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The solution of the St. Venant-Exner equations as a model for bed evolution is studied under conditions when the Froude number, F, approaches unity, and the quasi-steady model becomes singular. It is confirmed that the strict criterion for critical flow, the vanishing of a water surface disturbance celerity, is not met, yet the direction of propagation of a bed wave apparently changes depending on whether F>1 or F<1. An analysis of the linearized model problem for an infinitesimal bed wave under near-uniform conditions is performed, and qualitative features of the solution are brought out, Under appropriate sediment transport conditions, when F-2&RARR;1, two bed waves, one traveling upstream and the other traveling downstream, are found to develop from an initially single localized bed perturbation. Simulations of the full unsteady problem were performed with the Preissmann scheme to confirm the linear analysis and to study the effects of nonlinearity and friction. A transcritical case, in which a region where F-2<1 is succeeded by a region where F-2>1, is also investigated, and the solution exhibits an apparently different behavior than cases where the flow, is everywhere sub- or supercritical, but can be understood as a hybrid of the latter cases.

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... The governing equations for the morphodynamics of uniform sediment have been thoroughly studied by De Vries [1965], Lyn [1987], Lisle et al. [2001], and Lyn and Altinakar [2002]. According to their research, the bed wave travels in the downstream direction under subcritical flow, and in the upstream direction under supercritical flow. ...

... A secondary wave, which propagates in the direction opposite to the primary wave, can also develop under near-critical flow. The celerities of these bed waves can be estimated using a first-order perturbation method [Lyn, 1987;Lyn and Altinakar, 2002]. ...

... The former is a by-product of advection driven by grain size-selective transport, whereas the latter is dominated by a nonlinear diffusion process. Here we did not implement the linearized analysis employed by Seminara et al. [1996], Lyn and Altinakar [2002], Fasolato et al. [2009], and Stecca et al. [2014], because in our numerical runs the hydrograph varies from a very small value (0.6 m 2 /s) to a very large value (10 m 2 /s), thus invalidating the assumption of small perturbations. We can nevertheless obtain an explanation for the formation of bed load sheets based on the different forms of the governing equations for uniform sediment and sediment mixtures. ...

The evolution of gravel-bed rivers under cycled hydrographs and constant sediment supply is studied in this paper. With the aid of numerical modeling and flume experiments, previous research has indicated that under such water and sediment supply conditions, a river can reach a mobile-bed equilibrium characterized by a Hydrograph Boundary Layer (HBL) located near the upstream end of the river. The HBL defines the spatial region within which riverbed topography and grain size respond to the hydrograph, and downstream of which these bed characteristics are invariant. However, neither the governing physical mechanisms nor the general applicability of the HBL is yet well-understood. In this paper, we implement a 1D morphodynamic model for this problem. The model is first validated with data for both flume experiments and numerical simulations. It is then implemented at field scale with grain size distributions varying from uniform sediment to very poorly sorted sediment mixtures. Our results show that the idea of an HBL is applicable in the case of uniform sediment, but breaks down in the case of poorly-sorted sediment mixtures. In such cases, persistent low-amplitude bedload sheets emerge along the entire channel reach, in addition to HBL-like upstream boundary effects. Bedload sheets can be described as surface texture or grain size sorting waves in consonance with low-amplitude elevation variations, and are ascribed to nonlinear advection terms in the governing equations that vanish for uniform sediment, but become significant when sediment is very poorly sorted.

... Typically, the transport parameter ψ may be estimated as small as O(10 −5 )O(10 −3 ) [e.g. 12,15,17,18] therefore it seems reasonable to expand λ in powers of ψ as follows: ...

... The behavior of such curves is well known: for ψ 1 and thus for Fr 1 the celerity of a small amplitude bed wave is considerably smaller than that of small amplitude hydrodynamic waves [18]. Therefore, the bed interacts only weakly with the water surface, thus justifying an approach in which the equations governing hydrodynamics are solved separately from those governing morphodynamics. ...

... From Eqs. (14) emerge that, if ψ 1, z behaves like a characteristic variable of the morphodynamic problem. This behavior can be illustrated considering the linearised problem proposed by Lyn and Altinakar [18]. It consists on the evolution of a small (innitesimal) erodible hump due to a nearly uniform water ow in a straight channel. ...

Morphological accelerators, such as the MORFAC (MORphological acceleration FACtor) approach, are widely adopted techniques for the acceleration of the bed evolution, which reduce the computational cost of morphodynamic numerical simulations. In this work we apply an acceleration to the one-dimensional morphodynamic problem described by the de Saint Venant-Exner model by multiplying all the spatial derivatives related to the mass or momentum ux by an acceleration factor ≥ 1 which may be dierent for each equation. The goal is to identify the best combination of the accelerating factors for which i) the bed responds linearly to hydrodynamic changes; ii) a decrease of the computational cost is obtained. The sought combination is obtained by studying the behaviour of an approximate solution of the three eigenvalues associated with the ux matrix of the accelerated system. This approach allows to derive a new linear morphodynamic acceleration technique, the MASSPEED (MASs equations SPEEDup) approach, and the a priori determination of the highest possible acceleration for a given simulation. In this new approach both mass conservation equations (water and sediment) are accelerated by the same factor, differently from the MORFAC approach where only the sediment mass equation is modified. The analysis shows that the MASSPEED gives a larger validity range for linear acceleration and requires smaller computational costs than that of the MORFAC approach. The MASSPEED approach is then implemented using an adaptive approach that applies the maximum linear acceleration similarly to the implementation of the Courant-Friedrichs-Lewy stability condition. Finally, numerical simulations have been performed in order to assess accuracy and efficiency of the new approach. Results obtained in the long-term propagation of a sediment hump demonstrate the advantages of the new approach. The validation of the method is performed under steady or quasi-steady flow conditions, whereas further investigation is needed to extend the use of morphological accelerators to fully unsteady flows.

... Although it could be tempting to discuss the behaviour of the morphodynamic eigenvalues in terms of the flow (h±) waves plus a new, slower, 'bed' wave, this would imply a change of sign of two roots at Fr = 1 . In fact, as shown in Fig. 5b, in the transcritical region none of the celerities vanishes and consequently there is no change of sign as in the hydrodynamic case: for any given Froude number, including unity, (m1) and (m2) propagate downstream and (m3) upstream [39]. Moreover, moving away from criticality, (m2) and (m3) tend to the quasi-steady (b) eigenvalue in the subcritical and supercritical regime, respectively. ...

... Hence, the condition (46) ensures that the system of PDEs is hyperbolic for any value of the parameters. For this reason, the roots of (44), as well as their behaviour in the transcritical region, have been thoroughly discussed in the past due to the role this equation plays in the definition of the characteristic curves of the SVE hyperbolic system [39][40][41]. ...

... The relative amplitudes are plotted in Fig. 7, together with the forced solution given by (39). ...

Linear stability analysis is used to investigate the behavior of small perturbations of a uniform flow in a straight channel with an erodible bed composed by a unisize sediment. A shallow-water flow model is employed and bedload sediment transport is assumed. The mathematical structure of the linear problem, in terms of the eigenvalues and their associated eigenvectors is explored in detail and information is gathered on the wavespeed and growth rate of the perturbations as a function of their wavelength and of the relevant flow and sediment parameters. Several aspects of the solution are discussed, with particular focus on the behaviour in the transcritical region where the Froude number approaches unity. An approximate solution for the roots of the eigenrelationship is presented, which is not uniformly valid in the transcritical region, leading to the appearance of an unphysical instability. A regular perturbation expansion is then introduced that allows for the elimination of this singularity.

... Cordier et al. (2011) conducted numerical experiments demonstrating that the decoupled approach may fail, producing unphysical instabilities, even using a robust and well-balanced numerical scheme for shallow-water system. From a physical point of view the decoupled approach is justified when the bed weakly interacts with the hydrodynamic waves, a condition that holds only for situations far from critical conditions De Vries (1965); Lyn and Altinakar (2002), i.e. when the Froude number (Fr)≪1 or ≫1. On the contrary, the coupled approach can be applied in all conditions at the price of having a governing system of PDEs written in non-conservative form (e.g. ...

... is a measure of the intensity of total bedload in the flow usually in the Vries 1965;Lyn and Altinakar 2002). ψ is obtained from differentiating the sediment transport formula and thus depends on the bedload closure relationship adopted. ...

... From a physical point of view, under sub-or supercritical conditions (Fr < 1 or Fr > 1) the bed interacts only weakly with the water surface and small bottom perturbations propagate at a slower pace compared with the hydrodynamic waves, whereas under near-critical conditions (Fr ≃ 1) the interactions between the bed and hydrodynamic waves are quite strong. For background on the hyperbolicity of the SVE, the eigenvalues behavior and the physical behavior of small bed perturbations see (Cordier et al. 2011;Lyn and Altinakar 2002), for example. ...

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vázquez-Cendón (2012) and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.

... Equations (1)-(3) form the Saint-Venant-Exner model, e.g., [34,35] which governs one-dimensional morphodynamics when a single sediment fraction is considered (unisize-sediment case The active layer approach [26,27] provides additional continuity equations for mixed sediment. The sediment mixture is discretised into N fractions, each one characterized by one representative grain diameter d k , where k is an index in the range [1, N ]. ...

... which measures the intensity of bedload in the flow in the linearised analyses of morphodynamic models, e.g. [54,35,32,34,46,22], for the reference state assumes the value ψ R = 0.0075, indicating high sediment transport [54,34,32]. Therefore, all fractions are mobile. ...

