Higher-order homogenised riblet boundary conditions

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 as a consequence it cannot predict other than a linear dependence of drag reduction upon $s^+$; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion coefficients and higher-order equivalent boundary conditions on a virtual flat surface. While the majority of our results, though nonlinear in $s^+$, remain linear in velocity, and therefore we shall not directly address the shape of the drag-reduction curve, this procedure will also allow us to explore the way nonlinearities of the Navier-Stokes equations first enter the $s^+$-expansion, with somewhat surprising negative results.