows of interest have been captured and suitably resolved using the tools available to the researcher?

  5. The minimum state This work defines a threshold criterion for DNS, experiments, and complementary theoretical modelling • Focus attention on time-dependent evolution of the energy- and scalar variance- containing scales • Provide an argument and criterion on how an extremely highReynoldsnumber problem can be scaled to amanageableone • Distinctive from: • LES, which typically requires that the resolved scale contains 80% of energy (Pope, NJP, 04); therefore, LES may be restrictive in Reynolds number • Euler scaling relates the astrophysical problems to high energy density laboratory experiments (Ryutov et al., Ap J., 1999) • “Mixing transition” (Dimotakis,JFM,2000) Euler scaling & Mixing transition will be reviewed

  6. The minimum state The starting point of our approach is to establish a more precise definition of the energy- and scalar variance-containing scales Velocity field: 1. The traditional definition of the inertial range: • Free from the external agencies at large scales • Free from the dissipation process ( is the outer scale) ( is the Kolmogorov scale) 2. A more precise definition of the inertial range: L-T << The Liepmann-Taylor scale: L-T 5 Re -1/2  Theinner viscous scale:   50 Re -3/4  The inertial range of the scalar field: L-T <  <  L-T5 Pé-1/2 .  50 Pé-3/4 . Dimotakis, JFM, 2000; see also Zhou et al., Phys. Rev. E; 2003; Phys. Plasma, 2003

  7. The minimum state The minimum state:the energy-containing scales of the flow and scalar fields under investigation will not be contaminated by interaction with the (non-universal) velocity dissipation and scalar diffusivity Original Problem:Very High Re Turbulent Flow E(k) KC=KL-T Scaled Problem: Manageable high Re Turbulent Flow Nonlocal interactions 2KC Local interactions KL-T The Liepmann-Taylor wavenumber of the scaled problem Dissipation scale K

  8. The minimum state Normalized energy flux demonstrated both scale similarity inertial range and -4/3 interacting scales The modes smaller than the Liepmann-Taylorwavenumber kL-T would essentially not interact beyond 2kL-T Domaradzki, this workshop; A scale disparity parameter defined to measure the locality of scale interactions S= max(k,p,q)/min(k,p,q) S-4/3 S-4/3 Zhou, Phys. Fluids A, 1993 Gotoh and Watanabe, JoT, 2007 S S

  9. The minimum state The minimum state:the lowest Re flow and Pe scalar field that the scaled problem would capture the same physics in the energy and scalar variance- containing scales of the problem of practical interest Θ(ҝ) Original Problem:Very High Pe scalar field ҝC Scaled Problem:Manageable high Pe scalar field Nonlocal interactions 2ҝC local interactions ҝL-T The Liepmann-Taylor wavenumber of the scaled problem Scalar diffusion scale ҝ

  10. The minimum state For a scalar field,the modes smaller than the Liepmann-Taylorwavenumber kL-T would essentially not interact beyond 3kL-T scalar field flow field S-2/3 S-4/3 Gotoh and Watanabe, JoT, 2007 S Normalized scalar variances flux demonstrated both scale similarity inertial range and -2/3 interacting scales

  11. The minimum state The minimum stateis appropriate because of the data redundancy in the inertial range, which can be demonstrated using a self-similarity scaling law Transfer function of different resolutions (with the lengths of the inertial range scaled; this figure answers a question by W. David McComb; this workshop ) Triadic interaction at different wavenumber in the inertial range T(k,p,q) = a3T(ak,ap,aq) (Kraichnan, JFM, 1971; Domaradzki, this workshop; Zhou, Phys. Fluids A 1993a, b) An ideal Kolmogorov inertial range can be constructed from the datasets of different resolutionss

  12. The minimum state The minimum state: the energy-containing scales of the flow and scalar fields will not be contaminated by interaction with the (non-universal) dissipation/ diffusivity scales The requirement of 2KL-T= Kνdetermines the Reynolds number of the minimum state Velocity field: L-T 5 Re -1/2  2  = L-T or Re = 1.6 105   50 Re -3/4  The requirement of 3 KL-T= Kνdetermines the Peclet number of the minimum state Scalar field:  50 Pé-3/4  3  = L-T or Pe = 8.1 105 L-T5 Pé-1/2 .

