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 94551-0808 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
Transcript
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 94551-0808
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. 3Kane et al., Astrophys. J. 564, 896 (2002)
Muller, Fryxell, and Arnett A&A (1991)
1 Problem of practical interest: Very High Reynolds number: Re~1010
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
• Focus 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
r
The Liepmann-Taylor scale:
Dimotakis, JFM, 2000; see also Zhou et al., Phys. Rev. E; 2003; Phys. Plasma, 2003
2. 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
E(k)Original Problem: Very High
Re Turbulent Flow
KL-T
KC=KL-T
2KC
The minimum state
The Liepmann-Taylor wavenumber of the scaled problem
Nonlocal interactions
Local interactions
Zhou & Buckingham. 8
),,(),( ;),,(),(),(
,,
qpqp
qpkTskTqpkTtkEkt
tkE 2
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 SNormalized energy flux demonstrated both scale
similarity inertial range and -4/3 interacting scales
The modes smaller than the Liepmann-Taylor wavenumber kL-T would essentially not interact beyond 2kL-T
Zhou, Phys. Fluids A, 1993 Gotoh and Watanabe, JoT, 2007
The minimum state
S-4/3
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
Θ(ҝ) Original Problem: Very High Pe scalar field
ҝ L-T
ҝC
2ҝ C
The minimum state
The Liepmann-Taylor wavenumber of the scaled problem
local interactions
Nonlocal interactions
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For a scalar field, the modes smaller than the Liepmann-Taylor wavenumber kL-T would essentially not interact beyond 3kL-T
Normalized scalar variances flux demonstrated both scale similarity inertial range and -2/3 interacting scales
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
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The minimum state
The requirement of 2KL-T= Kν
determines the Reynolds number of the minimum state
The requirement of 3 KL-T= Kν
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 = 1.6 105
3 = L-T or Pe = 8.1 105
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The critical Re of the minimum state is 1.6x105 and the critical Pe of the minimum state for passive scalar field is 8.1105
Viscous effects
Large-scale effects
Log Re
Lo
g s
pat
ial
scal
es
Redundant data
(1/2) [L-T =(5/2) Re-1/2 ]
Kolmogorov scale, K = Re-3/4
Lower bound: Inner viscous scale, = 50Re-3/4
Inertial range, n < < L-T
(1/3) [L-T = (5/3) Pe-1/2]
Due to the different scaling with Reynolds numberan uncoupled (inertial) range appears for Re > 104 Dimotakis, JFM, 2000
The minimum state
Zhou & Buckingham. 14
In current practice, the Euler scaling1 relates the astrophysical problems to high energy density laboratory experiments
1Ryutov et al., Ap J., 1999; Ap J.S., 2000; Phys. Plasma 2001; Remington, 05
0
0
up
putt
p
ut
puut
u
,astro ~ , ~ , ~labastrolablabastro uuurrr
2~~ ~ ,~
~upppp
u
rtt lablabastrolabastro
Euler number (Eu): 2~~~ up
Euler scaling:
Euler equation:
The Euler scaling
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Unfortunately, the Euler scaling could not consider the distinctive spectral scales of high Re number turbulent flows
Parameters SN1987a Laboratory experiments
r (cm) 91010 5.310-3
u (cm/s) 2107 1.3105
(g/cm3) 7.510-3 4.2
Eu 0.29 0.34
Re 2.61010 1.7106
Energy containing scales, external forcing
Data from Remington, Ryutov
The Euler scaling
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“Mixing transition” was proposed2 at Reynolds number Re ≥ 1-2 104
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)?
AWE Rocket-Rig RT experiments
David Youngs, this workshop talk
Around transitionBefore transition After transition
3Zhou 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
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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 1.6 105 and Pe 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:
