Linearized boundary conditions at a rough surface

Linearized boundary conditions are a common numerical tool in any flow problems where the solid wall is nominally flat but the effects of small roughness of height $\epsilon$ are being investigated. Typical are receptivity problems in aerodynamic transition prediction or turbulent flow control. However, two distinguished mathematical limits have to be considered: a ``shallow" limit, where the linearized boundary condition properly applies, involving a family of surfaces that become smoother and smoother as $\epsilon\rightarrow 0$, and a ``small" limit, more closely representative of usually encountered roughness, whose family of surfaces remain geometrically similar to themselves (in particular, retain their slope) as $\epsilon\rightarrow 0$. A connection between the two limits will be established through an analysis of their asymptotic behaviour. As a result, the correct effect of the surface in the ``small'' limit, obtained through a numerical solution of the Stokes equation, will be recast as an equivalent linearized boundary condition modified by a suitable {\it protrusion coefficient} (related to the {\it protrusion height} used years ago in the study of riblets). Quantitative numerical examples of such protrusion coefficients will be provided.