Scaling of secondary flows with surface parameters: A linear approach

Abstract

Secondary flows are generated when a lateral variation of the topography, such as streamwise aligned ridges, is imposed upon a turbulent wall-bounded flow. In this case, the time-averaged flow field is characterized by streamwise vortices known as Prandtl’s vortices of the second kind (Prandtl, 1952). As demonstrated in previous experimental and numerical works, the strength and flow organization of these vortices depend on the ridge shape. In this paper, the effect of the ridge geometry on the generation of secondary flows is investigated using the linearized RANS-based model proposed by Zampino et al. (2022). The model is derived from the assumption that the ridges are shallow, with height smaller than any other length scale, e.g. the viscous length scale. Symmetric channels with rectangular, triangular, elliptical and trapezoidal ridges are studied. The model predicts that the strength of secondary flows can be scaled with the mean ridge height, regardless of the ridge shape, when the ridges are narrower than the half channel height and isolated, i.e. when the lateral separation between the ridges is much larger than the ridge width. Finally, the appearance of tertiary flows and the effect of the ridge shape on the flow organization is studied in detail for trapezoidal geometries. It is observed that tertiary flows emerge for ridge configurations where the scaling behaviour does not hold.