An investigation of the linear response of a turbulent flow to infinitesimal wall waviness was started some years ago in connection with the problem of ripple formation in a sandy bottom, by means of an analytical approximation (Luchini & Charru, J. Fluid Mech. 665 2010). A specific quantity of interest is the phase of the wall shear stress modulation relative to the channel bottom, when the latter is given a small sinusoidal undulation, as this phase is critical to the stability or instability of the sediment bed. This study later led to questioning the turbulent eddy-viscosity models commonly adopted in the field, and more recently to direct numerical simulations in which the wavy bottom is accounted for through an immersed-boundary technique. Unexpected nontrivial, and possibly singular, behaviour was observed for waves much longer than the periodic box size of 4 pi usually accepted for DNS. In addition to explaining some earlier findings of Luchini & Russo (AIMETA Conference 2011), these results point at the existence of very long structures able to resonate with the oscillations of the wall and whose study will deserve to be continued.
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