Dam-Break Flow for Arbitrary Slopes of the Bottom

Abstract

The dam-break flow problem in the shallow-water approximation on an inclined bed for arbitrary slopes of the bottom is considered. An analytical solution for the spreading of the water fronts at the initial stages is given. A self-similar solution asymptotically valid at large time is also found. For intermediate times the problem is solved numerically by the method of characteristics.

Nonparallel spatial stability of shallow water flow down an inclined plane of arbitrary slope

“…As an example on the formation of roll waves in a nonuniform flow, we have considered the dam-break problem in an inclined channel (Fernandez-Feria 2006;Bohorquez and Fernandez-Feria 2006). Actually, our interest in the present stability problem originated from the observation of roll waves in some numerical simulations for that non-uniform flow problem.…”

An exact solution for ideal dam‐break floods on steep slopes

“…At time t = 0, the dam collapses instantaneously and unleashes a flood of finite volume down the slope. An important difference between our formulation and that of Fernandez-Feria [2006] lies in the initial configuration of the flow, because Fernandez-Feria [2006] investigated the case of a vertical dam. Although a vertical dam is more similar to some real-world scenarios, it leads to significant mathematical difficulties when the method of characteristics is employed owing to singular behavior of the front and rear (both u and h being zero there).…”

“…Such assumptions typically lead to a kinematic wave approximation, which enables substantial simplification because the mass and momentum balances making up the shallow-water equations are transformed into a single nonlinear diffusion equation [Hunt, 1983;Porporato, 2004a, 2004b;Chanson, 2006]. Exact solutions of the shallow-water equations for steep slopes have been obtained for infinite-volume dam-break floods [Shen and Meyer, 1963;Mangeney et al, 2000;Karelsky et al, 2000;Peregrine and Williams, 2001], and the case of a finitevolume flood has been investigated by Dressler [1958] and later by Fernandez-Feria [2006], who provided a partial solution by computing the position and velocity of the surge front and rear. Savage and Hutter [1989] constructed two similarity solutions known as the parabolic cap and M-wave, but these differ from the long-time asymptotic solution of the problem investigated here.…”

Savage‐Hutter model for avalanche dynamics in inclined channels: Analytical solutions

“…Some extensions of the classical dam break problem for the avalanche on an inclined plate have been done. Fernandez‐Feria [2006] extended the 1‐D dam break solution to more complicated initial condition different from (32); the avalanche shape is not constant at x < 0. In fact, Fernandez‐Feria [2006] found an analytical solution only in the vicinity of the avalanche edge, of which the motion does not depend on the avalanche shape behind the front, at least for small times.…”

“…Some attempts have been made to simulate submarine landslides using the full three‐dimensional (3‐D) Navier‐Stokes equations [ Heinrich et al , 1998, 1999; Mangeney et al , 2000b]. Similar models developed to describe dry subaerial avalanches, pyroclastic and debris flows including possible sources of tsunami, have aroused much interest in the last decade [see also Gray et al , 1999; Mangeney et al , 2000b; Heinrich et al , 2001; Bouchut et al , 2003; Mangeney‐Castelnau et al , 2003, 2005; Le Friant et al , 2006; Fernandez‐Feria , 2006; Rudenko et al , 2007; Mangeney et al , 2007a; Bouchut et al , 2008; Pirulli and Mangeney , 2008; Pelanti et al , 2008; Yu et al , 2009; Luca et al , 2009]. Most of such models of landslides and avalanches are based on homogeneous shallow water flows that are deformed during propagation.…”