Uriel Frisch - Department of Physicsrahul/ictsweb/uriel-precision... · 2011-12-28 · Uriel Frisch...

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IS DIRECT NUMERICAL SIMULATION OF TURBULENCE ENTERING INTO THE HIGH-PRECISION ERA? Uriel Frisch Observatoire de la Côte d’Azur, Nice Spectral methods: history and accuracy Making use of high-precision data: van der Hoeven asymptotic extrapolation A case study where higher precision beats higher resolution Sunday, December 11, 2011

Transcript of Uriel Frisch - Department of Physicsrahul/ictsweb/uriel-precision... · 2011-12-28 · Uriel Frisch...

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The scientific community lost a deeply original individual with the passing of Steven AlanOrszag on May 1, 2011. He had a profound influence in fluid mechanics where he tackledchallenging problems in turbulent flow using mathematical and computational methods.

Steve was born in New York in 1943 and at 19 obtained his B.S. in Mathematics fromMIT. He spent the academic year 1962-63 as a Henry Fellow at St John’s in CambridgeUniversity where he did Part III of the Mathematical Tripos (now a Masters degree program).Having obtained a strong mathematical foundation for the study of fluid flows he wentto Princeton as a National Science Foundation graduate fellow where in three years hecompleted his PhD in the Department of Astrophysical Sciences, a summer of which hespent as a fellow in the Geophysical Fluid Dynamics Program in Woods Hole. Already asa graduate student his work was characterized by a combined approach of mathematicalfoundation and computational exposition to capture the reality of complex flow phenomena.Indeed, his mentors, Martin Kruskal, Lyman Spitzer and Bengt Stromgren, provided a blendof theory and observation into which Steve’s numerical predictions ideally dovetailed. Hisearliest paper on the atmospheres of neutron stars used state of the art computing but theevident intransigence of astrophysical flows quickly led him to examine the underlying fluidmechanisms governed by the Navier-Stokes equations. His thesis, “Theory of Turbulence”,led to a series of papers with Kruskal and Robert Kraichnan over the next several yearswhile a member of the Institute for Advanced Study and as a visitor at the National Centerfor Atmospheric Research.

His growing understanding of both the mathematical foundation and broad relevance ofturbulent flow provided the rostrum for his research at MIT where he was an applied math-ematics professor from 1967-84 and a Sloan Foundation Fellow. There he and Carl Benderconceived of lecture courses out of which their widely used text “Advanced Mathematical

May 26, 2011 1 Physics Today (with the editors)

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IS DIRECT NUMERICAL SIMULATION OF TURBULENCE ENTERING INTO THE HIGH-PRECISION ERA?

Uriel FrischObservatoire de la Côte d’Azur, Nice

••

Spectral methods: history and accuracy

Making use of high-precision data: van der Hoeven asymptotic extrapolation

A case study where higher precision beats higher resolution

Sunday, December 11, 2011

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470 J . YON N E U M A N N

settled directly by computation. Thus, an extensive exploration of the stability

properties and the first important i~nstable modes (if any) of the Couette flow and

the Poise~~ille flow should be undertaken. There is reason to believe that these

proble~ns would actually not constitute a major task for really modern high-speed

n ~ a c h i nes. (b) The development and stability of the statistical energy-frequency-spectra

which appear i 11 the theories of Kolmogoroff-Onsager-Weizlcker (intermediate

frequency range. cf. Chapter VII) and of Heisenberg (high frequency range, cf.

Chapter IX) should be evaluated numerically. In contrast to (a) above, this is

likely to be a major problem for any machine now in sight. The main reason for

this i s that the problem will have to be solved in 3 dimensiqns. (Regarding the

inadequacy of t he 2-dimensional approach. cf. 3.3 and 2.4.) Under these conditions,

even a linear resolution of3 : 8 would require following 83 = 512 spatial points

and a linear resolution of 1 : 20 would already call for 203 = 8,000 points. The

machines which one can hope to use in the immediate future will hardly be able

to exceed the first limit and none that is in sight for several years to come is likely

to get 111uch beyond the second one. There are ways to mitigate t6e effects of this

discouragingly low resolution, but it is clear that the procedure will be a difficult

one and a great deal of analytical ingenuity will be needed to make such compu-

tations fruitful. ?

I t has been occasionally suggested that in such a computational model turbulence

will develop because of the perturbations that are introduced by the "arithmetical

noise" of the computation itself, that is. by the round-off errors of the calculation.

The author is inclined to believe that while this source of "noise" is unquestionably

present. its effects are likely to be different. These effects are of great importance

and will have to be studied. They do, however, introduce perturbations in the high

frequencies (indeed, in the highest ones that are available, those defined by the

computational resolution itself), and there can be little.doubt that turbulence, as

we know it, is excited by perturbations introduced in the low frequency range.

(Cf. 6.1 and 8.2.) (c) The models of J. M. Burgers, referred to in latter part of 5.2, generate

phenomena which are in many ways similar to turbulence and may actually be a

manifestation of the same underlying principle. If this is correct, then Burgers'

models are of the greatest importance from the computational point of view, too.

Indeed, they are 1-dimensional, and therefore the main. limitation of the program

of (b) above disappears in their case. The work in the sense of (b) should, therefore,

certainly be extended to Burgers' I-dimensional models.

There may be other mathematical tricks, besides those developed by Burgers,

to render I - or 2-dimensional flows turbulent. Thus, suitable vorticity increasing

influences might provide the turbulence catalyzing factor which appears to be

missing in ordinary 2-dimensional flow (cf. 2.4).

1 1.4. A detailed formulation of the mathematical problems which should in the

author's opinion be attacked in the sense of this program would conflict with the

character and limitations of this report. Such formulations, as well as estimates of

the computational requirements that they entail, will be given on another occasion.

RECENT THEORIES OF TURBULENCET

By J. VON NEUMANN

Introduction

DURING August, 1949, the author visited several European centers of research in hydrodynamics and attended a conference on "Problems of Motion of Gaseous Masses of Cosmical ~imensions," (organized by International Union of Theoretical and Applied Mechanics and International Union of Astronomy), held from August 16th to 19th, 1949, in Paris. (P.a.)$ These trips were undertaken in connection with Contract N7-onr-388, and, more specifically, with the work performed under that contract in various fields of hydrodynamics and of the application of high-speed computational methods to that subject and related ones. An important aspect of this program embraces various forms of the theory of turbulence.

This report summarizes the author's present views on this subject. It is the resultant of several factors: Evaluation of the recent literature of certain phases of the subject of turbulence, discussions that took place during the visits referred

'

to, and some discussions that took place in this country. Among the latter the author is particularly indebted to a series of conferences on turbulence given by S. Chandrasekhar in Princeton in the spring of 1949 and several personal dis- cussions with E. Teller and with W. Elsasser in the fall of 1949.

I t has been attempted in the presentation that follows, to work out a reasonably integrated whole. The distribution of emphases is, nevertheless, subjective and the survey of the literature does not aim at completeness. In addition, the discussion represents a merger of all the influences enumerated above, so that the separation of the various factors would only be possible by going back to the sources.

The emphasis in the discussion that follows is mainly on strictly hydrodynamical turbulence. The Paris conference was equally devoted to questions of magneto-

? Editor's note: This report to the Office of Naval Research, written by von Neumann in 1949, was not considered by the author as suitable for publication for reasons which will be given below. However, it is considered by a number of workers in the field to be one of the most illuminating discussions of turbulence extant. It is therefore being printed in the collected works.

In letters transmitting copies of this report von Neumann has stated: "It was not written for publication, and it therefore contains a number of inadequacies which

I would certainly correct before I let it circulate widely or go into print. The main corrections are: "1. The references to, and the assessments of, the literature require a very careful going over. "2. If I wrote the paper now, I would discuss the latest work of Burgers (Proceedings of'rlre

Royal Dlrtclt Academy of Sciences, Vol. 53 (1950) No. 2, page 122, No. 3, page 247, No. 4, page 393) much more fully. It seems to me that i t is the most promising development along Burgers' line, that has occurred to date, and I think in particular, that the first one of the above references has considerable potentialities.

"3. The remarks at the end of the report regarding magneto-hydrodynamic turbulence should be revised and corrected, in view of the recent work of Batchelor and Chandrasekhar."

A.H.T. $ Throughout the text, references such as F., or C.a., or A. I ., or K.b.2., are to the Bibliography.

John von Neumann’s 1949 “secret paper”

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Steve Orszag 1969: Let us go (pseudo) spectral

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Forerunner, co-developer and many extremists...

J . Fluid Mech. (1983), vol. 130, pp. 411-452

I’rintrd in &eat Britain 41 1

Small-scale structure of the Taylor-Green vortex

By M A R C E. BRACHETt, D A N I E L I. MEIRON, STEVEN A. ORSZAG,

Massachusetts Institute of Technology, Cambridge, MA 02139

B. G. NICKEL, University of Guelph, Guelph, Ontario

R U D O L F H. MORF R.C.A. Laboratories. Zurich, Switzerland

AND U R I E L FRISCH CNRS, Observatoire de Nice, 06-Nice, France

(Received 5 February 1982 and in revised form 14 ,June 1982)

The dynamics of both the inviscid and viscous Taylor-Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier-Stokes equations (with up to 2563 modes) and by power-series analysis in time.

The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance cf(ct) of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that b(t) decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place a t later times, possibly leading to a real singularity (6 = 0) a t a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag Rr. Frisch (1980) analysis from order t4* to order P o . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t,,, at which the maximum energy dissipation is achieved. Early-time, high-R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t,,, the small scales of the flow are nearly isotropic provided that R 1000. Various features characteristic of fully developed turbulence are observed near t,,, when R = 3000 and R, = 110:

(i) a k-n inertial range in the energy spectrum is obtained with n z 1.G2.2 (in contrast with a much steeper spectrum a t earlier times) ;

t Present address: CNRS, Observatoire de Nice, WNice, France.

