1

Observations on the scaling of the streamwise velocity component in

wall turbulence

Jonathan Morrison

Beverley McKeon

Dept. Aeronautics, Imperial College

Perry Symposium, Kingston May 2004Perry Symposium, Kingston May 2004

2

Synopsis• Self-similarity – what should we be looking for?• Log law: self-similar scaling - • Local-equilibrium approximation as a self-similar

energy balance in physical space.• Spatial transport – interaction between inner and

outer regions.• Self-similarity of energy balance appears as inertial

subrange in spectral space.• Consistent physical- and spectral- space views.• Self-similarity as a pre-requisite for universality.

uy,

3

What is self-similarity?

• Simultaneous overlap analysis for , indicates motion independent of inner and outer lengthscales

• Therefore the constant in the log argument is merely a constant of integration and may be freely chosen.

• It is usually taken to be the dominant imposed lengthscale so that its influence on B or B** is removed: as

• Overlap analysis indicates is universal: but self-similarity is a pre-requisite.

6.5, By

**ln1

,ln1

Bk

y

u

UB

yu

u

U

5.8**, Bk

y

Ryu

4

The local-equilibrium approximation

• Application of self-similar scaling to the energy balance gives P =

• Therefore expect log-law and local-equilibrium regions to be coincident.

• Inertial subrange is self-similar spectral transfer, T(k), as demonstrated by simultaneous overlap with inner and outer scaling.

• Wavelet decomposition (DNS data, JFM 491) shows T(k) much more spatially intermittent than equivalent terms for either P or

• Therefore T(k) is unlikely to scale simply.• Even then, energy balance at any point in space is an

integration over all k – so P = will only ever be an approximation.

• Usefulness of a “first-order” subrange (Bradshaw 1967)?

uy,

5

Self-similarity of the second moment• Examine self-similarity using distinction between inner

and outer influences in wall region.

• Examine possibility of self-similarity in .

• If not self-similar, then is very unlikely to be either.

• Comparison of Townsend’s 1956 ideas with those of 1976 – are outer-layer influences “inactive”?

• Use these ideas to highlight principal differences between scaling of in pipes and boundary layers, and even between different flows at the same R+.

)( 111 k)( 111 k

)1( yuy

)( 111 k

2u

)( RuR

6

“Strong” asymptotic condition: R+=∞

• As , and , “large eddies are weak” (Townsend 1956).

• “Neglecting this possibility of outside influence”:

where is a “universal” function.• Therefore, provided is independent of y,

collapse on inner variables alone is sufficient to demonstrate self-similarity.

• Then:

R 0/ Ry

)(),,( 12

111 ykyuuyk

11

211

ku

11

7

“Strong” asymptotic condition• Neglecting streamwise gradient of Reynolds stress:

• Write with

• Last term negligible (scale separation again) and removal of cross product linearizes the outer influence.

• “Inactive motion is a meandering or swirling made up from attached eddies of large size……” (Townsend 1961).

ouuu i

2

2)(

y

Uwvuv

yyz

xωu

0oω

xioxiiuvy

ωuωu

)(

8

Conclusions from“strong” asymptotic condition

• Write :

• Blocking means that

• Therefore, and are, to first order, only.

• But: , and outer influence

appears as a linear superposition.

• Therefore, at the same R+, internal and external flows

are the same.

ouuu i

uv

2v yF

0

20

22 2 uuuuu ii

xy

uv 00

yFu ii

2 RFu oo

2

0

9

What’s wrong with this picture?Superpipe

100 101 102 103 104 1050

1

2

3

4

5

6

7

8

9

10

ReD

y+

___

u2 +

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

WALL SCALING

10-3 10-2 10-1 1000

1

2

3

4

5

6

7

8

9

10

____

u2+

y/R

ReD

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

OUTER SCALING

10

What’s wrong with this picture ?Self-similar structure

• implies hierarchy of self-similar, non-interacting attached wall eddies that makes valid the assumption of linear superposition.

