Canestrelli, Alberto (2009) Numerical Modelling of Alluvial Rivers by Shock Capturing Methods. [Ph.D. thesis]

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## Abstract (english)

The problem of modelling both the unsteady hydrodynamics and the bed morphological variations in natural channels is generally performed by solving the De Saint Venant balance equations for the liquid phase together with the Exner continuity equation for the sediments carried as bed-load. This thesis focuses on the development of an high-order accurate centred scheme of the finite volume type for the numerical solution of the coupled De Saint Venant-Exner system. A new scheme, called PRICE-C, is proposed. It solves the system of equations in a non-conservative form, however it has the important characteristic of reducing automatically to a conservative scheme if the underlying PDE system is a conservation law. It is applied to the shallow water equations in the presence of either a fix or a movable bed. The scheme is first introduced in a one-dimensional framework, and it is then extended to the two-dimensional case. The extension is not straightforward in the case of an unstructured mesh, since averages over suitable edge-based control volumes have to be performed. The scheme is extended to high order of accuracy in space and time via the ADER-WENO and MUSCL technique respectively for the one- and twodimensional case. The well-balanced property of the scheme is proven, i.e. the ability to reach steady states also in the presence of discontinuous water surface or discontinuous bottom profile. The scheme can deal with subcritical and supercritical flows, as well as transcritical situations. Moreover the proposed approach leads to a correct estimate of the celerity of surface discontinuities as well sediment bores and small bottom perturbations. The main characteristic of the scheme is its simplicity: it is based on a simple centred approach, that means that the knowledge of the eigenvalues of the matrix of the system is not required. This is important since the interaction between sediment transport and water flow not always admits detailed knowledge of the eigenstructure. Hence this scheme can be useful to engineers since they need simple numerical tools that can be easily used without entering in the mathematical detail of the differential hyperbolic system under consideration. Moreover the centred strategy gives generality to the scheme: in fact, it can be applied without modification to any kind of hyperbolic equations with non-conservative terms.

## Abstract (italian)

La modellazione dell’idrodinamica e delle variazioni orfologiche in canali naturali `e generalmente effettuata risolvendo numericamente le equazioni delle onde lunghe in acque basse, che regolano il moto della fase fluida, assieme all’equazione di Exner, che descrive l’evoluzione del fondo. L’argomento della presente tesi consiste nello sviluppo di un schema ai volumi finiti di tipo ”centrato” per la soluzione accoppiata di tale sistema di equazioni. Un nuovo schema, denominato PRICE-C, `e qui introdotto: esso risolve le equazioni in forma conconservativa, ma ha l’importante propriet`a di degenerare in uno schema conservativo se il sottostante sistema di equazioni ammette una forma conservativa. Lo schema `e applicato alle equazioni delle onde lunghe in acque basse sia nel caso di fondo fisso che di fondo mobile, dapprima in un ambito unidimensionale e successivamente in quello bidimensionale. L’estensione non `e immediata nel caso
in cui il reticolo di calcolo sia non-strutturato, dal momento che le equazioni differenziali devono essere mediate su opportuni volumi di controllo. Lo schema `e poi esteso ad alti ordini di accuratezza nello spazio e nel tempo attraverso le procedure ADER-WENO e MUSCL rispettivamente per il caso unidimensionale e bidimensionale. Inoltre si dimostra come lo schema proposto verifichi la ”well-balanced property”, che consiste nella capacit`a di raggiungere soluzioni stazionarie, anche in presenza di discontinuit`a della superficie libera e del fondo. Condizioni di corrente lenta e rapida, come pure condizioni di tipo transcritico vengono correttamente risolte. Inoltre lo schema in grado di riprodurre le celerit`a di propagazione di discontinuit`a della superficie e fronti di sedimenti al fondo, cos`? come la celerit`a di propagazione di piccoli disturbi del fondo. Caratteristica principale dello schema `e la sua semplicit`a: `e basato su un semplice approccio di tipo centrato, cio`e non necessita la conoscenza degli autovalori
della matrice del sistema. Questa `e un’importante caratteristica dal momento che non sempre autovalori e autovettori sono calcolabili analiticamente, in particolare nel caso di complesse formule di chiusura per il trasporto al fondo. Quindi questo schema pu`o rivelarsi utile per l’ingegnere che spesso necessita di un semplice strumento numerico che possa essere applicato ad un sistema di equazioni differenziali di tipo iperbolico senza dover entrare nel dettaglio delle propriet`a atematiche del sistema stesso. Data la sua generalit`a, infatti, lo schema pu`o essere applicato ad ogni tipo di sistema iperbolico contenente termini non-conservativi.

EPrint type: Ph.D. thesis D'Alpaos, Luigi and Defina, Andrea and Lanzoni, Stefano Ciclo 21 > Scuole per il 21simo ciclo > SCIENZE DELL'INGEGNERIA CIVILE E AMBIENTALE 02 February 2009 02 February 2009 Finite volume method. Exner equation. High order scheme. Centred schemes. Area 08 - Ingegneria civile e Architettura > ICAR/01 Idraulica Dipartimenti > pre 2012 - Dipartimento di Ingegneria Idraulica, Marittima, Ambientale e Geotecnica 1968 02 Feb 2009
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