14 F L M 130

Energy dissipation rate and energy spectrum in high resolution directnumerical simulations of turbulence in a periodic box

Yukio Kaneda and Takashi IshiharaDepartment of Computational Science and Engineering, Graduate School of Engineering,Nagoya University, Nagoya 464-8603, Japan

Mitsuo YokokawaGrid Technology Research Center, National Institute of Advanced Industrial Science and Technology,Umezono 1-1-1, Tsukuba 305-8568, Japan

Ken’ichi Itakura and Atsuya UnoEarth Simulator Center, Japan Marine Science and Technology Center, Kanazawa-ku,Yokohama 236-0001, Japan

!Received 14 November 2002; accepted 2 December 2002; published 8 January 2003"

High-resolution direct numerical simulations !DNSs" of incompressible homogeneous turbulence ina periodic box with up to 40963 grid points were performed on the Earth Simulator computing

system. DNS databases, including the present results, suggest that the normalized mean energy

dissipation rate per unit mass tends to a constant, independent of the fluid kinematic viscosity # as#!0. The DNS results also suggest that the energy spectrum in the inertial subrange almost follows

the Kolmogorov k!5/3 scaling law, where k is the wavenumber, but the exponent is steeper than

!5/3 by about 0.1. © 2003 American Institute of Physics. $DOI: 10.1063/1.1539855%

Direct numerical simulation !DNS" of turbulent flowsprovides us with detailed turbulence data that are free from

experimental ambiguities such as the effects of using Tay-

lor’s hypothesis, one dimensional surrogates, etc.1 However,

the degrees of freedom or resolution necessary for DNS in-

creases rapidly with the Reynolds number. The maximum

resolution is obviously limited by the available computing

memory and speed. To date, DNSs of incompressible turbu-

lence using spectral methods have been limited to a maxi-

mum of 10243 grid points.

The recently developed Earth Simulator !ES", with apeak performance and main memory of 40 TFlops and 10

TBytes, respectively, provides a new opportunity in this re-

spect. We recently performed a series of DNSs of incom-

pressible turbulence that obeys the Navier–Stokes equations

using an alias-free spectral method and up to 40963 grid

points on the ES. This Letter reports some of the results with

a special emphasis on the mean energy dissipation rate &'(per unit mass and the energy spectrum E(k).

The numerical method used for the DNS is similar to

that reported in our previous studies.2,3 In particular, the total

energy E was maintained at an almost time-independent con-

stant ("0.5) by introducing negative viscosity in the wave-number range k#2.5. The minimum wavenumber of the

DNS was 1. A preliminary report of the DNS with an em-

phasis on the parallel computing aspect was presented in

Ref. 4.

We performed two series of runs, Series 1 and 2, in

which the maximum wavenumber kmax and the kinematic

viscosity # were chosen so that kmax)*1 and 2, respectively,where ) is the Kolmogorov length scale defined by )"(#3/&'()1/4. In each series, we first executed a low resolu-

tion preliminary run (2563) using an appropriate random ini-

tial flow field. The preliminary run was continued until the

simulated field reached a statistically quasistationary state,

judged by monitoring several one-point statistics. We then

reduced the kinematic viscosity # and started a new DNS

with twice the resolution in each direction, using the final

state of the lower resolution DNS for the initial conditions.

We repeated this process until the resolution reached 40963

or 20483. The conditions for each run are listed in Table I.

We used double precision arithmetic for all of the runs, ex-

cept Run 4096-1 in which we used single precision arith-

metic for the time integration and double precision arith-

metic for the convolution sums when evaluating the

nonlinear terms in wavevector space. Our preliminary test at

a resolution of 10243 suggested that the lower arithmetic

precision has no significant influence on the energy spec-

trum. Figure 1 shows the Taylor-scale Reynolds number R+

versus time t .

DNS data may be used to answer some fundamental

questions in turbulence research. Among these is a question

about the normalized mean energy dissipation rate D

,&'(L/u!3: ‘‘Does D tend to a constant independent of the

kinematic viscosity # in the limits as # tends to zero?’’ Here,u! is the characteristic velocity of the energy-containing ed-dies given by 3u!2"2E and L is the integral length scale

defined by

L"-

2u!2 ! k!1E!k "dk .

The independence of D from # for large Reynolds numberflows is a basic premise in the phenomenology of turbulence;

its significance has been emphasized in the literature, as

noted by Sreenivasan.5 He examined the DNS data available

PHYSICS OF FLUIDS VOLUME 15, NUMBER 2 FEBRUARY 2003

L211070-6631/2003/15(2)/21/4/$20.00 © 2003 American Institute of Physics

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http://www.oca.eu/etc7/bray-phd1966.pdf

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Spectral methods can be exponentially accurate

RECENT THEORIES OF TURBULENCE 461

Dimensionally: k-' vk -- cm2 sec- ', hence i(k")dkW - cm2 sec- ' and I(k") - cm3 sec- '. Now F(kN)dk" - energylunit mass - cm2 s e f 2 , hence F(k") .-

cm3 ~ e c - ~ , and k" - cm-l. Hence there exists only one combination F(k3, k"

which agrees dimensionally with I&") : ~(k")%k" -%. Therefore, necessarily :

where C, is a dimensionless absolute constant, presumably of moderate size.

