1. Implications of recent Ekman-layer DNS for near-wall similarity
Gary Coleman*, Philippe Spalart**, Roderick Johnstone*
*University of Southampton
**Boeing Commercial Airplanes
– UK Turbulence Consortium – x

2. Turbulent (pressure-driven) Ekman layer:
Balance between pressure gradient, Coriolis and “friction”
 3D boundary layer…
Defining parameter: Reynolds number Re=GD/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
Hodograp:
v/G Re
-P
u/G

3. Parameters: Re = 1000, 1414, 2000 and 2828 d+ ~ Re1.6
(Neglecting “mid-latitude” effects: Wh=0)

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 don’t matter (cf. Utah atmospheric data). 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 (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 changes skin friction by 2% (well, assuming unchanged S-A RANS model in outer layer)

6. Expected qualitative behavior in channel flow
Karman Measure
Expected qualitative behavior in channel flow
S-A model, for illustration only (Mellor-Herring buffer-layer function)
Increasing Re z+

7. 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

9. 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

10. Karman Measure in Ekman-Layer DNS
d log ( y+ ) / dU+
Chauhan-Nagib-Monkewitz
Fit to experiments
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 )/dU+
Shifting to ln ( z ) magically creates a plateau at 0.38!
(The experimental results would not “line up” exactly using the shift.)

12. Surface-stress similarity test: magnitude
k=0.38, a+=7.5 offset
u*/G Re

13. Surface-stress similarity test: direction
k=0.38, a+=7.5 offset
a0 (deg)
High-Re theory,
k=0.38, no offset Re

14. 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 > 0)
Ekman DNS does not contradict the boundary-layer experiments:
The log law is established only for z+ > 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 ) 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
It’s 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”…

15. Mean velocity defect versus Re
cross-shear
shear-wise
(<u>-G) / u*
Re=1000
1414
2000
2828
zf/u*

16. Reynolds shear stress versus Re (surface-shear coordinates)
<u’w’>/u*^2
Re=1000
1414
2000
2828
t / u*^2
<v’w’>/u*^2
zf/u*