Velocity intermittency in turbulence : how to objectively characterize it ?

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Velocity intermittency in turbulence : how to objectively characterize it ?

Antoine Naert, Laurent Puech, Benoît Chabaud, Joachim Peinke, Bernard Castaing, Bernard Hebral

To cite this version:

Antoine Naert, Laurent Puech, Benoît Chabaud, Joachim Peinke, Bernard Castaing, et al.. Veloc-

ity intermittency in turbulence : how to objectively characterize it ?. Journal de Physique II, EDP

Sciences, 1994, 4 (2), pp.215-224. �10.1051/jp2:1994124�. �jpa-00247956�

J. Phys II France 4 j1994) 215-224 _FEBRUARY 1994, PAGE 215

Classification Physics Abstracts

47.25

Velocity intermittency in turbulence

_:

how to objectively

characterize it ?

Antoine Naert, Laurent Puech, Benoit Chabaud, Joachim Peinke, Bemard

Castaing

and

Bernard Hebral

Centre de Recherches _sur (es Trds Basses Tempdratures (*), CNRS, B-P- 166, 3804? Grenoble Cedex 9, France

(Received 22 Marc-h J993, re>,ised 2J september J993, accepted 25 October J993j

R4sum4. L'dvolution d'un certain paramdtre A^~ ddfini dans l'article _a dtd utilisde pour tenter de trancher _entre diffdrents moddles de l'intermittence _en turbulence. Le but de cet article est de valider _une mdthode

e~pdrimentale

de ddterrnination de _ce paramdtre,

inddpendamment

^de tout

meddle, h partir ^des densitds de

probabilitd

de diffdrences de vitesse (pdo. Les conditions

expdrimentales

qui

perrnettraient

de pousser plus loin la caractdrisation de _ces pdf sent dgalement discutdes.

Abstract. The evolution of A_{~, a} parameter ^{whose definition is recalled in the} paper, has been used in order to test various interrnittency models in turbulence. This paper aims to validate _an experimental, ^model independent, method for

extracting

this parameter ^{from the}

velocity

difference

probability density

functions

(pdo.

We also discuss the charactenstics of _an experiment which could push ^{further the} analysis of these pdf.

1. Introduction.

Statistics of

velocity

difference between _two

points

^is the main tool for

characterizing developed

turbulent flows

[1, 21. Recently

^the

interejt

shifted from the moments of the

distribution of these

velocity

differences _to the

study

^{of the whole}

probability density

function

(pdf~ [3-51.

Models _were

proposed [4-91,

based _on various

physical grounds,

each _one

giving

a

good

agreement ^{with the observed}

pdfs.

Thus the

problem

is less in this agreement than in the

insight

the

corresponding analysis gives

into the

underlying physics.

Most of the

proposed

models _can be

presented

in _a unified way

connecting

the

pdf

at _a scale

r to the

large

scale _one

[10, 1].

If &u is the considered component ^{of the}

velocity

difference

&u

=

u,(x

+

r) u~(x)

(*) Laboratoire associd h l'Universitd Joseph Fourier.

and

P~

^~~ its

probability density

function

(m)

=

(&u~) ),

these models _assume that

"r "r

)Pr~~) "iGr~~)PL()j$ _(I)

r r r

where L is the

large integral

scale and G _a distribution which

depends

_on the model.

In this paper we show the interest of

using

a

log

normal ansatz for G~:

~

ln~ «/«~

G~ cc exp

~

(2)

"r 2 A

(in general

_«~, ^which

gives

the maximum of

G,

is different from

«~).

This interest does _not

only

lie in that this should be the

limiting

form for G~ at a small _r

lO].

The

major

interest is that the parameter ^{A~ obtained in}

fitting

the

experimental P~

^with

equations (1, 2),

_measures the

mean

squared

deviation

((&

In

«)~),

_even if G~

significantly

differs from _a

log

normal distribution. This is

important

for the

following

_reasons. The

existing

^theories on

intermittency

in

developed

turbulence

predict

both different

shapes

for G and different behaviours for

((&

^In

«)~)

_{ve;sus r,}

Physically (

(& ^In

«)~)

_measures the

progression

of

intermittency along

the scales

(number

of steps ^{in the cascade}

models).