From this

The expression for the energy flow (per unit mass) in question is, therefore,

This expression is not proposed by Heisenberg as a rigorously correct one, and

even a certain amount of qualitative criticism against it seems justified. It is never-

theless a very interesting first attempt, and its exploitation by Heisenberg is ,

instructive.

Various experimental and quasi-theoretical considerations indicate that the

constant C, lies between 1 and 0.5. (H.a.2, p. 640, top, p. 657, equation (100)

el seq.)

9.2. If the T~ expression of 9.1 is accepted, then the condition of a stationary

(turbulent) state is that T~ be constant (in k ) and equal to W. This condition of

stationarity is compatible with certain frequency intervals being entirely "un-

excited", but if' we exclude this, as being implausible in a structure of power laws,

then the familiar k-% law obtains for k g k, (cf. 7.2) and a k-' law for k % k,.

The rigorous solution was determined by Chandrasekhar, it is

with

(C.a.) Note, that these k,*, A* differ only slightly from the k,, A of the simpler,

qualitative theory, as obtained at the end of 7.2.

9.3. Regarding the k-' law, this may be said :

(a) A k-' law of energy implies, that the velocity u possesses 1st and 2nd,

probably no 3rd, and certainly no 4th and higher derivatives. That this should

indeed be so, is quite unlikely (as pointed out by Heisenberg himself): Burgers'

work, as well as any intuitive appraisal of the situation, makes it amply plausible,

that the flow in the dimensions g~~~ should be absolutely smooth, i.e. essentially

analytical-hence F(k) should decrease exponentially. (B.c.2, p. 32.) Indeed, it

What von Neumann tell us in his 1949 paper (p. 461)

Quickly it became clear that spectral methods can be much more accurate than finite-differences. As pointed out in the book by Gottlieb and Orszag: the way accuracy increaseswith resolution depends solely on the smoothness of the solution to the equations (because the latter controls the fall-off of the Fourier coefficients at high wavenumbers).

Many hydrodynamical problems with periodic boundary conditions have proven or conjectured analytic solutions

For such “analytic” problems, truncation errors decrease exponentially with themaximum wavenumber as , where is the distance to the real space domain of the nearest complex-space singularity

exp(!!Kmax) !

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Precision needed for testing theoretical ideasImplementing asymptotic interpolation for the Burgers equationwith single-mode initial condition

• Inviscid Burgers equation

!tu + u (!xu) = 0, u(0, x) = u0(x) = !12

sinx

• Fourier–Lagrangian representation (Fournier–Frisch)

u(t, x) =!

k=±1,±2,...

eikxuk(t), uk(t) = ! 12i"kt

" 2!

0e!ik(a+tu0(a))da

• Explicit solution uk(t) = 1iktJk(kt/2) in terms of the Bessel function Jk of order k

gives via asymptotique (Debye) expansion

uk(t) " 1it

1#2"

$1! (t/2)2

k!32 e!k(arccosh 2

t!#

1!(t/2)2)

%

&1 +"!

n=1

#n( 1#1!(t/2)2

)

kn

'

( ,

where #n are known polynomials (see e.g. Abramowitz–Stegun)

A good candidate for testing asymptotic interpolation. Fourier coefficients calculatedwith high precision (80 digits) at t = 1 and |k| $ 1000.

5

Let us start with an academic example•

• Given a large but finite number of such Fourier coe!cients, determined

from a numerical simulation, how can one reconstruct such an expansion?

More generally, given a function f(n) known numerically for n = 1, 2, . . . , Nhow can one find its asymptotic expansion? Precision is crucial!

Sunday, December 11, 2011

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Journal of Symbolic Computation 44 (2009) 1000–1016

Contents lists available at ScienceDirect

Journal of Symbolic Computation

journal homepage: www.elsevier.com/locate/jsc

On asymptotic extrapolation!

Joris van der Hoeven1

CNRS, Département de Mathématiques, Bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France

a r t i c l e i n f o

Article history:Received 7 July 2008Accepted 5 January 2009Available online 31 January 2009

Keywords:ExtrapolationAsymptotic expansionAlgorithmGuessing

a b s t r a c t

Consider a power series f !R[[z]], which is obtained by a precisemathematical construction. For instance, f might be the solutionto some differential or functional initial value problem or thediagonal of the solution to a partial differential equation. In caseswhen no suitable method is available beforehand for determiningthe asymptotics of the coefficients fn, but when many suchcoefficients can be computed with high accuracy, it would beuseful if a plausible asymptotic expansion for fn could be guessedautomatically.

In this paper, we will present a general scheme for the designof such ‘‘asymptotic extrapolation algorithms’’. Roughly speaking,using discrete differentiation and techniques from automaticasymptotics,we strip off the termsof the asymptotic expansion oneby one. The knowledge of more terms of the asymptotic expansionwill then allow us to approximate the coefficients in the expansionwith high accuracy.

