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IntroductionOpen-channel turbulent flows often present considerably complex aspects so that ordinary hydraulic formulas such as the pipe resistance law and logarithmic velocity distribution along the normals to the walls are unfortunately not directly applicable.It was learned that to determine the velocity field and the friction factor in open-channels with sufficient accuracy, it is necessary to evaluate the effects both of the shape of the cross-section and of the non-uniform distribution of the shear-stresses over the wetted perimeter . These factors in fact determine a threedimensional flow in open channels. The flow can be considered to be defmed by the superimposition of a primary flow and a secondary flow. The primary flow actually moves the fluid in the longitudinal direction of the channel. The secondary flow can be separated, according to the classification formulated by Prandtl [11], into a secondary flow of the fust kind (a result of variations in the cross-section) and a secondary flow of the second kind (showing closed streamlines that can be grouped in several cells). The latter motion is due to a non-homogeneous distribution of the turbulent velocity fluctuations in the boundarylayer.

In the present paper we propose a three-dimensional model describing the local evolution of any two- or three-dimensional disturbance superimposed on a basic shear flow. A numerical investigation, using a truncated form of the evolution model, was performed to select resonant tree-dimensional disturbance in a transitional cylinder wake. A truncation criterion of non-linear triadic resonance, which is a generalisation of the standard linear-resonance condition, allowed us to locate some triadic sets of linear eigenmodes. Numerical integration of the corresponding low-dimensional dynamical systems showed interesting results that are in qualitative agreement with those performed by Williamson and Prasad [j. Fluid Mech. 256 (1993) 269]. This seems to validate the truncation criterion which, together with the theoretical model, proved to be an appropriate tool for describing the local evolution of large-scale structures in shear flows.