Applied Mathematics and Mechanics

(English Edition, Vol. 21, No. 9, Sep 2000)

Published by Shanghai University, Shanghai, China

Article ID : 0253-4827(2000)09-1016-08

E X T E N D E D S E L F - S I M I L A R SCALING LAW OF M U L T I - S C A L E

E D D Y S T R U C T U R E IN WALL T U R B U L E N C E *

JIANG Nan ( ~ ~r WANG Zhen-dong (2E~3~, ) l ' a , SHU Wei ( ~ t-~) I'~"

( 1. Institute of Mechanics LNM, Chinese Academy of Science, Beijing 100080, P R China;

2. Department of Mechanics, Tianjin University, Tianjin 300072, P R China)

(Communicated by Li Jia-chun)

Abstract : The Iongitudinal fluctuating velocity of a turbulent boundary layer was measured

in a water channel at a moderate Reynold~ number. The extended selfl simiIar scaling law of

structure function proposed by Benzi was verified. The longitudinal fluctuating velocity in

the turbulent boundary layer was decomposed into many multi-scale eddy structures by

wavelet transform. The extended self-similar scaling law of structure function for each scale

eddy velocity was investigated. The conclusions are 1 ) The statistical properties of

turbulence could be self-similar not only at high Reynolds number, but also at moderate and

low Reynolds number, and they could be characterized by the same set of scaling exponents

~ t ( n ) = n/3 and ~2( n) = n/3 of the fully developed regime. 2) The range of scales

where the extended self-similarity valid is much larger than the inertial range and extend~ far

deep into the dissipation range with the same set of scaling exponents. 3 ) The extended self-

similarity is applicable not only for homogeneous turbulence, but also for shear turbulence

such as turbulent boundary layers.

Key words: wavelet transform; eddy; scaling law

CLC number: 0357 Document code: A

Introduction

Much work has been devoted in the last few decades to the measurement and modeling of the

scaling law of structure function of turbulent flows. The so-called " velocity structure funcdon of

order n " for turbulent flows is def'med as ( & V ( r ) " ) , where/', V ( r ) = V( x + r) - V( x ) is the

velocity component increment parallel to the relative displacement r of two positions separated by

a distance of r in the flow field.

Let us remember that the research of structure function scaling law is usually limited by the

following assumptions: 1) in the full developed turbulent flow so that the Reynolds number is

inf'mite; 2) local homogeneous and isotropic; 3) for r in the inertial range.

* R e c e i v e d d a t e : 1999-09-03 ; R e v i s e d d a t e : 2000-06-08

F o u n d a t i o n i t e m : the National Natural Science Foundation of China (19732005); Doctoral

Program Foundation of the Education Committee of China (97005612); the National Climbing Protect

Biography: JIANG Nan (1968 ~ ) , Associate Professor, Doctor

1016

Self-Similar Scaling Law of Multi-Scale Eddy Structure 1017

The expectation of the scaling law is that

(AV(r) " ) ~ r~"> ('7<< r<< L), (a) where 7/is the dissipate length, L is the integral scale and ~(r~) is called scaling exponent. The

scaling law is an indication of the existence of scale invariance in turbulence.

For the third-order structure function, one can deduce the following Kolmogorov relation

within above assumptions from the Navier-Stokes equations:

4 ? ( A V ( r ) z ) (2) ( A V ( r ) s) = - ~ e r + 6v ?r '

where V is the kinematics viscosity, (} stands for ensemble averaging and e is the average rate of

energy dissipation per unit mass. Within the inertial range, the second term on the right-hand side

in Eq. (2) can be neglected

4 ( ~ v ( , ) 3) = _ 3 - ~ , .

This means that: ~(3) = 1.

The classical Kolmogorov theory predicts that: ~(r~) -

(3)

(4)

'~ (5 ) 3

Benzi et al ~l] recently showed evidence for the so-called extended self-similarity in their

measurements of turbulence generated either by a wake flow past a cylinder or by a jet at moderate

Reynolds number:

( A V ( r ) " ) = A n I ( A V ( r ) 3 ) I ~(~) = B~<] A V ( r ) 3 [)~'-(~), (6)

where Am and B, are two different sets of constant independent of r . Since the third-order structure

function ( A V ( r ) 3 ) is proportional to r , instead of plotting the nth-order structure function

( A V ( r ) " ) against r , they plotted the nth-order structure function ( A V ( r ) " ) against the third-

order structure function ( A V ( r ) 3 ) . Eq. (6) is valid not only in the fully developed turbulence

but also at moderate and low Reynolds number, even if no inertial range is established.