© 2009 Published by Elsevier Ltd

1. Introduction

Consider an infinite sequence f0, f1, . . . of real numbers. If f0, f1, . . . are the coefficients of a formalpower series f ! R[[z]], then it is well-known (Pólya, 1937;Wilf, 2004; Flajolet and Sedgewick, 1996)that a lot of information about the behaviour of f near its dominant singularity can be obtained fromthe asymptotic behaviour of the sequence f0, f1, . . .. However, if f is the solution to some complicatedequation, then it can be hard to compute the asymptotic behaviour using formal methods. On theother hand, the coefficients f0, f1, . . . of such a solution f can often be computed numerically up to ahigh order (van der Hoeven, 2002). This raises the question of how to guess the asymptotic behaviourof f0, f1, . . ., on the basis of this numerical evidence.

! The paper was originally written using GNU TEXMACS (see www.texmacs.org). Unfortunately, Elsevier insists on the use ofLATEX with its own style files. Any insufficiencies in the typesetting quality should therefore be imputed to Elsevier.

E-mail addresses: [email protected], [email protected]: http://www.math.u-psud.fr/"vdhoeven.

1 Tel.: +33 1 69 15 31 71; fax: +33 1 69 15 15 60.

0747-7171/$ – see front matter© 2009 Published by Elsevier Ltddoi:10.1016/j.jsc.2009.01.001

Asymptotic extrapolation

••••

Suppose we have determined numerically with high precision the values of a function for integers n up to some high value N.Goal: to obtain as many terms as possible in the asymptotic expansion of for high n.

Trying to fit the function by a guessed leading-order asymptotic form with a few free parameters gives a poor accuracy for these parameters.

Lacking such information, the asymptotic extrapolation method achieves the goal by applying to the data a sequence of suitably chosen transformations that strip off successively the leading and subleading terms.

With a priori information about the structure of the expansion, an expression containing leading and subleading terms (with free parameters) can give a much better fit.

At certain stages in the sequence of transformations, the processed data may allow simple extrapolation.

By undoing the transformations, starting from the highest extrapolation stage, one obtains the asymptotic expansion of the original data, up to some order which depends on the precision of the data and on the value of N.

The process can be continued as long as the transformed data is free from conspicuous rounding noise.

Gn

Gn

••

Sunday, December 11, 2011

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The machinery of asymptotic extrapolationJoris van der Hoeven’s asymptotic interpolationJ. van der Hoeven, Algorithms for asymptotic interpolation, submitted to J. Symbolic Computation, 2006

• Interpolate the sequence Gn in the “most asymptotic” region n = L, ..., N

• Transformations:

I Inverse: Gn !" 1Gn

R Ratio: Gn !" GnGn!1

SR Second ratio: Gn !" GnGn!2G2

n!1

D Difference: Gn !" Gn !Gn!1

• Going down (assuming Gn > 0):

– Test 1: if Gn < 1 apply I

– Test 2: Does Gn grow faster than n5/2?# Yes: if the growth is exponential apply SR, otherwise R# No: apply D

– Continue untill obtaining data which are easy to interpolate and clean enough

• Go back by inverting transformations I, R, SR and D

4

Sunday, December 11, 2011

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1.00004

1.00008

1.00012

1.00016

200 400 600 800 1000

Gn(1)

0

1e-06

2e-06

3e-06

4e-06

200 400 600 800 1000

Gn(2)

0

1e+08

2e+08

3e+08

4e+08

200 400 600 800 1000

Gn(3)

0

2e+05

4e+05

6e+05

8e+05

1e+06

200 400 600 800 1000

Gn(4)

500

1000

1500

2000

200 400 600 800 1000

Gn(5)

1e-10

1e-08

1e-06

1e-04

200 400 600 800 1000

Gn(6)

- 2

Ck!!e!"k SR!" 1 +!

k2

!D!" 2!

k3

I!" k3

2!D!" 3k2

2!D!" 3k

!D!" 3

!

6

Testing asymptotic interpolation on Burgers(W. Pauls and U. Frisch, 2007 J. Stat. Phys. 127, 1095–1119)

Burgers equation. u(x, 0) = ! 12 sinx. Fourier coe!cients at t = 1 calculated

with 80-digit precision up to K = 1000. Reconstruction up to 6th subleading term

Sunday, December 11, 2011

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Results: leading order and six subleading terms

• The interpolation procedure can be continued to improve accuracy on !, " and todetermine more subdominant terms

• Transformations applied

SR, -D, I, D, D, D, D, I, D, D, D, D, D

• Comparing numerical and theoretical results

! " C

6 stages 1.49999999993 0.4509324931404 0.4286913791

13 stages 1.49999999999999995 0.450932493140378061868 0.4286913790524959

Theor. value 3/2 0.450932493140378061861 0.42869137905249585643

#1 #2 #3

6 stages !0.17641252 0.17295 !0.401

13 stages !0.17641258225238 0.172968106990 !0.406446182

Theor. value !0.176412582252385 0.1729681069958 !0.4064461802

#4 #5 #6

13 stages 1.384160933 !6.192505762 34.5269751

Theor. value 1.3841609326 !6.1925057618568063655 34.526975286449930956

7

with 80-digit precision seven more stages of transformation can be carried out to improve

acuracy on ! and " and determine more subleading terms

Sunday, December 11, 2011

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High precision important for understanding theory

44 W. Pauls et al. / Physica D 219 (2006) 40–59

in terms of which (15) reads

[!"#] + v · !# + # = 0, (21)

with the boundary conditions (for y1, y2 " #$)

v %!

