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On the scaling of the slip velocity in turbulent flows over
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Superhydrophobic surfaces can significantly reduce hydrodynamic skin drag by
accommodating large slip velocity near the surface due to entrapment of air bubbles
within their micro-scale roughness elements. While there are many Stokes flow
solutions for flows near superhydrophobic surfaces that describe the relation between
effective slip length and surface geometry, such relations are not fully known in the
turbulent flow limit. In this work, we present a phenomenological model for the
kinematics of flow near a superhydrophobic surface with periodic post-patterns at
high Reynolds numbers. The model predicts an inverse square root scaling with solid
fraction, and a cube root scaling of the slip length with pattern size, which is
different from the reported scaling in the Stokes flow limit. A mixed model is then
proposed that recovers both Stokes flow solution and the presented scaling,
respectively, in the small and large texture size limits. This model is validated
using direct numerical simulations of turbulent flows over superhydrophobic posts
over a wide range of texture sizes from L+ ≈ 6 to 310 and
solid fractions from ϕs = 1/9 to 1/64. Our report also
embarks on the extension of friction laws of turbulent wall-bounded flows to
superhydrophobic surfaces. To this end, we present a review of a simplified model for
the mean velocity profile, which we call the shifted-turbulent boundary layer model,
and address two previous shortcomings regarding the closure and accuracy of this
model. Furthermore, we address the process of homogenization of the texture effect to
an effective slip length by investigating correlations between slip velocity and
shear over pattern-averaged data for streamwise and spanwise directions. For
L+ of up to O(10), shear stress and
slip velocity are perfectly correlated and well described by a homogenized slip
length consistent with Stokes flow solutions. In contrast, in the limit of large
L+, the pattern-averaged shear stress and slip
velocity become uncorrelated and thus the homogenized boundary condition is unable to
capture the bulk behavior of the patterned surface.
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