Moreover, it has been shown that the range of scales where Eq. (6) valid is much larger than the

inertial range and extends far deep into the dissipation range.

Stolovitzky (1993)[z] repeated the experiments of Benzi and pregented their experimental

results. They measured the time series of the fluctuating velocity at a moderate Reynolds number

in a turbulent boundary layer over a flat plate and investigated the extended self-similarity of the

structure function. They revealed that, for low-order moments, a single scaling law with the

same scaling exponents not far from ~ ( n ) = n /3 for dissipate as well as inertial range.

However, as the order of the moment increases, the scaling law in the dissipate region and in the

inertial region separates out. Within the dissipate region, the scaling exponents are nearly given

by ~ ( n ) = n /3 with ~1(8) = 2 .66 , ~ ( 8 ) = 2 .42 . For r in the inertial range, the plot of

versus consists of another linear region of slope ~1 (8) = 2 .05 , ~z (8) = 2.12 slightly less than

~ ( n ) = n / 3 , joined by a smooth transitional region. The difference between the two regimes

becomes increasingly apparent for higher n .

1 Experimental Apparatus and Techniques

Experiments were conducted in a full-developed turbulent flow of a free-surface water

channel. Velocity measurements in the water channel were taken by TSI anemometer system with

1018 JIANG Nan, WANG Zhen-dong and SHU Wei

a TSI model 1210-20W single-sensor hot-fi lm probe and a TSI model 1218-20W single-sensor

hot-f i lm boundary probe. The coming stream velocity and intermediate Reynolds number were

U~. = 0 . 2 8 m / s and Re�9 = 2570 respectively. The hot-Trim probe was located at y+ = 16 above

the lower wall o f the channel. Fig. 1 shows the longitudinal fluctuating velocity signal obtained

from the hot-f i lm probe located in the near wall region of a turbulent boundary layer.

0.6

0 v

" - 0 . 6 0 1 2 3 4 5 6 7 8 9

t t/s

Fig. 1 Longitudinal fluctuating velocity in the near wall region of

a turbulent boundary layer (y+ = 16)

E x t e n d e d Self-Similar i ty of F luc tua t ing Veloci ty S t ruc tu re Func t ion in

Wall Turbulence

In F ig .2 , we show the structure function log I (AV( r ) " > l against log I ( A V ( r ) 3 ) I in inertial

range and in dissipation range where it is shown that the nth-order structure function has apparently

different scaling laws in dissipation range and in inertial range. The slopes in dissipation range are

much smaller than those in inert ia/range. Fig .3 is a plot of ~1 ( n ) in (6) against n in inertial range

where the scaling exponents ~1 ( n ) are aligned on two curves of which one is larger than ~1 ( n ) =

n/3 for odd-order and the other is smaller than ~l ( n ) = n/3 for even-order.

0.01

0.001

.~ 0.0001

<3 v

1E - 005

1 E - 0 0 6

/ / - -

,, -- 2 f / /~ / /

/ /

/ / / / /

r t = 4 ~ . . / /

/

/ , v /

5"5 I �9 5.0 ' 4.5 : " 0 � 9 4.0 i 3.5 i . . . . 00

3.0 ...... ! - �9 ~ �9 v

2.5 2.0 1.5 1.0 0.5

0 . t . . . . . . . . i . . . . , . . . .

l E - 0 0 6 1 E - 0 0 5 0 . 0 0 0 1 0 . 0 0 1 0.01

<av(~) ~) Hg.2 Structure functionlog I (AV(r) ~) I Hg.3

against log I ( A V ( r ) 3) I

. . . . . �9 ' O

: " O Q

Q t 1 '

. . . . i . . . . . . , . . . . ; ~ , , n , , ,

2 4 6 8 10 12 14 16 n

Plot of the scaling exponents

~l ( n ) against n

In F i g . 4 , we show the structure function log(I A V ( r ) ~ I> against log(I A V ( r ) 3 1) . It is

shown that the nth-order structure function has apparently different scaling laws in dissipation range

and in inertial range. The slopes in dissipate range are much smaller than those in inertial range.

