Why do only some riblets promote spanwise rollers?

Linear-stability modelling suggests that all sufficiently large riblets promote maximally growing spanwise rollers (García-Mayoral & Jiménez 2011 J. Fluid Mech. vol. 678, 317–347), yet direct numerical simulations (DNS) have shown that this is not the case (Endrikat et al. 2021 J. Fluid Mech. vol. 913, A37) some riblet shapes do not form spanwise rollers at all. Thus, the drag-reduction breakdown across all riblet shapes cannot be solely attributed to maximally growing spanwise rollers, prompting a reappraisal of the modelling. In this paper, comparing DNS data with riblet-resolving linear-stability predictions shows that the spanwise rollers are actually marginal modes, not maximally growing instabilities. This riblet-resolved linear analysis also predicts that not all riblet shapes promote spanwise rollers, in agreement with DNS, and unlike earlier linear-stability modelling, which relied on a one-dimensional (1-D) mean flow and on an over-simplified effective wall-admittance boundary condition. These riblet-resolved calculations further inform how to capture the effect of the riblet shape in a 1D model. Once captured, predictions with an effective boundary condition match riblet-resolved results, but still do not indicate what features of the riblet geometry promote the roller instability. Thus, the wall admittance is measured near the riblet crests, in both the riblet-resolved linear analysis and DNS, to show that the in-groove dynamics is dominated by a balance between the overlying pressure and unsteady inertia, and not viscous diffusion, as previously assumed. This pressure–unsteady-inertia balance sets the linear scaling of the wall admittance with riblet size, as observed in DNS, and is a key factor in setting the streamwise wavelength of the spanwise rollers. Furthermore, modelling this pressure–unsteady-inertia balance in the wall admittance reveals the role of riblet slenderness in promoting spanwise rollers, which provides the missing link in previous correlations between the riblet geometry and the presence or lack of rollers.