Implications of recent Ekman-layer

DNS for near-wall similarity

x

– UK Turbulence Consortium –

Gary Coleman*, Philippe Spalart**, Roderick Johnstone*

*University of Southampton**Boeing Commercial Airplanes

Turbulent (pressure-driven) Ekman layer:• Balance between pressure gradient, Coriolis and “friction” 3D boundary layer…

• Defining parameter: Reynolds number Re=GD/, where G freestream/geostrophic wind speedD = (2/f)1/2 viscous boundary-layer depthf = 2v Coriolis/rotation parameter =/ kinematic viscosity

u/G

v/G

Re

Hodograph

v

-P

Parameters:

• Re = 1000, 1414, 2000 and 2828: + ~ Re1.6

• (Neglecting “mid-latitude” effects: h

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). Recall hodograph is nearly straight for 80% of Ue

The Quest for the Law of the Wall• Expectations: for “unperturbed” turbulent boundary layer:

– Mean velocity U = U(z,w,,) U+ = f(z+), for large z+ and small z/andU+ = (1/) ln(z+) + C defines the log layer

– Impartial determination: “Karman measure” (z+) = d ( ln z+ ) / d U+

– If expectations valid, then (z+) constant in the “logarithmic region”

• History:– Until 70s: classical experiments, Coles.

• Probable range: from 0.40 to 0.41 (although k- 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: controls extrapolation of drag to other Reynolds numbers… Going to Rex = 108, changing from 0.41 to 0.385 changes skin friction

by 2% (well, assuming unchanged S-A RANS model in outer layer)

Karman Measure Expected qualitative behavior in channel flow

S-A model, for illustration only (Mellor-Herring buffer-layer function)

Increasing Re

z+

Looking for the Karman Constant in DNS

Expected qualitative behavior “High”-Reynolds-number DNS

Increasing Re

Oh dear…

z+ z+

Ekman-Layer DNS at Re = 2828

• Coriolis term allows BL homogeneous in x, y and t

• Pressure gradient, equivalent to channel at Re = 1250

• Boundary-layer thickness 5000/u

• 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.soton.ac.uk)

Ekman Reynolds numbers from 1000 to 2828: + scales like Re1.6

velocity aligned with wall stressvelocity magnitude (3D effect)

Log Law in Ekman-Layer DNS?

Re = 1000

14142000

2828

velocity orthogonal to wall stress

• 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…

Karman Measure in Ekman-Layer DNS

Chauhan-Nagib-MonkewitzFit to experiments

d lo

g ( y

+ ) /

dU+

Re

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

*Karman Measure in Ekman-Layer DNS with Shift*d

ln(z

+ +

7.5

)/dU

+

Surface-stress similarity test: magnitude

Re

u*/G

=0.38, a+=7.5 offset

Surface-stress similarity test: direction

=0.38, a+=7.5 offset

Re

0 (deg)

High-Re theory,=0.38, no offset

Summary• Channel and Ekman DNS are racing for Reynolds numbers

– An order of magnitude gained over Kim et al (1987), but 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 is different with pressure gradient (i.e. non-constant (z)

profiles) new Couette-Poiseuille DNS now underway (to have d/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 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 = 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”…

Mean velocity defect versus Re

zf/u*

cross-shear

shear-wise

uG/ u*

1414Re=1000

20002828

Reynolds shear stress versus Re(surface-shear coordinates)

Re=1000

20002828

1414

<v’w’>/u*^2

/ u*^2

zf/u*

<u’w’>/u*^2