... Within this subset, we particularly focus on the waves which carry most morphodynamic changes, namely the "bed" wave and the N − 1 = 4 "sorting" waves as defined in [54]. The "bed" wave under well-developed subcritical conditions is described by any morphodynamic model based on the Exner equation (3), even when a single sediment size is considered [22,35], while the "sorting waves" are specifically related to the introduction of active layer equations (6) for mixed sediment [54,49,46]. ...

We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [55ss]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, i) we empirically assess the balancedness of vertical mass fluxes under degradation; ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [55], which is here employed for the first time to assess a mixed-sediment model; iii) we reproduce Ribberink’s E8-E9 flume experiment [47].

... Typically, the transport parameter ψ may be estimated as small as O(10 −3 -10 −5 ) [e.g. 12,15,17,18] therefore it seems reasonable to expand λ in powers of ψ as follows: ...

... The behavior of such curves is well known: far from the critical state (i.e. Fr 1) the celerity of a small amplitude bed wave is considerably smaller than that of small amplitude hydrodynamic waves [18]. ...

... We also assume ψ L = 0.01 and h L = 1 m to which correspond Fr L = 0.33. The solution of system (44) can be obtained analytically by using characteristic variables [28] as done by Lyn and Altinakar [18]. More details on how the linear analytical solution is obtained are given in Appendix A. bed elevation, to their unperturbed initial values h L , q L and z L are given in Fig. 6b, Fig. 6c and Fig. 6d, respectively. ...

Morphological accelerators, such as the MORFAC (MORphological acceleration FACtor) approach, are widely adopted techniques for the acceleration of the bed evolution, which reduces the computational cost of morphodynamic numerical simulations. In this work we apply a non-uniform acceleration to the one-dimensional morphodynamic problem described by the de Saint Venant-Exner model by multiplying all the spatial derivatives by an individual constant (>1) acceleration factor. The final goal is to identify the best combination of the three accelerating factors for which i) the bed responds linearly to hydrodynamic changes; ii) a consistent decrease of the computational cost is obtained. The sought combination is obtained by studying the behaviour of an approximate solution of the three eigenvalues associated with the flux matrix of the accelerated system. This approach allows to derive a new linear morphodynamic acceleration technique, the MASSPEED (MASs equations SPEEDup) approach, and the a priori determination of the highest acceleration allowed for a given simulation. In this new approach both mass conservation equations (water and sediment) are accelerated by the same factor, differently from the MORFAC approach where only the sediment mass equation is modified. The analysis shows that the MASSPEED gives a larger validity range for linear acceleration and requires smaller computational costs than that of the classical MORFAC approach. The MASSPEED approach is implemented within an example code, using an adaptive approach that applies the maximum linear acceleration similarly to the Courant-Friedrichs-Lewy stability condition. Finally, numerical simulations have been performed in order to assess the accuracy and efficiency of the new approach. Results obtained in the long-term propagation of a sediment hump demonstrate the advantages of the new approach.

... Cordier et al. [11] conducted numerical experiments demonstrating that the decoupled approach may fail, producing unphysical instabilities even using a robust and well-balanced numerical scheme for shallow-water system. From a physical point of view the decoupled approach is justified when the bed weakly interacts with the hydrodynamic waves, a condition that holds only for situations far from critical conditions [7,14,27], i.e. when the Froude number (F r) 1 or 1. On the contrary, the coupled approach can be applied in all conditions at the price of having a governing system of PDEs written in non-conservative form [e.g. ...

... is a measure of the intensity of total bedload in the flow usually in the range 0 < ψ < ξ of order O (−2) [14,27]. ψ is obtained from differentiating the sediment transport formula and thus depends on the bedload closure relationship adopted. ...

... From a physical point of view, under sub-or supercritical conditions (F r < 1 or F r > 1) the bed interacts only weakly with the water surface and small bottom perturbations propagate at a slower pace compared with the hydrodynamic waves, whereas under nearcritical conditions (F r 1) the flow is close to critical conditions and the interactions between the bed and hydrodynamic waves are quite strong. For background on the hyperbolicity of the SVE, the eigenvalues behaviour and the physical behaviour of small bed perturbations see [11,27], for example. ...

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vazquez-Cend\`on and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.

... The latter is a measure of the intensity of total bedload in the ow (see, e.g., De Vries, 1965;Lyn and Altinakar, 2002;Stecca et al., 2014a), ranging between 0 (no sediment transport, xed bed) and O (10 −2 ) (high transport). ...

... For the unisize-sediment case (N = 1 fractions), the analysis of Lyn (1987), Lyn and Altinakar (2002), who generalized the previous study of De Vries (1965), gives approximations to three characteristic speeds. These are given in dimensionless form as ...

... It is positive under subcritical ow, denoting a wave which travels in the downstream direction, and negative under supercritical ow, indicating an upstream-traveling wave. The speeds λ * 1,2 (14), which are equal to those of the xed-bed Saint-Venant model, identify the celerity of waves mainly carrying perturbations of the ow variables at much faster pace (Lyn and Altinakar, 2002). ...

In this chapter we review the state of the art of theoretical and numerical developments in modeling mixed-sediment morphodynamics in gravel bed rivers using the active layer approach (Hirano, 1971, 1972). We perform a detailed mathematical analysis and apply accurate numerical solution techniques to problems of practical relevance. We consider the one-dimensional hydro-morphodynamic model which arises by coupling the Saint-Venant equations for free-surface flow with the active layer model for mixed sediment. By analytical manipulations, we show that the model propagates disturbances induced by spatial gradients in the volumetric fraction content in the active layer through distinct “sorting” waves, and that gradients in the grain size distribution of the active layer are able to trigger significant bed elevation changes. These changes propagate i) in the downstream direction with a propagation velocity inversely proportional to the active layer thickness and ii) at faster pace than the morphodynamic changes governed by the Saint-Venant-Exner model in the case of unisize sediment. We also show that the change of the mathematical nature of the problem, from hyperbolic to elliptic, may occur considering a standard setting of the parameters under conditions of practical relevance. This results in an ill-posed mathematical problem, which may lead to nonphysically oscillating numerical solutions. When the model is well-posed, numerical simulations of laboratory experiments show that numerical accuracy is a key ingredient to correctly reproduce the speed of morphodynamic processes, i.e., of streamwise sorting and its interplay with the evolution of bed elevation. Finally, we discuss the limitations of the model and future research directions.

... Analogously, the "bed celerity " (i.e., the speed of the wave related to changes in bed elevation ( Lyn and Altinakar, 2002;Morris and Williams, 1996;Stecca et al., 2014;De Vries, 1965 )) is generally slow compared to the celerities associated with perturbations of the flow. This fact has encouraged the use of a "morphodynamic acceleration factor " in morphodynamic modeling to reduce the computational time ( Latteux, 1995;Lesser et al., 2004;Ranasinghe et al., 2011;Roelvink, 2006 ). ...

... The regularization strategy is not limited to a particular range of parameter settings. Yet, when using the value of derived in this section, the Froude number cannot be in the transcritical region, as in this case the quasi-steady approximation is not valid ( Cao and Carling, 2002b;Colombini and Stocchino, 2005;Lyn, 1987;Lyn and Altinakar, 2002;Sieben, 1999 ). In the following section we consider unsteady flow, which extends the regularization technique to the transcritical region. ...

... We have developed the numerical research code Elv to model mixedsize sediment river morphodynamics ( Blom et al., 2017a;2017b ) which solves the equations for flow, bed elevation, and the bed surface grain size distribution in a decoupled manner (i.e., in series and not as a coupled system of equations). Thus, our code is not appropriate for solving transcritical situations ( Lyn, 1987;Lyn and Altinakar, 2002;Sieben, 1999 ) or cases with a high sediment concentration ( Cao and Carling, 2002a;Morris and Williams, 1996 ). ...

A notable drawback in mixed-size sediment morphodynamic modeling is that the most commonly used mathematical model in this field (i.e., the active layer model (Hirano, 1971)) can become ill-posed under certain circumstances. Under these conditions the model loses
its predictive capabilities as negligible perturbations in the initial or boundary conditions produce significant differences in the solution. In this paper we propose a preconditioning method that regularizes the model to recover well-posedness by altering the time scale of
the sediment mixing processes. We compare results of the regularized model to data from 4 new laboratory experiments conducted under conditions in which the active layer model is ill-posed. The regularized active layer model captures the change of bed elevation and surface
texture averaged over the passage of several bedforms. Small scale changes due to individual bedforms are not accounted for by neither the original nor the regularized active layer models.

... In fact, the de Saint Venant-Exner model (1) has three characteristic variables: the characteristic variable related to the smallest eigenvalue (in absolute value) can be mainly associated with the bed elevation and the others with the water flow [12,25]. Moreover, the intermediate Riemann wave in the dSVE model is related to the same smallest eigenvalue. ...

... Figure 3: Results of Test 2. Panels a, b and c: comparison between analytical, Eq. (32), and numerical solutions (for h, q and z, respectively) at the end of the simulation (with grid size nc = 100 for the DOT scheme; nc = 300 for the PRICEC and A-DOT schemes). Panels d, e and f : CPU time versus Eϕ, Eq. (30), with ϕ = h, q and z, respectively, by using: nc = [25,50,100,300,600] for DOT; nc = [100,300,600,900,1800] for PRICE-C and A-DOT. ...

... respectively) at the end of the simulation (with grid size nc = 300 for the DOT scheme; nc = 1200 for the PRICE-C and A-DOT schemes). Panels d, e and f : CPU time versus Eϕ, Eq. (30), with ϕ = h q, and z, respectively, by using: nc = [25,50,100,300] for DOT; nc = [100,300,600,1200] for PRICE-C and A-DOT. A preliminary steady-state solution over a fixed bed (A g = 0.0) is computed starting from the following initial condition: ...

