Whither turbulence computations?
by
K.R. Sreenivasan
New York University
A commentary on the work of
Victor Yakhot
Boston University
Jörg Schumacher
University of Ilmenau
P.K. Yeung
Georgia Institute of Technology
Diego Donzis
Texas A&M
perhaps others
Indian Institute of Science
Tuesday
December 13, 2011
From Herman Winick, SLAC
Massive parallelism, with O(105) CPU cores; so
doing simulations has become a big task in itself.
New paradigms, new
architecture, etc, yet…
GMR
Rl
Earth Simulator
Kaneda et al. (2003)
160 nodes, each node with 8 vector-type
processors (total of 1280 processors); peak
performance per processor is ~100 GFlops.
Total peak performance is 130 TFlops.
My collaborators have used:
(1) Kraken at NICS/U of Tennessee (with
112,896 computer cores, peak
performance of 1.17 Pflops and a memory
of 147 TB) and (2) Jaguar at Oak Ridge
(224,256 cores, peak performance of 1.75
Pflops, memory of 360 Tb)
In the mean time …
the 10 PetaFlop barrier has been broken by a
Fujitsu machine and
ExaFlops, 100 million (?) cores (~25MW)
are on their way by ~2018:
What is the hydrogen atom of turbulence?
Phys. Rev. Lett. 28, 76 (1972)
Box turbulence
L = integral scale
N = number of grid points
hK = Kolmogorov scale
Rl = microscale Reynolds
number (√Re)
Dx = a1hK
a2L
For “standard” conditions,
a1 = 2, a2 = 5, we have
Rl  0.5 W1/6, Rl  4.5 N2/3
Rl
If the Earth Simulator can
compute
N = 4096, Rl  1200 (Re  105)
with L/hK = O(1000)
Exaflop machines can handle:
N = 32,768
Rl  4,000 (Re  106)
L/hK = O(10,000) or 4 decades
This ought to happen by the end
of the decade
But it won’t simulate anywhere
as large a Reynolds number!
e
surrogate
dissipation
higher Re
hK = (n3/<e>)1/4
h = (n3/e)1/4
Local dissipation scale can be far smaller.
ν = η u ( x  η )  u ( x )
Duh.h/n = 1. Duh has large fluctuations.
Thus, h can often be less than hk.
Distribution of length scales
probability density of h/<h>
Schumacher, Yakhot
log10 (h/<h>)
• From the distribution of length scales, we
have <td>= <hB2>/k  10 <hB>2/k
• Eddy diffusive time/molecular diffusive
time  Re1/2/100;exceeds unity only for
Re  104 ( mixing transition mentioned by
Narasimha yesterday)
Scales smaller than h (and hB) clearly
exist, so …
–How much of the data acquired
from resolutions of the order h is
reliable? The question becomes
more relevant at high Reynolds
numbers.
–What do we miss if we don’t
resolve the sub-Kolmogorov
scales?
–How critical is it to resolve the
sub-Kolmogorov scales for the
inertial range (for example)?
–How much better should be the
grid resolution for the discrete
version to remain “truthful” to the
continuum equations?
(in units of the mean)
The 1283 box represents “standard resolution”
From J. Schumacher
Sn  rn by Taylor’s expansion near r = 0
Jörg Schumacher1, Herwig Zilken2, Katepalli R. Sreenivasan3,4
1
Department of Physics, Philipps University Marburg, D-35032 Marburg, Germany
Visualization Laboratory, Central Institute for Applied Mathematics, Research Center Jülich, D-52425 Jülich, Germany
International Centre for Theoretical Physics, I-34014, Trieste, Italy
4 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
2
3
hB  hB
h  h
grid
grid
hB  hB
0
Pseudospectral simulation of scalar
mixing for a Schmidt number of 32 in a
homogeneous isotropic turbulent flow.
The left picture shows a slice through
the instantaneous scalar dissipation
field. The color coding runs
logarithmically from 0.00001 (blue) to
100 (red) in units of the mean scalar
dissipation rate. Magnifications of the
black frames in the left panel are plotted
on the right. The Kolmogorov scale and
Batchelor scales are indicated in each
case. The grid resolution is also shown
with N=1024 for a total box length of
L=2p. In both magnifications scale
variations around the Batchelor scale are
excited and indeed observable because
the grid resoluion in the simulations is
better than the Batchelor scale.
h  h
100
Isosurfaces of the scalar dissipation field
at a level of 11 in units of the mean scalar
dissipation rate. The iso-surfaces are
colored with respect to a flow property that
is a measure of local vorticity [1,2]. This
information is deduced from an eigenvalue
analysis of the velocity gradient tensor at
each grid point [1]. Green represents pure
straining motion and red corresponds to
the vorticity dominated motion. The
picture illustrates that both flow topologies
contribute to the steepening of intense
dissipative fronts. Additionally, the grayshaded cutting plane is shown.
Support by the Deutsche Forschungsgemeinschaft (DFG) and the US National
Science Foundation (NSF) is gratefully
acknowledged. Computations were done
on the IBM-JUMP cluster at the John von
Neumann-Institute for Computing.
0
References
[1] J. Schumacher and K. R. Sreenivasan, Phys. Rev. Lett. 91, 174501 (2003)
[2] J. Zhou, R. J. Adrian, S. Balachandar and T. M. Kendall, J. Fluid Mech. 387, 353 (1999)
Another view of the same thing
measurable differences
Low scalar
dissipation
k max η B = 1.5
k max η B = 6
no conspicuous difference
Theory for hsmallest
Yakhot, Phys. Rev. E 63, 026307 (2001)
Kurien & Sreenivasan, PRE 64, 056302 (2001)
Yakhot & Sreenivasan, J. Stat. Phys. 121, 823 (2005)
Schumacher, Sreenivasan & Yakhot, New J. Phys. 9, 89 (2007)
•
•
•
•
Derive exact dynamical equations for structure functions of all
orders
Model pressure terms (or use the point splitting technique), and
determine inertial scaling analytically
Match this inertial scaling with the smooth behavior for very small
scales (which are analytic)
Pick the scale corresponding to moments of infinite order
hsmallest/L = Re1
(instead of Re3/4, as for the standard Kolmogorov scale)
N = Re3
(instead of the standard Re9/4 relation)
A practical consequence
Rl  0.5 W1/8, Rl  4.5 N1/2
(instead of Rl  0.5 W1/6, Rl  4.5 N2/3)
Or, computational grid for a given
Reynolds number Re  Re4
(instead of the traditional Re3
estimate from Landua & Lifshitz)
present/traditional = O(Re)
For Rl = 103, Re  105, ratio = O(105)
A 40963 box can resolve all scales only
up to Rl  300
(not 1200 as previously thought)
A 32,7683 box can resolve all scales
only up to Rl  1000
(not 4,000 as we might have projected)
Previous work
Mathematical
Constantin, Foias, Manley and Temam, JFM 150, 427 (1985)
The authors show that the degrees of
freedom N of a 3-D turbulent flow obey
N ~ (L/h)3 ~ Re3,
and argue that the conventional estimate of
Re9/4 (e.g., Landau & Lifshitz 1959), based
on the Kolmogorov scale determined by the
average dissipation rate, is optimistic. If the
wavenumber spectrum varies as a powerlaw bounded on both sides with the roll-off
rate of n, they show that
(L/h)3 ~ Re6/(n+1),
giving Re9/4 for n = 5/3.
Phenomenological
(a) From the measurements of Meneveau &
Sreenivasan (1988+): Re4
(b) From Paladin & Vulpiani (1988): Re3b, b >0
(c) From the She-Leveque model (1994): Re3.6
(d)
δr v
= vx 
r
2p
vx
r
 v 
  r 
 r 
2
v xx 
=
2p
2p
vx
p ( 2 p  1)
hk
p2
r
vx
2
h
p2
 v
  r 
 r 
= Re
3
1
 
