Long waves in channels of arbitrary cross-section

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Patricio Winckler, Philip Liu

Thursday 2 july 2015

9:00 - 9:20h at Antarctica (level 0)

Themes: IAHR/COPRI Symposium on Long Waves and Relevant Extremes

Parallel session: 10D. COPRI Symposium: Long waves and relevant extremes

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A series of 1D cross-sectional averaged theories of long waves in channels have been developed (e.g. Teng & Wu 1992). However, none of them is able to account simultaneously for 1) arbitrary cross-sections, 2) rapid changes in cross-sectional geometry along the axis channel, 3) curvature in the horizontal plane and 4) branching; all features which may be found in natural streams. A new theory is constructed by properly integrating the 3D equations for irrotational flows over a channel cross-section and applying the boundary conditions on the free surface and along the channel bottom. The governing equations are expressed in terms of the cross-sectionally averaged velocity in the direction of wave propagation and the spanwise averaged free surface elevation. The resulting governing equations are of the Boussinesq-type for weakly nonlinear weakly dispersive waves with coefficients that depend on the configuration of the channel cross-section, which can be precomputed once the bathymetry is prescribed. Velocity components, pressure and surface elevation on a cross-section are recovered from higher order approximations in the perturbation scheme and can be calculated once Boussinesq-type equations are solved. The theory may be used in a variety of problems. As an example, for long distance propagation of landslide tsunami in fjords, travel-times and maximum wave heights can be rapidly estimated from 1D governing equations, making the present theory suitable for warning systems. The theory would also be applicable to river dynamics, flood and tidal waves in estuaries, among other phenomena. To illustrate the capability of the present theory, analytic and numerical examples for channels of uniform, slowly- and rapidly-varying cross-sections are presented. The effect of curvature and branching is studied by means of simple channel geometries.