A Mathematical And Numerical Study Of Roll Waves
Author(s)
S. J. D. D’Alessio & J. P. Pascal
Abstract
A mathematical and numerical study of
roll waves
S. J. D. D’Alessio1 & J. P. Pascal2
1Department of Applied Mathematics, University of Waterloo, Canada
2Department of Mathematics, Ryerson University, Canada
Abstract
In this paper we analyze the gravity-driven laminar flow of a shallow fluid layer
down an uneven incline with the principal objective of investigating the effect of
bottom topography on the instability of the flow. The equations of motion are
approximations to the Navier-Stokes equations which exploit the assumed relative
shallowness of the fluid layer and fast laminar flow. The explicit dependence
on the cross-stream coordinate is eliminated by depth-integrating the equations
of motion. A linear stability analysis of the steady flow is carried out by taking
advantage of Floquet-Bloch theory. A numerical scheme is devised for solving the
nonlinear governing equations and is used to calculate the evolution of the perturbed
steady flow. The results are used to confirm the analytical predictions and
to investigate the structure of roll waves. The effect of bottom topography for cases
where the bottom profile corresponds to periodic undulations is discussed.
Keywords: roll waves, bottom topography, linear stability analysis, nonhydrostatic
pressure, integral-boundary-layer equations.
1 Introduction
The instability of a free-surface gravity-driven flow down an incline leads to the
formation of a wave pattern consisting of a series of rapid surface variations followed
by slowly varying sections. This phenomenon is referred to as roll waves
and occurs in many artificial and natural settings [1]. The formation of roll waves
can have devastating consequences. For example, in mud flows they can have suf-
ficient force to cause serious property damage while in open conduits they can
damage flow-rate measuring devices. The potential impact of roll waves can be
reduced, or even eliminated, by appealing to theoretical predictions.Mathematical
Keywords
roll waves, bottom topography, linear stability analysis, nonhydrostatic
pressure, integral-boundary-layer equations.
Related Book
Other papers in this volume
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