 4  2
2

2
v xx
2p
 O (r )
4
vx
 vx
vx
v xx =
2
h
2
2
so that
p
2p
h2
4
p
vx
Use
6
( )
v xxx  O r
12
v xx =
vx
2
p
vx
Put
r
p ( 2 p  1)  r
 2
=
h
12
 2
2




=1
(if

  O (r 4 )


 2 = 2 / 3)
Now, let r = Dx = chosen resolution. We then have
2p
vx

( r v r ) p
2p
vx
 Dx

h
 k




2
=
p (2 p  1)
12
Re
3
2
 2 
2
3
(2  2 )
<en>  Redn
d1
d2
d3
d4
theory
DNS
RSH
0
0.157
0.489
0.944
0
0.152
0.476 ± 0.009
0.978 ± 0,034
0
0.173
0.465
0.844
DNS are in the direction of the theory, but need to go
to higher moments to be certain
More detailed comparisons in
Schumacher, KRS and Yakhot, New J. Phys. 9, 89 (2007)
Reynolds number barrier?
Improvements are happening with respect to
• Size of transistors (now ~20nm)
• Speed of communication (now ~10-4 c)
• Density of information
• Watts/CPU, etc, etc
• Petascale → Exascale → Zettascale
• Algorithms are improving and PK may own one
zettacale machine for full-time simulations for about
10 years ((instead of for a few months as now), but
Limit of computability: Rl = 10,000
(only new paradigms (e.g., biomolecular transistors,
quantum computing, etc) can push past this barrier
Rl = 10 - 100 has come easy
100-1000 has come with some
difficulty
1000 – 10,000 will come with
extreme difficulty
As said yesterday:
Steve Orszag, a pioneer in DNS
turbulence simulations saw the
turbulence problem mostly as one of
computability.
With some luck, we will “soon” know
everything worth knowing about box
turbulence within a decade, and
declare the problem as solved.
Unfortunately, it will have to wait for
some more time.