evidence for a mixing transition in fully-developed pipe flow beverley mckeon, jonathan morrison...
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![Page 1: Evidence for a mixing transition in fully-developed pipe flow Beverley McKeon, Jonathan Morrison DEPT AERONAUTICS IMPERIAL COLLEGE](https://reader036.vdocuments.net/reader036/viewer/2022062309/56649d445503460f94a21a49/html5/thumbnails/1.jpg)
Evidence for a mixing transition in fully-developed pipe flow
Beverley McKeon, Jonathan MorrisonDEPT AERONAUTICSIMPERIAL COLLEGE
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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
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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
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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
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Variation of wake factor with Re
From Coles (1962)
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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
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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)
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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 /
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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
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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
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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
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Scaling of streamwise fluctuations u2
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Similarity of the streamwise fluctuation spectrum I
Perry and Li, J. Fluid Mech. (1990). Fig. 1b
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Similarity of the streamwise fluctuation spectrum II
y/R = 0.10
y/R = 0.38
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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
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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
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Inner mean velocity scaling
y+
= 0.421
= 0.385?
Power law
= U+ - 1/ ln y+
A B C
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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
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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