Conditional velocity pdf in 3-D turbulence.

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Conditional velocity pdf in 3-D turbulence.

Yves Gagne, Muriel Marchand, Bernard Castaing

To cite this version:

Yves Gagne, Muriel Marchand, Bernard Castaing. Conditional velocity pdf in 3-D turbulence.. Journal

de Physique II, EDP Sciences, 1994, 4 (1), pp.1-8. �10.1051/jp2:1994104�. �jpa-00247941�

Classification Physics Abstracts 47.25C 92.60E

Conditional velocity pdf ⁱⁿ 3-D turbulence.

Yves

Gagne (~),

Muriel Marchand

(~)

and Bernard

Castaing _(~)

(~)

Institut de

M6canique

de Grenoble,

L-E-G-I-I U-J-F-I I-N-P-G-I C-N-R-S-,

B-P-

53X,

38041

Grenoble

Cedex,

France

(~)

Centre de Recherches _sur les TrAs Basses

Temp6ratures, C-N-R-S-,

B-P-

166X,

38042

Grenoble

Cedex,

France

(Received

23

July1993,

received in final form 19 October 1993,

accepted

25 October

1993)

Abstract. It is shown that the

probability density

functions of

velocity

increments at

small scale in turbulent flows turn to _an universal

(Gaussian) shape

when conditioned to _a

precisely

defined _energy transfer rate Et The standard deviation

«(Et

of this distribution de- pends _{on Et}

following

_a

Kolmogorov

like relation

a~

=

CEt

-I with _a Reynolds number

dependent

coefficient C.

Introduction.

One of the main

properties

of

fully developed

turbulence is the small scale

intermittency-

We

usually

consider that its

qualitative signature

is the non-Gaussian

shape

of the

probability density

function

(hereafter

noted

p-d-f)

of the

velocity

increments defined

by 6u(1)

=

u(x

₊

I) u(x)

for I

lying

within the inertial _or

dissipative

_range

(for clarity,

_we shall consider the I

parallel _component

of 6u:

dull)).

This stretched

exponential-like

tail

shape

of the

p-d-f

of

6u(1) changes

when the distance decreases from the

largest

inertial scales to the smallest

dissipation

_ones. In other

words,

the

shape

of the

p.d,f.

of

6u(1)

normalized

by

its standard deviation

a&u(o

^{is not} self-similar with I

iii.

When

analyzing

via the

scaling exponents (p

^defined

by

<

6u(1) >~

r- l~P

,

the consequence of this

change

is the non-linear behaviour of

(p

with p in constrast, ^for

instance,

with the

predictions

of the

p-model

_[2]-

Another

problem

_concerns the _use of the relation between the statistics of the

velocity

increments

6u(1)

^and the local energy transfer rate

averaged

_{over a} ball of size

I,

_Et

~3

6J ~j

(~)

l

However,

for _some

authors, II

_can be taken _as

a relation between _{two random} variables if _vi is the

typical velocity

of structures of size [3] For others

(Frisch _[4]),

vi is

6u(1)

but the

equality

2 JOURNAL DE

PHYSIQUE

II N°1

only

_means that the two

quantities

^{"have the} same

scaling properties,

I-e- that moments of the

same order have the _same

scaling exponents".

Is there _a

possibility

to

experimentally clarify

this

point?

The

objectives

of this _{paper are,}

first,

to

study

the statistics of the

velocity

increment

6u(1)

conditioned to the statistics of the _energy transfer rate et in the whole _range of inertial and

dissipation

scales

and, second,

to test

statistically

the

previous

relation.

In section

I,

we

give experimental

results _on conditional

p-d-f-

of

6u(1)

measured in two different

Reynolds

number turbulent flows. Section 2 is devoted to the

dependence

of the

averaged

value of the conditional variable

6u(1)

/~, ^{on its standard deviation} a&u(i)~ ^and

finally,

e<

in section 3 _we

give experimental

data in order to test the

Kolmogorov

relation

(I).

