J. Fluid Mech. (1985), vol. 157, pp. 44-70 Printed in &eat Britain 449 A unified bar-bend theory of river meanders By P. BLONDEAUX AND G. SEMINARA Istituto di Idraulica, Facoltk di Ingegneria, Universitk di Genova, Genoa, Italy (Received 3 April 1984 and in revised form 27 November 1984) A two-dimensional model of flow and bed topography in sinuous channels with erodible boundaries is developed and applied in order to investigate the mechanism of meander initiation. By reexamining the problem recently tackled by Ikeda, Parker t Sawai (1981), a previously undiscovered ‘resonance ’ phenomenon is detected which occurs when the values of the relevant parameters fall within a neighbourhood of certain critical values. It is suggested that the above resonance controls the bend growth, and it is shown that it is connected in some sense with bar instability. In fact, by performing a linear stability analysis of flow in straight erodible channels, resonant flow in sinuous channels is shown to occur when curvature ‘forces’ a ‘natural ’ solution represented by approximately steady perturbations of the alternate bar type. A comparison with experimental observations appears to support the idea that resonance is associated with meander formation. 1. Introduction Theoretical attempts to give a mechanistic justification of fluvial meandering have proliferated in the last two decades. Most of the theories, which derived from the original works of Hansen (1967) and Callander (1969), actually treated the formation of alternate bars in a straight alluvial channel with non-erodible banks. The works of Adachi (1967), Hayaahi (1971), Sukegawa (1971), Engelund t Skovgaard (1973), Parker (1975, 1976), Hayashi t Ozaki (1976, 1980) and Fredsrae (1978) have developed an increMingly deeper understanding of the instability process that leads to the flow winding about bars with the channel axis keeping straight. In particular, Fredsm’s theory, which appears to be the most successful attempt, shows that the ingredients necessary to explain the basic bar-instability mechanism are friction, inertial effects involving the transverse velocity, and sediment transport evaluated by taking into account the effect of transverse bed slope. Furthermore, the theory can predict whether an alluvial stream remains straight, tends to develop alternate bars or tends to braid. I n all the above contributions the ability of a stream to develop alternate bars is taken as implying incipient meandering. Recently Ikeda, Parker t Sawai (1981) following ideas originally put forward by Ikeda, Hino & Kikkawa (1976), tackled the problem of meander formation from an apparently different point of view. By relaxing the restraint of fixed sidewalls, Ikeda et al. (1981) investigated the stability of channels with sinuous erodible banks and found conditions for the lateral bend amplitude to grow. The main conclusion of this theory is that ‘bar’ and ‘bend’ instabilities operate at similar wavelengths when sinuosity is not too large. This would provide justification for the assumption, implicit in previous works, that alternate bar formation eventually leads to a meandering channel with an initial wavelength close to that of alternate bars.