#1, #12

", # % 1

2ey1 # 2e2y2 . (22)

For other initial conditions, only the boundary condition (22)must be modified. The " -derivative term has been put withinsquare brackets since we are only interested in the steady-statesolution. Note that the pseudo-hydrodynamic formulation inthe y-plane is that of a quasi-two-dimensional flow in a 3Dcontainer with bottom friction producing a Rayleigh drag. Inthis formulation " " +$ as we approach the initial instant.An alternative interpretation is to define " as ln t , to avoidreversing the course of time, and then to change the signs ofv and of # and replace the Rayleigh drag by an instability.

In the pseudo-hydrodynamic formulation it is now obviousthat the problem is invariant under an arbitrary translationh = (h1, h2) in y-space. By (14), such a translation amountsto a factor ek·h on the Fourier coefficients F(k). It follows,as noted in MBF, that the set of initial conditions !0(x) =eh1 cos x1 +e2h2 cos 2x2 is equivalent to the SOC as long as h iswithin the analyticity domain. Similarly, a translation in k-spacewith integer components (n1, n2) is equivalent to multiplyingF($+i y1, i y2) by the exponential factor en1 y1+n2 y2 in y-space.The exponential being an entire function, this changes neitherthe positions nor the nature of the singularities at finite distance.

3. Numerical investigation of scaling laws in Fourier space

We shall show in this section that the solution of theEuler equation in the short-time asymptotic regime defined inthe previous section has remarkably clean scaling propertiesin Fourier space. By this we mean that the wavenumberdependence of the Fourier coefficients is represented as adecreasing exponential multiplied by an algebraic prefactorwhose exponent can be measured very accurately. Such afunctional form is not surprising. In fact the exponential is thesignature of the location of a singularity while the prefactorencodes the nature of the singularity. For one-dimensionalanalytical functions with isolated singularities in the complexspace this is well known: a singularity at z% of the form (z#z%)

&

has a signature in the modulus of the Fourier transorm athigh wavenumbers k of the form C |k|#&#1e#'|k|, where ' isthe distance of z% to the real axis (see, e.g., Ref. [24]). Suchasymptotic results have been extended in the 1990’s to theFourier transforms of periodic analytical functions of severalcomplex variables when the wavevector k tends to infinity witha fixed rational slope tan ( = k2/k1 = p/q , where p and q arerelative prime integers [25–27].

When the Fourier coefficients are obtained numerically,there is a maximum wavenumber kmax. Unless it is taken verylarge, there will be very few points on the line of slope p/qas soon as q is not a very small integer. But a large value ofkmax entails extremely small Fourier coefficients because of theexponential decrease with the wavenumber. Thus, as stressed in

Fig. 1. Fourier coefficients of the stream function F along two lines of differentslopes as a function of k & |k| in lin–log coordinates.

Fig. 2. Same as in Fig. 1 after division by exp(#'k) (compensated Fouriercoefficients) in log–log coordinates. Most of the points are in the asymptoticpower-law regime, at least visually.

MBF, very high precision may be needed to avoid swampingby the rounding errors. Truncation errors are not an issue in theshort-time asymptotic regime since the Fourier coefficients canbe calculated from (12) with arbitrary accuracy.

The data obtained for the SOC initial condition in MBFhad wavenumbers k & |k| up to 1000 or 2000, dependingon the direction and were calculated with 35-digit accuracy.10

Most of the results presented here are based on the 35-digitcalculation. Additional calculations are also presented here withvarious initial conditions, with up to 100-digit precision andwavenumbers which can reach 4000 in particular directions. Wenote that the MPFUN90 package for high-precision calculationused in MBF, here and in Ref. [19] makes use of fast Fouriertransform techniques. Thus the CPU time per multiplication, asa function of the number of digits N , is proportional to N log N[28].

We now show that it is quite easy in principle to observescaling by analyzing the behavior of the Fourier coefficients indirections of rational slope. Figs. 1 and 2 give two examples ofthe analysis of Fourier coefficients along straight lines through

10 In MBF it was stated that, when using only double-precision (15-digit)accuracy, unacceptably large errors are obtained beyond wavenumber 800.Actually, as pointed out by Zimmermann (private communication), the double-precision calculation can be modified in such a way that, up to wavenumber1000, the relative error on Fourier modes does not exceed 10#5.

•••

Run 2048-1. Figure 4 shows !(k) at various times in Run2048-1. The range over which !(k) is nearly constant isquite wide; it is wider than the flat range of the correspond-

ing compensated-energy-spectrum "see Fig. 5#. The station-arity is also much better than that of lower resolution DNSs

"figures omitted#, and !(k)/$%& is close to 1. In the study ofthe universal features of small-scale statistics of turbulence,

if there are any, it is desirable to simulate or realize an iner-

tial subrange exhibiting "i#–"iii# rather than "i#– "iii#. Thepresent results suggest that a resolution at the level of Run

2048-1 is required for such a simulation. Such DNSs are

expected to provide valuable data for the study of turbulence,

and in particular for improving our understanding of possible

universality characteristics in the inertial subrange.

