Laminar boundary layers over small-scale textured surfaces

Abstract

We develop a model for steady, laminar boundary layers over small-scale textured surfaces. Although the texture is small relative to the boundary-layer thickness, it modifies the flow via a slip length. We use matched asymptotic expansions to simplify the problem, dividing the flow into outer, boundary-layer and inner regions. The far-field behaviour of the inner problem yields a slip boundary condition for the boundary layer. We derive an asymptotic solution valid when the slip length is small. For arbitrary slip lengths, we develop a numerical method combining Chebyshev collocation and finite differences. We apply this framework to canonical small-scale textured surfaces, including superhydrophobic surfaces and riblets, and utilise existing analytical slip formulae. However, the framework is expected to extend to liquid-infused, porous, compliant or deformable surfaces with a variety of regular or random textures. We demonstrate how slip modifies the boundary layer's velocity field, wall shear stress and displacement thickness across a range of surface configurations, and examine the linear stability of the resulting slip-modified boundary layers. Our approach enables computationally inexpensive modelling of a wide range of small-scale textured surfaces within laminar boundary-layer flows, providing predictive capability for drag, boundary-layer growth and transition across applications ranging from microfluidics to turbo-machinery and marine transport.