A model based on Hirano-Exner equations for two-dimensional transient flows over heterogeneous erodible beds

In order to study the morphological evolution of river beds composed of heterogeneous material, the interaction among the different grain sizes must be taken into account. In this paper, these equations are combined with the two-dimensional shallow water equations to describe the flow field. The resulting system of equations can be solved in two ways: (i) in a coupled way, solving flow and sediment equations simultaneously at a given time-step or (ii) in an uncoupled manner by first solving the flow field and using the magnitudes obtained at each time-step to update the channel morphology (bed and surface composition). The coupled strategy is preferable when dealing with strong and quick interactions between the flow field, the bed evolution and the different particle sizes present on the bed surface. A number of numerical difficulties arise from solving the fully coupled system of equations. These problems are reduced by means of a weakly-coupled strategy to numerically estimate the wave celerities containing the information of the bed and the grain sizes present on the bed. Hence, a two-dimensional numerical scheme able to simulate in a self-stable way the unsteady morphological evolution of channels formed by cohesionless grain size mixtures is presented. The coupling technique is simplified without decreasing the number of waves involved in the numerical scheme but by simplifying their definitions. The numerical results are satisfactorily tested with synthetic cases and against experimental data.     

Bed load transport in braided gravel-bed stream models

Bed load transport rate was measured in ten self-formed small-scale gravel braided streams developed in a laboratory flume at several different values of steady discharge and flume gradient. The streams are approximate Froude models of typical prototype braided streams but of no particular river. Slight viscous effects may be present in the models because particle Reynolds numbers are close to 70. Total bed load discharge was measured every fifteen minutes throughout each 60 hour run. In addition, 80 channel cross-sections were measured in each run to establish the average channel geometry. Total bed load transport rate correlates well with total discharge and total stream power, although at a given stream power bed load discharge is greater when braiding is less intense and the width/depth ratio is lower. Analysis using unit stream power and cross-section average bed shear stress reveals that the laboratory data conform to existing empirical bed load transport relationships. However, comparison with field data from gravel-bed rivers shows discrepancies that may be due to differences in bed material size gradation and bed sediment structure. At constant discharge, wide fluctuations in bed load discharge occur with some regularity. Periods range from 2 to 10 hours in the models, which is equivalent to several tens of hours in a prototype. The presence of these long-period fluctuations compounds the problems of field measurement of bed load in braided streams.     

High-order balanced CWENO scheme for movable bed shallow water equations

In this work the numerical integration of 1D shallow water equations (SWE) over movable bed is performed using a well-balanced central weighted essentially non-oscillatory (CWENO) scheme, fourth-order accurate in space and in time. Time accuracy is obtained following a Runge–Kutta (RK) procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained using WENO reconstructions of conservative variables and of flux and bed derivatives. An original treatment for bed slope source term, which maintains the established order of accuracy and satisfies the property of exactly preserving the quiescent flow (C-property), is introduced in the scheme. This treatment consists of two procedures. The former involves the evaluation of the point-values of the flux derivative, considered as a whole with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the expected regularity of the free surface elevation. The high accuracy of the scheme allows to obtain good results using coarse grids, with consequent gain in terms of computational effort. The well-balancing of the scheme allows to reproduce small perturbations of the free surface and of the bottom otherwise of the same order of magnitude of the numerical errors induced by the non-balancing. The accuracy, the well-balancing and the good resolution of the model in reproducing free surface flow over movable bed are tested over analytical solutions and over numerical results available in literature.     

Geometry Relations with Bed Load Formulae for Alluvial Boundaries

Employing bed load formulae hydraulic geometry relations were derived for stream width, sediment transport velocity, and bed slope. The relations were examined in terms of friction factor, bed load discharge, bed load diameter, and water discharge. Two fundamental approaches to the prediction of hydraulic geometry have been developed. The first and most widely adopted approach is based on empirical equations whereas the second is based on solution of the governing equations of flow. The applied bed load formulae belong to different authors. Here, the comparison with the other derived relations is presented.     

Fully coupled mathematical modeling of turbidity currents over erodible bed

Turbidity currents may feature active sediment transport and rapid bed deformation, such as those responsible for the erosion of many submarine canyons. Yet previous mathematical models are built upon simplified governing equations and involve steady flow and weak sediment transport assumptions, which are not in complete accordance with rigorous conservation laws. It so far remains unknown if these could have considerable impacts on the evolution of turbidity currents. Here a fully coupled modeling study is presented to gain new insights into the evolution of turbidity currents. The recent analysis of the multiple time scales of subaerial sediment-laden flows over erodible bed [Cao Z, Li Y, Yue Z. Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling. Adv Water Resour 2007;30(4):715–29] is extended to subaqueous turbidity currents to complement the fully coupled modeling. Results from numerical simulations show the ability of the present coupled model to reproduce self-accelerating turbidity currents. Comparison among the fully and partially coupled and decoupled models along with the analysis of the relative time scale of bed deformation explicitly demonstrate that fully coupled modeling is essential for refined resolution of those turbidity currents featuring active sediment transport and rapid bed deformation, and existing models based on simplified conservation laws need to be reformulated.     

