Anomalous scaling of velocity structure functions in turbulence: a new approach

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Anomalous scaling of velocity structure functions in turbulence: a new approach

G. Ruiz Chavarría

To cite this version:

G. Ruiz Chavarría. Anomalous scaling of velocity structure functions in turbulence: a new ap- proach. Journal de Physique II, EDP Sciences, 1994, 4 (7), pp.1083-1088. �10.1051/jp2:1994186�.

�jpa-00248029�

J.

Pliys.

II FYance 4

(1994)

1083-1088 JULY

1994,

PAGE1083

Classification

Physics

Abstracts 47.25

Short Communication

Anomalous scaling of velocity structure functions in turbulence:

a new

approach

G. Ruiz Chavarria

(*)

Laboratoire de

Physique (**),

Ecole Normale

Supdrieure

de

Lyon,

46 Allde d'Italie, 69364

Lyon,

France

(Received

13

April

1994,

accepted

in final form 24

May 1994)

Abstract In this letter _we report

experimental

evidence about the

possibility

to derive the n-order structure function

(in fully developed turbulence)

in terms

oflocally averaged dissipation

rate _cr and the third order structure function. The results

presented

here _{are a}

generalization

of the 1962 Oboukhov

hypothesis

about the

scaling

in the inertial _range. The results _are in _accor-

dance with the Extended Self

Similarity (ESS),

a concept

recently

introduced in the literature.

In fact _we

reproduce

the n-order structure function not

only

in the

region

where ESS is valid but in _a much broader _range.

The statistical behaviour of three-dimensional

fully developed

turbulence at small scales has been

intensively investigated

in the last five decades. A _{common way} to board this

problem

is

through

the

velocity

structure functions. The nth order structure function is defined _as:

<

bu(r)"

> ₌ <

(u(x

₊

r) viz))"

>

where _u is the

component

^{of the}

velocity along

the direction of _r and _{< >} stands for _a statistical average.

Kolmogorov's original theory iii predicts

that at very

large Reynolds

_num-

bers the

scaling

<

bu(r)"

> m

~"/3

_{holds for}

r in the inertial range

(q

_{< r} <

L,

where q =

(v3/

< _c

>)~H

is the

dissipative

scale and L is the

integral scale). Many experimental

and theoretical studies do not

support

the

Kolmogorov theory _{[2, 3],}

and the deviations _are

usually

related to

intermittency

ⁱⁿ the _energy

dissipation.

In 1962 Oboukhov and

Kolmogorov [4,

5]

proposed

_an inertial range

scaling

^of the n-order structure function which includes the

intermittency

in the energy

dissipation

rate:

<

bu(r)"

_>

mJ <

cf/~

_>

^r"/~ (l)

(*)

On leave from:

Departamento

de

Fisica,

Facultad de

Ciencias,

UNAM, ^Ciudad

Universitaria,

04510 Mdxico D-F-, Mexico.

(**)

CNRS. URA 1325.

1084 JOURNAL DE

PHYSIQUE

II N°7

where _cr is the _average of the energy

dissipation

rate c over a volume of linear dimension _r and _{s =}

lsv(8u/8x)~

for

homogeneous

^and

isotropic

turbulence.

Equation (I)

also

implies

_a

power law behaviour of <

cll~

> in the inertial range.

In this work _{a new}

scaling

which extends

beyond

the inertial range is shown. This is the basis to

reproduce

_an

arbitrary velocity

structure function for all scales in terms of _<

cl/~

_>

and _a universal function

f:

~

n/3

<

bu(r)"

_>

=

Cn

<

cf/~

_{> r}

f n~~ (2)

where

Cn

and

f

are

dimensionless,

the latter

being

related to the third order structure function that is:

~

f ~))

^"

<)>

^~

^~~i~~

^~ _~~~

In order to define

f(r/~)

_we have used <

(bu(r)(~

> rather than <

bu(r)3

_> because the

computation

of the latter is much _more

demanding,

but the two

quantities

exhibit _a linear

dependence

in _a broad _{range, as} shown in reference [6]. The _constant D _was introduced to normalize this

function,

that

is, f

must go ^to I in the inertial range. On the other

hand,

the

continuity

in

velocity implies

<

(bu(r)(3

> m

r3

_when

r lies in the

vicinity

of the

Kolmogorov

scale _q, then

f

becomes

proportional

to

r~.

_At first

sight

the

right

^hand side of

equation (2) reproduces

the characteristics of _a structure function in two

regions.

The first _one is

the

dissipative

_range where the power law behaviour is

completely

dominated

by

the _term

(r fir In) )"/3,

the

quantity

<

sll~

>

going

to _a constant value when small scales _are considered.

