  Secondary flows are generated when a lateral variation of the topography,
such as streamwise aligned ridges, is imposed to a turbulent wall-bounded flow.
In this case, the flow field is characterized by vortical structures developing
along the streamwise direction known as Prandt's vortices of the second kind
(Prandtl 1965). As demonstrated in previous experimental and numerical works,
the strength and flow organization of these turbulent structures depend on the
ridge shape. In this paper, the effect of the ridge geometry on the generation
of secondary currents is investigated using the linearised RANS-based model
proposed by Zampino (2022). Here, symmetric channels with rectangular,
triangular and elliptical ridges are studied and the secondary flows are
compared in order to highlight the main differences and similarities. The
analogies of the flow organisation between the three geometries suggest that
the secondary currents do not depend on the ridge shape when the ridges are
small and isolated, and the strength of secondary flows collapses when properly
scaled with the mean ridge height. Finally, the generation of secondary flows
and the effect of the ridge shape on the flow organisation is studied in detail
for the complex geometries. In particular, for trapezoidal ridges, we observed
that tertiary flows emerge for the ridges where the scaling behaviour does not
hold.