For instance the

Kolmogorov

Obukhov

theory [2],

which _can be _{seen as one}

particular

multifractal model,

predicts

a

logarithmic dependence

1(&

in

«)~)

=

(

_in ^~

On the other

hand,

the variational model [71

gives

_{a power} law

dependence

(

(b In _«

)~)

_cc r~ ^P

Our message is that

discriminating

between these models

by looking

at the behaviour of

((&

In _«

)~)

versus r is much easier than

by determining

the

shape

of

G,

to which the

pdf

^of

3u is

poorly

sensitive,

We demonstrate this

point

in sections 2 and 3

by constructing

an artificial

pdf

^for

&u

using equation (I)

where G is

deliberately

different from _a

log

normal distribution, In section 3 we use a binomial distribution

recently proposed

as

corresponding

to the random beta model, a multifractal type one, In each case we show that the behaviour of A^~ well reflects that of

(

In _« )~

)

In section 4 _we address the

question

^of the

shape

of G. We discuss the characteristic of _an

experiment

which could

adequately

discriminate between different

proposed shapes.

We show that _a

large

number of

significant

measurements are necessary, of the order of 10?

points.

2. A first

example.

As discussed in the introduction _{we now construct an} artificial

pdf using equation ii)

P~~(x)

=

~~

~~~

i~~

exp

~~~

(l + b

In~

«

~

exp

~~

~

(3)

(2

_ar « 2 A 2

«

where _x stands for ^~~ and _« for

I

_and

A~ ^{is the} normalisation factor. We denote it

«~ «~

N° 2 MEASURING INTERMITTENCY IN TURBULENCE 217

P~~

because _we shall _treat it _{as an}

experimental

distribution and search for the parameter A which makes the _« theoretical _» distribution

~~~~~

(2

A

$

~~~

)~A~

~~P

~

~~~

having

a

shape

the closest

possible

to the

P~~

one. We then compare ^{A ~} with _:

n ~r

~~~

~~l

Ae (in

_{« )2 ~~~}

in2

fi

« ^{~~ ~ ~ ~~~ "}

~

d in _« ~~~

We have thus first _to

adequately

determine the difference in

shape

between

P~~

and

~ih

We

begin by normalizing

them to the _{same mean}

squared

value,

namely1, Pe(X')

" "e

Pex("eX~) (6)

p,(x<)

= «~

p~(«~x<) (7)

with

+JJ

+JJ

x'~ P~(x')d~<'=

_j

(a~x')~P~~(a~x') d(a~,<')

=

-« £Ye

~-«

and

+~J

+~

X~~

~t(X~)

i~~

~ j

(£Yt~')~ ~th(£YtX~) d(£YtX~)

_~ l

-oJ £Yt _-~

In _a real

experiment,

the

pdf

is constructed

by accumulating

the _events in

histograms

where x' has _a

given

value

xl

within A. Let

n~ be the number of _events such _as

X,' ~,t~

~ X,' ^{+ d.}

If N is the total number of measurements, P

(x,')

₌ ²

A

The difference between _two models for the

pdf

has to be

compared

to the difference _we

expect ^between two realisations of the _same

experiment,

which is

given by

(n~

h,)~

_{= h~}

and thus

(P (x,') ^> (x;'))~

~

~

In a first

approach,

a measure of the difference between

P~~

and P~~ ^{could thus be the}

integral

~+"

(Pe(x') P,(x'))~

~,

-m

Pt(x')

P~(P

_~) is the normalised version of

P~,(P~~).

See

equations

(6,

7).

However such _an

expression

could not converge if

P~

and

P~

^has very different

asymptotic

behaviour when x'

~ ± oo

(see

Sect.

4).

We thus

prefer

to define

~

+°~

~

(Pe(x'l-Pt(x'l)~

~, ~

~

_~

P~(.<')+P~(x')

Let _{us now} compare

P~~ given by equation (3)

with

A~

=

0.49 _to P

~~

(Eq. (4)).

The

integrals giving P~(x')

and

P,(x')

have been calculated

by

the

trapezoidal

rule with _an absolute

precision

of 10-~°

((3

In

«)~)

_{has been evaluated} within 10-5 of relative

precision

and

X~(A

~) was calculated within 10-2 of relative

precision [12].

The minimum of

X~(A

~) was estimated

by

the method of Brent

[121.

