3,916 research outputs found
A numerical method for junctions in networks of shallow-water channels
There is growing interest in developing mathematical models and appropriate
numerical methods for problems involving networks formed by, essentially,
one-dimensional (1D) domains joined by junctions. Examples include hyperbolic
equations in networks of gas tubes, water channels and vessel networks for
blood and lymph in the human circulatory system. A key point in designing
numerical methods for such applications is the treatment of junctions, i.e.
points at which two or more 1D domains converge and where the flow exhibits
multidimensional behaviour. This paper focuses on the design of methods for
networks of water channels. Our methods adopt the finite volume approach to
make full use of the two-dimensional shallow water equations on the true
physical domain, locally at junctions, while solving the usual one-dimensional
shallow water equations away from the junctions. In addition to mass
conservation, our methods enforce conservation of momentum at junctions; the
latter seems to be the missing element in methods currently available. Apart
from simplicity and robustness, the salient feature of the proposed methods is
their ability to successfully deal with transcritical and supercritical flows
at junctions, a property not enjoyed by existing published methodologies.
Systematic assessment of the proposed methods for a variety of flow
configurations is carried out. The methods are directly applicable to other
systems, provided the multidimensional versions of the 1D equations are
available
Wave dynamics on networks: method and application to the sine-Gordon equation
We consider a scalar Hamiltonian nonlinear wave equation formulated on
networks; this is a non standard problem because these domains are not locally
homeomorphic to any subset of the Euclidean space. More precisely, we assume
each edge to be a 1D uniform line with end points identified with graph
vertices. The interface conditions at these vertices are introduced and
justified using conservation laws and an homothetic argument. We present a
detailed methodology based on a symplectic finite difference scheme together
with a special treatment at the junctions to solve the problem and apply it to
the sine-Gordon equation. Numerical results on a simple graph containing four
loops show the performance of the scheme for kinks and breathers initial
conditions.Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Time domain simulations of dynamic river networks
The problem of simulating a river network is considered. A river network is considered to comprise of rivers, dams/lakes as well as weirs. We suggest a numerical approach with specific features that enable the correct representation of these assets. For each river the flow of water is described by the shallow water equations which is a system of hyperbolic partial differential equations and at the junctions of the rivers, suitable coupling conditions, viewed as interior boundary conditions are used to couple the dynamics. A different model for the dams is also presented. Numerical test cases are presented which show that the model is able to reproduce the expected dynamics of the system.
Other aspects of the modelling such as rainfall, run-off, overflow/flooding, evaporation, absorption/seepage, bed-slopes, bed friction have not been incorporated in the model due to their specific nature
Numerical modelling of open channel junctions using the Riemann problem approach
The solution of an extended Riemann problem is used to provide the internal
boundary conditions at a junction when simulating one-dimensional flow through
an open channel network. The proposed approach, compared to classic junction
models, does not require the tuning of semi-empirical coefficients and it is
theoretically well-founded. The Riemann problem approach is validated using
experimental data, two-dimensional model results and analytical solutions. In
particular, a set of experimental data is used to test each model under
subcritical steady flow conditions, and different channel junctions are
considered, with both continuous and discontinuous bottom elevation. Moreover,
the numerical results are compared with analytical solutions in a star network
to test unsteady conditions. Satisfactory results are obtained for all the
simulations, and particularly for Y-shaped networks and for cases involving
variations in channels' bottom and width. By contrast, classic models suffer
when geometrical channel effects are involved
Modeling and analysis of pooled stepped chutes
We consider an application of pooled stepped chutes where the transport in
each pooled step is described by the shallow--water equations. Such systems can
be found for example at large dams in order to release overflowing water. We
analyze the mathematical conditions coupling the flows between different chutes
taken from the engineering literature. We present the solution to a Riemann
problem in the large and also a well--posedness result for the coupled problem.
We finally report on some numerical experiments.Comment: 17 pages, 31 figure
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