I- As is said in the

introduction,

_we have studied the statistical distribution of the

velocity

increments

6u(1)

at _a

given

scale

I,

conditioned to the value of _Et

Using

the

homogeneity

and

isotropy assumptions

at small

scale,

the energy

dissipation

rate at _a fixed

point

x is defined

by

6~(~,t)"15~ j

~~~~

Following Kolmogorov

and Obukhov

[5],

^{the local} average of the _energy

dissipation

_rate

e)(x, t)

taken _{over a} ball of size centered _{at x} is

ej(x,t)

~-

j j e*(x',t)d3x'

~,_~ ~i

By

also

using

the

Taylor hypothesis

to

get spatial gradient

from time derivative

signal, experimentalists

follow the ID version of this definition _{to measure} the local _energy

dissipation

rate

El averaged

over a linear interval of size

I,

for I

lying

in the inertial _or

dissipation

_range.

If I

belongs

to the inertial range,

El roughly corresponds

to the energy transfer rate E,

averaged

_over the _same interval I. When is in the

dissipation

_range, the local transfer _{E, can} be estimated _as

~'~~

~~ ^" ~~~

I _~~~ Ill

^~ _~~

^~l~~)~j

in which ~"~~~ ^~

takes into account the

dissipation

which _occurs at the scale I itself.

Clearly,

I

this _term increases when I goes down to the

Kolmogorov

scale _q and it appears that its value is _no

longer negligible

at such _a scale.

We have calculated this ersatz of the local transfer rate 5, on three different

velocity signals

obtained _on the axis of _a

laboratory jet

flow

(R>

₌

428)

and in the wind tunnel of O-N-E-R-A- in

Modane,

both _on the axis

(R>

₌

2720)

and _near the wall

(Rx

=

1689)

_[6]. The ratio of the

sampling

interval /hz _over the

Kolmogorov

scale q was about 0.15 in the

jet

and

respectively

12 and 19 in the wind tunnel which _means that the

spatial

resolution is

good

in the

jet

and very poor in the two flows at

high Reynolds

number.

practically,

_5, has been calculated from the

velocity

data

samples

in the

following

_way:

~f ^u~+j

ui+j-i

~l(u~+m

u~ ^~

~' ~

i

_{/hz m/hz}

J=1

in which I

= m/hz.

Figure

I

gives

a

comparison

between the

shapes

of the

global p-d-f

of

6u(1)

and the conditional

one at a

"given

value" of _5, which _means

practically

_an interval of values centered at 5, with _a constant width that _we have chosen

equal

_{to < 5, >}

/6.

In

figure I,

the scale I

belongs

to the inertial range,

respectively lo

= 40 in the

jet (Fig. la),

120 _on the axis of the wind tunnel

(Fig.

lb and 190 _near the wall

(Fig. lc).

The

given

value of _e, is

equal

to < Et >.

o s

aP<r)

35

$ j

s

s

s s

&

a a)

o5 ~ 5

~ ⁱ5

GP(~) ~~(t)

25

25

35 35

~ ~~'

~

, j.~

i$I

55 ^'"'

o s o s

io s o s

& ~

a °

b) c)

Fig.

I. Global

p-d-f

of

but (dashed line)

and

p-d-f

of

but

conditioned to < Et >

compared

to _a Gaussian

shape (parabolic

in

Iin-log coordinates),

for

lying

in the inertial range.

a): jet,

R> ₌ 428,

= 40~.

b):

axis of the wind tunnel, R>

= 2720, 1= 120~.

c): "boundary layer"

of the wind

tunnel, Rx

= 1689, 1= 190~.

Figure

2

gives

the _same

comparison

for _a scale chosen in the

dissipation

_range:

lo

₌ 10

only

in the

jet (in

the wind

tunnel,

the

dissipation

_range has _not been

reached).

As is well-

known,

the

global p-d-f

of

6u(1) changes

from _an

exponential-like

tail

shape

to a

sharper

and

more stretched _one when _goes down from the inertial to the

dissipation

_range. On the

contrary,

^the conditional

p-d-f