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

... Even in the simplest case of the one-dimensional Saint-Venant equations, substantial numerical difficulties arise when coupling the classical Exner equation with algebraic bed load discharge equations. For instance, two of the three system eigenvalues vanish [60], which requires a careful treatment of critical conditions and the absence of sediment motion. Schemes used for strongly coupled or decoupled numerical schemes are also the object of intense debates [29,45,66,72]. ...

... The transition from subcritical to supercritical flow can be successfully reproduced using the same entropy fix method as for the shallow water equations. No modification is required by our model equations in contrast with coupled numerical methods based on the classical Exner equation (5) that run into difficulties in the presence of critical conditions [60]. Also the current Riemann solver can be employed to solve (5) for the mean particle concentration instead of equation (6) for the stochastic Poisson density, simply replacing b by γ in (A.1)- ...

... Sediment transport capacity can be understood as the maximum amount of sediment that can be transported by a flow in a particular steady state. Models based on this assumption have been proposed to compute many experimental and real-scale sediment transport problems [2][3][4][5][6]. Nevertheless, in the two last decades, a new approach accounting for the time and space lag between the actual solid fluxes and the local hydrodynamic properties has received increasing attention [7]. ...

... Previous efforts in this sense and dealing with bed load and suspended load were reported in [48,49]. [2] and [3] proposed expressions to approximate the hyperbolic system eigenvalues for the 1D erosive bed load problem, but they did not consider irregular topography neither the influence of the cross-section update mechanism on the characteristic wave celerities. ...

This work is focused on a numerical finite volume scheme for the coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expressions modeling the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled equations can be rewritten as a non-conservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. Moreover, the wave celerities for the coupled morpho-hydrodyamical system depend on the erosion-deposition mechanism selected to update the channel cross-section profile. This influence is incorporated into the system solution by means of a new parameter related to the channel bottom variation celerity. Special interest is put to show that, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge in presence of arbitrary shape channels. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver, first order accurate in time and space, dealing with solid transport flux variations caused by the channel geometry changes. Channel cross-section variations lead to the appearance of a new solid flux source term which should be discretized properly. The stability region is controlled by wave celerities together with a proper reconstruction of the approximate local Riemann problem solution, enforcing positive values for the intermediate states of the conserved variables. Comparison of the numerical results for several analytical and experimental cases demonstrates the effectiveness, exact well-balancedness and accuracy of the scheme.

... We remark that this resonance appears only when approximate roots of (4.7) are sought, because the expansion (4.9) is not uniformly valid in the transcritical Fr 1 region as k → ∞. Indeed, in the complete analysis of Lyn & Altinakar (2002), the celerities associated with the three eigenvalues are always two positive and one negative and there is no sign of this spurious resonance, the physical meaning of which will become clear in the next subsection. Note also that if the normal stress T n1 is included in the analysis, its damping, stabilizing effect modifies the behaviour of the solution in the short-wave range, as in the roll-wave example cited before ...

... The spurious resonance in the transcritical region is still present, but is amenable in this context of a more physical explanation: the system being decoupled, the slow Sorting and bed waves 885 A46-17 bed disturbance is felt by the flow as an external forcing which excites a 'natural' frequency of the flow itself, thus leading the system to resonate. We remark once more that the resonance disappears once the complete coupled morphodynamic problem is solved (Lyn & Altinakar 2002). ...

Sorting and bed waves in unidirectional shallow-water flows - Volume 885 - Marco Colombini, Costanza Carbonari

... The celerity of the perturbations associated with the flow variables (i.e. h, q x and q y ) is orders of magnitude larger than the celerity of perturbations in bed elevation if the Froude number is sufficiently small (Fr 0.7 (De Vries 1965, 1973Lyn & Altinakar 2002)). Under this condition we can decouple the system and consider steady flow to study the propagation of perturbations in bed elevation (i.e. ...

... Parameter ψ j (−) represents the sediment transport intensity (e.g. De Vries 1965;Lyn & Altinakar 2002;Stecca et al. 2014) and ranges between 0 (no sediment transport) and O(10 −2 ) (high sediment discharge): ...

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

... The last test case presented here is an extension of the analytical linearized solution proposed by Lyn and Altinakar [14] and already tested in [3]. Because of the near-critical condition of the water flow (i.e., Fr = Table 2: L 1 , L 2 and L ∞ norms of the pointwise error that the numerical model committed at the end of the simulation shows in Figure 3, using the analytical eigensystem of section 2.2 and its approximation proposed in section 2.3. ...

... Numerical test with analytical and numerical DOT Riemann solver upon the linearized analytical solution proposed by Lyn & Altinakar[14], at t = 20 [s]. We omitted the water discharge because the result are analogous at what it is shown here for the water depth and the bottom topography. ...

We consider the Shallow Water Equations (SWE) coupled with the Exner equation.
To solve these balance laws, we implement a P0P2-ADER scheme using a path conservative method for handling the non-conservative terms of the system.
In this framework we present a comparison between three different Dumbser-Osher-Toro (DOT) Riemann solvers.
In particular, we focus on three different approaches to obtain the eigensystem of the Jacobian matrix needed to compute the fluctuations at the cell edges.
For a general formulation of the bedload transport flux, we compute eigenvalues and eigenvectors numerically, analytically and using an approximate original solution for lowland rivers (i.e. with Froude number Fr<<1) based on a perturbative analysis.
To test these different approaches we use a suitable set of test cases.
Three of them are presented here: a test with a smooth analytical solution, a Riemann problem with analytical solution and a test in which the Froude number approaches unity.
Finally, a computational costs analysis shows that, even if the approximate DOT is the most computationally efficient, the analytical DOT is more robust with about 10% of additional cost.
The numerical DOT is shown to be the heavier solution.

... 329 Various variability aspects have been studied using field observations, laboratory experiments, and numerical models , Cui et al. 1996, Cao et al. 2002, Papanicolaou et al. 2004, Wu et al. 2004, Cui and Parker 2005, Curran and Wilcock 2005, Wright and Parker 2005a, 2005b. Analytical discussions of these issues have been equally valuable (Seminara 1997, Repetto et al. 2002, Lyn and Altinakar 2002, Lanzoni et al. 2006. ...

... The perturbations created by the variable boundary conditions at the up-and downstream ends of a river reach propagate downstream and upstream along it, following the characteristics of the set of governing equations. A discussion of the characteristics to evaluate the general behavior of these perturbations was initiated almost a century ago for fixed-bottom channels and reconsidered later for uniform grain size material (De Vries 1965, 1973, Lyn 1987, Correia et al. 1992, Morris and Williams 1996, Lyn and Altinakar 2002. The analysis was subsequently expanded to include aspects of nonuniform grain size, adaptation length of suspended particles, or two-dimensional (2D) equations, for example to determine the limits of applicability of different model assumptions (Sieben 1997). ...

La solution analytique d'un modele morphodynamique unidimensionnel d'un bief de riviere, avec un materiau non-uniforme en tailles de grain, en ecoulement instationnaire, est discutee. Le modele de "fleuve harmonique" suppose une propagation instantanee de l'ecoulement de l'eau le long du bief, une variation sinusoidale du temps aux deux extremites, ainsi que de la largeur de canal dans l'espace, avec deux classes de taille de grain de materiau. Independamment de la propagation instantanee de l'ecoulement de l'eau, toute l'information concernant les conditions aux limites sont repercutees le long du bief en trois vagues. Excepte pour les nombres de Froude extremement bas qui peuvent se produire typiquement pour les fleuves barres, et pour des ondes de periode extremement longue (echelles de temps geologiques), la deuxieme et troisieme onde peuvent etre negligees car pratiquement toute l'information est transmise par la premiere onde vers l'aval. D'ailleurs, les conditions aux limites dominantes affectant le bief sont le taux de transport et la composition des sediments transportes.

... This allows for solving the hydrodynamic and morphodynamic equations consecutively instead of simultaneously. It has been shown that such decoupling of the equations is acceptable for values of the Froude number smaller than 0.7 approximately De Vries, 1965;Lyn, 1987;Lyn & Altinakar, 2002), which is typically considered a valid approximation for lowland rivers. ...

... Regarding (i), the decoupling of the equations is acceptable under conditions where the Froude number is smaller than approximately 0.7 De Vries, 1965;Lyn, 1987;Lyn & Altinakar, 2002). Such a low Froude value implies that the bed celerity and the celerities associated with the hydrodynamic behavior are of a different order of magnitude and therefore interact only weakly. ...

Recent analysis of equilibrium and quasi-equilibrium channel geometry in engineered (fixed-width) rivers has successfully shown that two temporal scales can be distinguished, with quasi-static (long-term) and dynamic (short-term) components. This distinction is based on the fact that channel slope cannot keep pace with short-term fluctuations of the controls. Here we exploit the distinction between the two temporal scales to model the transient (so time-dependent) phase of channel response, which is the phase wherein the channel approaches its new equilibrium. We show that (a) besides channel slope, also the bed surface texture cannot keep pace with short-term fluctuations of the controls, and (b) mean transient channel response is determined by the probability distributions of the controls (e.g., flow duration curve rather than flow rate sequence). These findings allow us to set up a rapid numerical method that determines the mean transient channel response under stochastic controls. The method is based on distinguishing modes (i.e., sets of controls) and takes the probability density of each mode into account. At each time step, we compute the mode-specific flow, sediment transport rate, and corresponding change in bed level and surface texture. The net change within the time step is computed by weighting the mode-specific changes in bed level and surface texture with the probability density of each mode. The resulting mean transient channel response is a deterministic one, despite the controls being stochastic variables. We show that the proposed method provides a rapid alternative to Monte Carlo analysis regarding the mean time-dependent channel response.

