Ekman layers near wavy boundaries
David Gérard-Varet1
and Emmanuel Dormy2,3
1DMA/CNRS, Ecole Normale Supérieure, 45, rue
d'Ulm, 75230 Paris Cedex 05, France.
2LPS/CNRS, Département de Physique, Ecole Normale Supérieure,
24, rue Lhomond, 75231 Paris Cedex 05, France.
3IPGP, 4 place Jussieu, 75252 Paris Cedex 05, France.
J. Fluid Mech. (2006), vol. 565, pp. 115-134.
Abstract.
We investigate the effect of boundary roughness on the dynamical properties
of the flow in laminar Ekman boundary layers. The study considers wavy
boundaries having both horizontal wavelength and vertical extent comparable
in size with the boundary layer width. In the case of flat boundaries,
Ekman layers are known to be active, i.e. to affect significantly the
dynamics of the mainstream flow. We show how the layer modelling needs to
be modified to account for such wavy boundaries. In particular, nonlinear
terms enter the laminar description. This model can be linearized in the
limit of small Reynolds numbers. The resulting equations are studied using
both asymptotic expansions and full numerical simulations. We find that
small-scale roughness significantly alters energy dissipation in the
boundary layer. This can result in either a reduction or an increase of
dissipation, depending on, in particular, the orientation of the mainstream
flow with respect to boundary modulation. Agreement is obtained between
theoretical and computational results.