of

6u(1)

has _a

quasi

Gaussian

shape

which remains constant

4 JOURNAL DE

PHYSIQUE

II N°1

when is

decreasing; "quasi"

_means that these

p-d-f

_are not

symmetric

but

always

have _a weak

negative

skewness like all the classical

longitudinal velocity

increments.

Probably,

in the _case of transversal

velocity increments,

these conditional

p-d-f

would be

purely

Gaussian. In

figures

16 and

lc,

the tails of the

experimental

conditional

p-d-f-

_{move away} from _a

parabolic shape,

which

suggests

_a

remaining intermittency.

That feature

clearly

_comes from the poor

spatial

resolution

(/hz

_m 12 and

19)

in the estimation of _Et

ap(r)

£ j .h

~Jl'~ /

~ />)

.I '

io s o 5

K a

Fig.

2. Same _as

figure

I, with I

lying

in the

dissipation

_range, for the

jet. R>

₌ 428, 1= 101

This

quasi

Gaussian

shape

of the

p-d-f

conditioned to Et has

already

been obtained

by Stolovitsky

et al. [7]

only

for inertial scales. In _{contrast to} us, these authors

got

^bimodal

conditioned

p-d-f

of

6u(1)

for

dissipative

scales. It is due to the fact that

,

in their _paper, the

velocity

fin

(I)

has been conditioned to the

quantity El

and not to _{Et as we} did. We have checked their results

using El

instead of _Et and have found the _same bimodal conditional distribution in the

dissipative

_range.

The results of

figure

I

suggest

that the

right quantity

to condition the statistics of the

velocity

field is the _energy transfer rate and not the

dissipation

_one. If _{a non} Gaussian

shape

is

a statistical

signature

of

intermittency,

_{as many}

people think,

then those results show that the small scale

intermittency

is

fully

contained in the distribution of the energy transfer rate and

disappears

at fixed _Et

Moreover,

this result is in very

good agreement

^{with the main}

physical assumption

of the

empirical

model of

velocity p-d-f proposed by Castaing

et al.

ill

which is

theoretically supported by

the variational

approach (Castaing _[8]).

2. One of the consequences of the model

quoted

above is that the calculated distribu- tions of

6u(1)

^have a non zero average value.

Obviously,

this

point

is in contradiction with

experimental

data for which the _mean value <

6u(1)

> is

always

_zero. In order to understand how this average of

6u(1)

is restored to zero, we have studied the

dependence

of the _mean value

on the standard deviation of the conditional distribution of

6u(1)

_/~,,

namely

<

6u(1)

/~~ > us.

a&u(tj~ Figure

3 shows such _a behaviour at _a

given lo

= 60 in the _case of the wind tunnel.

The

e~ierimental _points correspond

to different chosen values of _et

equally

distributed from

zero to 2x < e, > with _a

step

^of < e, >

16.

^This

figure clearly displays

_a linear behaviour

of <

6u(1)

_/~~ > on a&u(,)~~ _,

decreasing

^from

positive

to

negative

values _as

a&u~,)~~ ^increases.

This remarkable feature

o~curs

_for

any scale and _any flow and leads to _a linear

cleilicient

_K

apparently independent

of the scale I. A

study

of its

dependence

with the

Reynolds

number asks for _more than _our three

examples

but note anyway that K is

roughly

^the same for the three of them.

0.05

A

A

0 03

~b"</6f ^~

0 01

k ~j

o.oi a-

0.03 ~-a

a

o.05

0.15 0.2 0.25 0.3 0.35 0.4

03ui>n

Fig.

3. <

bit,

_/~~ > _versus a&u~_~~ ^{on the axis of the wind tunnel.} R> ₌ 2720, 1=

60q

W

~

i

ik,

j '

Fig.

4.

p-d-f-

of

bu(I)

obtained