Comparison of Ray Methods with the Exact Solution in the 1-D Anisotropic “Simplified Twisted Crystal” Model

The exact analytical solution for the plane S-wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, is compared with four different approximate ray-theory solutions. The four different ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. The comparison is carried out numerically, by evaluating both the exact analytical solution and the analytical solutions of the equations of the four ray methods. The comparison simultaneously demonstrates the limits of applicability of the isotropic and anisotropic ray theories, and the superior accuracy of the coupling ray theory over a broad frequency range. The comparison also shows the possible inaccuracy due to the quasi-isotropic perturbation of travel times in the equations of the coupling ray theory. The coupling ray theory thus should definitely be preferred to the isotropic and anisotropic ray theories, but the quasi-isotropic perturbation of travel times should be avoided. Although the simplified twisted crystal model is designed for testing purposes and has no direct relation to geological structures, the wave-propagation phenomena important in the comparison are similar to those in the models of the geological structures.In additional numerical tests, the exact analytical solution is numerically compared with the finite-difference numerical results, and the analytical solutions of the equations of different ray methods are compared with the corresponding numerical results of 3-D ray-tracing programs developed by the authors of the paper.     

Coupled simulation of transient bed evolution in alluvial channels

Abstract

In dealing with the transient sediment transport problem, the commonly used uncoupled model may not be suitable. The uncoupling technique is intended to separate the physical coupling phenomenon of water flow and sediment transport into two independent processes. Very often, as a result, severe numerical oscillation and solution instability problems appear in the simulation of transient sediment transport in alluvial channels. The coupled model, which simultaneously solves water flow continuity, momentum and sediment continuity equations, gives fewer numerical oscillation and solution instability problems. In this article, a coupled model using a matrix double-sweep method to solve the system of nonlinear algebraic equations has been developed. Several test runs designed on the basis of a schematic model have been performed. The numerical oscillation and solution instability problems have been investigated through a comparison with those obtained from an uncoupled model. Based on the proposed case studies, it can be concluded that, for transient bed evolution, the performance of the coupled model is much better than that of the uncoupled model. The numerical oscillation is reduced and the solution is more stable. This newly developed coupled model was also applied to the Cho-Shui River in Taiwan. This application study implied that the effect of the peaky flood wave propagation on the bed evolution could be simulated better by the coupled model than by the uncoupled model.     

2D stream hydrodynamic,sediment transport and bed morphology model for engineering applications

A 2D depth‐averaged hydrodynamic, sediment transport and bed morphology model named STREMR HySeD is presented. The depth‐averaged sediment transport equations are derived from the 3D dilute, multiphase, flow equations and are incorporated into the hydrodynamic model STREMR. The hydrodynamic model includes a two‐equation turbulence model and a correction for the mean flow due to secondary flows. The suspended sediment load can be subdivided into different size classes using the continuum (two‐fluid) approach; however, only one bed sediment size is used herein. The validation of the model is presented by comparing the suspended sediment transport module against experimental measurements and analytical solutions for the case of equilibrium sediment‐laden in a transition from a rigid bed to a porous bed where re‐suspension of sediment is prevented. On the other hand, the bed‐load sediment transport and bed evolution numerical results are compared against bed equilibrium experimental results for the case of a meander bend. A sensitivity analysis based on the correction for secondary flow on the mean flow including the effect of secondary flow on bed shear stresses direction as well as the downward acceleration effect due to gravity on transverse bed slopes is performed and discussed. In general, acceptable agreement is found when comparing the numerical results obtained with STREMR HySeD against experimental measurements and analytical solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