The second _one is the inertial _range where the power law behaviour is

composed

of _a

dependence

on

r"/3

_and

a correction related to the energy

dissipation

rate. At intermediate scales the relation also holds as shown

by

the

experimental

data. It must be noticed that

equation (I)

is

a

particular

_case of

equation (2).

In _a

previous

^letter

by

Benzi et al. [6] an extended self

scaling

is found when _a structure function is

plotted

_versus another _one. The self

scaling

is observed _{on a} broad

region

that

begins approximately

at

S~ [3, 6],

^{earlier than the} appearance of any inertial range and at

Reynolds

numbers for which _no

scaling

is

expected.

The anomalous

exponent ((n)

associated with the n-order structure function is calculated _{as a}

slope

in a

log-log plot

of _<

bu(r)"

>

versus <

(bu(~)(3

>. Now _a self

scaling

broader than that observed in ESS is shown.

This letter is intended to discuss _a transformation that allows two different

velocity

structure functions to be

together

and also the influence of the

intermittency

to be shown in all the

scales,

not

only

in the inertial range.

Table 1.

R

R, L(cm) ~(~m)

11000 170 10 440

300000 800 12.S 100

Sonde size _: Diameter 25 pm.

Length

0.25 _mm.

The

experimental set-up

^{consists of} a wind tunnel with _{a cross} section 50 _x 50

cm~

and 3 _m

long.

^Turbulence was

generated by

a

cylinder

_{or a}

jet,

^the

integral

scales L _were 10 and

N°7 SCALING OF STRUCTURE FUNCTIONS IN TURBULENCE 1085

12.5 _cm

respectively.

The

Reynolds

numbers based _on L _were 11 000 for the

cylinder

and 300 000 for the

jet. Furthermore,

the

Reynolds

numbers based _on the

Taylor

scale _were 170 and 800

respectively (see

Tab.

I).

It is

interesting

to notice that the

only

_case when _a inertial

range was

clearly

detected

corresponds

to the measurements in the

jet (Re

= 300

coo),

_{as we}

can see in

figure I,

where the structure function _<

(bu(r)(3

> vs.

r/~

is

plotted

in

log-log

scale. The instantaneous

velocity

_was measured with _a hot _wire whose dimensions _are

given

in table I. The data _were

digitalized

with _a 16 bits

a/d

converter. The

probe

_was

placed

at 25 times the

integral

scale downstream. In order to

compute

the structure functions and _<

sll~

>

the

Taylor hypothesis

_was

used, assuming

that the eddies _are convected

by

the _mean flow. In other words _we

replace

r

by Ut,

where U is the _mean

velocity,

and the derivative

8u/8x

is

~~P~~~~~

~Y

j I

40

20

~ O-O

j~

~i

~l

20 fi

~ _A

a

40 ^~

~ o o

~~l

-0 O-o 1.0 2 0 S-O 40

Log r/q

Fig.

1.

Log

< (bu~( > _vs.

Log (r/~).

The data

concerning

two different

Reynolds

numbers _are

plotted:

Re

= 11 000

(o)

and Re ₌ 300 000

(/h). Only

in the latter _{case a} inertial range is

clearly

detected.

In

isotropic

and

homogeneous

turbulence:

cr =

~~"

(8u/8x)~dz (4)

r

~

For

experimental

convenience _we take _a one-dimensional _average rather than _a three- dimensional _one.

According

to other authors this

change

does not

produce

_any modification in the results [7]. The evaluation of

c)/~

_involves

a derivative and _an

integral.

The derivative

was calculated

using

_a difference schema with the two

neighbouring points

_[8]:

~

~

~~~

~

~2h~~~

_~~

~~~

lo86 JOURNAL DE

PHYSIQUE

II N°7

The function

~"

was

interpolated

with the

spline technique

and the

integration

_{was per-} ax

formed

using

this

interpolation.

The number of data _was

equal

to lo~ for the

cylinder

_case, whereas in the

jet

2.S _x

lo~

_data

were recorded. With this amount of data it is

possible

to evaluate with statistical _accuracy until the six order structure function. In

figure

2 the function

<

s)/~

_> and _<

s$/~

>

(normalized by

<

c"/3 >)

_are

plotted

vs.