Fig .5 is a plot o f ~2 ( n ) in (6) against n in inertial range where it is shown that ~2 ( n ) = n /3 .

3 Multi-scale Eddy Structure Extended Self-slmilariW Scaling Law

As far as turbulence is concerned, wavelet has special physical meaning. " E d d y " provides

�9 Self-Similar Scaling Law of Multi-Scale Eddy Structure 1019

0.01 n =2

i . / 0.001 '

~" n = 4/.,J"" / e l / o.ooo ~ ~ . ; , , , - ~ i . . . . . .

�9 rt = 6 / + " t I I ;

IE - 005 i/ . . .

" o ' . | ' " o . o l ' o .

<<xV(r) 3)

5.0 4.5 4.0 3.5

,-~ 3.0 "~ 2.5

2,0 1,5 1.0 0.5

0

I

2 4 6 8 10 12 14 I1

16

Fig.4 Structure function log( I A V(r)" I ) Fig.5 Plot of the scaling exponent

against log(I AV(r) 3 I ) ~z(n) against n

the most suitable elementary decomposition of turbulence and is first introduced by Tennekes and

Lumley [3] . According to them, eddies are fairly broad and self-similar contributions in the

spectral domain that, unlike "waves" , correspond to localized contributions in the physical

space. They are localized both in spatial space and in time space. Fourier Transform does not

take into account localized eddies in physics and therefore does not suit to decompose turbulence.

This leads the physical meaning of Fourier Transform lost. Wavelet representation provides the

decompositions of turbulence into eddy modes and the wavelet projection could be a very good

alternative. "Eddies" are to turbulence study, what wavelets are. Fig .6 shows the typical shape

of an "eddy" self-correlation function proposed by Tennekes and Lumley based on turbulence

interpretation. Fig.7 shows the shape of an "eddy" self-correlation function of wall turbulence

obtained by wavelet decomposition from experimental measured signals. As a new tool, wavelet

transform can be devoted to the use for decomposing turbulence into eddies modes instead of

Fourier Transform. Fig. 8 shows the reconstructed single scale eddy velocity by wavelet

transformation of the sampled fluctuating velocity signal shown in Fig. 1.

/ / " ~ \ \

/ / / / I / \\\\

2 /k 0.1k 0.6k x 1.6k 10k Ink I i

Fig. 6 An eddy seN-correlation function of wave number k and wavelength 27r/k [31

In Fig .9 , we show the structure function log I ( A V l ( r ) ~) I against log I ( A V l ( r ) 3 ) I where

the nth-order structure function is aligned on a single line of slope ~l ( n ) = n / 3 . Fig. 10 is a

plot of ~ l ( n ) in (6) against n where it is shown that ~ l ( n ) -- n/3.

1020 JIANG Nan, WANG Zhen-dong and SHU Wei

1.5

1.0

0 .5

0 .0

- 0 . 5

- 1 . 0

o.6

1 . l i E - 01-[ . I ]

8 . l i E - 02:

6 . l i e - 02'

4 . l i E - 02:

2 . l i e - 02:

0 . l i e + l i ,

F i g . 7 An eddy sel f -correla t ion funct ion ob ta ined by wavele t t r ans fo rm

0.06~ " ll

_

1000 2000 3000 3000

0 1000 2000 3000 4 0 0 0

0 i5 'Q 1000 2000 3000 4 0 0 0

0 I 0 0 0 ' 2000 3000 4 0 0 0 0 .2

0

- 0 " 2 0 1 000 2 000 4 000 0 . 1 5 / 3 0 ~

0.12'10 1 l iO 2 000 3 000 4 000

0 1000 2000 3000 4 0 0 0

Fig . 8

0.01

0.001

0.000 l

~- I E - 005

<1 1E - 006

1E - 007

1E - 008

Recons t ruc ted veloci ty s ignal for each single scale eddy by wave le t decompos i t ion

I E - 008 1 E - 005i

0.1 0.01

0.001

0.0001

1E - 005

IE - 006

1E - 007 1E - 008

0.01 I

. . . . . o.ooo t )

