This paper reports on laboratory experiments concerning frictional rotating turbulent boundary layer in spin-up flow over flat horizontal bottom. Stereoscopic Particle Image Velocimetry technique is used to obtain two-dimensional three components fluctuating velocity fields. Velocity profiles measured in laminar regime show a remarkable agreement with the Ekman theoretical predictions. Results obtained in turbulent regime confirm the measurement method validity and allow to plan an extensive analysis of the turbulent Ekman layers

SOMMERIA, Joël

SOUS, Damien

poster CFM2007-1190International audienceThis paper reports on laboratory experiments concerning frictional rotating turbulent boundary layer in spin-up flow over flat horizontal bottom. Stereoscopic Particle Image Velocimetry technique is used to obtain two-dimensional three components fluctuating velocity fields. Velocity profiles measured in laminar regime show a remarkable agreement with the Ekman theoretical predictions. Results obtained in turbulent regime confirm the measurement method validity and allow to plan an extensive analysis of the turbulent Ekman layers.Cette communication présente les résultats d'une campagne de mesures en fluide tournant sur des couches limites turbulentes sur fond plat horizontal. Une techique de PIV stéréoscopique est utilisée pour obtenir les trois composantes des champs de vitesse dans un plan. Les profils de vitesse obtenus en régime laminaire sont en très bon accord avec les prédictions théoriques d'Ekman. Les résultats obtenus en régime turbulent valident la méthode de mesure et permettent d'envisager une étude extensive des couches d'Ekman turbulentes

Sous, Damien

Sommeria, Joël

Hal - Université Grenoble Alpes

Experimental study of a turbulent Ekman layer

I-Revues

18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Experimental study of a turbulent Ekman layer
Damien Sous1,2 & Joël Sommeria2
1 MAE
Arizona State University, Tempe, USA
2 Coriolis/LEGI
Grenoble, France
sous@coriolis-legi.org
Résumé :
Cette  communication  présente  les  résultats  d'une  campagne  de  mesures  en  fluide  tournant  sur  des  
couches limites turbulentes sur fond plat horizontal. Une techique de PIV stéréoscopique est utilisée pour  
obtenir les trois composantes des champs de vitesse dans un plan. Les profils  de vitesse obtenus en  
régime laminaire sont en très bon accord avec les prédictions théoriques d'Ekman. Les résultats obtenus  
en régime turbulent valident la méthode de mesure et permettent d'envisager une étude extensive des 
couches d'Ekman turbulentes.
Abstract :
This paper reports on laboratory experiments concerning frictional rotating turbulent boundary layer in  
spin-up flow over flat horizontal bottom. Stereoscopic Particle Image Velocimetry technique is used to  
obtain  two-dimensional  three  components  fluctuating  velocity  fields.  Velocity  profiles  measured  in  
laminar regime show a remarkable agreement with the Ekman theoretical predictions. Results obtained 
in turbulent regime confirm the measurement method validity and allow to plan an extensive analysis of  
the turbulent Ekman layers.
Key-Words : 
Ekman layer, turbulence, SPIV
1 Introduction
The role of friction in geophysical flows is closely related to the structure of the 
boundary layer which appears on the bottom. In its pioneering study of boundary layers in 
presence  of  background  rotation,  V.W.Ekman  (1905)  shown  that  the  rupture  of  the 
geostrophic balance due to velocity defect induces a rotation of the velocity vector near the 
bottom, leading to the well-know Ekman spiral. 
Although numerous field and laboratory observations have confirmed the general 
features of the Ekman model in the atmospheric and oceanic bottom boundary layer, few 
detailed  laboratory  experiments  have  been  able  to  quantitatively  analyse  the  rotating 
boundary layers, particularly in the turbulent case ( Greenspan 1968). Stability studies (Lilly 
1966) have shown that the Ekman layer becomes unstable for boundary Reynolds numbers 
around Reδ  = 55; here ( ) 2/120 f/u2Re ν=δ , where  0u is the velocity outside the boundary 
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18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
layer,  f=2Ω the  Coriolis  parameter  and  ν the  kinematic  viscosity.   Using  laboratory 
experiments, Caldwell and van Atta (1970) gave support for the theoretical Lilly criterion 
and found that the transition threshold to fully turbulent regime does not occur until Reδ  = 
148. Some measurements in fully turbulent rotating boundary layers have been obtained on 
smooth and rough surfaces (Howroyd and Slawson 1975, Ferrero et al 2005), and compared 
with atmospheric boundary layer theory (Garratt  1992) in terms of scaling laws,  surface 
stress and fluxes. 
However,  our  understanding  of  the  physical  processes  governing  the  turbulent 
Ekman layer is still limited by the lack of reliable data on local turbulence properties. The 
present  experimental  work  aims  to  employ  recent  measurement  techniques  such  as 
Stereoscopic Particle Image Velocimetry (SPIV) to enhance our knowledge of the bottom 
Ekman layer, particularly focusing on the turbulent regime.
2 Experimental set-up
The  experimental  campaign  took  place  on  the  hydrodynamic  rotating  tank  of 
Coriolis/LEGI  (Grenoble).  The  tank  is  filled  with  salted  water  of  constant  density  1,02 
kg/m3. The depth at rest is 85cm, while during rotation the free surface is paraboloid. The 
anticlockwise rotation period T of the tank was impulsively decreased from T=53s to T=40s 
to generate a relative spin-up flow. Coriolis parameter is f=0,31 and the Rossby number 
defined by /=0,33
Quantitative measurements of the two-dimensional three-components velocity fields are 
provided  by  Stereoscopic  Particle  Image  Velocimetry  (SPIV).  This  technique  requires  two 
cameras  to record simultaneous but distinct off-axis views of the same acquisition area. Two 
CCD cameras (1024x1024 pixels, B&W 12 bits) are placed outside the tank, viewing the same 
acquisition  area  through  a  Plexiglas  window  in  the  vertical  tank  rim  and  recording 
simultaneously two series of images at adjustable time rate. The acquisition area, which size is 
about 9x9cm², is illuminated by a vertical continuous Argon laser sheet. The angle between 
laser sheet and azimuthal direction is  8° and the cameras are placed at -14° and +30° with 
respect to the axis view. The homogeneous fluid is uniformly seeded with by 30µm neutrally 
buoyant particles. 
Measurements in a spin-up seeded flow have thus been performed, consisting in the 
simultaneous  acquisition  of  images  series  for  both  camera.  In-plane  displacements  of  the 
particle patterns between two successive images are calculated by direct cross-correlation of the 
image  luminosity  using  Correlation  Image  Velocimetry  (CIV,  Fincham and Delerce 2000). 
Specific mesh grids are used to get higher points density in the boundary layer. The out-of-plane 
component of the velocity is finally reconstructed from the two in-plane displacements fields 
using an original stereoscopic method, based on a linear approximation around the acquisition 
plane. 
3 Results
The three components of the velocity vector are defined as:  u azimuthal clockwise ,  v 
radial inward, and w vertical upward. Experimental parameters are choosen so as to generate a 
fulle turbulent boundary layer slowly decaying toward laminar regime.
We present first the results obtained in the laminar regime which can easily be compared 
with the Ekman theoretical solutions for a homogeneous rotating bottom boundary layer. 
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18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
))/sin((
))/cos(1(
/
/
E
z
E
z
zeUv
zeUu
E
E
δ
δ
δ
δ
−
−
=
−=
where U is the geostrophic velocity, z the distance above the bottom and E=2/ f 
1/2  is 
the Ekman layer depth. 
Figure  1  presents  the  comparison  between  instantaneous  experimental  and  analytical 
Ekman profiles for spin-up case for boundary layer Reynolds number Re=60. One notes the 
very  good  agreement  between  measured  velocity  profiles  and  theoretical  predictions.  The 
azimuthal  velocity  is  constant  above  the  boundary  layer,  i.e.  for  z>3  cm,  slightly 
increases in the upper part  of the boundary layer and then decreases toward 0.  The 
radial velocity is very weak for z>2cm and then increases to reach its maximal value 
around z=0,5cm. This reveals clearly the presence of a radial inward Ekman transport in 
the boundary layer, due to rupture of the geostrophic balance, meaning that the Coriolis 
force is no more balanced by the pressure gradient. 
FIG. 1 –Comparison between measured laminar profiles and Ekman theoretical solutions
Figure 2 shows the evolution of the profiles of the three components of velocity during the 
complete spin-up process. Profiles have been averaged over 50 successive velocity fields to 
remove the influence of turbulent fluctuations. One notes the influence of turbulence on the 
profiles, which tends to smooth the velocity gradients.
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18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
FIG. 2 – Evolution of the velocity profiles during the spin-up process
Turbulent fluctuations are thus calculated as the local deviance to the mean value, and 
averaged  in  space  and  time  to  obtain  the  turbulent  momentum fluxes.  Figure  3  shows the 
evolution of the profile of vertical flux of azimuthal momentum. One notes that the turbulence 
in essentially localized in the boundary layer, slowly decreasing during time. The negative sign 
is consistent with a downward flux of azimuthal momentum. The usual assumption of a constant 
flux in the turbulent layer is not verified.
FIG. 3 – Evolution of the profiles of vertical flux of azimuthal momentum
During the spin-up process, the azimuthal velocity adjustement of a vertical fluid column 
is equal to the total momentum flux, i.e. the sum of viscous flux, turbulent flux and the Coriolis 
force on the radial component of velocity:
HdU
dt
=f∫0
z
v−〈u'w '〉 ∂U
∂ z
The  right  hand  term  is  theoretically  constant  in  the  fluid  column.  Figure  4  plots  the 
comparison between the left  ahnd and the right hand term of the momentum equation. The 
results presented are obtained at z=1cm above the bottom, i.e. in the boundary layer, but very 
close results are obtained for different heights (not shown here).
One notes the very good agreement between the momentum decay and momentum flux in 
the boundary layer. This confirms the validity of the measurements by relating local turbulent 
data to mean flow features.
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18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
FIG. 4 – Comparison between momentum decay and momentum flux in the boundary layer
4 Conclusions
Spin-up  experiments  of  homogeneous  fluid  have  been  performed  on  the  Coriolis 
turntable to study the development of frictional rotating « Ekman » layer on a flat horizontal 
solid  bottom.  The  experimental  parameters  have  been  chosen  so  as  to  generate  turbulent 
boundary layer slowly decaying toward laminar regime. Accurate measurements of turbulence 
properties are performed using Stereoscopic Particle Image Velocimetry. An original method 
based  on  a  linear  approximation  has  been  used  to  reconstruct  of  the  out-of-plane  velocity 
component from the two off-axis views.
The measurements  have been first  successfully  validated by comparing the  velocity 
profiles in the laminar regime with the theoretical prediction of V.W.Ekman. Small negative 
vertical velocity has been measured near the solid bottom, which can be related to the presence 
of Ekman suction in this anti-cyclonic spin-up flow. Vertical velocity profiles show a smoothing 
due to turbulent diffusion. Computations of turbulent fluctuations have thus been performed to 
give  access  the  turbulent  properties  of  the  flow.  A  clear  downward  flux  of  streamwise 
momentum has been observed in the boundary layer, but the assumption of quasi-constant flux 
is  verified  only  over  a  small  part  of  the  layer.  Finally,  a  relation  relating  local  turbulent 
measurements  with  mean  flow  decay  in  terms  of   momentum  decay  have  confirmed  the 
reliability of the measurement method.
These  conclusions  indicate  that  the  measurement  method  presented  is  useful  for 
analysing the turbulent Ekman layer and further experimental campaigns are planned. From an 
experimental  point  of  view, spatial  and temporal  resolutions will  be improved to  allow the 
tracking of instabilities inside and outside the boundary layer. More physically, it will of great 
interest to analyse the influence of external parameters (rotation rate, bottom roughness, flow 
direction,  topography  or  depth)  on  the  turbulence  dynamics  and  friction  processes,  and  to 
compare  experimental  results  with  the  similarity  laws  used  by  atmospheric  boundary  layer 
theory. Due to its ubiquitous influence on geophysical flows, a particular attention will be paid 
to the fluid stratification. 
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18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Références
Ekman, V.W. 1905 On the influence of the earth's rotation on ocean currents.  Arkiv. Matem., 
Atr. Fysik, Stockholm 2(11)
Ferrero, E., Longhetto, A., Montabone, L., Mortarini, L., Manfrin, M., Sommeria, J., 
Didelle, H., Giraud, C., Rizza, U. 2005,Physical simulations of neutral boundary layer in 
rotating tank, Nuovo Cimento C 28:1-17
Fincham, A. Delerce,  G.  2000 Advance optimization of correlation imaging velocimetry 
algorithms Exp. Fluids. Suppl. 29:13-22
Howroyd, G.C., Slawson, P.R .1975 The characteristics of a laboratory produced turbulent 
Ekman layer. Boundary-Layer Meterolog.  8(2):201-219.
Lilly, K.D. 1966 On the Instability of Ekman Boundary Flow. J. Atmos. Sci. 23:481-494. 
Prasad, A.K. 2000 Stereoscopic particle image velocimetry. Exp. Fluids 29:103-116.
Raffel, M., Willert, C., Kompenhaus, J. 2002, Particle image Velocimetry: a practical guide. 
Berlin: Springer
Tsai, R.Y. 1986 An Efficient and Accurate Camera Calibration Technique for 3D Machine 
Vision. Proceedings of IEEE Conference on Computer Vision and Pattern Recognition. 
Miami Beach, FL, pp. 364–374
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Experimental study of turbulent Ekman layer

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Experimental study of turbulent Ekman layer

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