... We reproduced the movable bed test case, as described by Lyn and Altinakar [18], of the propagation of a sediment hump. The simulated test case is the one termed as " Near Critical, High-Transport Case " in the paper. ...

... The volume fraction was chosen to be φ = 0.6. Finally, the critical flow velocity u cr was set in order to adjust the nondimensional bedload transport parameter, used in Lyn and Altinakar [18] to linearize the set of equa- tions: ...

... One-dimensional modeling of unsteady sediment transport and bed evolution in alluvial channels is most often performed by supplementing the St. Venant equations, describing fluid continuity and flow momentum conservation, with Exner equation, describing sediment continuity (Lyn and Altinakar, 2002). 2D morphodynamic models are commonly built as well with the 2D for of the St. Venant equations. ...

River flow is complicated because it depends on many different local conditions and on the interaction with the environment. Therefore, river flow is directly associated with turbulence, seepage and sediment transport among other natural phenomena. Thus, this dissertation had involved the three above mentioned phenomena:
. Turbulence is considered one of the most interesting and non understood phenomena in environmental sciences. Due to the fact that practically all natural streams flow under turbulent regime, it results interesting to analyze many of the roles that turbulence plays in river hydrodynamics.
. Seepage can be defined as the motion of any fluid through a porous medium. In water engineering, seepage is equivalent to the percolation of water through the soil from unlined canals, ditches and watercourses. Nevertheless, seepage is commonly neglected by river specialists in the analysis of water and environmental engineering issues.
. Sediment Transport. Bed erosion (scouring), deposition and entrainment of material or river morphology modification are examples of the consequences that sediment transport can provoke. That is the reason why since last century; plenty of scientists had devoted their lives to research the phenomena associated with sediment transport in order to enhance the quality of life of people living close to rivers and other water
bodies.
The main objective of this thesis was to analyze the influence of seepage on river dynamics (above all its impacts on turbulence and sediment transport mechanisms) at the laboratory scale. The contents of this contribution can be divided in three general parts:
1. Introduction to the problem; state of the art and theoretical background of the phenomenon that is analyzed.
2. The description of the experimental works and analysis of their output.
3. The theoretical frameworks that are used in fluid mechanics and hydro-engineering are briefly described. A proposal on how is possible to model river dynamics with seepage is stated.
The first part of the thesis (Chapters I-II) presents at a glance the theoretical background of open-channel flow turbulence, sediment transport and its association with groundwater flow. In the first chapter, the introduction and motivation of the thesis is described. The final part of the first chapter introduces the main problem that is discussed; namely the influence of seepage on open-channel turbulent flowand in the water-sediment interaction. The theory of open channel hydrodynamics is presented in the second chapter: from fundamental laws of fluid mechanics to the statistical description of turbulence. This chapter includes as well the basic theory concerning sediment transport mechanisms. The second part (Chapter III) is a summary of the laboratorial works that were carried out. The objective of these experiments was to investigate the effects of seepage (upward) not only on water-sediment interaction, but also on open channel hydrodynamics. This chapter can be divided in two parts: At a first stage, concentration profiles were determined due to (and only due to) sediment entrainment provoked by flow turbulence for established hydrodynamic conditions (no seepage was considered). At the second stage, the experimental set-up was modified to allow groundwater flow through the flume’s bed. Seepage’s effects on bottom changes (bed and suspended transport modes) and on open-channel turbulence were analyzed for different hydraulic conditions and seepage intensities. The third part (Chapter IV-V) presents the introduction to further research; A proposal on how is possible to analyze seepage’s influence in river dynamics is stated. This proposal includes three alternatives; Namely, to include groundwater flow as a new Dirichlet boundary condition in an existing morphodynamic model, to change the conditions of incipient motion or to couple a quasi-laminar flow module (to analyze seepage) with a turbulent flow module (to analyze open-channel flow). At the end of the thesis, the output of this researching is summarized and discussed.
As a brief conclusion, it is possible to state that regardless the small magnitude of the groundwater flow; seepage affects sediment transport mechanisms as demonstrated by Herrera-Granados (2008b, 2008c, 2010a); because it is influencing the instantaneous velocity field of the open channel flow. Therefore, seepage is influencing open-channel flow turbulence as described by Herrera-Granados (2011) and Herrera-Granados and Kostecki (2011). Hence, the onset ofmotion of sediment particles is modified and consequently, sediment transport rates change.

... Considering well-developed sub-critical (Fr<0.8) or super-critical (Fr>1.2) Froude regions, such models decouple the hydrodynamics and the morphodynamics (De Vriend et al., 1993;Juez et al., 2013), when considering both sediment (Lyn, 1987;Lyn & Altinakar, 2002) and graded sediment (Stecca et al., 2014). Therefore, it is possible to adopt the 1-D sediment continuity (eq. 1) developed by Exner (1920), and the mass balance for each grain-size fraction of the active layer (2), computed following Hirano (1971). ...

The paper presents a 0-D model of an alluvial watercourse schematized in two connected reaches, evolving at the long time-scale and under the hypothesis of Local Uniform Flow. Each reach is defined by its geometry (constant length and width, time-changing slope) and grain-size composition of the bed, while the sediment transport is computed using a sediment rating curve. The slope evolution is provided by a 0-D mass balance and the evolution of the bed composition is computed by a 0-D Hirano equation. A system of differential equations, solved with a predictor-corrector scheme, is derived and applied to the schematic watercourse to simulate the morphological response to changing initial conditions, and the evolution towards long-term equilibrium conditions. Differently from a single-reach 0-D schematization with uniform grain-size, besides the simplifications adopted, the model proposed here simulates the behaviour of alluvial rivers in a physically-based way, showing a grain-size fining in the downstream direction accompanied by milder slopes, and a tendency to develop concave longitudinal profiles.

... Subcritical Fr < 1 values indicate tranquil flows with downstream controls. However, the presence of a movable bed makes the identification of sub-and super-critical regimes less obvious, as additional phenomena come into play (Lyn, 1987; Lyn and Altinakar, 2002). Figure 8. Streamwise scenario for a convexo-concave landscape topography, from runoff initiation to the main rivers, across flow typologies (Overland: O; High-gradient: Hg; Bedforms: B; or Fluvial: F) and spatiotemporal scales (L, T , H ). All sketches and drawings for the Highgradient and Bedforms typologies were taken from Montgomery and Buffington (1997). ...

This review paper investigates the determinants of modelling choices, for numerous applications of 1-D free-surface flow and morphodynamic equations in hydrology and hydraulics, across multiple spatiotemporal scales. We aim to characterize each case study by its signature composed of model refinement (Navier–Stokes: NS; Reynolds-averaged Navier–Stokes: RANS; Saint-Venant: SV; or approximations to Saint-Venant: ASV), spatiotemporal scales and subscales (domain length: L from 1 cm to 1000 km; temporal scale: T from 1 s to 1 year; flow depth: H from 1 mm to 10 m; spatial step for modelling: δL; temporal step: δT), flow typology (Overland: O; High gradient: Hg; Bedforms: B; Fluvial: F), and dimensionless numbers (dimensionless time period T*, Reynolds number Re, Froude number Fr, slope S, inundation ratio Λz, Shields number θ). The determinants of modelling choices are therefore sought in the interplay between flow characteristics and cross-scale and scale-independent views. The influence of spatiotemporal scales on modelling choices is first quantified through the expected correlation between increasing scales and decreasing model refinements (though modelling objectives also show through the chosen spatial and temporal subscales). Then flow typology appears a secondary but important determinant in the choice of model refinement. This finding is confirmed by the discriminating values of several dimensionless numbers, which prove preferential associations between model refinements and flow typologies. This review is intended to help modellers in positioning their choices with respect to the most frequent practices, within a generic, normative procedure possibly enriched by the community for a larger, comprehensive and updated image of modelling strategies.

... On the other hand, the above were possibly the reasons for which the numerical model gave unsatisfactory results when applied to T3. In the literature, it has been argued that trans-critical flows with a mobile bed can be handled by a conventional numerical approach [39] and examples of numerically simulated mixed-flow regime can be found [40,41]. However, most of the discussed case studies deal with conditions remaining constant at the boundaries while, to the best of our knowledge, the numerical modelling of a trans-critical flow in both space and time was only performed by Reference [40]. ...

This communication explores the use of numerical modelling to simulate the hydro-morphologic response of a laboratory flume subject to sediment overloading. The numerical model calibration was performed by introducing a multiplicative factor in the Meyer–Peter and Müller transport formula, in order to achieve a correspondence with the bed and water profiles recorded during a test carried out under a subcritical flow regime. The model was validated using a second subcritical test, and then run to simulate an experiment during which morphological changes made the water regime switch from subcritical to supercritical. The “relationship” between physical and numerical modelling was explored in terms of how the boundary conditions for the two approaches had to be set. Results showed that, even though the first two experiments were reproduced well, the third one could not be modeled adequately. This was explained considering that, after the switch of the flow regime, some of the boundary conditions posed into the numerical model turned out to be misplaced, while others were lacking. The numerical modelling of hydro-morphologic processes where the flow regime is trans-critical in time requires particular care in the position of the boundary conditions, accounting for the instant at which the water regime changes.

