A 2-dimensional--3-component model of turbulent flow over riblets

Riblets are streamwise-aligned grooves that are designed to reduce drag by modifying the near-wall flow with respect to that of the smooth wall. Nevertheless, drag reduction breaks down when the viscous-scaled square root of the groove area $\ell_g^+>11$, and this breakdown has been attributed to the formation of time-averaged secondary flows over riblets, among other mechanisms. Here we propose to predict these secondary flows by adapting the 2-dimensional--3-component (2D--3C) model of Gayme \textit{et al.} (\textit{J. Fluid Mech.}, vol. 665, 2010, pp. 99--119), in which a sustained turbulent flow is obtained by modelling the incoherent turbulent fluctuations as random forcing. We conduct 2D--3C simulations of flow over several riblet geometries and sizes and compare the results with minimal direct numerical simulations. The 2D--3C model captures the onset and the topology of the secondary flows, suggesting that they are generated by a preferential distribution of near-wall turbulence pinned by the riblet grooves. The model can be used to predict the slip velocity at the riblet crest, providing a better estimate than Stokes (purely viscous) calculations for riblets of moderate sizes, $\ell_g^+<20$.