The effect of asymmetric dune roughness on tidal asymmetry in the Weser estuary

The bed of estuaries is often characterized by ripples and dunes of varying size. Whereas smaller bedforms adapt their morphological shape to the oscillating tidal currents, large compound dunes (here: asymmetric tidal dunes) remain stable for periods longer than a tidal cycle. Bedforms constitute a form roughness, that is, hydraulic flow resistance, which has a large-scale effect on tidal asymmetry and, hence, on hydrodynamics, sediment transport, and morphodynamics of estuaries and coastal seas. Flow separation behind the dune crest and recirculation on the steep downstream side result in turbulence and energy loss. Since the energy dissipation can be related to the dune lee slope angle, asymmetric dune shapes induce variable flow resistance during ebb and flood phases. Here, a noncalibrated numerical model has been applied to analyze the large-scale effect of symmetric and asymmetric dune shapes on estuarine tidal asymmetry evaluated by residual bed load sediment transport at the Weser estuary, Germany. Scenario simulations were performed with parameterized bed roughness of symmetric and asymmetric dune shapes and without dune roughness. The spatiotemporal interaction of distinct dune shapes with the main drivers of estuarine sediment and morphodynamics, that is, river discharge and tidal energy, is shown to be complex but substantial. The contrasting effects of flood- and ebb-oriented asymmetric dunes on residual bed load transport rates and directions are estimated to be of a similar importance as the controls of seasonal changes of discharge on these net sediment fluxes at the Lower Weser estuary. This corroborates the need to consider dune-induced directional bed roughness in numerical models of estuarine and tidal environments.     

Effect of advective and diffusive sediment transport on the formation of local and global bottom patterns in tidal embayments

Bathymetric field data of tidal basins reveal two main classes of bottom patterns: (1) tidal bars, located near the entrance of the basin (length scale determined by the embayment width) and (2) global channel-shoal patterns which scale with the basin length. Previous models were able to describe only either one of these patterns. In this paper it is shown that both of them can be investigated within the framework of an idealised model of a rectangular tidal embayment, with fixed side walls and an erodible bed. The water motion is described by the depth-averaged shallow-water equations and is forced by a prescribed vertical tide at the seaward entrance. Sediment is transported as suspended load and only realistic values of the bottom friction parameter are considered. By assuming the ratio of embayment length over tidal wave length to be small, the model allows for a morphodynamic equilibrium, characterised by a spatially uniform tide moving over a bottom which slopes upwards toward the landward boundary. This equilibrium is unstable for a range of values of the model parameters, such that growth of bedforms occurs. Both global and local bottom patterns are found. In this study particular emphasis is laid on the mechanism governing the growth of a new type of localised bottom pattern. These patterns consist of small bars located near the entrance of the basin, resembling multiple row bars, and are found when advective sediment fluxes prevail over diffusive sediment fluxes. The formation process of these new bedforms is discussed in detail. The results agree well with field data. Comparison of the results with those obtained with a process-based, numerical model shows that, although the idealised model is strongly simplified, it is capable of producing the essential morphodynamics. Therefore, the idealised model is a useful tool to investigate mechanisms of bottom pattern growth.Responsible Editor: Iris Grabemann     

Critical movement of large rocks in currents and waves

Cobbles, boulders, and rocks often are used in a bed protection layer near a structure to protect the underlying sand bed against erosion by combined current and waves. The design of a bed protection layer consisting of loose rocks (rubble mound) requires knowledge of the stability and movement (as bed load) of very coarse materials. If some movement (or damage) is acceptable, the rock diameter can be designed to be smaller. This paper addresses the stability and movement of very coarse materials (cobbles, boulders, and rocks) based on the concept of the critical Shields mobility number. It is shown that the bed load transport of large cobbles, boulders, and rocks can be described by the equations of Meyer-Peter and Mueller (MPM) and Cheng. Both are valid for relatively small Shields mobility numbers. New and general equations for the design of a bed protection layer (including some permissible damage) in conditions with a current with or without waves are proposed based on the Shields mobility parameter and the bed load transport equation of Cheng. Laboratory and field data of critical velocities for pebbles, cobbles, boulders, and rocks have been analyzed and compared to the computed results of the proposed equations. Practical applications are given to demonstrate the general applicability of the proposed equations.     

Coupled and splitting bedload sediment transport models based on a modified flux-wave approach

Numerical modeling of free-surface flow over a mobile bed with predominantly bedload sediment transport can be done by solving the shallow water and Exner equations using coupled and splitting approaches.The coupled method uses a coupling of the governing equations at the same time step leading to a non-conservative solution.The splitting method solves the Exner and the shallow water equations in a separate manner,and is only capable of modeling weak free-surface and bedload interactions.In the current study,an extended version of a Godunov-type wave propagation algorithm is presented for modeling of morphodynamic systems using both coupled and splitting approaches.In the introduced coupled method the entire morphodynamic system is solved in the form of a conservation law.For the splitting technique,a new wave Riemann decomposition is defined which enables the scheme to be utilized for mild and strong interactions.To consider the bedload sediment discharge within the Exner equation,the Smart and Meyer-Peter&Müller formulae are used.It was found that the coupled solution gives accurate predictions for all investigated flow regimes including propagation over a dry-state using a Courant-Friedrichs-Lewy(CFL)number equal to 0.6.Furthermore,the splitting method was able to model all flow regimes with a lower CFL number of 0.3.     