... The SVE system is an extension of the shallow-water system, where a new dynamical variable is introduced that describes the bed evolution of the bottom topography. The SVE system has been mainly considered for the description of model bedload sediment transport phenomena that occur in large time-and space-scale fluid dynamics systems [32][33][34][35][36]. In the following, we apply Lie's algorithm to determine the Lie point symmetries for the SVE system and then to determine the one-dimensional optimal system. ...

We present the Lie symmetry analysis for a hyperbolic partial differential system known as the one-dimensional Saint-Venant–Exner model. The model describes shallow-water systems with bed evolution given by the Exner terms. The sediment flux is considered to be a power-law function of the velocity of the fluid. The admitted Lie symmetries are classified according to the power index of the sediment flux. Furthermore, the one-dimensional optimal system is determined in all cases. From the Lie symmetries we derive similarity transformations which are applied to reduce the hyperbolic system into a set of ordinary differential equations. Closed-form exact solutions, which have not been presented before in the literature, are presented. Finally, the initial value problem for the similarity solutions is discussed.

... Relevant investigations have been carried out by means of laboratory experiments or instantaneous local flow features and can be formulated by different empirical closure relations found in literature ( Yang, 1996 ). Models based on this assumption are commonly called equilibrium or capacity transport models ( Tingsanchali and Chinnarasri, 2001;Cao et al., 2002;Lyn and Altinakar, 2002;Hudson and Sweby, 2003;Goutière et al., 2008;Castro-Díaz et al., 2008;Murillo and Garcia-Navarro, 2010;Juez et al., 2014;Gunawan and Lhébrard, 2015;Martínez-Aranda et al., 2019 ). On the other hand, in non-capacity models, the actual transport rates are computed through advection and mass exchange with the static erodible bed. ...

Finite-depth sediment layers are common in natural water bodies. The presence of underlying bedrock strata covered by erodible bed layers is ubiquitous in rivers and estuaries. In the last years, the development of models based on the non-capacity sediment transport assumption, also called non-equilibrium assumption, has offered a new theoretical background to deal with complex non-erodible bed configurations and the associated numerical problems. Bedload non-capacity sediment transport models consider that the actual solid transport state can be different from the equilibrium state and depending on the temporal evolution of the flow. The treatment of finite-depth erodible bed layers, i.e. partially erodible beds, in bedload models based on the equilibrium approach has usually been made using numerical fixes, which correct the unphysical results obtained in some cases. Generally , the presence of a finite-depth erodible layer implies the introduction of a kind of non-equilibrium condition in the bedload transport state. Nevertheless , this common natural bed configuration has not been previously considered in the development of numerical models. In this work, a finite volume model (FVM) for bedload transport based on non-capacity approach and dealing with finite-depth erodible layers is proposed. New expressions for the actual bedload transport rate and the net exchange flux through the static-moving bed layers interface are used to develop a numerical scheme which solves the coupled shallow water and non-capacity bedload transport system of equations. The reconstruction of the intermediate states for the local Riemann problem at each intercell edge is designed to correctly model the presence of non-erodible strata, avoiding the appearance of unphysical results in the approximate solution without reducing the time step. The new coupled scheme is tested against laboratory benchmarking experiments in order to demonstrate its stability and accuracy, pointing out the properties of both equilibrium and non-equilibrium formulations.

... With specific reference to slit-check dams, the aggradation process is characterized by a deposit front that rises in the subcritical region close to the hydraulic jump [ Fig. 3(a)] and advances both upstream and downstream [ Fig. 3(b)]. Under these conditions, the water level discontinuity is smaller than the hydraulic jump in fixed beds and transcritical conditions occur all over the deposit with Froude numbers close to 1 (Sieben 1999;Lyn and Altinakar 2002;Castro-Orgaz et al. 2008). ...

Sediment trapping behavior of slit-check dams and related design criteria are usually evaluated assuming the hypothesis of steady-state hydraulic conditions associated with the peak of the design flood. A novel approach based on the use of a 1D unsteady flow numerical model to simulate in detail transport processes upstream from slit-check dams during flood events is presented in this paper. The adopted model is validated by comparison with results of laboratory experiments from the literature showing the ability to reproduce accurately the temporal evolution of the sediment deposit behind the dam. The presented model-based approach enabled a realistic evaluation of the sediment trapping efficiency of the dam during the flood, i.e., the percent trapped portion of sediment volume transported by the flood, for different sizes of the dam slit. Especially for narrow slits, simulations revealed lower trapping efficiencies of slit-check dams at the flow peak condition compared to an approach from literature based on the steady-state flow assumption. Also, simulations allowed for the evaluation of the behavior of the dam at the end of the flood event, revealing the potential of the model-based approach to analyze the morphological evolution of the river downstream of the dam. (C) 2014 American Society of Civil Engineers.

... The computation of the eigenvalues is relatively easy when the solid flux is given by the law (1.7), but this is not the general case. This is the technique used in [94]. In [78], the authors use a flux-limited version of Roe's scheme to solve several formulations of the Shallow Water-Exner system. ...

The present thesis deals with the modeling and numerical simulation of complex geophysical ﬂows. Two processes are studied: sediment transport, and variable density ﬂows. For both ﬂows, the approach is the same. In each case, a reduced vertically-averaged model is derived from the 3D Navier-Stokes equations by making a speciﬁc asymptotic analysis. The models verify stability properties. Attention is paid to preserving these properties at the discrete level, in particular the entropy stability. The behavior of both models is illustrated numerically. Concerning the sediment transport model, the sediment layer is ﬁrst studied alone. Then, a coupled sediment-water model is presented and simulated. The inﬂuence of a viscosity term in the model for the sediment layer is investigated. Due to this viscosity term, the sediment ﬂux is non-local. A transport threshold is added to the model. The water layer is modeled by the Shallow Water equations. Adding some non-locality to the model allows to simulate dune growth and propagation. In the variable density ﬂow model, the density is a function of one or several tracers such as temperature and salinity. The model derivation consists in removing the dependence of the density on the pressure. A layer-averaged formulation of the model is proposed, which is subsequently used to propose a numerical discretization. The numerical simulations emphasize the diﬀerences between this model and a model relying on the classical Boussinesq approximation.

... In the work by Lanzoni et al. [2006], a perturbation analysis on the solutions of coupled and decoupled systems is performed, in order to show that the latter ones fail in computing solutions, especially in the case of fairly short perturbations in the neighborhood of critical states of flow. The main reason is that for Froude numbers close to unity, perturbations of bed level are propagated both upstream and downstream with comparable intensities (see Lyn and Altinakar [2002]), as opposite to the cases of the Froude number being much higher or lower than unity, in which such perturbations are mostly propagated upstream or downstream, respectively. Another interesting analysis on the matter can be found in Cordier et al. [2011]. ...

This work focuses on the implementation of a Shallow Water-Exner model for compound natural channels with complex geometry and movable bed within the finite volume framework. The model is devised for compound channels modeling: cross-section overbanks are treated with fixed bed conditions, while the main channel is left free to modify its morphology. A capacitive approach is used for bedload transport modeling, in which the solid flow rates are estimated with bedload transport formulas. The model equations pose some numerical issues in the case of natural channels, where bedload transport may occur for both subcritical and supercritical flows and geometry varies in space. An explicit path-conservative scheme, designed to overcome all these issues, is presented in the paper. The scheme solves liquid and solid phases dynamics in a coupled manner, in order to correctly model near critical currents/channel interactions and is well-balanced, that is able to properly reproduce steady states. The Roe and Osher Riemann solvers are implemented, so as to take into account the spatial geometry variations of natural channels. The scheme reaches up to 2nd order accuracy. Validation is performed with fixed and movable bed test cases whose analytical solution is known, and with flume experimental data. An application of the model to a real case study is also shown. This article is protected by copyright. All rights reserved.

... In brief, we solve the backwater equation in combination with the flux form of the Exner (1920) and active layer equations in a decoupled manner. This implies that the equations are assumed to weakly interact with each other, which is acceptable if the Froude number is below approximately 0.7 (Lyn, 1987;Lyn & Altinakar, 2002;Sieben, 1999). The backwater equation is solved using the standard fourth-order Runge-Kutta method, and the Exner (1920) and active layer equations are solved using a first-order upwind scheme (i.e., FTBS (Forward in Time, Backward in Space); Long et al., 2008;Sonke et al., 2003;Zima et al., 2015) using a variable time step to guarantee a CFL (Courant-Friedrichs-Lewy) number (Courant et al., 1928; see, e.g., Toro, 2009) equal to 0.9. ...

The active layer model (Hirano, 1971) is frequently used for modelling mixed‐size sediment river morphodynamic processes. It assumes that all the dynamics of the bed surface are captured by a homogeneous top layer that interacts with the flow. Although successful in reproducing a wide range of phenomena, it has two problems: (1) it may become mathematically ill‐posed, which causes the model to lose its predictive capabilities, and (2) it does not capture dispersion of tracer sediment. We extend the active layer model by accounting for conservation of the sediment in transport and obtain a new model that overcomes the two problems. We analytically assess the model properties and discover an instability mechanism associated with the formation of waves under conditions in which the active layer model is ill‐posed. Numerical simulations using the new model show that it is capable of reproducing two laboratory experiments conducted under conditions in which the active layer model is ill‐posed. The new model captures the formation of waves and mixing due to an increase in active layer thickness. Simulations of tracer dispersion show that the model reproduces reasonably well a laboratory experiment under conditions in which the effect of temporary burial of sediment due to bedforms is negligible. Simulations of a field experiment illustrate that the model does not capture the effect of temporary burial of sediment by bedforms.