• Then:

• Even atmospheric surface layer show that this is not the case: the absence of direct viscous effects is an insufficient condition.

112

1

)(

)()(ˆyy

yv

yvyvRvv

11k

11

Hunt et al. (Adv. Turb 2, Springer 1989)

Linear superposition:

12

Hunt et al. (Adv. Turb 2, Springer 1989)

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What’s wrong with this picture? Wall-pressure fluctuations

• Integration of spectrum

in the approximate range

gives:

where B ≈ 1.6 and C > 0.

• Therefore, even consideration of the active motion alone shows that:

1. wall-pressure fluctuations increase with R+,

2. large scales penetrate to the wall: the near-wall region is not “sheltered”.

1

2

1)(k

Ak wpp

30)1.0( 11 ukR

CRBp

w

w ln2

2

14

A “weak” asymptotic condition: R+→∞

• “Superpipe” data show that outer influence:

1. is not “inactive” and interacts with inner component

2. increases with R+

3. increases with decreasing distance from the wall.• Therefore, linear decomposition is not possible, i.e:

• Therefore to demonstrate complete similarity, we must have simultaneous collapse on inner and outer variables.

• “It now appears that simple similarity of the motion is not possible with attached eddies and, in particular, the stress-intensity ratio depends on position in the layer”

(Townsend 1976).

);();(2 RyGRyyFu

15

Superpipe spectra

10-1 100 101 102 103 1040

0.5

1

1.5

k1R

k1R (k1R)

u2

ReD=3.1M, R+=5.4x104

ReD=5.7M, R+=1.0x105

y/R y+

0.030 1.6x103

0.051 2.7x103

0.096 5.2x103

0.030 3.0x103

0.051 5.1x103

0.096 9.6x103

_________

- - - - -

10-3 10-2 10-1 100 101 102 1030

0.5

1

1.5

k1y

k1y (k1y)

u2

ReD=3.1M, R+=5.4x104

ReD=5.7M, R+=1.0x105

y/R y+

0.030 1.6x103

0.051 2.7x103

0.096 5.2x103

0.030 3.0x103

0.051 5.1x103

0.096 9.6x103

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Laban’s Mills surface layer (Högström)

10-1 100 101 102 103 1040

0.5

1

1.5

2

3.1m

6.3m

"Sorbus" - Outer Scaling

8 hours sampling

z0=1.2 cm

500 m

k1

k1 (k1 )

u2

10-3 10-2 10-1 100 101 1020

0.5

1

1.5

2

3.1m

6.3m

"Sorbus" - Inner Scaling

8 hours sampling

z0=1.2 cm

500 m

k1y

k1y (k1 y )

u2

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Observations

• Both the “strong” and “weak” asymptotic conditions lead to complete similarity.

• spectra in both the “superpipe” and the atmospheric surface layer show only incomplete similarity. In both cases, R+ is too low to show complete similarity.

• As the Reynolds number increases, the receding influence of direct viscous effects has to be distinguished from the increasing influence of outer-layer effects, because the inner/outer interaction is non-linear:

)( 111 k

);(2 RyyFu

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Nature of the inner-outer interaction

• Streamwise momentum:

• The mesolayer defined by

• Balance of viscous and inertial forces gives length scale

• The energy balance in the mesolayer involves turbulent and

viscous transport, as well as production and dissipation.

• Since , a mesolayer exists at any R+.

• The lower limit to the log law should be expected to increase

approximately as .

21 Rm

Uwvuvy

yz2)(

0)(

uvy

RR 2

11

21

R

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The mesolayer:locus of outer peak in

0 50000 1000000

200

400

600

800

1000

yp+

R+

1.8(R+)0.52

uv 5.0)(5.1 RLocus of :

(momentum equation and log law)

First appearance of log law

2u

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Nature of the inner-outer interaction

• The occurrence of a self-similar range cannot be

expected in or below the mesolayer.

• The full decomposition

shows how ideas concerning widely separated wavenumbers

can be misleading – inner/outer interaction is a more

important consideration if looking for self-similarity,

• Does inner/outer interaction preclude self-similarity of

inertial-range statistics?

