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

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

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

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

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)

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+

Looking for the Karman Constant in DNS Expected qualitative behavior “High”-Reynolds-number DNS Oh dear… Increasing Re z+ z+

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.soton.ac.uk)

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

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…

*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.)

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

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

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+ + 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 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 cross-shear shear-wise (<u>-G) / u* Re=1000 1414 2000 2828 zf/u*

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*