Implications of recent Ekmanlayer DNS for nearwall similarity

1
Implications of recent Ekman-layerDNS for
near-wall similarity
Gary Coleman, Philippe Spalart, Roderick
Johnstone University of Southampton Boeing
Commercial Airplanes

• x
• UK Turbulence Consortium

2
Turbulent (pressure-driven) Ekman layer

• Balance between pressure gradient, Coriolis and
  friction
• ? 3D boundary layer
• Defining parameter Reynolds number ReGD/n ,
  where
• G ? freestream/geostrophic wind
  speed
• D (2n/f)1/2 ? viscous boundary-layer depth
• f 2Wv ? Coriolis/rotation parameter
• m/r ? kinematic viscosity

Wv
Hodograph
v/G
Re
u/G
-?P
3
Parameters

• Re 1000, 1414, 2000 and 2828 d Re1.6
• (Neglecting mid-latitude effects Wh0)

4
Relevance

• Flow over swept-wing aircraft, turbine blades,
  within curved ducts, etc
• Planetary boundary layer
• Canonical near-wall turbulence
• ideal test case for near-wall similarity
  theories, i.e. laws of the wall
• Q. But what about rotation, skewing, FPG?
• A. If Re is large enough, we assume that these
  dont matter (cf. Utah
  atmospheric data). Recall hodograph is nearly
  straight for 80 of Ue

5
The Quest for the Law of the Wall

• Expectations for unperturbed turbulent
  boundary layer
• Mean velocity U U(z,tw,r,m) ? U f(z), for
  large z and small z/d,
• and U (1/k) ln(z) C ? defines the log
  layer
• Impartial determination Karman measure k(z)
  d ( ln z ) / d U
• If expectations valid, then k(z) ? constant in
  the logarithmic region
• History
• Until 70s classical experiments, Coles.
• Probable range k from 0.40 to 0.41 (although
  k-e was higher)
• 80s and 90s channel and ZPG boundary layer DNS
• DNS was not yet strong enough
• 00s pipe and BL experiments, channel and Ekman
  DNS
• Cold War started range now 0.38 to 0.436!
  (Oh dear)
• Q. Is DNS strong enough now? (A. well, sort of)
• Industrial impact
• k controls extrapolation of drag to other
  Reynolds numbers
• ? Going to Rex 108, changing k from 0.41 to
  0.385 changes skin friction by 2 (well, assuming
  unchanged S-A RANS model in outer layer)

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Karman Measure

• Expected qualitative behavior in channel flow
• S-A model, for illustration only
  (Mellor-Herring buffer-layer function)

Increasing Re
z
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Looking for the Karman Constant in DNS

• Expected qualitative behavior
  High-Reynolds-number DNS

Oh dear
Increasing Re
z
z
8
Ekman-Layer DNS at Re 2828

• Coriolis term allows BL homogeneous in x, y and t
• Pressure gradient, equivalent to channel at Ret
  1250
• Boundary-layer thickness
• d ? 5000n/ut
• Fully spectral Jacobi/Fourier BL code
• 768 x 2304 x 204 (360M) quadrature/collocation
  points
• Patch over 15,0002 in wall units, i.e. 150
  streaks side-by-side!
• Observe the mega-patches also
• To appear in Spalart et al (2008), Phys. Fluids
  (preprints from GNC data at www.dnsdata.afm.ses.s
  oton.ac.uk)

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Log Law in Ekman-Layer DNS?
2828
2000
1414
Re 1000
velocity aligned with wall stress velocity
magnitude (3D effect)
velocity orthogonal to wall stress

• Ekman Reynolds numbers from 1000 to 2828 d
  scales like Re1.6

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Karman Measure in Ekman-Layer DNS
Re
Chauhan-Nagib-Monkewitz Fit to experiments
d log ( y ) / dU

• Confirms U figure Law of the Wall is coming
  in
• At this level of detail, the BL experiment
  disagrees slightly with DNS
• Plateau waits until 300

11
Karman Measure in Ekman-Layer DNS with Shift
d ln(z 7.5)/dU

• Shifting to ln ( z 7.5 ) magically creates a
  plateau at 0.38!
• (The experimental results would not line up
  exactly using the shift.)

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Surface-stress similarity test magnitude
k0.38, a7.5 offset
u/G
Re
13
Surface-stress similarity test direction
k0.38, a7.5 offset
a0 (deg)
High-Re theory, k0.38, no offset
Re
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Summary

• Channel and Ekman DNS are racing for Reynolds
  numbers
• An order of magnitude gained over Kim et al
  (1987), but k is no more certain than it was!
• The experimental Karman constant is also
  uncertain
• The Superpipe gives at least 0.42
• The IIT and KTH ZPG BL experiments give 0.384
• The law of the wall itself is not under attack
• Or is it? Some claim k is different with
  pressure gradient (i.e. non-constant t(z)
  profiles) ? new Couette-Poiseuille DNS now
  underway (to have dt/dz gt 0)
• Ekman DNS does not contradict the boundary-layer
  experiments
• The log law is established only for z gt 200 at
  best
• U first overshoots the log law, and blends in
  from above
• And k is around 0.384
• Ekman DNS likes the idea of a shift
• ln( z 7.5 ) instead of ln( z )
• It makes a perfect log layer, blending simply
  from below, with k 0.38!
• It is within the law of the wall, i.e.,
  independent of the flow Reynolds number
• Its not the easiest thing to explain physically,
  but nothing rules it out
• Does not agree with experiment perfectly, at this
  level of detail, but U versus Re behaviour
  collapses, and is converging to something
  rational

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Mean velocity defect versus Re
cross-shear
shear-wise
(ltugt-G) / u
Re1000
1414
2000
2828
zf/u
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Reynolds shear stress versus Re(surface-shear
coordinates)
ltuwgt/u2
Re1000
1414
2000
2828
t / u2
ltvwgt/u2
zf/u