Whither turbulence computations?by

K.R. SreenivasanNew York University

A commentary on the work of

Victor YakhotBoston University

Jörg SchumacherUniversity of Ilmenau

P.K. Yeung Georgia Institute of Technology

Diego DonzisTexas A&M

perhaps others

Indian Institute of Science

Tuesday

December 13, 2011

From Herman Winick, SLAC

R

New paradigms, new architecture, etc, yet…

Massive parallelism, with O(105) CPU cores; so doing simulations has become a big task in itself.

GMR

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

= Kolmogorov scale R= microscale Reynolds

number (√Re) 2L

x = 1

For “standard” conditions, 12 = 5, we have

R 0.5 W1/6, R 4.5 N2/3

R

If the Earth Simulator can compute

N = 4096, R 1200 (Re 105)

with L/ = O(1000)

Exaflop machines can handle:

N = 32,768

R 4,000 (Re 106)

L/ = 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!

surrogatedissipation

Local dissipation scale can be far smaller.

u = 1. u has large fluctuations.

Thus,can often be less than .

higher Re

xuηxuην

Distribution of length scales

log10 (

prob

abili

ty d

ensi

ty o

f /

<>

Schumacher, Yakhot

• From the distribution of length scales, we have <d>= <

2>/<>2/

• Eddy diffusive time/molecular diffusive time Re1/2/100;exceeds unity only for Re 104 ( mixing transition mentioned by Narasimha yesterday)

Scales smaller than (and B) clearly exist, so …

–How much of the data acquired from resolutions of the order 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?

The 1283 box represents “standard resolution”

From J. Schumacher

(in units of the mean)

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, Germany2 Visualization Laboratory, Central Institute for Applied Mathematics, Research Center Jülich, D-52425 Jülich, Germany3 International Centre for Theoretical Physics, I-34014, Trieste, Italy4 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

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)

BB

grid

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=2. 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.

BB

grid

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 gray-shaded cutting plane is shown.

0

Support by the Deutsche Forschungs-gemeinschaft (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

100

Another view of the same thing

Low scalar dissipation

1.5ηk Bmax 6ηk Bmax

no conspicuous difference

measurable differences

Theory for smallest

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

smallest/L = Re

(instead of Re, as for the standard Kolmogorov scale)

N = Re3

(instead of the standard Re9/4 relation)

A practical consequence R 0.5 W1/8, R 4.5 N1/2

(instead of R 0.5 W1/6, R 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 R= 103, Re 105, ratio = O(105)

A 40963 box can resolve all scales only up to R 300

(not 1200 as previously thought)A 32,7683 box can resolve all scales

only up to R 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/)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 power-law bounded on both sides with the roll-off rate of n, they show that

(L/)3 ~ Re6/(n+1),

giving Re9/4 for n = 5/3.

Phenomenological

(a) From the measurements of Meneveau & Sreenivasan (1988+): Re

(b) From Paladin & Vulpiani (1988): Re3 >0(c) From the She-Leveque model (1994): Re3.6

(d)

22 2

3

2

2

3

2

2

2

Re12

12v

vv

pp

x

r

k

px

pr

px

42

v6

v2

vv

rOrr

r

δxxxxxx

r

)(v

vv

12

)12(

v

vv

4

2

22

2

2

2

rOrppr

px

xxpx

px

p

rpx

22

22v

vvv

v

px

xxpx

xxx

)(12

)12(

v

vv

422

2

2

2

rOrppr

px

p

rpx

2

1

4

3

2 2Re

k

1 (if )3/22

Put so that

Use

Now, let r = x = chosen resolution. We then have

<n> Redn

theory DNS RSH

d1 0 0 0

d2 0.157 0.152 0.173

d3 0.489 0.476 ± 0.009 0.465

d4 0.944 0.978 ± 0,034 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: R= 10,000

(only new paradigms (e.g., biomolecular transistors, quantum computing, etc) can push past this barrier

R = 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.