Transport of suspended sediment under the dam‐break flow on an inclined plane bed of arbitrary slope

The problem of transport of suspended sediment after the break of a dam on an inclined bed is considered. To that end we use the shallow‐water approximation for arbitrary, constant slopes of the bottom, taking into consideration the effect of friction. The numerical technique and the frictional model are validated by comparison with available experimental data and asymptotic analytical solutions, with special attention to the numerical solution near the wetting front. The transport of suspended sediment down the inclined bed is obtained and discussed as a function of the slope of the bed for different values of the parameters characterizing the sediment and its transport properties. For sufficiently large times we always find the formation of roll waves near the water front, which affects the transport of sediments significantly. These strong oscillations are accurately computed with the numerical method used. The relative importance of the bed load (to the suspended load) sediment transport is also discussed as a function of the size of the sediment particles and the slope of the bed for different models on the initiation of sediment suspension from bed load. We also check the dilute sediment approach and characterize the conditions for its failure. Finally, the results of the present simplified model are intended to be used as tests of more complex numerical models. Copyright © 2007 John Wiley & Sons, Ltd.     

Assessing the impact of distributed snow water equivalent calibration and assimilation of Copernicus snow water equivalent on modelled snow and streamflow performance

Accuracy of the Copernicus snow water equivalent (SWE) product and the impact of SWE calibration and assimilation on modelled SWE and streamflow was evaluated. Daily snowpack measurements were made at 12 locations from 2016 to 2019 across a 4104 km² mixed-forest basin in the Great Lakes region of central Ontario, Canada. Sub-basin daily SWE calculated from these sites, observed discharge, and lake levels were used to calibrate a hydrologic model developed using the Raven modelling framework. Copernicus SWE was bias corrected during the melt period using mean bias subtraction and was compared to daily basin average SWE calculated from the measured data. Bias corrected Copernicus SWE was assimilated into the models using a range of parameters and the parameterizations from the model calibration. The bias corrected Copernicus product agreed well with measured data and provided a good estimate of mean basin SWE demonstrating that the product shows promise for hydrology applications within the study region. Calibration to spatially distributed SWE substantially improved the basin scale SWE estimate while only slightly degrading the flow simulation demonstrating the value of including SWE in a multi-objective calibration formulation. The particle filter experiments yielded the best SWE estimation but moderately degraded the flow simulation. The particle filter experiments constrained by the calibrated snow parameters produced similar results to the experiments using the upper and lower bounds indicating that, in this study, model calibration prior to assimilation was not valuable. The calibrated models exhibited varying levels of skill in estimating SWE but demonstrated similar streamflow performance. This indicates that basin outlet streamflow can be accurately estimated using a model with a poor representation of distributed SWE. This may be sufficient for applications where estimating flow is the primary water management objective. However, in applications where understanding the physical processes of snow accumulation, melt and streamflow generation are important, such as assessing the impact of climate change on water resources, accurate representations of SWE are required and can be improved via multi-objective calibration or data assimilation, as demonstrated in this study.     

Excavation of subglacial bedrock channels by seasonal meltwater flow

Subglacial water flow drives the excavation of a variety of bedrock channels including tunnel valleys and inner gorges. Subglacial floods of various magnitudes – events occurring once per year or less frequently with discharges larger than a few hundred cubic metres per second – are often invoked to explain the erosive power of subglacial water flow. In this study we examine whether subglacial floods are necessary to carve bedrock channels, or if more frequent melt season events (e.g. daily production of meltwater) can explain the formation of substantial bedrock channels over a glacial cycle. We use a one‐dimensional numerical model of bedrock erosion by subglacial meltwater, where water flows through interacting distributed and channelized drainage systems. The shear stresses produced drive bedrock erosion by bed‐ and suspended‐load abrasion. We show that seasonal meltwater discharge can incise an incipient bedrock channel a few tens of centimetres deep and several metres wide, assuming abrasion is the only mechanism of erosion, a particle size of D=256 mm and a prescribed sediment supply per unit width. Using the same sediment characteristics, flood flows yield wider but significantly shallower bedrock channels than seasonal meltwater flows. Furthermore, the smaller the shear stresses produced by a flood, the deeper the bedrock channel. Shear stresses produced by seasonal meltwater are sufficient to readily transport boulders as bedload. Larger flows produce greater shear stresses and the sediment is carried in suspension, which produces fewer contacts with the bed and less erosion. We demonstrate that seasonal meltwater discharge can excavate bedrock volumes commensurate with channels several tens of metres to a few hundred metres wide and several tens of metres deep over several thousand years. Such simulated channels are commensurate with published observations of tunnel valleys and inner gorges. Copyright © 2018 John Wiley & Sons, Ltd.     