1 E - 006 ~- ] , - , - i E _ 0 0 7 [ 1 E T , 0 0 8 ' , t E - O 0 5 ,,

IE - 009 I E - 007 0.000 1 1 E - 0 0 9 q E - 007: 0.0001 1 E - 006 0.001 1E - 006 . 0.001 1 E - 006

( •

a = 8 a = 16 a = 32

F i g . 9 The n t h - o r d e r structure funct ion log( I &Vt ( r ) " I ) aga ins t

log( I A Vz ( r ) 3 I ) for each scale eddy

1E - 009 1 E - 007 0.000 0.00t

Self-Similar Scaling Law of Mult i -Scale Eddy Structure 1021

0.01

0.001

0.000 1

1E - 005

I E - 006

1E - 007 IE - 009 I E - 007

I E - 008 1E - 005

0.0001

1

0.1

0.01 ---- - - ~

0.001 - - ~

0.0001 / j ' ' ' /

1E- 005 'l l z - 008 IE - 005~ �9 1 E - 0 0 6

1E - 009 1E - 00T 0.000 IE - 006 0.001

a = 6 4

1E - 006

a = 128

1

0.1

0.01

0.001

0.000 l

1E - 005

IE - 006

0.t301

E: oo8 !z-,,oo5 1 E - 009 1 E - 007. 0.0001

tE - 0t36 0.00t

a = 256

1

0.1

0.01

0.00l

0.0001

1E - 005 1E - 009 1E - 007

IE - 006

f

, /

. - ~ / " j

1E - 008 1E - 005

0.0001 0.001

0.01

0.001

0.000 1

1E - 005

1E - 006 IE - 008 1E - 005 i ,i i , i ,~

1 E - 009 1 E - 007 ' 0.000 1 1 E - 006 0.1301

Fig . 9

a = 512 a = 1024

The n th-order structure function log( I AVi ( r ) " I ) against

log( I A V z ( r ) 3 I ) for each scale eddy

3.5 3.0 2.5

~" 2.o 1.5 1.0 0.5

0

3.5 3.0 2.5 2.0

1.5 1,0 0.5

0

i , ; ' I I i

2 3 4 5 6 7 8 9 1 0 n

a = 8

./ ./

/ : / / "

/. �9 //

X / /

z i i i , , i i

2 3 4 5 6 7 8 9 1 0

3 . 5

3.0 / /

2 .0 1.5 . / 1.0 . . . . . . . 0.5 / / . ,

0 ~ ', , , , i , L

1 2 3 4 5 6 7 8 9 1 0

3 . 5 ' 3.0 2.5 2.0 1.5 1.0 0.5

0

?l

a = 1 6

3.5 3.0 2.5 2.0 1.5 1.0

"l 3.5 . . . . . [ / " 3.0

: / - 2.5 / , / 2.0

/ " l . 5

1 . 0

: / 0 . 5 0 , . . . . , i i

2 3 4 5 6 7 8 9 1 0 rt n

, / /"

/

i i i i , , , i

2 3 4 5 6 7 8 910 TL

a = 3 2

//[ . / / / / ]

/ / /

I . : (

2 3 4 5 6 7 8 9 1 0

a = 64 a = 128 a = 256

Fig . 10 Plot o f ~t ( n ) against n for each scale eddy

1022 J1ANG Nan, WANG Zhen-dong and SHU Wei

3.0 2.5 2.0 . . . . . . . . : ~

1.5 1 1 .o : 0 . 5 . . . . . . . .

0 . . . . . i . . . . . 2 3 4 5 6 7 8 9 1 0

a = 512

3.5 : . . , / 3.0 - / 2.5 2.o i / /

1.5 /

1.0 / 0.5 .

0 . . . . . , , ,

2 34 5 6 7 8 910 11

a = 1024

H g . 10 Plot of ~t ( n ) against n for each scale eddy

F ig . 11 is a p lo t that l o g ( I & V t ( r ) ~ I ) versus l o g ( I A V L ( r ) 3 I ) w h e r e the n th -o rde r

structure funct ion i s a l igned on a s ingle l ine o f s lope ~_ ( n ) = n / 3 . F i g . 12 is a p lo t o f ~,_ ( n ) in

( 6 ) against n where it is also shown that ~_

0.001 0.1

/ 0.01

0.0001 " " ~ ' / " ......... 0.001

IE - 005 IE o08 IE-'oo~ o.ooo x

a = 8

n) = n / 3 .

