Higher-order equivalent boundary conditions for riblets and the role of nonlinearities

The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter s⁺, and therefore it cannot predict other than a linear dependence of drag reduction upon s⁺, i.e. the initial slope of the drag-reduction curve. Extensions have been proposed in the literature, to account for the transpiration due to longitudinal variation of the prevalent velocity and other higher-order effects; these extensions, however, do not generally account for the full set of effects of a given order. Here a formal asymptotic expansion is proposed using matched asymptotics, which consistently provides higher-order protrusion heights and equivalent boundary conditions on a virtual flat surface to any desired order. The expansion is explicitly carried out up to third order, and examples are given for specific geometries. This procedure also allows us to explore nonlinearities of the Navier-Stokes equations and the way they enter the s⁺-expansion, with somewhat surprising results.