  13. The minimum state The critical Re of the minimum state is 1.6x105 and the critical Pe of the minimum state for passive scalar field is 8.1105 (1/2) [lL-T =(5/2) d.Re-1/2 ] Large-scale effects (1/3) [L-T = (5/3) d.Pe-1/2] Log spatial scales Inertial range, ln < l < lL-T Redundant data Viscous effects Lower bound: Inner viscous scale, ln = 50d.Re-3/4 Kolmogorov scale, lK = d.Re-3/4 Log Re Due to the different scaling with Reynolds number an uncoupled (inertial) range appears for Re > 104Dimotakis, JFM, 2000

  14. The Euler scaling In current practice, the Euler scaling1 relates the astrophysical problems to high energy density laboratory experiments Euler equation: 1Ryutov et al., Ap J., 1999; Ap J.S., 2000; Phys. Plasma 2001; Remington, 05 Euler scaling: Euler number (Eu):

  15. The Euler scaling Unfortunately, the Euler scaling could not consider the distinctive spectral scales of high Re number turbulent flows Energy containing scales, external forcing Data from Remington, Ryutov

  16. Mixing transition “Mixing transition”was proposed2 at Reynolds number Re ≥ 1-2 104 Turbulent, fully atomically mixed Turbulent, but not atomically mixed Turbulent, fully atomically mixed Chaotic Shear layer Re ≈ 2.5 x 103 Re ≈ 104 Outer envelope Interior Re ≈ 1.75 x 103 Re ≈ 2.3 x 104 Liquid-phase, round jet Liquid-phase, planar shear flow 2 P.E. Dimotakis, J. Fluid Mech. 409, 69 (2000)

  17. Mixing transition However, “mixing transition” does not answer the question: Is it enough for an experiment or a simulation to have just passed the mixing transition (Re = 1-2  104)? 3Zhou et al., PRE 2003; Phys. Plasma 2003; Robey et al. Phys. Plasma, 2003 shock l = 50mm h h h t = 8 ns t = 12 ns t = 14 ns AWE Rocket-Rig RT experiments David Youngs, this workshop talk # 19731 # 19732 After transition Before transition Around transition Time-dependent mixing transition is developed to indicate when flow in a laboratory experiment become turbulent

  18. Euler scaling and mixing transition There are some outstanding issues that cannot be answered by the Euler scaling and mixing transition • The Euler scaling • Does not have viscosity • Does not know at what size the spectral space can be scaled accurately Ryutov and Remington (Phys. Plasma, 2003) have suggested several experimental studies At what spatial scale can the astrophysical phenomena be reproduced in laboratory experiments ? • The time dependent mixing transition Is flow that just passed the mixing transition enough to capture all the physics of energy-containing scales? If not, how high must the Reynolds number be?

  19. Minimum state, Euler scaling and mixing transition The minimum state offers a perspective that unifies the Euler scaling and mixing transition • The minimum state: • The Reynolds and Peclet numbers must be high enough to capture three-dimensional, time-dependent evolution of the energy-containing and passive scalar variance-containing scales • Scaling of the astrophysical phenomena to a laboratory experiment An extremely high Reynolds number flow CAN BE SCALED to a flow with Reynolds number at or above that of the minimum state. The same method applied to an extremely highPeclet numbers scalar field

  20. Summary and conclusion • A minimum state is proposed so that the energy-containing scales of the flow and scalar fields under investigation • 1. will not be contaminated by interaction with the (non-universal) velocity dissipation and scalar diffusivity • 2. should reproduce significant energy containing and passive scalar variance-containing scales • 3. The critical Re of 1.6  105andPe of 8.1 105 are needed for the minimum state • We have reviewed two concepts that are relevant to studying astrophysical problems in a laboratory setting • 1. Flow that just passed the mixing transition is not sufficient • 2. The spectral information cannot be captured by the Euler similarity scaling • 3. We have unified and extended the concepts of both mixing transition and similarity scaling:

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