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Turbulent Scaling in the Viscous SublayerTurbulent Scaling in the Viscous Sublayer

Dmitrii Ph. SikovskyDmitrii Ph. Sikovsky

Novosibirsk State University, Laboratory for Nonlinear Wave Processes,Kutateladze Institute of Thermophysics, Laboratory for Fundamentals of Energy Technologies

SOLITONS, COLLAPSES AND TURBULENCEVI-TH INTERNATIONAL CONFERENCE, NOVOIBIRSK, RUSSIA, 4-8 JUNE 2012

Turbulence in Channel

,eU , ,wdP

dx

,w

vFriction velocity

( ) ?U y ( ) ?n m ku w y v

GoverningParameters: ,eU ,v , Reynolds Number: Re ,eU

,

v

2

2?w

fe

cU

Recent debate concerning the mean-velocityprofile of a turbulent wall-bounded flow

Derivation of Logarithmic Law of the Wall:Similarity hypothesis: Th. von Karman (1930)

Mixing-length hypothesis: L.Prandtl (1932)Dimensional reasoning: L.D.Landau (1944)

Questions and DoubtsNikuradse, J., 1932. Gesetzmabigkeiten der turbulenten Stromung in glatten Rohren. VDI-Fortschritt-Heft, No. 356.Nunner, W., 1956. Warmeubergang und Druckabfall inrauhen Rohre. VDI-Forschungsheft, No. 455.Simpson RL. Characteristics of turbulent boundary layersat low Reynolds numbers with and without transpiration. J Fluid Mech 1970;42:769–802.Malkus WVR. Turbulent velocity profiles from stability criteria. J Fluid Mech 1979;90:401–14.Barenblatt GI. Similarity, self-similarity, and intermediate hypothesis. New York: Plenum; 1979.Long RR, Chen T-C. Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 1981;105:19–59.Wei T, Willmarth WW. Reynolds-number effects on the structure of a turbulent channel flow. J Fluid Mech 1989; 204:57–95.George WK, Castillo L, Knecht P. The zero-pressure gradient turbulent boundary layer revisited. In: Reed XB, Patterson GK, Zakin JL, editors. Thirteenth symposium on turbulence. Rolla, MO: University of Missouri; 1992.Bradshaw P. Turbulence: the chief outstanding difficulty of our subject. Exp Fluids 1994;16:203–16.Purushothaman K. Reynolds number effects and the momentum flux in turbulent boundary layers. PhD dissertation, Department of Mechanical Engineering, Yale University, New Haven, CT, 1993.Smith RW. Effect of Reynolds number on the structure of turbulent boundary layers. PhD thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1994.Afzal N. Power law and logarithmic law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers. Acta Mech 2001;151:195–216.Buschmann M.H., Gad-el-Hak M. Recent developments in scaling of wall-bounded flows. Prog. Aero. Sci. 2007. 42:419-67

Plan of the presentation

1. Classical approach from the viewpoint of method of matched asymptotic expansions

2. Deficiencies of classical scaling

3. New scaling for the viscous sublayer

Formulation of the Problem22

12

d U d u dP

dy dxdy

vvReynolds averaged equationfor mean momentum transport:

Boundary conditions:at the wall,

2

0, 0,dU

U udy

vv

at the centerline,

0 :y

:y , 0, 0e

dUU U u

dy v

2 1y dU

udy

v v

Viscosity is a small parameter at higher derivative.

Singular perturbation problem! (K.Yajnik, 1970)

0

Matched Asymptotic Expansions

2 1y dU

udy

v vOuter asymptotic expansion (K.Yajnik, 1970)

in the outer layer

20 1 ( )u O v 2

0 1y

v 1

dU

dy

20 1 ( )U U U O

0 does not satisfy the wall boudary condition 0y 20 v

(1)y O

0 ?U 1 ?U

Viscosity does not affect Reynolds shear stress in the leading order of approximation!

Extension to mean velocity, whole Reynolds stress tensor and higher-order moments of velocity fluctuations leads to Townsend’s Reynolds Number Similarity Principle (A.Townsend, 1956)

M.Van Dyke. Perturbation Methods in Fluid Mechanics.N.Y.:Acad. Press, 1964.

