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We conduct direct simulations of turbulent channels imposing differentvirtual origins for all three velocities using Robin, slip-like boundaryconditions to study the effect of displacing the origins perceived by differentflow components, a mechanism common to small-textured surfaces. We also explorethis effect using opposition control. For riblets, Luchini et al. (1991)proposed that their effect could be reduced to the offset between thestreamwise- and spanwise-velocity origins, the latter being the originperceived by turbulence. Later results on superhydrophobic surfaces suggestthat the apparent wall-normal-velocity origin could also play a role. Ourresults support that the relevant parameter is the offset between the originsperceived by the mean flow and by turbulence. The former is determined by thestreamwise slip length, but the latter depends on both the wall-normal andspanwise slip lengths. The slip on the streamwise velocity fluctuations, inturn, has a negligible effect on the virtual origin for turbulence, at least inthe drag-reduction regime. This suggests that the origin perceived by thequasi-streamwise vortices, which induce the cross velocities at the surface, iskey in determining the virtual origin for turbulence, while that perceived bythe near-wall streaks, which are associated with the streamwise velocityfluctuations, plays a secondary role. In this framework, the changes inturbulent quantities typically reported in the flow-control literature areshown to be only a result of the choice of origin and are absent whenreferenced to the origin experienced by turbulence. Other than this shift inorigin, we show that turbulence remains essentially smooth-wall-like, and asimple expression can predict the virtual origin for turbulence. The effect ondrag can be reproduced a priori by introducing the virtual origins into asmooth-wall eddy-viscosity framework.

The smooth-wall-like behaviour of turbulence over drag-altering surfaces: a unifying virtual-origin framework