• Is this most likely in the local-equilibrium region?

11k

xooxioxoixiiuvy

ωuωuωuωu

)(

not small

R

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A first-order inertial subrange• Bradshaw (1967) suggested that a sufficient condition for a

“first-order” subrange is that T(k) >> sources or sinks.

• This occurs in a wide range of flows for

• No local isotropy: , but decreasing rapidly as

• Local-equilibrium region is a physical-space equivalent, where small spatial transport appears as a (small) source or sink at each k (JFM 241).

• Saddoughi and Veeravalli (1994) show two decades of -5/3: lower one, : higher one for

• How does the requirement of self-similar T(k) fit in?

100R

0)( 112 kR1k

0)( 112 kR1500R

0)( 112 kR

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A self-similar inertial subrange• Simultaneous collapse

• and can be estimated from local-equilibrium approximation and the log law: y/R=0.096

• No specific requirement for local isotropy:

1112

11112

1 35

353

2

kk

Cyku

yk

uv

ReD R

55k 105

75k 140

150k 210

230k 270

1.0m 575

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Inertial subrange scaling: u

outer scale

10-2 10-1 100 101 102 1030

0.1

0.2

0.3

0.4

0.5

0.6

ky 10-4 10-3 10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.655k75k150k230k1.02m

k

1562 y4 kHz

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Outer velocity scale – second moment

10-2 10-1 1000

2

4

6

8

10

ReD

y/R

___

u2

4.9x103

1.0x104

1.8x104

2.5x104

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

Outer velocity scaleSuperpipe with data of den Toonder

u

10-2 10-1 1000

0.1

0.2

0.3

0.4

0.5

ReD

y/R

___

u2

4.9x103

1.0x104

1.8x104

2.5x104

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

Outer velocity scaleSuperpipe with data of den Toonder

Ucl-U__

25

Outer velocity scale – fourth moment

10-2 10-1 1000

25

50

75

100

125

150

175

ReD

y/R

___

u4

4.9x103

1.0x104

1.8x104

2.5x104

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

Outer velocity scaleSuperpipe with data of den Toonder

u

10-2 10-1 1000

0.1

0.2

0.3

0.4

0.5

ReD

y/R

___

u4

4.9x103

1.0x104

1.8x104

2.5x104

5.5x104

7.5x104

1.5x105

2.3x105

4.1x105

1.0x106

3.1x106

5.7x106

Outer velocity scaleSuperpipe with data of den Toonder

Ucl-U__

26

scaling

11122

121112

1 35

353

2

kkC

ykUU

yk

uv cl

UUcl

28.456.4

u

UUcl

27

Inertial subrange scaling: outer scale

10-2 10-1 100 101 102 1030

0.01

0.02

0.03

ky

UUcl

10-4 10-3 10-2 10-1 100 1010

0.01

0.02

0.0355k75k150k230k1.02m

k

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Conclusions - I

• Statistics in boundary layers at short fetch and high velocity will not be the same as those at long fetch and low velocity.

• and in pipes and boundary layers at the same R+ are not the same.

• “fully-developed pipe flow” does not appear to be a universal condition.

• But, self-similarity does lead to universal properties (log law, inertial subrange?), but R+=constant does not.

)( 111 k2u

29

Conclusions - II

• Inner/outer interaction dominates: “top-down” influence increases with increasing R+, and decreasing y/R.

• Mesolayer determines lower limit to log region.

• is a better velocity scale for

• The pressure velocity scale is only a second-order correction:– at R+ = 5000, 7%.

21~ R

UUcl 31075Re D

31

2

R

up

pu

30

Conclusions III • In local-equilibrium region, self-similar inertial

subrange appears above• Departures from self-similarity: retain -5/3 scaling,

but relax condition = constant (Lumley 1964):

where .• Then, if • Need to look in outer region: larger spatial transport,

but larger.

500R

21

31

)(~

kkE

k

kT

k

25

23

)()( kkEkT

1,)( kT

R