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
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
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.
Zhou & Buckingham. 2
Introduction
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.
1
Problem of practical interest:
Very High Reynolds number: Re~1010
•
•
High energy density physics,
Supernovae
1987
Supernovae and other astrophysical
applications
•
Turbulent mixing of materials
Muller, Fryxell, and Arnett
A&A (1991)
2.
Currently available facility:
Moderate to high Reynolds number
Re~105 -- 106
•
•
•
Omega Laser
Laboratory experiments
Laser facilities
Simulations
Kane et al., Astrophys. J.
564, 896 (2002)
Zhou & Buckingham. 3
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?
Zhou & Buckingham. 4
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 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)
Euler
scaling &
Mixing
transition
will be
reviewed
Zhou & Buckingham. 5
The minimum state
The starting point of our approach is to establish a more
precise definition of the energy- and scalar variancecontaining scales
Velocity field:
  r  
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:
The inner viscous scale:
 L-T  5 Re -1/2 
  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
Zhou & Buckingham. 6
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
E(k)
KC=KL-T
Original Problem: Very High
Re Turbulent Flow
Nonlocal
interactions
2KC
Local
interactions
KL-T
The Liepmann-Taylor
wavenumber of the
scaled problem
Scaled Problem:
Manageable high Re Turbulent Flow
Dissipation
scale
K
Zhou & Buckingham. 7
The minimum state
The modes smaller than the Liepmann-Taylor wavenumber
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)
E ( k , t )
 k 2 E ( k , t )   T ( k , p, q );
t
p ,q
T ( k , s )   T ( k , p, q )
p ,q
S-4/3
S-4/3
Zhou, Phys. Fluids A, 1993
S
Gotoh and Watanabe,
JoT, 2007
S
Normalized energy flux demonstrated both scale similarity
inertial range and -4/3 interacting scales
Zhou & Buckingham. 8
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)
ҝC
Original Problem: Very High
Pe scalar field
Nonlocal
interactions
2ҝ C
local
interactions
ҝ L-T
The Liepmann-Taylor
wavenumber of the
scaled problem
Scaled Problem: Manageable high Pe scalar field
Scalar
diffusion
scale
ҝ
Zhou & Buckingham. 9
The minimum state
For a scalar field, the modes smaller than the Liepmann-Taylor
wavenumber kL-T would essentially not interact beyond 3kL-T
flow
field
S-4/3
scalar
field
S-2/3
Gotoh and Watanabe, JoT, 2007
S
Normalized scalar variances flux demonstrated both scale
similarity inertial range and -2/3 interacting scales
Zhou & Buckingham. 10
The minimum state
The minimum state is 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
Triadic interaction at different
wavenumber in the inertial range
(with the lengths of the inertial range scaled;
this figure answers a question by W. David
McComb; this workshop )
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 Zhou & Buckingham. 11
from the datasets of different resolutionss
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:
The requirement of
2KL-T= Kν determines
the Reynolds number
of the minimum
state
2  = L-T
or Re = 1.6 105
 L-T  5 Re -1/2 
  50 Re -3/4 
Scalar field:
The requirement of
3 KL-T= Kν
determines the
Peclet number of
the minimum state
3  = L-T
or Pe = 8.1 105
  50 Pé -3/4 
L-T  5 Pé -1/2 .
Zhou & Buckingham. 12
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) [L-T =(5/2) .Re-1/2 ]
Log spatial scales
Large-scale effects
(1/3) [L-T = (5/3) .Pe-1/2]
Inertial
range,
n <  < L-T
Viscous effects
Log Re
Due to the different scaling with Reynolds number
an uncoupled (inertial) range appears for Re > 104
Dimotakis, JFM, 2000
Lower bound:
Inner viscous scale,
 = 50.Re-3/4
Kolmogorov scale,
K = .Re-3/4
Zhou & Buckingham. 13
The Euler scaling
In current practice, the Euler scaling1 relates the astrophysical
problems to high energy density laboratory experiments
Euler equation:
 u

 u  u   p
 t



   u   0
t
p

p

 u  p   u    0
t
t

Euler scaling:

Euler number (Eu):
1Ryutov
rastro  ~
r rlab , uastro  u~ ulab , astro  ~ lab,
~
r
t astro  tlab ~ ,
pastro  plab ~
p  plab ~u~ 2
u
~
p  ~u~ 2
et al., Ap J., 1999; Ap J.S., 2000; Phys. Plasma 2001; Remington, 05 Zhou & Buckingham. 14
The Euler scaling
Unfortunately, the Euler scaling could not consider the
distinctive spectral scales of high Re number turbulent flows
Parameters
SN1987a
Laboratory
experiments
r (cm)
91010
5.310-3
u (cm/s)
2107
1.3105
 (g/cm3)
7.510-3
4.2
Eu
0.29
0.34
Re
2.61010
1.7106
Energy containing scales, external forcing
Data from Remington, Ryutov
Zhou & Buckingham. 15
Mixing transition
“Mixing transition” was proposed2 at Reynolds
number Re ≥ 1-2 104
Shear layer
Re ≈ 1.75 x 103
Turbulent, but not
atomically mixed
Turbulent, fully
atomically mixed
Re ≈
2.5 x 103
Re ≈ 104
Interior
Turbulent, fully
atomically mixed
Outer envelope
Chaotic
Re ≈ 2.3 x 104
Liquid-phase, planar shear flow
2
Liquid-phase, round jet
P.E. Dimotakis, J. Fluid Mech. 409, 69 (2000)
Zhou & Buckingham. 16
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)?
shock
=
50m
h
h
t = 8 ns
t = 12 ns
h
t = 14 ns
AWE Rocket-Rig
RT experiments
19732
##19731
Before transition
Around transition
After transition
David Youngs,
this workshop
talk
Time-dependent mixing transition is developed to indicate when flow
in a laboratory experiment become turbulent
Zhou & Buckingham. 17
3Zhou
et al., PRE 2003; Phys. Plasma 2003; Robey et al. Phys. Plasma, 2003
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?
Zhou & Buckingham. 18
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 threedimensional, 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
Zhou & Buckingham. 19
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:
Zhou & Buckingham. 20