J

7 / o.oool /

~E- oo5 ~E-005 , 0.ooi 1E-008 0 . 0 0 0 1 0.01

0.1

0.01 f /

O. 001 . / "

. / i "

0.0001

~E-oo5 i z - o o s ,, o.om 1E- 008 0.0001 0.01

a = 1 6 a = 3 2

0.1

o.01

0.001

0.0o01

1E - 005

f /

/

I E - 005 0.001 �9 I , -)- ,

0.0001

a = 6 4

0.01

0.1

0.01 . . . . . . - /

/ / 0 . 0 0 1

/ / / 0.000 1

[E - 005 0 . ( 3 0 l IE IE-008005 , 0 .0~1

t

0.01

a = 128

0.11 I

o.01

0.001

0.o0o 1

IE - 005

/ i /

/ . / *

/ -

1E -005 i 0.(301 1E-008 0.0001

a = 256

0.01

0.1

0.01

0.001 i

/ o.0oo 1 . . . . i

1E- 0 0 5 1 E - 005 0.001 1 E - 0 0 8 0.01301 0.0l

0.1

0.Ol

o.ool

0.OOOl

IE - 005 1E - 008

/ /

o.00Ol o.01

Fig. 11

a = 512 a = 1024

The nth-order structure function log( I A Vz ( r ) " I )

versus log( I A Vt ( r)3 I ) for each scale eddy

Self-Similar Scaling Law of Multi-Scale FAdy Structure 1023

3.5 3.0 2.5 2.0 1.5 1.0 0.5

3.5 3.0 2.5 2.0 1.5 1.0 0.5

0

/ / " / *

/

.f

s

2 34 56 7 8910 n

a = 8

i / / i

2 3 4 5 6 7 8 9 1 0 II

a = 6 4

3.51 3.0: 2 . 5 2.0 1.5 1.O 0.5

0

3.5 3.0 2.5 2.0 1.5 1.0 0.5

0

3.5 3.0 2.5 2.0 1.5 1.0 0.5

0

2 3 4 5 6 7 8 9 1 0 n

a = 512

Fig. 12

2 3 4 5 6 7 8 9 1 0

a = 1 6

"l : �9 / /

/ / / ' / "

2 3 4 5 6 7 8 9 1 0 /l

3.5 3.0 2.5 2.0 1.5 1.0 0.5

0 ; i ; i , i ~ ; 2 3 4 5 6 7 8 9 1 0

n

a = 3 2

3 5 3.0 . , : :

2.5 " //

2.0 �9 / / ' : l 1.01"5 /./// i 0.5

2 3 4 5 6 7 8910 rL

a = 128

3.5 1 3.0 . . . . ;,,/"

7 / / 2.0 . /

i / 1.0 71 / / /

0.5 0

2 3 4 5 6 7 8 9 1 0 I1

a = 1024

Plot ~2 ( n ) against n for each scale eddy

a = 256

4 C o n c l u s i o n s

1) The statistical properties of turbulence could be self-similar not only at high Reynolds

number, but also at moderate and low Reynolds number , and they could be characterized by the

same set of scaling exponents of the fully developed regime.

2) The range of scales where the extended serf-similarity valid is much larger than the

inertial range and extends far deep into the dissipation range.

3) The extended self-similari ty is applicable not only for homogeneous turbulence, but also

for shear turbulence such as turbulent boundary layers .

R e f e r e n c e s :

[1] Benzi R, Ciliberto S, Tripiccione R, et al. Extended self-similarity in turbulence f l o w s [ J j .

Physical Review E, 1993,48 ( 1 ) : 29 ~ 32.

[2] Stolovitzky G, Sreenivasan K R. Scaling of structure functions[ J ] . Physical Review E, 1993,48

(1) : 3 3 - 36.

[3] Tennekes H, Lumley J L. A First Course in Turbulence [ M ] . Cambridge Massachusetts and

London, England: M1T Press, 1972.