Three A^~ values _are chosen in the

neighbourhood

of the

expected optimum

: A

),

A

j,

and _a

parabola

^{is fitted with}

the three

corresponding X~' _X~(A)),

_{The minimum of this}

_{parabola gives Al,}

_{The process is}

then iterated with

Al

and the two closest of the

preceding Al,

It is

stopped

when two successive minima differ

by

^less than 10-3 _{in absolute}

precision,

6 0" ^~

x2

4

0'~

2 0" ^~

b

0 0.2 0.4 0.6 0.8

Fig.

I. The difference X~ _versus the deformation parameter b.

In

figure

I _we have

plotted

the smallest

X~ (Eq, (8))

_versus b which _measures the difference between the

«

experimental

_» G distribution and _a

log

normal distribution,

The

X~

values _are

rising

rather

rapidly

with b, Then

they

saturate at a

relatively

small b value. This _can be understood. When b is

large

the width of the distribution G becomes _very small. G is then similar to a Dirac

distribution,

whatever its

shape

^{and the}

corresponding pdf

goes to a Gaussian

shape (see Eqs,

(Ii and

(4)),

Thus

X~

should go ^{to zero} when

b goes to

infinity,

For b

= loo,

X~

is

equal

to 1,7 _x

10~~.

The small values of

X~

_are coherent with the small difference observed between

A^~ and

(

(3 In _«

)~) (Eq. (5)),

This is shown in

figure

2, The decrease in A^~

corresponds

to the

above mentioned

sharpening

of G when b grows,

Looking

at different

A~

_values

(A~

=

0.49,

0.7 and

I)

_we have been able to summarize the

X~

behaviour

by

an

approximate scaling (Fig. 3).

Within _a few percent,

X~/A~

behaves _{as a} universal function of bA. This

scaling

is

probably

^fortuitous and not even true in the

neighbourhood

of b

=

0. However the values _we have

spanned

are the

experimentally interesting

ones

[7, 1II

and this

approximate scaling

could

help

in

estimating

_an effective b value from the measured

X~

and A ² values.

The conclusion of this section is thus clear _: the parameter ^{~ obtained from the}

experimental

method _we discuss

always gives

a

good

estimate of

((31n«)~)

_even if the real

G function is

significantly

different from _a

log-normal

distribution. As this result has been

N° 2 MEASURING INTERMITTENCY IN TURBULENCE 219

o.5

,-

0.45

0. 4

0.35

0 3

<ln20~

0.25

~2 0.2

0 0.2 0.4 0,6 O-S I

b Fig. ?. Comparison between and

((6

In _rr

)~).

0 0 0 6 _{, ~ -. .i T --1} .-~-r i~

xz/~4

0.004

0.002

b~

0

0 0.1 0.2 0.3 0.4 0.5

Fig. 3. The appro~imate sc311ng _~ummanzing the X~ behaviour _1,e13us and b.

obtained

through

a very

peculiar

and ad hoc class of G function. _we shall

strengthen

the argument

by using

a more credible class of function,

recently propo~ed.

This will in

particular

introduce the effect of the asymmetry of G _iersiis In _r,, Another conclusion is that the value of

X~

if

;tati,tically ~ignificant,

could

help

ⁱⁿ

revealing

_a difference between the _real G and a

log-normal

distribution. This last

point

^{will be}

developed

in the fourth section, 3. The binomial distribution.

Recently

Benzi et al.

[81

derived _a formula for the

pdf

of

velocity

differences within the framework of the random beta model

[131.

Their result for the

pdf

at the scale _r is

P~(&v

cc

~ ~j

^~ a ^"

~(

l _a

)~

B~^~~~

ii

^"~

_exp[- ^CB~

^~'~

ii ~'~(3v

_)~

(9)

~~

K~

where

f,,

=

'

=

2~" C

=

(2 Vi j~' _{Vo being}

_the characteristic

velocity

difference _at

large

L

~cale. _a

=

7/8 and B

=

1/2 _are chosen such _i to

reproduce

the

(~

^values obtained

by fitting

the

experimental p-order

moment

((3uy)

with _a power ^law in _I

((&CY)

_CC i~~

I<fi ~'ii Iii i'ili'i'jt it -T ' lL3W'Rh1>»~ '

With the notation of _our

equation (1)

_we have

" "

V0 ~i~ ^B~~~

" "n

B~~~ (lo)

The distribution G is

given by

G(I

cc

( _{[~j a~~~(I}

-a)~B~3(In ~i _{+~lnB), (11)}

~n K=0

~ ~n 3

Thus

~3

In ^"

~)