... In this sense, it is a complete Riemann solver. This feature allows accurate simulation of phenomena strongly influenced by the intermediate waves, such as morphodynamic evolution.In fact, the de Saint Venant-Exner model (1) has three characteristic variables: the characteristic variable related to the smallest eigenvalue (in absolute value) can be mainly associated with the bed elevation and the others with the water flow[12,25]. Moreover, the intermediate Riemann wave in the dSVE model isrelated to the same smallest eigenvalue. ...

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with an analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

... Finally, the sediment mass flux can be obtained by writing the jump relations between the intermediate states . Eventually, by assuming that the sediment-related information is only contained in the two first characteristics as suggested by Lyn and Altinakar (2002), both in subcritical and supercritical regimes, we can express the flux as follows: ...

We propose a finite-volume model that aims at improving the ability of 2D numerical models to accurately predict the morphological evolution of sandy beds when subjected to transient flows like dam-breaks. This model solves shallow water and Exner equations with a weakly coupled approach while the fluxes at the interfaces of the cells are calculated thanks to a lateralized HLLC flux scheme. Besides describing the model, we ran it for four different test cases: a steady flow on an inclined bed leading to aggradation or degradation, a dam-break leading to high interaction between the flow and the bed, a dam-break with a symmetrical enlargement close to the gate and a dam-break in a channel with a 90° bend. The gathered results are discussed and compared to an existing fully coupled approach based on HLLC fluxes. Although both models equally perform regarding water levels, the weakly coupled model looks to better predict the bed evolution for the four test cases. In particular, its results are not affected by an excessive numerical diffusion encountered by the coupled model. Moreover, it usually better estimates the amplitudes of the maximum deposits and scours. It is also more stable when subject to high bed–flow interaction.

... TVD Mac Cormack scheme obtained a good result on discontinuity compared with the Classical Mac Cormack Scheme. In recent times, many options are available on solving the flow discontinuity using a SWE model [3,4,5,6]. ...

... In the case of bed load models (Γ = 0), strict hyperbolicity has previously been demonstrated for various cases. A number of earlier papers derived formulae for the system characteristics by expanding 1 in terms of parameters that are small when the bed dynamics is slow compared with the hydraulic variables (Lyn 1987;Zanré & Needham 1994;Lyn & Altinakar 2002;Lanzoni et al. 2006). From this perspective, the degeneracy of the system characteristics that underpins ill posedness in the → 0 limit is already well appreciated, since it causes naive asymptotic formulae for 1 to break down near Fr = 1 (Lyn 1987;Zanré & Needham 1994). ...

We analyse the linear stability of uniform steady morphodynamic flows using an extended shallow-water model that permits material to be exchanged between a suspended sedimentary mixture and its underlying bed. Any physical closures are left as arbitrary functions of the flow variables, so that our conclusions apply to a wide class of models used in engineering and geosciences. The inclusion of morphodynamics modifies the usual threshold for roll-wave instability by introducing a singularity into the linearised system at the critical Froude number $Fr = 1$. This leads to unbounded growth of short-wave disturbances and corresponding ill-posedness of the governing equations, which may be traced to a resonance between stability modes associated with the flow and the bed. By incorporating a suitable physical regularisation, we show that ill-posedness may be removed without affecting the location of the underlying instability. Alternatively, the inclusion of a bed load flux layer, common in fluvial models, can be sufficient to avoid ill-posedness under modest constraints. Implications of our analyses are considered by employing simple closures, including a drag law that switches between fluid and granular characteristics, depending on the sediment concentration. Steady layers are shown to bifurcate into two states: dilute flows which are stable at low $Fr$ and concentrated flows which are always unstable to disturbances in concentration. Finally, properties of the morphodynamic instability and the effects of regularisation are examined in detail by computing growth rates of the linear modes across a wide region of parameter space.

... There have been interesting works analyzing the time scales of fluvial processes (Lyn 1987;de Vries 1965de Vries , 1973de Vries , 1975. Closely related to these are the analyses on the fundamental features of celerity and long-wave propagation in erodible channels (Sieben 1999;Lyn and Altinakar 2002;Lanzoni et al. 2006). These studies provide insight into the interaction between fluvial flow and bed deformation, but because these studies invoke a priori the assumption of sediment transport capacity, they cannot reveal if capacity models for sediment transport are applicable. ...

Fluvial bed load transport is often considered to assume a capacity regime exclusively determined by local flow conditions, but its applicability in naturally occurring unsteady flows remains to be theoretically justified. In addition, mathematical river models are often decoupled, being based on simplified conservation equations and ignoring the feedback impacts of bed deformation to a certain extent. So far whether the decoupling could have considerable impacts on the fluvial processes with bed load transport remains poorly understood. This paper presents a theoretical investigation of both issues. The multiple time scales of fluvial processes with bed load sediment are evaluated to examine the applicability of bed load transport capacity and decoupled models. Numerical case studies involving active bed load transport by highly unsteady flows complement the analysis of the time scales. It is found that bed load transport can sufficiently rapidly adapt to capacity in line with local flow because sediment exchange with the bed overwhelms the advection of bed load sediment by the mean flow. The present work provides theoretical justification of the concept of bed load transport capacity in most circumstances, which is underpinned by existing observations of bed load transport by flash floods. For fluvial processes with bed load transport, the feedback impacts of bed deformation are limited; therefore, decoupled modeling is, in this sense, appropriate. DOI: 10.1061/(ASCE)HY.1943-7900.0000296. (c) 2011 American Society of Civil Engineers.

... Relevant investigations have been carried out by means of laboratory experiments or instantaneous local flow features and can be formulated by different empirical closure relations found in literature ( Yang, 1996 ). Models based on this assumption are commonly called equilibrium or capacity transport models ( Tingsanchali and Chinnarasri, 2001;Cao et al., 2002;Lyn and Altinakar, 2002;Hudson and Sweby, 2003;Goutière et al., 2008;Castro-Díaz et al., 2008;Murillo and Garcia-Navarro, 2010;Juez et al., 2014;Gunawan and Lhébrard, 2015;Martínez-Aranda et al., 2019 ). On the other hand, in non-capacity models, the actual transport rates are computed through advection and mass exchange with the static erodible bed. ...

... The numerical solution of the fully coupled Saint-Venant-Exner equations, as in the present case, is fraught with difficulties in determining the eigenvalues of the Jacobian matrix, which represent the wave celerities required for computing the interface flux. 45 Moreover, the performance of numerical models using the equilibrium formulation also depends upon the specific numerical scheme used. Castro D ıaz et al. 46 suggested that lower order schemes fail to capture the sediment evolution appropriately due to numerical diffusion. ...

Erosional failure of granular dams by an overtopping body of water is investigated using a depth-averaged morphodynamic model. The transport of sediment by the flow assumes the sediment flux to remain in equilibrium with the local bed shear stress. Accordingly, the shallow-water hydrodynamic equations are coupled with the Exner equation for mass conservation of the sediment. The system of equations is solved using a fully coupled well-balanced finite volume method, second-order accurate in time and space. The effect of the steep bed slope of a dam face is incorporated into both the hydrodynamics and sediment transport equations, leading to improved predictions. Comparison with results obtained from nonequilibrium sediment transport models indicates that such models perform poorly while predicting the bed evolution near the toe of an eroding dam. Observations from experimental studies demonstrate that the amount of sediment entrained by the flow is not significant, except during the initial moments of failure. This suggests that the vertical exchange of mass between the bed and the flow layer, as assumed by the nonequilibrium models, may not be completely valid during the failure. The equilibrium model results, reproducing the key flow features of the overtopping failure process, are validated by experimental measurements. The study provides fresh insights into the sediment transport processes associated with the erosion of a granular dam by overtopping, establishes the appropriateness of the equilibrium approach for its numerical modeling, and proposes a well-balanced second-order accurate solution technique for solving the resulting coupled equations of flow and sediment transport. Published under license by AIP Publishing. https://doi.

... Mais la nature de ces caractéristiques et leur attribution à l'un ou l'autre des paramètres pour un écoulement critique n'est pas trivial et mérite d'être considéré plus en détail (Lyn and Altinakar, 2002). ...

Modelisation of breach formation in earth dam is very complicated because it is a progressive phenomenon.
Indeed, while the calculated flow depends on the size of the breach, the velocity of the breach formation depends of the erosion rate imposed by the flow itself. Open-channel hydraulics deals reasonably well with two types of flow : slowly evolving alluvial flows, on the one hand, and transient flows in rigid geometries on the other hand. Dam-break induced geomorphic flows combine the difficulties of these two types of flow.
In this report, we discuss the capacity of tools provided by river engineering to simulate earth dam break. In a
first time, breach formation and corresponding main parameters are presented. Then all the equations and
hypothesis used in mechanistics models are recalled and their validity in such a context is discussed. It appears that, if such models should be able to simulate in a satisfactory way the flow and the water level, it should under estimate the erosion rate, because all the processes are not taken into account in the equations. A test case from laboratory experiments is then simulated with three programs from Cemagref (a simplified model using Bernouilli’s equation, a 1D and a 2D models using Saint Venant’s equations). We noticed some numerical instabilities, which highlighted the coupling of water velocity and geometry evolution in breach formation. But results are in fair agreement with the analysis of equations we provided first.

... where the last two terms represent the Froude numbers of the water and solid phases, respectively. Following a classical way to represent the eigenvalues for free-surface flows (Lyn & Altinakar 2002;Garegnani et al. 2013), we plot the behaviour of the differentλ as a function of the water Froude number Fr w by keeping fixed the values of the other parameters. A typical behaviour of these plots is presented in figure 4. It allows us to make three general considerations. ...

