Linear stability of the 1D Saint-Venant equations and drag parameterizations

Olivier Thual (1) , Louis-Romain Plumerault (2) , Dominique Astruc (1)

1 IMFT - Institut de mécanique des fluides de Toulouse
2 LASAGEC - Laboratoire de Sciences Appliquées au Génie Civil et Côtier

The stability of the homogeneous and steady flow based on the one-dimensional Saint-Venant equations for free surface and shallow water flows of constant slope is derived and displayed through graphs. With a suitable choice of units, the small and large drag limits, respectively, correspond to the small and large spatio-temporal scales of a linear system only controlled by the Froude number and two other dimensionless numbers associated with the bottom drag parameterization. Between the small drag limit, with the two families of marginal and non-dispersive shallow water waves, and the large drag limit, with the marginal and non-dispersive waves of the kinematic wave approximation, dispersive roll waves are detailed. These waves are damped or amplified, depending on the value of the three control parameters. The spatial generalized dispersion relations are also derived indicating that the roll-wave instability is of the convective type for all drag parameterizations.

Mots clés: Convective instability; Kinematic-wave approximation; Roll-wave instability; Saint-Venant equations; Shallow water

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Ecologie, Environnement

Olivier Thual, Louis-Romain Plumerault, Dominique Astruc. Linear stability of the 1D Saint-Venant equations and drag parameterizations. Journal of Hydraulic Research, 2010, vol. 48, pp. 348-353. ⟨10.1080/00221686.2010.481837⟩. ⟨hal-00951007⟩