=

~~~

j~ ^((3K)~)

~n 3

=

n(

^~~~

_)~ ^"~~

^"~~

⁽¹²⁾

3 (~x +

B(1

_a

))~

Following

the _same process as in section 2 _we

identify

the

pdf given by equation (9)

with

P~~(3u).

We then normalize it

through equation (6),

and _we compare this normalized

pdf P~(x')

with the normalized

«

log

normal» _one

P~, searching

for the minimum of

X~. ^This minimum value and the

corresponding

A^~ value are

given

in

figures

4 and 5, for _a

large

_range of values of _n,

~ , ~. ⁴ X2

4

10"~

2

0'~

n

0 20 40 60 80 100

Fig, ^4, The difference X~ versus n for the binomial distribution,

0 3

~~ ~,

,

;

;

~" <ln2~

0 .2 ^'

o. i

n

0

0 20 40 60 80 100

Fig.

5.

Comparison

between A^~ and

((6

In

«)~)

for the binomial distribution.

N° 2 MEASURING INTERMITTENCY IN TURBULENCE 221

The difference between A^~ and

( (3

In _«

)~),

and the minimum values of

X~

_are

larger

here than in the

previous

_case. This is

probably

due _to the

large dissymmetry

in In _« of the

« binomial _» distribution,

This

dissymmetry

_appears when

comparing

_a

typical

« binomial _»

pdf

(n ₌ 15 with the

«

log

normal _{» one}

giving

the lowest

X~

(see

Fig. 6).

Our choice of

X~ (Eq. (8)) clearly

emphasizes

the center of the

distribution,

where the

experimental precision

is the best _one, The

discrepancy

_appears in the

wings.

lo(~P

3

5 P

/ P

(n=15)

',

,' b~

7

-lo .5 0 5 lo

x

Fig,

6.

Comparison

between the _« binomial

» pdf (n ₌ 15) (full line) and the best fitted _« log normal _»

one (dashed line), The experimental points _are given to

figure

a

typical uncertainty.

Here the number of

independent

measurements is 2 _x 10~. _A significant

comparison

needs _to take account of the skewness

through

the asymmetry of Pi (Eq. (I)). See reference [71.

We note however that the difference between

A~

_and

((3

In

«)~)

remains within 15 fl.

More

important,

A^~ appears as

nearly

linear in _n in the present range, which is _a

significant

characteristic of

( (3

In _«

)~)

^{in this} type ^{of models. The behaviour of A~ thus well} reflects the behaviour of

((3

In

«)~),

_even in this _{extreme case.}

In

particular

the _recent evidence

[7, 121

for _a power law behaviour _{versus r} for

A~ shows that

((3

In

«)~)

_{presents a} similar behaviour. In the cascade models like the

random-beta _one discussed in this

section,

the number of steps n should _not be taken _as linear in In _r but

proportional

to

r~P

4. The statistical _error.

The minimum

x~

_values obtained in the two

previous

sections _were

relatively

small. So the

question

^{raises of} the

experimental

_accuracy needed to

distinguish

between _a

log

normal distribution of _« and _a deformed _one for instance, In this section _we estimate the minimum

x~

_{to be}

_expected

if _one compares an

experimental pdf (with

N measurement

points)

with its

exact

shape, asymptotically

obtained with _an infinite number of measurements.

Let _us call J the size of _a class, like in section 2. We have to estimate

X~=2 ( ^~~~~~'~ _~'~~'~~~A= ( _{E, (13)}

,=-«

Pe(x,)+Pt(.«)

,=_«

where

P~

i~ _now the

experimental pdf

and

Pj

its

asymptotic

limit when N

~ oo. The number of

points

in the class is _n, N

~P~(,i,

) for _our

particular experiment

and it~ average

expectation

value is

h,

₌ N

APj(.i,).

Two

limiting

_ca~es have to be considered.

2

k,

~~ ~~'~ ~' ~~~~