... They are only determined by instantaneous local flow features and can be formulated by different empirical closure relations found in literature [148]. Models based on this assumption are commonly called equilibrium or capacity transport models [15,23,45,48,52,64,83,86,102,136]. On the other hand, in non-capacity models, the actual transport rates are computed through advection and mass exchange with the static erodible bed. ...

Among the geophysical and environmental surface phenomena, rapid flows of water and sediment mixtures are probably the most challenging and unknown gravity-driven processes. Sediment transport is ubiquitous in environmental water bodies such as rivers, floods, coasts and estuaries, but also is the main process in wet landslides, debris flows and muddy slurries. In this kind of flows, the fluidized material in motion consists of a mixture of water and multiple solid phases which might be of different nature, such as different sediment size-classes, organic materials, chemical solutes or heavy metals in mine tailings. Modeling sediment transport involves an increasing complexity due to the variable bulk properties in the sediment-water mixture, the coupling of physical processes and the presence of multiple layers phenomena. Two-dimensional shallow-type mathematical models are built in the context of free surface flows and are applicable to a large number of these geophysical surface processes involving sediment transport. Their numerical solution in the Finite Volume (FV) framework is governed by the particular set of equations chosen, by the dynamical properties of the system, by the coupling between flow variables and by the computational grid choice. Moreover, the estimation of the mass and momentum source terms can also affect the robustness and accuracy of the solution. The complexity of the numerical resolution and the computational cost of simulation tools increase considerably with the number of equations involved. Furthermore, most of these highly unsteady flows usually occur along very steep and irregular terrains which require to use a refined non-structured spatial discretization in order to capture the terrain complexity, increasing exponentially the computational times. So that, the computational effort required is one of the biggest challenges for the application of depth-averaged 2D models to realistic large-scale long-term flows. Throughout this thesis, proper 2D shallow-type mathematical models, robust and accurate FV numerical algorithms and efficient high-performance computational codes are combined to develop Efficient Simulation Tools (EST's) for environmental surface processes involving sediment transport with realistic temporal and spatial scales. New EST's able to deal with structured and unstructured meshes are proposed for variable-density mud/debris flows, passive suspended transport and generalized bedload transport. Special attention is paid to the coupling between system variables and to the integration of mass and momentum source terms. The features of each EST have been carefully analyzed and their capabilities have been demonstrated using analytical and experimental benchmark tests, as well as observations in real events.

... Since our purpose is focusing on non-uniform transport, we have chosen to start from the simplest mobile-bed model composed by the water mass and momentum balance (de St. Venant equations) plus the so called Exner equation. Garegnani et al. (2011) demonstrated that this model is nothing but a consistent asymptotic simplification, valid when the concentration is less than 1% (see also Garegnani et al. 2013), of a widely used quasi two-phase model ( Lyn & Altinakar 2002, Rosatti & Fraccarollo 2006, Wu & Wang 2008, Rosatti, Murillo, & Fraccarollo 2008, Armanini, Fraccarollo, & Rosatti 2009, among others). Moreover we consider a one-dimensional depth-averaged set valid for rectangular channels with unitary width. ...

One-dimensional mathematical modelling of non-uniform sediment transport in freesurface, mobile-bed flows, is commonly faced adding to the De St. Venant equations, the mass-balance equations for each grain-size class in which the statistical distribution of the bed material is divided on the basis of a given choice. This method is called Bed Material Fraction (BMF) approach. A possible alternative is the so-called Statistical Moment (SM) approach, in which the non-uniformity of the sediments is described in terms of the statistical moments of the distribution. In this work, we show how the two methods can be obtained as different approximations of the same differential equation for the bed material distribution in the space of the diameters. Moreover, because of the strong similitude of the relevant differential equation in the x-t space, the two methods can be treated in a unified way from a numerical point of view. Finally, an idealized test case allows comparing the performances of the two approaches.

New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

The one-dimensional numerical model presented in the present paper divides the flow in two fully coupled layers, a water layer and a water-sediment transport layer. Initially, this model was used to depict dam-break flows, which do not require a specific treatment of boundary conditions. The aim of the present research is to extend the model to fluvial flows requiring an appropriate boundary condition treatment. This treatment commonly relies on characteristics. However, in the frame of a two-layer model with five equations, those characteristics are not obvious to determine. This paper shows, how to extend numerically the common eigenstructure analysis to address the problem. Examples are presented and the boundary conditions treatment is illustrated on the particular case of a hydraulic jump over a mobile bed. Results of the numerical model including the adequate boundary conditions are favourably compared to experimental results.

A mathematical model of overtopping dam-break is developed using the coupled 1D shallow water equations on steep surface and the Exner equation. The model takes advantages of the Roe approximate Riemann solvers and the predictor-corrector scheme for the source term treatment. The issues of computational stability and accuracy on sediment erosion transport and bed evolution can thus be ensured. The result shows that the model is able to better simulate the process of non-cohesive earth dam breaches and provides a new approach for numerical modeling of overtop dam-break.

A new coupled finite-volume scheme based on the Augmented Roe solver adapted to simulate morphological evolution of arbitrary cross-sections is presented. In pure hydrodynamic conditions, the Augmented Roe scheme has proven to provide accurate results and a constant discharge in steady-flow conditions. Here, this scheme is extended to solve the one-dimensional Saint-Venant- Exner system of equations written for arbitrary cross-sections. Therefore, new eigenvalues and source-term calculations are proposed to account for the irregular shape of the cross-sections. The performances of the proposed scheme are assessed by comparison with three different onedimensional numerical models aimed at simulating morphological changes, with coupled or uncoupled approaches, and based on HLL or Roe-based flux calculations. Numerous test cases were examined, including water at rest, steady flows and transient flows for which experimental results exist. The results show that the proposed scheme provides stable and accurate results for a wider range of situations than is available with other classical models.

New finite-volume forms of the Saint-Venant equations for one-dimensional (1D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and serve to transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. The derivation introduces an analytical approximation of the free surface across a finite volume element (e.g. linear, parabolic) as well as an analytical approximation of the bottom topography. Integration of the product of these provides an approximation of a piezometric pressure gradient term that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, water surface elevations, and the channel bottom elevation (but without using any volume-averaged bottom slope). The new conservative form should be more tractable for large-scale simulations of river networks and urban drainage systems than the traditional conservative form of the Saint-Venant equations where it is difficult to maintain a well-balanced discretization for highly-variable topography.

A hybrid local time step/global maximum time step (LTS/GMaTS) method is proposed for computationally efficient modeling of hydro-sediment-morphodynamic processes. The governing equations are numerically solved on unstructured triangular meshes using a well-balanced shock-capturing finite volume method with the HLLC approximate Riemann solver. High computational efficiency is achieved by implementing the LTS to solve equations governing sediment-laden flows (i.e., the hydro-sediment part), and implementing the GMaTS to solve equations governing bed materials (i.e., the morphodynamic part). Two benchmark experimental dam-break flows over erodible beds and one field case of the Taipingkou waterway, Middle Yangtze River, are simulated to demonstrate the high computational efficiency and the satisfactory quantitative accuracy. It is shown that the computational efficiency of the new model can be faster by an order of magnitude than a traditional model of similar type but implementing the global minimum time step (GMiTS). The satisfactory quantitative accuracy of the new model for the present cases is demonstrated by the negligible L 2 norms of water level and bed elevation between the new model and the traditional model, as compared to the L 2 norms between the traditional model and the measured data.

In this paper, an extension of a second-order Godunov-type wave propagation algorithm is presented for modelling two-dimensional morphodynamic problems using a coupled approach. In this solution, the two-dimensional shallow water equations (SWEs) and bedload sediment mass balance laws are expressed in a coupled form. The proposed numerical solver treats the source term including the bedload variations as well as the friction terms within the flux-differencing of the finite-volume neighbouring cells. In order to solve the morphodynamic system in two-dimensions, the dimensional-splitting method is utilized. To consider the bedload sediment discharge within the Exner equation, the Smart and Meyer-Peter & Müller formulae are adopted. To verify the capability of the extended wave propagation solver in dealing with different flow regimes several numerical test cases are investigated. The numerical results show that for all examined cases, excellent agreement is achieved between the numerical results and the exact solutions and experimental data, confirming the effectiveness of the method.

The evolution of open-channel flow and morphology can be simulated by one-dimensional (ID) mathematical models. These models are typically solved by numerical or analytical methods. Because the behavior of variables can be explained by explicit mathematical determinations, compared to numerical solutions, analytical solutions provide fundamental and physical insights into flow and sediment transport mechanisms. The singular perturbation technique derives a hierarchical equation of waves and describes the evolutionary nature of disturbances in hyperbolic systems. The wave hierarchy consists of dynamic, diffusion, and kinetic waves. These three types of waves interact with each other in the process of propagation. Moreover, the Laplace transform is implemented to transform partial differential equations into ordinary differential equations. Analytical expressions in the wave front are subtracted by the approximation of kinetic and diffusion models. Moreover, an analytical solution consists of a linear combination of the kinetic wave front and the diffusion wave front expressions, pursuing to describe the physical mechanism of motion in open channels as completely as possible. Analytical solutions are presented as a combination of exponential functions, hyperbolic functions, and infinite series. The obtained analytical solution was further applied to the simulation of flood path and morphological evolution in the Lower Yellow River. The phenomenon of increased peak discharge in the downstream reach was successfully simulated. It was encouraging that the results were in good agreement with the observed data.