by

the

superposition

of

quasi-Gaussian

p-d-f- ^with several modal and standard deviation values.

As _we discussed

previously, Castaing

et al.

iii

have shown that the

p-d-f

of

6u(1)

can be fitted

by

_a

superposition

of Gaussian

p-d-f

with different standard deviations a&u~,)

increasing

from _very small values

corresponding

to the

top

of the

global p-d-f

to

large

_ones

taking

into

account the tails _as is shown in

figure

4.

6 JOURNAL DE

PHYSIQUE

II N°1

Figure

3 indicates that the narrowest Gaussian which contributes to the _most

probable

value of the

global 6u(1)

are

positively shifted,

whereas the

largest

Gaussian _are

negatively

shifted

leading

to the

negative

skewness of the

longitudinal p-d-f-

This result

clearly explains

what

was first _seen

by

_van Atta and Park

[9],

^that

is,

for _a

given

scale

I,

the most

probable

value of the

global

distribution of

6u(1)

is

always positive

and the

negative

skewness is

only

due to the

largest

and _rare

negative

fluctuations of

6u(1).

~&W

(<et>L)m

7

~ i~

~ ._.

I

o

0 0.2 0 4 0.6 0.8 2 4

(£j')in

«EL>L)in

a)

~~

a&tin

(<~~>~jin (<_EL>.L)^'~

4 35

3.5 u

30 ^._

~ ⁺

a

25 +±

~ +

2

20

1.5

~

15

io

o 5

o o

0 0.1 0.2 0.3 0A 0 5 0.6 0 0.2 0A 0.6 0.8 1.2 lA 1.6

fi

_(£10"

(<eL>z)"

b) _c)

Fig.

5. a&u~ ^versus

(E.I)~/~

normalized

by (<

_EL >

.L)~/~,

where L is the

integral

scale of each flow.

a):

axis

~)

_the wind tunnel.

R>

₌ 2720.

(0):

=

24q; (+):

=

60~; (A):

=

120~; (o);

= 250~;

(.);

= 2500~.

b):

"boundary

layer"

of the wind tunnel. Rx ₌ 1689.

(0):

= 38~;

(+):

= 95~;

(A):

1=

190~; (o);

₌

380~. c):

jet. R> ₌ 428.

(0):

1= 1.5q;

(+);

₌ 5.28q;

(A):

= 9.8~;

(o);

= 20q;

(x):

1= 40~;

IA):

₌ 60q;

(+):

1=

120~; (.):

= 240q.

3.

Knowing

the value of a&u(tj

coresponding

to each _{Et we} have studied their mutual behaviour in order to test relation

(I).

In

figure 5,

we have

plotted

a&u(i)~~ ^versus

(I.Et)~/~;

variables have been normalized with the

integral

scale L and the

correspondir~g

_energy transfer

rate < EL >. At _very

large Reynolds

numbers

(Fig. 5a, R>

= 2720 and

Fig. 5b, Rx

=

1689),

for different scales

ranging

in the inertial _range, relation

(I)

is well verified

by experimental data;

_we have

namely

a&ujij~~ =

C(I.st)~~~

where the coefficient C does not

depend

_on the scale I.

However~

the a&u(tj~~ decreases to a non zero value when the scale _{I goes} to zero.

Again,

it is due to the poor

estim(te

_of

Et at small scale I which overestimates the value of a&u(t)~~ ^The

experimental

values of the factor C _are

roughly equal

to:

C(2720)

_m 4.8 and

C(1689)~m

6.2. Even

though

these

C(R>)

coefficients

are

underestimated,

it _seems that

they

decrease with the

Reynolds

number since the factor

C(1689)

is _a

priori

_more reduced

by experimental

_error than the

C(2720)

_one.

In

figure

5c

(jet, R>

₌

428), experimental

data tends to the

origin

of the axis which denotes

a

good experimental

estimate of _Et at small scale I. But the

merging

of

experimental

data obtained at different values of the scale I is not _so

good,

in

particular

_we

get

_a

systematic

shift for each scale I. We have _no

satisfactory explanation

for this

systematic

shift. In

particular,

it

cannot be

fully explained by

the finite size of the _E, interval

correspr~iiding

to a

single point.

It could be

artificially

corrected

by

