Dam break in rectangular channels with different upstream-downstream widths

Abstract

The classic Stoker dam-break problem [1] is revisited in cases of different channel widths upstream and downstream of the dam. The channel is supposed to have a rectangular cross section and a horizontal and frictionless bottom. The system of the shallow water equations is enriched, using the width as a space-dependent variable, together with the depth and the unit discharge, which conversely depend on both space and time. Such a formulation allows a quasi-analytical treatment of the system, whose solution is similar to that of the classic Stoker solution when the downstream/upstream depth ratio is sufficiently large, except that a further stationary contact wave exists at the dam position. When the downstream/upstream depth ratio is small, the solution is richer than the Stoker solution because the critical state occurs at the dam position and the solution itself becomes resonant at the same position, where two eigenvalues are null and the strict hyperbolicity of the system is lost. The limits that identify the flow regime for channel contraction and channel expansion are discussed after showing that the nondimensional parameters governing the problem are the downstream/upstream width ratio and the downstream/upstream initial depth ratio.After the introduction of the previous analytical framework, a numerical analysis is also performed to evaluate a numerical method that is conceived to suitably capture rarefactions, shock waves and contact waves. A secondorder method is adopted, employing a Dumbser-Osher-Toro Riemann solver equipped with a nonlinear path. Such an original nonlinear path is shown to perform better than the classic linear path when contact waves of large amplitude must be captured, being able to obtain specific energy conservation and mass conservation at the singularity.The codes, written in MATLAB (MathWorks Inc.) language, are made available in Mendeley Data repository.

The exact solution to the Shallow water Equations Riemann problem at width jumps in rectangular channels

“…The Authors found that the problem may exhibit multiple solutions if the flow impinging a rapid contraction is supercritical, and connected this finding with the hydraulic hysteresis phenomenon (Defina and Susin 2006). Valiani and Caleffi (2019) replicated the dam-break solution by Cozzolino et al (2018b), giving additional mathematical details that were implicit in Cozzolino et al (2018b). Finally, Goudiaby and Kreiss (2020) considered a class of Riemann problems at sudden expansions with Borda-Carnot losses and subcritical flow, demonstrating that there are initial conditions for which this problem has no solution.…”

Dot-Type Schemes for Hybrid Hyperbolic Problems Arising from Free-Surface, Mobile-Bed, Shallow-Flow Models

“…This procedure provides an optimised approach, but it could requires a lot of mathematical effort to obtain the eigenstructure and can be used only with the specific set of PDEs. • Valiani and Caleffi [42] propose to use a path written in terms of the Riemann invariants. This choice could produce more complex and time-consuming integration since some models, such as the debris flow model, do not have an explicit version of the Riemann invariants.…”

“…𝜕𝐔 𝜕𝑥 (42) where 𝜕𝐖∕𝜕𝐔 = 𝐁 −1 𝑊 is the inverse of matrix 𝐁 𝑊 . Therefore, we can define the new matrix for the non-conservative terms as: Ĥ𝑊 = 𝐇 𝑊 𝐁 −1 𝑊 (43) and we can obtain the following pseudo-Combined form:…”

Optimal Use of Titanium Dioxide Colourant to Enable Water Surfaces to Be Measured by Kinect Sensors

“…It can be seen that KV2 exhibits a relatively flat spectrum, with no noise components above 0.2 mm in amplitude, whereas KV1 exhibits noise up to almost 1 mm, particularly below 1 Hz. This can be important for studies of low-frequency phenomena or single events such as dam breaks [ 54 , 55 , 56 , 57 ] where the behaviour of interest is in the order of mm. In these cases, it would be an order of magnitude more reliable to use a KV2 sensor.…”