~~~~'~

~

~~~"'~

NJ

(F~(v, P~(

i,

I

=

~ (ii~ fi,

I

=

~'

(14)

N- J- N-

J~

as~uming

a Poisson distribution for ii,. We get

2 (P

~(.t',

Pi (,ii

)~

~ l

~~ ~ P

~

(.t,

+ P (-I,^{) N}

iii

h,

« I. If _fi~ i~ zero we have

E,

k~

=

2 ~P i-r, ²

If ii, is _or

larger,

we can

neglect P~(,i,)

and

E,

n,

=

2

AP~(,1,

=

2

Calling ar(fi,)

the N

probability

to get _n, we have

E, =2r(0)j+2zar(fi,)(=2(1+ 7r(01)(=/=4AP~(_r,).

(16) Note that if

x~

were defined with

only P~

(and not

P~

₊

Pj)

in the denominator, the _sum of

equation

(13) would

diverge.

We shall

interpolate

between these _two

limiting

case~

taking E,

= 4

k,/N

for fi, w and

E,

=

for

k,

m Let _us call 2 i~ the

length

of the interval _on

which _{k~ m} For _a

large

N the

major

contribution to X~ comes fmm this interval and

4

~

2,ij,

X~=- (17)

We _can go a step ^further

assuming

a

particular shape

for

P,.

An

exponential shape

has often been used _to fit such

pdf.

It

gives

P,(.1)

=

~-exp(-

i-11

,'2)

, 2

(the

variance of

P~(,i)

is I). _~i~ is

given by

~~

exp(-.ij,

_,. 2)

=

, 2 ⁴

or .i~, =

~-

_{ln (2}

, 2 NJ1

,, 2 The contribution of _{j-v ~_<~} is

m

,/j

2 4

P,(~<)

d~< ₌ 4 exp (- _{iii ,} 2)

=

~

'i,

N° 2 MEASURING INTERMITTENCY IN TURBULENCE 223

Then

X~"~(l

+ In

(2,2NA)), (18)

The result

depends

on the size of the class _as too small _a class

gives large

statistical _errors in the

counting,

On the other hand _too

large

classes introduce _errors in

estimating

the

integrals,

We shall take J

= 1/30 _{as a}

compromise,

that it classes 30 times smaller than the root mean

square of 3u,

From the

preceding

section _we know that _an order of

magnitude

for

X~

which allows _some distinction between the

possible shapes

is lo ^~ This ask for N m

lo?

uncorrelated _measure-

ments. This is close to the

large~t samples

used yet

[71.

For

distinguishing

between two

proposed shapes

the strategy ^{is thus clear. First the number}

of measurements must be sufficient for the

« statistical

»

X~ being

smaller than that

giving

the difference between the two

shapes.

Second the

X~

must be estimated

using alternatively

the two

proposed shapes

_as the

« theoretical _{» one,} The smaller _X^~ is the be~t and must be of order of the

« statistical

» value to be the

good

_one.

5. Conclusion.

The conclusion of this

study

^{is that the}

shape

^of the

pdf

i~ _more sensitive to the width of the distribution G of In _« than _to its _exact

shape.

The parameter ^{~ obtained}

by using

a

log

normal

ansatz for G is

always

a reasonable _measure of

((3

In _jr

)2),

Physically

this parameter ^is an

objective

_measure of the evolution of the difference of

velocity pdfs,

In all models

referring

to an energy cascade, as for instance the random beta model discussed here in ~ection 3,

(

In _«

j~)

^is

proportional

_to the number of cascade steps.

Our

study

thus

fully

validate~ _an

experimental

_way of

measuring

this

important

_parameter,

This method ha~

already

shown that

((&

In

(,)~)

does _not behave _as In _{r, as} is

generally

assumed, but _{more as a} power law in _;.

The second

question

addressed in this paper concerned the

shape

^of the distribution of

i, ; G. We have shown that _a

significant

check of this

shape

asks for very

large

statistical

~ample,,

however not out of the present

experimental possibilitie~.

In

particular, distinguishing

between _a binomial di~tribution _as

proposed

in reference

(8]

and _a

log

normal _{one as}

proposed

in

[41

i~

certainly

feasible.

Acknowledgments.

Thi~ work has been

partly ,upported by

_a DRET _contract (N° 9?/105) and the

Rdgion

Rhbne-

Alpe,,

References

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[21 Monin A. S.~ Yaglom A, M., Statistical Fluid Mechanic~ (the MIT Press, 1975).

[3] Gagne Y., Thesis (19871 (Institut National Polytechnique de Grenoble, unpublished) Advances _in Turbulence 3

(Springer Verlag

1991).

[4j Castaing B., C-R At-ad. SC (Pa;is) 309 ~erie II (19891503.

(5] Kraichnan R., Pfi,vs Rei Lett. 65 j1990j 575 She Z. S., F/ilid Din. Research 8 j1991j 143.

[61 Andrews L. C.,

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