Modelling dam-break flows over mobile beds using a 2D coupled approach

Dam-break flows usually propagate along rivers and floodplains, where the processes of fluid flow, sediment transport and bed evolution are closely linked. However, the majority of existing two-dimensional (2D) models used to simulate dam-break flows are only applicable to fixed beds. Details are given in this paper of the development of a 2D morphodynamic model for predicting dam-break flows over mobile beds. In this model, the common 2D shallow water equations are modified, so that the effects of sediment concentrations and bed evolution on the flood wave propagation can be considered. These equations are used together with the non-equilibrium transport equations for graded sediments and the equation of bed evolution. The governing equations are solved using a matrix method, thus the hydrodynamic, sediment transport and morphological processes can be jointly solved. The model employs an unstructured finite volume algorithm, with an approximate Riemann solver, based on the Roe-MUSCL scheme. A predictor–corrector scheme is used in time stepping, leading to a second-order accurate solution in both time and space. In addition, the model considers the adjustment process of bed material composition during the morphological evolution process. The model was first verified against results from existing numerical models and laboratory experiments. It was then used to simulate dam-break flows over a fixed bed and a mobile bed to examine the differences in the predicted flood wave speed and depth. The effects of bed material size distributions on the flood flow and bed evolution were also investigated. The results indicate that there is a great difference between the dam-break flow predictions made over a fixed bed and a mobile bed. At the initial stage of a dam-break flow, the rate of bed evolution could be comparable to that of water depth change. Therefore, it is often necessary to employ the turbid water governing equations using a coupled approach for simulating dam-break flows.     

Effect of bed load supply on sediment transport in mountain streams

Mountain streams with their tributary torrents build the upper part of the fluvial network. They are important regarding the transfer of sediment from headwaters to lower basins. Channels are typically steep with wide grain size distributions, ranging from fine sand up to large boulders, and a stabilized bed surface. Mountain streams often are supply-limited with respect to mobile bed load, which needs to be addressed when bed load transport equations are applied to such streams. To better understand supply limitation, laboratory experiments highlighting the effect of bed load supply on incipient motion and bed load transport rate are discussed. Experimental tests were done in which fine bed load was supplied to a previously armored channel bed, with flow conditions ranging from one-third to twice the critical dis-charge for the bed surface. At flows not exceeding the critical discharge, the time series of the bed load transport rate at the downstream model boundary featured consistent patterns which are attributed to distinct phases: (i) a temporal lag, (ii) an equilibrium state, and (iii) a post-supply phase. Bed load transport occurred even at flows distinctly below that for incipient motion of the bed surface. But, with the mass of total bed load outflow approaching the supply amount, the mass did not exclusively consist of supplied grains. The coarser the supplied bed load, the more sediment was mobilized from the bed surface. At higher flows, processes differed. Total bed load outflow exceeded the supply amount and the break-up of the armor layer caused a refining of the bed surface.     

Bifurcation modelling in a meandering gravel–sand bed river

The Rhine bifurcation at Pannerden forms the major distribution point for water supply in the Netherlands, distributing not only water and sediment but also flooding risks and navigability. Its morphological stability has been a concern for centuries. We present experiences from more than two decades of numerical morphological modelling of this bifurcation with a gravel–sand bed and a meandering planform. Successive computations have shown the importance of upstream approach conditions, the necessity to include physical mechanisms for grain sorting and alluvial roughness, and the need to assume a thicker active layer of the river bed than is suggested by laboratory flume experiments using a constant discharge. The active layer must be thicker in the model to account for river bed variations due to higher‐frequency discharge variations that are filtered out in morphological modelling. We discuss limitations in calibration and verification, but argue that, notwithstanding these limitations, 2D and 3D morphological models are valuable tools, not only for pragmatic applications to engineering problems, but also for revealing the limitations of established knowledge and understanding of the relevant physical processes. The application of numerical models to the Pannerden bifurcation appeared to reveal shortcomings in established model formulations that do not pose particular problems in other cases. This application is therefore particularly useful for setting the agenda for further research. Copyright © 2012 John Wiley & Sons, Ltd.