The delayed response of fluvial rivers to external disturbances has been described by many researchers. To simulate such behavior, the rate law model (or the delayed response model) was developed by previous researchers, and has been applied to a series of river morphological problems. However, to date, the applicability of the rate law model has not been fully understood. In the current paper, a physically-based analysis of the rate law model is presented to assess the responses of bed elevation and the grain size of surface sediment, using the response of the Shi-ting River, China, after the 2008 Wenchuan Earthquake as an example. First, a physically-based river morphodynamic model is implemented to reproduce the post-earthquake adjustments of the Shi-ting River. Next, the mathematical properties of the Hirano–Exner equation, which describes the dynamics of both bed elevation and surface texture, are analyzed. It is shown that the dynamics of bed elevation are dominated by a diffusion process and the analytical solution for bed elevation has a similar mathematical formulation to that of the rate law model. Thus, the rate law and morphodynamic models similarly predict the adjustment of bed elevations. In contrast, the dynamics of bed surface texture are controlled by both advection and diffusion processes. The advection and diffusion processes dominate adjustments over the short (annual to decadal scale) and long (century to millenial scale) terms, respectively. These physics produce the multi-scale and non-monotonic character of the adjustment of bed surface texture. As a result, the rate law model is incapable of describing the adjustment of bed surface texture. These findings highlight the applicability and limitations of the rate law model in simulating river morphodynamic processes.

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver (cf. [A. Harten, P. D. Lax and B. van Leer, SIAM Rev. 25, 35–61 (1983; Zbl 0565.65051)] for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed.

Originally published: New York : Wiley, 1974. "A Wiley-Interscience publication." Front Matter -- Introduction and General Outline -- Hyperbolic Waves. Waves and First Order Equations -- Specific Problems -- Burgers' Equation -- Hyperbolic Systems -- Gas Dynamics -- The Wave Equation -- Shock Dynamics -- The Propagation of Weak Shocks -- Wave Hierarchies -- Dispersive Waves. Linear Dispersive Waves -- Wave Patterns -- Water Waves -- Nonlinear Dispersion and the Variational Method -- Group Velocities, Instability, and Higher Order Dispersion -- Applications of the Nonlinear Theory -- Exact Solutions; Interacting Solitary Waves -- References -- Index -- Pure and Applied Mathematics. 1. Introduction and General Outline -- 2. Waves and First Order Equations -- 3. Specific Problems -- 4. Burgers' Equation -- 5. Hyperbolic Systems -- 6. Gas Dynamics -- 7. The Wave Equation -- 8. Shock Dynamics -- 9. The Propagation of Weak Shocks -- 10. Wave Hierarchies -- 11. Linear Dispersive Waves -- 12. Wave Patterns -- 13. Water Waves -- 14. Nonlinear Dispersion and the Variational Method -- 15. Group Velocities, Instability, and Higher Order Dispersion -- 16. Applications of the Nonlinear Theory -- 17. Exact Solutions; Interacting Solitary Waves.

A MATHEMATIC MODEL FOR SIMULATING THE JOINT MECHANISM OF UNSTEADY SEDIMENT AND WATER FLOW IS PRESENTED.THIS MODEL IS BASED UPON A TREATMENT OF THE BASIC SET OF PARTIAL DIFFERENTIAL EQUATIONS FOR ONE-DIMENSIONAL, UNSTEADY, MOVABLE BED OPEN-CHANNEL FLOW WITH HOMOGENEOUS DENSITY.THE METHOD OF CHARACTERISTICS TRANSFORMS THE BASIC SET OF PARTIAL DIFFERENTIAL EQUATIONS INTO A SET OF ORDINARY DIFFERENTIAL EQUATION WITH THREE DIFFERENT SETS OF SLOPE.THESE THREE NEW EQUATIONS ARE SUBSEQUENTLY TREATED AS FINITE DIFFERENCE EQUATIONS, AND SOLUTIONS OF THE UNSTEADY PROBLEM ARE READILY POSSIBLE WITH THE AID OF THE KNOWN BOUNDARY OR PREVIOUS CONDITIONS.(A)

The nonlinear mathematical model for depth-averaged flow allows for discontinuous solutions that correspond with the dynamic behaviour of rapid changes in flow. Such discontinuities, that can fundamentally affect the predicted morphological response of a river, are analyzed theoretically and numerically. Hereto, the entropy conditions as defined by Lax and the Rankine-Hugoniot or shock relations are used to analyze propagation rates and stability. Apart from subcritical and supercritical flows, a transition regime can be identified for flows with mobile beds.

Spatial lag effects, which are viewed as the inability of an alluvial system to immediately overcome the presence of constrained sediment boundary conditions, are investigated under bed load sediment transport conditions. An equation which characterises spatial lag effects and two theoretical spatial lag coefficient relations are investigated experimentally. Values of the spatial lag coefficient obtained from experimental data are compared with the theoretical relations and one relation is recommended. A numerical model which incorporates the spatial lag equation is formulated. The performance of the model is tested against data measured by a previous investigator and the ability of the model to successfully predict the spatial and temporal variation of the mean bed elevation and bed load transport rate under steady flow, non-equilibrium conditions is verified.

The Preissmann scheme, often referred to as the four-point scheme, is a bidiagonal implicit finite-difference method for solution of the de St. Venant equations. It is unconditionally stable and extremely robust, and thus is one of the most widely used methods in free-surface one-dimensional subcritical numerical modeling. The purpose of this technical note is to discuss the limitations of Preissmann scheme when applied to transcritical flow. In particular, the analysis presented shows that the Preissmann scheme cannot be used to simulate transcritical flow using the through (shock capturing) method.

The standard one-dimensional equations of unsteady sedimenttransport are examined, and multiple time (or length) scales are identified. The existence of multiple scales may lead to a singularly perturbed behavior that should be taken into account in any general numerical model. Previous models, which reduce the number of conservation equations solved simultaneously from three to two, are seen to be unable to satisfy exactly either a general boundary condition or a general initial condition. Implications for numerical modeling are explored. Numerical results for the model problem of sediment-deposition upstream of a dam illustrate the analytical argument.

Open-channel hydraulics is a subset of shallow-water theory that, in turn, is a subset of hydrodynamics. The distinguishing feature of shallow-water theory is the assumption of hydrostatic pressure. One-dimensional, open-channel hydraulics goes further in that it averages the velocity in both the vertical and the horizontal directions. The derivation of the open-channel equations from the general hydrodynamics relationships displays the approximations. The hydrostatic approximation is well documented in the literature, but some confusion persists in the averaging process, a confusion that leads to erroneous definitions of critical depth, Froude number, and even Bernoulli's equation. Critical depth must be defined so that it displays the singularities in the unsteady and steady equations of motion. Bernoulli's equation stems from conservation of momentum and should not carry an energy correction factor for nonuniform velocity distribution. In most circumstances, the error made by neglecting momentum and energy correction factors is tertiary, smaller than errors of erroneous friction and nonhydrostatic pressure.

The relative magnitudes of the celerities at which disturbances on the water surface and the bed are propagated play a key role in the mathematical modeling of mobile bed flow. Earlier analyses assumed that the solids concentration of the flow was negligible. This assumption is inappropriate both for some natural streams and for many aqueous mine-waste disposal applications. Here, the relative celerities of mobile bed flow are determined for flows with finite solids concentrations. The solids concentration is assumed to be a function of the velocity and depth of flow. The analysis confirms and extends the results of earlier analyses. It is shown that the movement of the sediment and the water can be considered to be mathematically independent of each other only within very limited ranges of solids concentration and Froude number. This severely constrains the application of numerous existing mathematical models to mobile bed flows with relatively high solids concentrations.

In this study, effects of mountain-river conditions on one-dimensional mathematical models for river hydraulics and morphology are analysed. This analysis includes the applicability of simplified models, the mathematical stability in case of graded sediment and the behaviour of discontinous flows at mobile beds. The numerical model that is developed hereto is based on the Godunov method with simultaneous solution of changes in hydraulic and morphological variables. This model is applied on hypothetical cases and flume experiments.

Derivations and analyses of basic equations for I-dimensional sediment-laden flow (concentrations up to about 10% of volume) on a mobile bed are presented . Equations of mass and momentum conservation have been derived by means of a control section as well as by depth integration. Therefore a three-layer approach has been used (i.e., bed layer, bed-load layer, and suspended-load layer). Despite the assumptions of uniform sediment, fixed banks and constant width the derivations can easily be extended for more general models. Analysis of the basic equations b means of the method of characteristics showed that with increasing concentration wave celerities alter, and showed that critical flow occurs at Froude numbers less than unity. A stability analysis of the equations showed that the criterion for occurrence of roll waves in supercritical flow is also modified by the increased concentration. Due to increased concentrations roll waves can occur in sediment-laden flow at lower Froude numbers than in clear water flow.

Considerations about non-steady bed-load transport in open channels

- M De Vries
- ͑1965͒

de Vries, M. ͑1965͒. ''Considerations about non-steady bed-load transport in open channels.'' Proc., 11th Cong. Int. Assn. Hydraulic Research, Leningrad, 3.8.1–3.

Modeling of hydraulics and morphology in mountain rivers PhD thesis, also ''Communications on hydraulic and geotech-nical engineering

- J Sieben
- ͑1997͒

Sieben, J. ͑1997͒. ''Modeling of hydraulics and morphology in mountain rivers.'' PhD thesis, also ''Communications on hydraulic and geotech-nical engineering.'' Rep. No. 97-3, Delft Univ. of Technology, Delft, The Netherlands.