Classical Scaling: Outer Layer2~ ,u v v

, ( )u O v v

10 ( ) ( )eU U F O

v v

Since in the near-wall region then it is reasonable to assume that

( ) ( )eU U y O v

y

Outer layer expansions

1 12

1 (Re )u dF

Od

vv

10 1 ( ) ( )e

UF O

U

2e fU c v

Classical Scaling: Inner Layer

Inner expansions: Law of the Wall

( ) ( ),U y O v

2~ ~dU

udy v v

Rescale the variables in equation so that the viscous stress will be of the same order as Reynolds stress

If then the stretched varible isy

yl

l

v

12

1 ( )u df

Ody

vv

1( ) ( )U

U f y O

v

This can be derived from classical similarity hypothesis (Monin,Yaglom, 1965; Kader, Yaglom, 1980): friction velocity is a scale for the velocity fluctuations and mean velocity differences

Overlap Layer: Matchingy

yl

l y 0y

Overlap layer

( ) lim ( )y

f y f y

Introduce 0

0( ) lim ( )F F

10( ) ( ) ( )f F

Matching leads to Pexider’s functional equation

1( ) lnf y y B Solution is Logarithmic Law of the Wall

10 ( ) lnF B D Friction Law: 1 1 1( ) ln ( )D O

0.4, 5B

From experimental data: 50 0.1y Log layer observed in the range

Van Dyke Matching Principle: The m-term inner expansion of the n-term outer expansion = the n-term outer expansion of the m-term inner expansion

Von Karman constant:

Weakness of Classical Scaling1. Usage of friction velocity as a velocity scale for the outer flow2. Influence of the outer scales on the statistics in the viscous sublayer3. Possible incomplete similarity

DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

Alternative Theories:Power Law or Log Law?

0 ( )( ) dFdf yy

dy d

N.Afzal: No Reynolds number similarity in the overlap layer

3 5( )

2f y y

G.I.Barenblatt: No Reynolds number similarity

W.K.George: Reynolds number dependence of , B

3

2ln Re

i oC y C ln( )

lno iC C

Experimental Results: SuperPipe Data

McKeon et al., 2004 (JFM):

Logarithmic law is foundin the range 600 0.12y

Power law is found in the range 50 300y

1ln 5.6

0.421U y

0.142 1 78.48 ~U y y

?

Revisiting the Classical Scaling in the Viscous Sublayer

2~ ~dU

udy v v

There can be a continuum of possible scalings in the viscous sublayer. Viscous sublayer thickness can be chosen arbitrarily, if the mean velocity scale is chosen as

l2l

U

v

Consider the following exact conditions at the wall for the derivatives of mean velocity and Reynolds shear stress following from Navier-Stokes equation (Monin, Yaglom, 1975)

0,n

n

d u

dy

v 0 2n 3

36

d u

dy

v

2dU

dy

v 22

2

d U

dy

v 3

30

d U

dy

4

4

6d U

dy

2

00.5 ( / )( / )

yu y w y z

where

Revisiting the Classical Scaling in the Viscous Sublayer

New scale for viscous sublayer thickness

2 3 1 3l v

Introduce the new variables 1 3 2 3y y

v 1 3 8 3U U v 2/u v v

1 ( )dU

O ydy

0,n

n

d

dy

0 2n 3

36

d

dy

0U 1dU

dy

2

20

d U l

dy

3

30

d U

dy

4

46

d U

dy

Mean momentum transport:

Boundary conditions at the wall:

Universal scaling of mean velocityand Reynolds shear stress:

( )U U y ( )y

3~u y v 3 2~l v

l vinstead of

Comparison with DNS Data: Mean Velocity and Reynolds Shear Stress

180 2000 Comparison of classical (left) and new (right) scaling of Reynolds shear stresses and mean velocity (inserts). Curves – DNS data in the range , dashed lines mark off the rough border of viscous sublayer.

Comparison with DNS Data:Rms. Streamwise velocity

180 2000

Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

Comparison with DNS Data:Rms. Normal Velocity

Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

Comparison with DNS Data:Rms. Spanwise Velocity

Comparison of classical (left) and new (right) scaling of Reynolds stresses and mean velocity (inserts). Curves – DNS data. Lines – plane channel (Hoyas, Jimenez, 2006), dotted lines – circular pipe (Wu, Moin, 2008), dash dotted lines – zero pressure gradient boundary layer (Schlatter et al., 2009)

Overlap Layer: Matching 8 3 1 1 3 ( )e oU u F f y v

( ) ( ) ( )H G F f

By introducing the new variables 1 3 2 3 v 1 3 8 3

eH U v 1 3 8 3oG u v

this equations transformed into the Vincze’s functional equation

General solution is1

( )y

f y A C

1

( )F A D

( )G 1

( )H A C D

Power Law with Non-Universal Exponent1 3

1 3

ln( )

ln( )ou

Reynolds number dependence of parameter

ZPG TBL

Circular pipe

Plane channel

5 1 24 10

(Cenedese et al., 1998)

Conclusion

1) The new inner scaling is proposed, in which the new relevant parameter – the third-order wall-normal derivative of the Reynolds shear stress at the wall is added. Proposed scaling gives the better collapse for both the mean velocity and Reynolds stresses profiles in the viscous sublayer at different Reynolds numbers and types of flow.

2) It is found that the matching condition for the mean velocity in the overlap region is equivalent to Vincze’s functional equation and has the solution in the form of power law. At finite Reynolds numbers the power law exponent is not universal and may be dependent not only of Reynolds number but of the flow type.