These considerations motivate us to revisit another

simple but fundamental question of turbulence: ‘‘Does the

energy spectrum E(k) in the inertial subrange follow Kol-

mogorov’s k!5/3 power law at large Reynolds numbers?’’

Figure 5 shows the compensated energy spectrum for the

present DNSs "the data were plotted in a slightly differentmanner in our preliminary report4#. From the simulations

with up to N"1024, one might think that the spectrum in therange given by

E"k #"K0%2/3k!5/3 "1#

with the Kolmogorov constant K0"1.6–1.7 is in good

agreement with experiments and numerical simulations "see,for example, Refs. 1, 3, 9, and 10#. However, Fig. 5 alsoshows that the flat region, i.e., the spectrum as described by

"1#, of the runs with N"2048 and 4096 is not much widerthan that of the lower resolution simulations. The higher

resolution spectra suggest that the compensated spectrum is

not flat, but rather tilted slightly, so that it is described by

E"k #'%2/3k!5/3!(k, "2#

with (k)0.The detection of such a correction to the Kolmogorov

scaling, if it in fact exists, is difficult from low-resolution

DNS databases. The least square fitting of the data of the

40963 resolution simulation for (d/d log k)logE(k) to

(!5/3!(k)log k#b (b is a constant# in the range 0.008$k*$0.03 gives (k"0.10. The slope with (k"0.10 isshown in Fig. 5.

It may be of interest to observe the scaling of the second

order moment of velocity, both in wavenumber and physical

space. For this purpose, let us consider the structure function

S2"r#"$!v"x#r,t #!v"x,t #!2&,

where S2 may, in general, be expanded in terms of the

spherical harmonics as

S2"r#" +n"0

,

+m"!n

n

f nm"r #Pnm"cos -#eim..

Here, r"!r! and -,. are the angular variables of r in spheri-cal polar coordinates, Pn

m is the associated Legendre polyno-

mial of order n ,m , and f nm(" f n ,!m* ) is a function of only r ,

where the asterisk denotes the complex conjugate. The time

argument is omitted. For S2 satisfying the symmetry S2(r)

"S2(!r), we have f km"0 for any odd integer k . In strictlyisotropic turbulence, f nm must be zero not only for odd n ,

but also for any n and m except n"m"0. However, ourpreliminary analysis of the DNS data suggests that the an-

isotropy is small but nonzero. In such cases, f nm is also small

but nonzero, and S2 itself may not be a good approximation

for f 0" f 00 . To improve the approximation for f 0 , one

might, for example, take the average of S2 over r/r

FIG. 3. Normalized energy dissipation rate D versus R/ from Ref. 5 "dataup to R/"250), Ref. 3 "!,"#, and the present DNS databases "#,$#.

FIG. 4. !(k)/$%& obtained from Run 2048-1.

FIG. 5. Compensated energy spectra from DNSs with "A# 5123, 10243, and"B# 20483, 40963 grid points. Scales on the right and left are for "A# and "B#,respectively.

L23Phys. Fluids, Vol. 15, No. 2, February 2003 Energy dissipation rate and energy spectrum

Downloaded 24 May 2004 to 192.54.174.9. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

There was a consensus that double precision is enough for turbulence simulation and thatincreased computer power is best used to increase resolution/Reynolds numbers

With this strategy, it may be difficult to falsify our ingrained theoretical ideas

Consider for example the Earth Simulator data for compensated energy spectra

An example of successful falsification through high-precision simulation

The result can be viewed as a validation of our ideas on intermittency corrections and bottlenecks

•But they could suffer from numerical artefacts or from the effect of subleading terms messing up the scaling (cf. Aurell et al. JFM 1992)

Higher precision (in the future) would allow us to apply suitable data processing techniques (e.g. asymptotic extrapolation) andto obtain a validation or a falsification of current ideas

••

(Precision: 35 to 100 digits)

Type of complex singularities determined via the high-k behaviour of the Fourier transform of the stream function.

Is this a law? No! The exponent differs very slightly from -8/3 and actually changes (slightly) when varying the initial conditon, thereby falsifying the idea of universality.

k!8/3

Nature of complex singularities for the 2D Euler equation.Pauls et al. Physica D 2006

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Nelkin scaling: increased precision beats increased resolution(Chakraborty-Frisch-Pauls-Ray arXiv:1111.2999 [nlin.CD] and the presentation by Sagar Chakraborty)

(

Scaling properties of fully devloped turbulence can be analysed through structure functions••

••

or by looking at the Nelkin (1990) Reynolds-number scaling of moments of velocity gradients

!("u)p# $ R!p , R % &Schumacher-Sreenivasan-Yakhot (2007): such scaling is already seen at Reynolds number ~200.

For the Burgers case the exponents are known ( ). How well can they be retrieved with asymptotic extrapolation from spectral simulation data with various precisions/resolutions? asymptotic extrapolation?