r/~

in

log-log

scale for the two

Reynolds

numbers considered here. The _{curves go} to _a constant value when _r

- j

020

o 15

A o io

£~ v

I

o~

o 05

~ a

o.oo

o °

~~~l.0

O-O I-O 20 30 40

Log r/fl a)

O-O A n ~a~

o o _°

~§~

0.2 %

A

~&t

V 0.4

I

01

06

08

~'~l.0 _{0 0 0}

2.0 30 4.0

Log

r/~

b)

Fig.

2.

al

_<

e)/~

_>

vs.

r/~

and

b)

< e$~~ vs.

r/~,

Two

Reynolds

numbers _are

considered,

Re = 11 000

(o)

and Re ₌ 300 000

(/h). Generally speaking

<

el~~

_{> goes to}

a constant value when

r - q, The functions < e~~~ > and _< e~~~ > were normalized by <

e~/3

_> and _<

e~/3

_>

respectively.

N°7 SCALING OF STRUCTURE FUNCTIONS IN TURBULENCE 1087

In

figure

3 the main result of _this work is

shown, namely

_a

log-log plot

of <

sll~

>

~

n/3

rf

_vs. <

bu(r)"

> for _n

= 2 and 6. In all _cases the data _are

closely

distributed

~~~

along

_a

straight line,

_even in the

vicinity

^of the

Kolmogorov scale,

and the

slope

is

always

very

close

_to I. This

implies

_a direct

proportionality

between the two

quantities plotted

in the

graphs.

The

possibility

to write the n-order

velocity

structure function in terms of _<

sll~

_{> and}

2.0

o-o

£_

~

f

i~i~

01

~

~~60 Log

<bu~>

&)

1088 JOURNAL DE

PHYSIQUE

II N°7

f(r/~) gives

_an

insight

in the anomalous

scaling. First,

the

exponent

of <

bu(~)"

> in the inertial _range is

separated

in _a

regular part, given by

^the

Kolmogorov theory of1941,

and _a

correction due to

intermittency

ⁱⁿ s. A _new result _concerns the influence of

intermittency

at scales

lying

between the

dissipative

and the inertial _ranges.

According

to

figure

2 the

influer~je

of _<

s)/~

> increases when _r decreases. But the _same

applies

to the term

rf

^~

~~~

~

Only

when _{r cs ~} the influence of the

intermittency

decreases and in the

dissipative

_range the behaviour of the structure function is

completely

dominated

by

the

continuity

in

velocity.

In this letter _we have shown _a

generalization

^{of the Oboukhov}

hypothesis

^of 1962 that

applies

at all scales below the

integral

_one and _even when the

Reynolds

number is _so low

that _no inertial _range is

clearly

detected. To do this _a function

f(~/~),

not

depending

_on the

intermittency,

_was introduced. The

generalization

of the Oboukhov

hypothesis permits

_us to

clarify

the role

played by

the energy

dissipation

at different scales _as has been

pointed

out in

a

previous paragraph.

All that has been

pointed

here _seems to indicate that the main features

concerning

the statistics of _a turbulent flow _are

already _presents

_at low

Reynolds

number and

over a broader range than the inertial _one.

Acknowledgements.

The author thanks S.

Ciliberto,

R.

Benzi,

C.

Baudet,

^{J.F. Pinton and} R. Labbe for their valuable comments.

References

iii Kolmogorov

A-N-, ^{The local} structure of turbulence in

incompressible

viscous fluid for very

large Reynolds

number, DokJ, Akad. Nauk. SSSR 30

(1941)

299.

[2] ^Kraichnan

R-H-,

On

Kolmogorov's

inertial range theories, J. Fluid Mech. 62

(1974)

305.

[3] Anselmet

F., Gagne

Y.,

Hopfinger

E-J- and Antonia R-A-,

High-order

structure functions in

turbulent shear flows, J. Fluid Mecli. 40

(1984)

63.

[4] Oboukhov

A-M-,

Some

specific

features of

atmospheric

turbulence, J. Fluid Mecli. 13

(1962)

77.

[5]

Kolmogorov A-N-,

A refinement of

previous hypothesis concerning

the local structure of turbu- lence in _a viscous

incompressible

fluid at

high Reynolds number,

J. Fluid Mecli. 13

(1962)

82.

[6] Benzi R., Ciliberto S., Baudet C., Ruiz Chavarria G. and

Tripiccione R.,

Extended self similarity in the

dissipation

_range of

fully developed

turbulence,

Europliys.

Lent. 24

(1993)

275.

[7] ^Thorodsen

S-T-,

Van Atta

C-W-, Experimental

evidence

supporting Kolmogorov's

refined simi-

larity

hypothesis, PJiys.

Fluids A 4

(1992)

2592.

[8] ^Kuznetzov

V-R-, Praskovsky

A-A-, SabeInikov V-A-, Fine-scale turbulence structure of intermit-

tent shear flows, J. Fluid Mecli. 243

(1992)

595.