Evidence for a mixing transition in fully-developed pipe flow

Beverley McKeon, Jonathan MorrisonDEPT AERONAUTICSIMPERIAL COLLEGE

Summary

• Dimotakis’ “mixing transition”

• Mixing transition in wall-bounded flows

• Relationship to the inertial subrange

• Importance of the mixing transition to self-similarity

• Insufficient separation of scales: the mesolayer

Dimotakis’ mixing transitionJ. Fluid Mech. 409 (2000)

• Originally observed in free shear layers (e.g. Konrad, 1976)

• “Ability of the flow to sustain three-dimensional fluctuations” in Konrad’s turbulent shear layer

• Dimotakis details existence in jets, boundary layers, bluff-body flows, grid turbulence etc

Dimotakis’ mixing transitionJ. Fluid Mech. 409 (2000)

• Universal phenomenon of turbulence/criterion for fully-developed turbulence

• Decoupling of viscous and large-scale effects

• Usually associated with inertial subrange

• Transition: Re* ~ 104 or R~ 100 – 140

Variation of wake factor with Re

From Coles (1962)

Pipe equivalent: variation of • For boundary layers wake factor from

• In pipe related to wake factor

• Note that is ratio of ZS to traditional outer velocity scales

y

WByU Cln1

BU

ln

12

uUUCL /

)(Re

2

3ln

143 DCLCL CCBRUUU

Same kind of Re variation in pipe flow

0

1

2

3

4

5

6

0 200000 400000 600000 800000 1000000

ReD

Nikuradse smooth pipe data (1932)

uUUCL /

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

ReD

McKeon et al analysis of ZS data

den Toonder and Nieuwstadt(1997)

Identification with mixing transition

• Transition for R~ 100 – 140, or Re* ~ 104

• ReR* = 104 when ReD ~ 75 x 103 (R110 when y+ = 100, approximately

• This is ReD where begins to decrease with Reynolds number

• R varies across pipe – start of mixing transition when R

• Coincides with the appearance of a “first-order” subrange (Lumley `64, Bradshaw `67, Lawn `71)

uUUCL /

Extension to inertial subrange?

• Mixing transition corresponds to decoupling of viscous and y scales - necessary for self-similarity

• Suggests examination of spectra, particularly close to dissipative range

• Inertial subrange: local region in wavenumber space where production=dissipation i.e. inertial transfer only

• “First order” inertial subrange (Bradshaw 1967): sources,sinks << inertial transfer

Scaling of the inertial subrange

For

K41 overlap

In overlap region, dissipation

11

1 ky

Ryu

3/51

3/2111 )( kAk

2111

3/51

3/23/2

2111

3/51 )()()()(

v

kkuvuvC

u

ykyk

y

uuv

3

Inertial subrange – ReD = 75 x 103

2

3/5 )(

u

kyky

2

3/53/2

)(

v

kkuv

y+ = 100 - 200 i.e. traditionally accepted log law region

Scaling of streamwise fluctuations u2

Similarity of the streamwise fluctuation spectrum I

Perry and Li, J. Fluid Mech. (1990). Fig. 1b

Similarity of the streamwise fluctuation spectrum II

y/R = 0.10

y/R = 0.38

Outer velocity scale for the pipe data

u

UUCL UUUU

CL

CL

y/R

ReD < 100 x 103

100 x 103 < ReD < 200 x 103

ReD > 200 x 103

ZS outer scale gives better collapse in core region for all Reynolds numbers

Self-similarity of mean velocity profile requires = const.

• Addition of inner and outer log laws shows that UCL

+ scales logarithmically in R+

• Integration of log law from wall to centerline shows that U+ also scales logarithmically in R+

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

ReD

McKeon et al analysis of ZS data

den Toonder and Nieuwstadt(1997)

• Thus for log law to hold, the difference between them, , must be a constant

• True for ReD > 300 x 103

Inner mean velocity scaling

y+

= 0.421

= 0.385?

Power law

= U+ - 1/ ln y+

A B C

Relationship with “mesolayer”

• e.g. Long & Chen (1981), Sreenivasan (1997), Wosnik, Castillo & George (2000), Klewicki et al

• Region where separation of scales is too small for inertially-dominated turbulence OR region where streamwise momentum equation reduces to balance of pressure and viscous forces

• Observed below mixing transition• Included in generalized log law formulation

(Buschmann and Gad-el-Hak); second order and higher matching terms are tiny for y+ > 1000

0dy

uvd

Summary

• Evidence for start of mixing transition in pipe flow at ReD = 75 x 103. (Not previously demonstrated)

• Correspondence of mixing transition with emergence of the “first-order” inertial subrange, end of mesolayer

• Importance of constant for similarity of mean velocity profile (Reynolds similarity)

• Difference between ReD for mixing transition and complete similarity