This work was performed under the auspices of the Lawrence Livermore National Security, LLC, (LLNS) under Contract No. DE-AC52-07NA27344 Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA Zhou and Buckingham. 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

Zhou & Buckingham. 2 ● 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. For many turbulent problems of scientific and engineering interest, resolving all interacting scales will remain a challenge in the foreseeable future Introduction

Zhou & Buckingham. 3 Kane et al., Astrophys. J. 564, 896 (2002) Muller, Fryxell, and Arnett A&A (1991) 1 Problem of practical interest: Very High Reynolds number: Re~10 10 High energy density physics, Supernovae and other astrophysical applications Turbulent mixing of materials 2. Currently available facility: Moderate to high Reynolds number Re~ Laboratory experiments Laser facilities Simulations Supernovae 1987 Omega Laser 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. Introduction

Zhou & Buckingham. 4 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? Question: Is it enough to understand the physics of the turbulent flows of interest using methodologies available?

Zhou & Buckingham. 5 F ocus attention on time-dependent evolution of the energy- and scalar variance- containing scales  Provide an argument and criterion on how an extremely high Reynolds number problem can be scaled to a manageable one 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) This work defines a threshold criterion for DNS, experiments, and complementary theoretical modelling Euler scaling & Mixing transition will be reviewed The minimum state

Zhou & Buckingham. 6 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 The Liepmann-Taylor scale: Dimotakis, JFM, 2000; see also Zhou et al., Phys. Rev. E; 2003; Phys. Plasma, A more precise definition of the inertial range: (  is the outer scale) The inertial range of the scalar field: The inner viscous scale: (  is the Kolmogorov scale) The minimum state L-T  5 Re -1/2   50 Re -3/4   L-T  5 Pé -1/2 .   50 Pé -3/4 . L-T < <   L-T <  <  

Zhou & Buckingham. 7 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 Scaled Problem: Manageable high Re Turbulent Flow Dissipation scale KK E(k) Original Problem: Very High Re Turbulent Flow K L-T K C =K L-T 2K C The minimum state The Liepmann-Taylor wavenumber of the scaled problem Nonlocal interactions Local interactions

Zhou & Buckingham. 8 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 S Normalized energy flux demonstrated both scale similarity inertial range and -4/3 interacting scales The modes smaller than the Liepmann-Taylor wavenumber k L-T would essentially not interact beyond 2k L-T Zhou, Phys. Fluids A, 1993 Gotoh and Watanabe, JoT, 2007 The minimum state S -4/3

Zhou & Buckingham. 9 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 Scaled Problem: Manageable high Pe scalar field Scalar diffusion scale ҝ ҝ  Θ(ҝ)Θ(ҝ) Original Problem: Very High Pe scalar field ҝ L-T ҝCҝC 2ҝ C2ҝ C The minimum state The Liepmann-Taylor wavenumber of the scaled problem local interactions Nonlocal interactions

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

Zhou & Buckingham. 11 T(k,p,q) = a 3 T(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 The minimum state is appropriate because of the data redundancy in the inertial range, which can be demonstrated using a self-similarity scaling law Triadic interaction at different wavenumber in the inertial range Transfer function of different resolutions (with the lengths of the inertial range scaled; this figure answers a question by W. David McComb; this workshop ) The minimum state

Zhou & Buckingham. 12 The minimum state The requirement of 2K L-T = K ν determines the Reynolds number of the minimum state The requirement of 3 K L-T = K ν determines the Peclet number of 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 Velocity field: Scalar field: L-T  5 Re -1/2   50 Re -3/4   L-T  5 Pé -1/2 .   50 Pé -3/4  2 = L-T or Re = 1.6   =  L-T or Pe = 8.1  10 5

Zhou & Buckingham. 13 The critical Re of the minimum state is 1.6x10 5 and the critical Pe of the minimum state for passive scalar field is 8.1  10 5 Viscous effects Large-scale effects Log Re Log spatial scales Redundant data (1/2) [ L-T =(5/2)   Re -1/2 ] Kolmogorov scale, K =   Re -3/4 Lower bound: Inner viscous scale, = 50   Re -3/4 Inertial range, n < < L-T (1/3) [  L-T = (5/3)   Pe -1/2 ] Due to the different scaling with Reynolds number an uncoupled (inertial) range appears for Re > 10 4 Dimotakis, JFM, 2000 The minimum state

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

Zhou & Buckingham. 15 Unfortunately, the Euler scaling could not consider the distinctive spectral scales of high Re number turbulent flows ParametersSN1987aLaboratory experiments r (cm) 9   u (cm/s) 2   10 5  (g/cm 3 )7.5  Eu Re 2.6   10 6 Energy containing scales, external forcing Data from Remington, Ryutov The Euler scaling

Zhou & Buckingham. 16 “Mixing transition” was proposed 2 at Reynolds number Re ≥ 1-2  10 4 Chaotic Liquid-phase, round jet Liquid-phase, planar shear flow Turbulent, fully atomically mixed Turbulent, but not atomically mixed 2 P.E. Dimotakis, J. Fluid Mech. 409, 69 (2000) Re ≈ 2.5 x 10 3 Re ≈ 10 4 Re ≈ 1.75 x 10 3 Re ≈ 2.3 x 10 4 Shear layer Outer envelope Interior Mixing transition

Zhou & Buckingham. 17 # t = 8 ns # t = 12 nst = 14 ns shock h h h = 50  m 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  10 4 )? AWE Rocket-Rig RT experiments David Youngs, this workshop talk Around transition Before transition After transition 3 Zhou et al., PRE 2003; Phys. Plasma 2003; Robey et al. Phys. Plasma, 2003 Mixing transition Time-dependent mixing transition is developed to indicate when flow in a laboratory experiment become turbulent

Zhou & Buckingham. 18 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? Euler scaling and mixing transition

Zhou & Buckingham. 19 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 high Peclet numbers scalar field The minimum state offers a perspective that unifies the Euler scaling and mixing transition Minimum state, Euler scaling and mixing transition

Zhou & Buckingham. 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  10 5 and Pe of 8.1  10 5 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: