18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Experimental study of a turbulent Ekman layer
Damien Sous1,2 & Jo?l Sommeria2
1 MAE
Arizona State University, Tempe, USA
2 Coriolis/LEGI
Grenoble, France
sous@coriolis-legi.org
R?sum? :
Cette communication pr?sente les r?sultats d'une campagne de mesures en fluide tournant sur des
couches limites turbulentes sur fond plat horizontal. Une techique de PIV st?r?oscopique est utilis?e pour
obtenir les trois composantes des champs de vitesse dans un plan. Les profils de vitesse obtenus en
r?gime laminaire sont en tr?s bon accord avec les pr?dictions th?oriques d'Ekman. Les r?sultats obtenus
en r?gime turbulent valident la m?thode de mesure et permettent d'envisager une ?tude 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 Re? = 55; here ( ) 2/120 f/u2Re ?=? , where 0u is the velocity outside the boundary
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18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
layer, f=2? the Coriolis parameter and ? 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 Re? =
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 9x9cm?, 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 30?m 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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18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
))/sin((
))/cos(1(
/
/
E
z
E
z
zeUv
zeUu
E
E
?
?
?
?
?
?
=
?=
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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18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 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:
HdUdt =f?0z 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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18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 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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18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
R?f?rences
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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
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