4

102

103

10!5

10!4

10!3

10!2

10!1

Rmax

Relative E

rror

FIG. 2: (Color online) Relative error of Nelkin exponents !4

and !6 obtained by asymptotic extrapolation from pseudo-spectral calculations up to a maximum Reynolds numberRmax. Upper set of curves: double precision calculations (!4:red filled circles, !6: blue filled triangles); lower set of curves:quadruple precision (!4: red inverted triangles, !6: blue filledsquares).

order to truly give accurate scaling exponents. Of course,the higher the Reynolds number, the lower the relativesubdominant corrections will be. But, without enoughprecision, the simultaneous determination of dominantterms and subdominant corrections, say by asymptoticextrapolation, will be unable to handle more than veryfew such corrections and thus gives us substantial errorsin the final results. In order to be closer to more realis-tic models such as the multi-dimensional Navier–Stokesequations, in investigating the trade-o! between asymp-toticity and precision, we refrain from using the exactsolution of the Burgers equation and resort to time in-tegration by (pseudo-)spectral technique. We use doubleand quadruple precision, both combined with asymptoticextrapolation, so as to obtain the most accurate possi-ble parameters. We calculate the scaling exponents !4

and !6 of the fourth and the sixth gradmoments, whosetheoretical exact values are three and five, respectively.We determine how accurately we can predict these ex-ponents when applying asymptotic extrapolation (whichfor this purpose is substantially better than the aforemen-tioned ESS technique), using various maximum Reynoldsnumbers Rmax. In double precision we were able touse three stages and in quadruple precision eight stagesof the aforementioned transformations. The maximumwavenumber and the size of the time step are the sameas reported at the beginning of the paper. We checked, byfurther halving of spatial and temporal resolutions, that

they contribute negligible errors to the result. Figure 2shows the relative errors for the two types of precisionas function of Rmax. It is striking that, when doublingthe precision we can decrease the Reynolds number byabout a factor of eleven (from 1000 to 90) and still ob-tain a substantial decrease (by a factor of 3 to 10) inthe relative error. For accurate determination of scal-ing exponents, increasing the precision is here definitelymore e"cient than increasing the Reynolds number. Itremains to be seen if this result carries over to a muchbroader class of equations, including multi-dimensionalproblems displaying random behavior.

We are indebted to Joris van der Hooven, T. Mat-sumuto, D. Mitra, O. Podvigina, and V. Zheligovsky fora number of useful discussions. S.C. thanks academic andfinancial support rendered by NBIA (Copenhagen); andDanish Research Council for a FNU Grant No. 505100-50 - 30,168. The work was partially supported by ANR“OTARIE” BLAN07-2 183172. Some of the computa-tions used the Mesocentre de calcul of the Observatoirede la Cote d’Azur.

! Electronic address: [email protected]† Electronic address: [email protected]‡ Electronic address: [email protected]§ Electronic address: [email protected]

[1] M. Nelkin, Phys. Rev. A 42, 7226 (1990).[2] B. Mandelbrot, J. Fluid Mech. 62, 331 (1974).[3] G. Parisi and U. Frisch, in Turbulence and Predictabil-

ity in Geophysical Fluid Dynamics and Climate Dynam-ics, edited by M. Ghil, R. Benzi and G. Parisi (North-Holland, Amsterdam, 1985), p. 84.

[4] U. Frisch, Turbulence — The Legacy of A. N. Kol-mogorov, (Cambridge University Press, Cambridge,1995).

[5] J. Schumacher, K. R. Sreenivasan and V. Yakhot, NewJ. Phys. 9, 89 (2007).

[6] S. Chakraborty, U. Frisch and S. S. Ray, J. Fluid Mech.649, 275 (2010).

[7] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Mas-saioli and S. Succi, Phys. Rev. E 48, R29 (1993).

[8] J. van der Hoeven, J. Symb. Comput. 44, 1000 (2009).[9] W. Pauls, and U. Frisch, J. Stat. Phys. 127, 1095 (2007).

[10] S. M Cox and P. C. Matthews, J. Comp. Phys. 176, 430(2002).

[11] E. Hopf, Comm. Pure Appl. Math. 3, 201 (1950).[12] J. D. Cole, Quart. Appl. Math. 9, 225 (1951).[13] D. O. Crighton and J. F. Scott, Phil. Trans. Roy. Soc.

London 292, 101 (1979).[14] D. H. Bailey, “High-Precision Arithmetic in Scientific

Computation”, Computing in Science and Engineer-ing, May-June, 2005, pg. 54-61; LBNL-57487. See alsohttp://crd.lbl.gov/ !dhbailey/

[15] http://www.kurims.kyoto-u.ac.jp/ !ooura/!t.html.[16] This technique can be readily extended to three-

dimensional Navier–Stokes experiments and simulations.[17] R - 1 means applying R and then subtracting unity.

!p = p! 1

Relative errors for the Nelkin exponents (red) and (blue) from pseudo-spectral calculations up to a maximum Reynolds number . Upper curves: double precision. Lower curves: quadruple precision

!4 !6

Rmax

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Conclusion: trade-offs between resolution and precision

Computers become more powerful (Moore’s law...), but so do algorithms

Better not to put all one's eggs in one basket

Depending on the nature of the problem (especially for scaling laws), it may be more efficient to increase precision and not just to boost resolution and Reynolds numbers

Sunday, December 11, 2011