Modelling spanwise heterogeneous roughness through a parametric forcing approach

Inhomogeneous rough surfaces in which strips of roughness alternate with smooth-wall strips are known to generate large-scale secondary motions. Those secondary motions are strongest if the strip width is of the order of the half-channel height and they generate a spatial wall shear stress distribution whose mean value can significantly exceed the area-averaged mean value of a homogeneously smooth and rough surface. In the present paper it is shown that a parametric forcing approach (Busse & Sandham, J. Fluid Mech., vol. 712, 2012, pp. 169–202; Forooghi et al., Intl J. Heat Fluid Flow, vol. 71, 2018, pp. 200–209), calibrated with data from turbulent channel flows over homogeneous roughness, can capture the topological features of the secondary motion over protruding and recessed roughness strips (Stroh et al., J. Fluid Mech., vol. 885, 2020, R5). However, the results suggest that the parametric forcing approach roughness model induces a slightly larger wall offset when applied to the present heterogeneous rough-wall conditions. Contrary to roughness-resolving simulations, where a significantly higher resolution is required to capture roughness geometry, the parametric forcing approach can be applied with usual smooth-wall direct numerical simulation resolution resulting in less computationally expensive simulations for the study of localized roughness effects. Such roughness model simulations are employed to systematically investigate the effect of the relative roughness protrusion on the physical mechanism of secondary flow formation and the related drag increase. It is found that strong secondary motions present over spanwise heterogeneous roughness with geometrical height difference generally lead to a drag increase. However, the physical mechanism guiding the secondary flow formation, and the resulting secondary flow topology, is different for protruding roughness strips and recessed roughness strips separated by protruding smooth surface strips.

Phase-conjugate mirror for water waves driven by the Faraday instability

The Faraday instability appears on liquid baths submitted to vertical oscillations above a critical value. The pattern of standing ripples at half the vibrating frequency that results from this parametric forcing is usually shaped by the boundary conditions imposed by the enclosing receptacle. Here, we show that the time modulation of the medium involved in the Faraday instability can act as a phase-conjugate mirror––a fact which is hidden in the extensively studied case of the boundary-driven regime. We first demonstrate the complete analogy with the equations governing its optical counterpart. We then use water baths combining shallow and deep areas of arbitrary shapes to spatially localize the Faraday instability. We give experimental evidence of the ability of the Faraday instability to generate counterpropagating phase-conjugated waves for any propagating signal wave. The canonical geometries of a point and plane source are implemented. We also verify that Faraday-based phase-conjugate mirrors hold the genuine property of being shape independent. These results show that a periodic modulation of the effective gravity can perform time-reversal operations on monochromatic propagating water waves, with a remarkable efficiency compared with wave manipulation in other fields of physics.