Second order discontinuous Galerkin scheme for compound natural channels with movable bed. Applications for the computation of rating curves
Introduction
In the present work, a higher order 1D Discontinuous Galerkin numerical scheme for the propagation of flood hydrograph over a compound channel with movable bed is presented. The scheme is designed in order to model the complex geometry variations occurring in natural channels, which are represented by source terms in the governing equations.
The model equations are represented by the coupled system of balance laws (SBL) formed by the 1D Shallow Water equations and the Exner sediments continuity equation, written for a compound natural channel. A bedload transport equation is used as closure condition for sediments dynamics.
Source terms appearing in the shallow water equations for natural channels are related to channel width, slope and friction. The presence of abrupt geometry variations, which, for instance, is common in cross-sections near stage measuring gauges in rivers, may lead to stiff source terms. The characteristic speed associated to such terms is much larger than the one associated to the flux term. If an explicit numerical scheme does not feature some appropriate treatment for stiff source terms, the numerical solution can be inaccurate and asymptotically inconsistent (see Dumbser et al. [3]) with wrong advection speed estimates or even instabilities on coarse grids.
In order to solve an SBL, a widely used method is the so-called source term splitting (see Toro [2]). It consists in splitting the SBL onto two sub-problems to be solved in sequence: a homogeneous problem, in which the source term in the original SBL is not taken into account, and an ODE containing only the source term and the time derivatives of state from the original SBL. The numerical method to solve the ODE can feature some appropriate treatment for stiff source terms if needed. This method is however not optimal when a scheme which is also capable of effectively reproducing stationary solutions is sought, as splitting methods may lead to oscillations near steady states.
The main reason that motivated the present work is the one of devising a scheme capable of treating moderately stiff and stiff source terms, while preserving other desired properties such as the well-balanced property, which allows steady states to be reproduced with a certain accuracy.
The scheme has been developed within the framework of the ADER-Discontinuous Galerkin (DG) methods, as proposed by Dumbser et al. [1]. As it will be clear from Section 3, it is necessary to employ a higher order scheme (a scheme with order of accuracy higher than the ), in order to preserve the coupling between the flux and the source term. Also, the nature of the model equations requires the use of a path-conservative formalism in order to treat liquid and solid dynamics in a coupled way.
In the literature, 1D shallow water numerical models have been recently proposed both using order (e.g. Catella et al. [4], Audusse et al. [5]) and higher order (e.g. Caleffi et al. [6], Siviglia et al. [7] and Canestrelli et al. [8]) schemes.
It would still be possible to use a shallow water scheme not featuring a special treatment for source terms in order to simulate abrupt geometry changes in natural rivers, by refining the computational grid. However, the use of an ADER-DG strategy in such cases proves to be effective with no need for grid refinement and the resulting scheme is very stable and can be easily extended to even higher orders of accuracy.
The use of an ADER-DG strategy in the case of shallow water equations for natural channels has never been investigated in the scientific literature, to the knowledge of the authors.
The proposed numerical scheme can in principle reach any order of space–time accuracy. For the applications in the present work, the order accuracy has been found sufficient and represents a good compromise between accuracy and calculation time. This is partly due to information on river geometry being often low in accuracy.
The model is validated against several benchmarks: (i) water at rest in a non-prismatic channel, (ii) dam break problem with a moving strong shock, (iii) steady flow in a Venturi-type flume, (iv) subcritical flow in an irregular channel, and (v) propagation of a sediment hump near critical conditions. Moreover, the model is further validated against field measurements of water level-flow discharge during a flood in the Ombrone Pistoiese river in Tuscany (Italy).
The implications of modelling moderately stiff or stiff source terms in the case of flow rating curves are discussed. Importantly, conditions of non-uniformity are found in cross-sections just upstream of bridges where, typically, water-level gauges are installed for flow monitoring by making use of stage-discharge rating curves. Abrupt geometric changes, due to a rapid cross-section narrowing, can deeply affect the flow, leading for instance to backwater effects and transition to a supercritical state. As a result, rating curves may considerably deviate from the classical power law function assuming non-trivial shapes. Moreover, flow unsteadiness can produce a hysteretic behavior (see Schmidt and Yen [9] and Francalanci et al. [10]). In these conditions, reliable rating curves need to be developed by coupling filed measurements with 1D hydraulic numerical modelling.
The model is applied to the case of abrupt geometry changes, where flow can be more effectively reproduced by using a higher order scheme featuring some kind of treatment for stiff source terms; a order scheme would lead to erroneous results, unless a drastic grid refinement was performed. Flow rating curves are derived in a schematic channel subject to a flood wave forcing in the cases of movable and fixed bed with a local width constriction (such as in the case of bridge piers). Various numerical tests are carried out considering different degrees of channel narrowing for given boundary conditions (i.e. input hydrograph, upstream bed level and downstream water level). Results are shown considering the present higher order scheme and its order version. It appears that in these geometrical conditions, rating curves are better reproduced only when a higher order numerical scheme is considered. Importantly, a order scheme can lead to wrong/inaccurate results although it does not become unstable.
The numerical scheme is introduced in Section 3, and its validation and application to the computation of rating curves are shown in Section 4. A discussion of the results and the conclusions are shown in Section 5.
Section snippets
Model equations
The model features the shallow water equations for a natural channel coupled with the Exner equation describing sediments mass conservation. A capacitive approach is used in the present work: the solid flow rate in the channel is assumed to be coincident with the solid flow rate as predicted by bedload transport formulas.
The three governing equations can be written as an SBL having the following form: where (x, t) are the space–time coordinates, U(x, t) is the state vector,
Numerical scheme
In order to solve a hyperbolic SBL in the form of Eq. (1), the main idea is the one of writing it as an homogenous nonconservative system. The following trivial equation is therefore added to the system: and an extended state vector is defined: If then .
The following homogeneous hyperbolic system of equations is considered: If Eq. (19) represents a nonconservative system (NCS), then the term cannot be written as the gradient of some
Validation of the numerical model
Validation of the present model is carried-out by performing numerical simulations of (i) water at rest in a non-prismatic channel, (ii) dam break problem with a moving strong shock, (iii) steady flow in a Venturi-type flume, (iv) subcritical flow in an irregular channel, (v) propagation of a sediment hump near critical conditions, and a real flood wave in the Ombrone Pistoiese river in Tuscany (Italy).
Discussion and conclusions
In the present work a 1D model based on a higher order numerical scheme for simulating flood wave propagation on a natural channel with a movable bed has been presented. An important feature of the scheme is represented by its ability at treating the moderately stiff or stiff source terms that may appear in the governing equations due to the occurrence of abrupt changes in the channel geometry.
After the validation of the scheme, its application to the computation of flow rating curves in a
Acknowledgments
The authors are grateful to the Tuscany regional authority ”Centro Funzionale di Monitoraggio Meteo Idrologico - Idraulico, Servizio Idrologico Regionale” for having contributed to the present research by providing the Ombrone river data used in the present work. Part of such data is not of public domain and was provided to the University of Florence under specific and broader agreements. The Editor Dr. Andrea Rinaldo, the Guest Editor Dr. Annunziato Siviglia and two anonymous referees are
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