_a different

dependence

of _{a uersw}

(1°

^with o >

1/3).

Note

however that the

dispersion

of the

points

is of the order of the shift which

yields

to think to a

sampling

artifact. The

slopes

of the different

pieces

of this _{curve are}

roughly

the _same, at

least for inertial scales and their _common value is about

C(428)

= 15.

Thus,

relation

(I)

is

experimentally

well verified

only

for inertial scales _as assumed

by Kolmogorov.

This

suggests

^{that the} _energy ^transfer rate E,

averaged

_{over an} inertial

separation

is indeed _an inertial

quantity

^and not _an inertial and

dissipative

_one as

proposed by

Kraichnan

[10]).

Concluding

remarks.

We have shown that the

velocity

field

6~(1)

at small scale is

intrisically

Gaussian

(within

the skewness of the

longitudinal case)

and the intermittent _non Gaussian

shape

of its

p-d-f

is

only

due to the

intermittency

^{of the} energy transfer rate E,. This result confirms that

quantitative intermittency

can be defined

only

from

energetical quantities

_as assumed in the variational

approach proposed by Castaing

_[8].

As these conditional

p-d-f

_are

Gaussian, they

are

completely

characterized

by

their stan- dard

deviation,

_even for the

longitudinal

_case, where _we have shown that the _mean values

<

dull)

/~~ > are not zero but

depend

_{on a}

unique

_way on the standard deviation. The

scaling

law of the standard deviations a&u(,)~~ ^on the value of _E, is

experimentally,

^for inertial

scales,

in

good agreement

^with the

well-kno~in

_{relation of}

Kolmogorov.

Our results

clearly

indicate that this

scaling depends

on the

Reynolds

number value. In the inertial range, our

experimen-

tal data lead to similar results than those

reported by Praskovsky ill],

Thoroddsen and _van Atta

[12]

^{and Chen et al.}

[13].

^{Since in these} papers the

velocity

increment

6u(1)

has been

conditionaly averaged

over a fixed value of E)

(and

not of _{E, as} done

by us),

_no

comparison

can

be made for scales I

lying

in the

dissipation

_range.

8 JOURNAL DE

PHYSIQUE

II N°1

Acknowledgements.

This work has benefited of D.R.E.T _contracts

N°92/105

and

92/083.

We thank M.

Vergassola

and U. Frisch for fruitful discussions _on the conditional

p-d-f

and the

Editor,

A. Arneodo for his comments which

permitted

_us to

improve

the _paper.

References

iii Castaing B., Gagne

Y. and

Hopfinger E-J-, Physica

D 46

(North-Holland, 1990)

177-200.

[2] ^Frisch

U.,

Sulem P-L- and Nelkin

M.,

J. Fluid Mech. 87

(1978)

719-736.

[3] ^{Meneveau C. and} Sreenivasan K-R-, Nuclear

Phys. (Suppl.)

82

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U.,

Turbulence and Stochastic Processes:

Kolmogorov's

ideas 50 _{years on}

(The Royal

Society, London, 1992)

_p. 89.

[5]

Kolmogorov A-N-,

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[6]

Castaing

B., Gagne ^Y. and Marchand M.,

Physica

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published.

[7]

Stolovitsky G.,

Kailasnath P. and Sreenivasan

K-R-, Phys.

Rev. Lent. 69

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l178-l181.

[8]

Castaing

B., J. Phys. France 50

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[9] ^Van Atta C-W- and Park J., Statistical Models and

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[10] ^Kraichnan

R.H.,

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A.

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