Depth-averaged Model Research Articles

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane

A laminar flow of a thin layer of mud down an inclined plane under the action of gravity is considered. The instability of a film flow and the formation of finite amplitude waves are studied within the framework of both two-dimensional governing equations of a power-law fluid and its depth-averaged hyperbolic simplification. The conditions of the existence of roll waves for these models are formulated in terms of the Whitham criterion. A free surface evolution and the development of roll waves are numerically calculated. The amplitude of roll waves obtained by the 2D equations is slightly larger than that for the 1D model. Moreover, for certain flow parameters, the small perturbations of the basic solution grow for the 2D equations and decay for the depth-averaged model. A two-parameter class of exact piecewise-smooth solutions of the 1D model is obtained and a comparison with a numerical solution is made. In the region of these parameters, diagrams of the roll waves existence are constructed.

Flood risk assessment in vegetated lower Asahi River of Okayama Prefecture in Japan using airborne topo-bathymetric LiDAR and depth-averaged flow model

This paper describes application of airborne LiDAR bathymetry (ALB) with near-infrared and green pulsed lasers for gathering distributed vegetation conditions and topo-bathymetric data for rivers. For the lower Asahi River of Okayama Prefecture in Japan, the ALB data validity was verified using field observation data. This study also examined the applicability of ALB data for numerical simulations of the lower Asahi River flooding in early July 2018 in Japan, comparing simulated and observed data. Results demonstrated that the methodology for this study works well for parameterization of distributed vegetation on a reach scale. This study also applied numerical tests to investigate the effects of vegetation establishment on flood control plans for the lower Asahi River using parameters validated for flood flow simulations. Results demonstrate that the predicted water level markedly exceeds the high water level because of thick vegetation presently established along few-kilometer-long upstream sections of the targeted river reach. Therefore, we conclude that the present findings can support cost-effective management tasks for vegetated rivers.

Movement process analysis of the high-speed long-runout Shuicheng landslide over 3-D complex terrain using a depth-averaged numerical model

Flow-like landslide is one of the most catastrophic types of natural hazards due to its high velocity and long travel distance. In 2019, a large catastrophic landslide was triggered by heavy rainfall and occurred in Shuicheng County, Guizhou, China. The Shuicheng landslide was characterized by a short slip time, high speed, and long sliding distance, causing significant damages to the downstream communities and properties. Depth-averaged models have been widely used to predict the velocity and runout distance of flow-like landslides. However, most of the existing depth-averaged models have various shortcomings for application in real-world simulations. In this study, a high-performance depth-averaged model taking into account the effects of topography-related vertical acceleration and centrifugal force was used to examine the influence of complex 3-D terrain on the landslide movement process. The simulation results were in satisfactory agreement with the field observations. This work reveals the landslide movement process at different stages, including acceleration, diversion, secondary acceleration, impact, and deposition. The maximum average velocity was predicted to be 35 m/s, with a local maximum velocity exceeding 50 m/s. The seismic records obtained from the adjacent seismic stations and the predicted kinetic energy and velocity of the landslide event revealed a dual acceleration and obstruction process. It was also found that the movement process and final deposit morphology were strongly influenced by the complex terrain and were sensitive to the surface friction coefficient. This may also be the reason for the survival of some houses in the middle of the slope during the event. This study provides a reference for investigating long-runout, high-speed, flow-like landslides.

Non-Hydrostatic Transitional Open-Channel Flows from a Supercritical to a Subcritical State

Abstract In this study, a depth-averaged numerical model was employed to investigate the two-dimensional flow features of transitional open-channel flows from a supercritical to a subcritical state. Compared to a shallow-water model, the proposed model incorporates supplementary terms to account for the effects of non-uniform velocity and non-hydrostatic pressure distributions. The model equation was solved numerically by means of the Adams–Bashforth–Moulton scheme. A wide variety of transitional open-channel flow problems such as hydraulic jumps was considered for assessing the suitability of the numerical model. The results of the model for the free-surface profile, pressure distribution, and characteristics of the first wave of an undular jump were compared with the experimental data, and the agreement was found to be satisfactory. Despite the effects of the three-dimensional characteristics of the flow and the bulking of the flow caused by air entrainment, the model performed reasonably well with respect to the simulations of the mean flow characteristics of the curvilinear turbulent flow problems. Furthermore, the results of this investigation confirmed that the model is more suitable for analyzing near-critical turbulent flow problems without cross-channel shock waves.

A Numerical Study of the Critical Threshold for Landslide Dam Formation Considering Landslide and River Dynamics

Landslide dam formation can be influenced by the erosive capacity of river flow and the dynamic characteristics of the landslide. When the deposition rate of a landslide that reaches a river is higher than the erosion rate of river flow, the landslide can form a dam by blocking the channel. Hence, in this paper, a dimensionless discharge threshold for landslide dam formation considering landslide and river dynamics is established and studied numerically. A two-layer depth-averaged model coupled with an erosion term is presented to simulate river and landslide movements and their interactions. Several numerical cases are simulated to study the influence of landslide and river dynamics on the critical threshold for dam formation by considering some key factors, such as landslide velocity and the angle between the river and landslide transport directions. Through the simulations, three types of landslide intrusion into river can be reflected: a dam forms quickly, a dam forms or does not form close to a critical state, and no dam forms. The results show that these factors together affect the process of dam formation if the difference between the landslide and river discharges is relatively small. All results are helpful to further clarify the formation of such dams for natural hazard prevention under future climate change conditions.

Asymptotic analysis of the eigenstructure of the two-layer model and a new family of criteria for evaluating the model hyperbolicity

Two-layer and multi-layer depth-averaged models have become popular for simulating exchange flows, seawater currents and geophysical flows. The partial differential equation systems associated with these models are similar to the single-layer shallow-water model. Yet, their eigenstructures are more complex owing to the pressure coupling between the layers. Such models occasionally lose their hyperbolic character, which may lead to numerical issues. A physical explanation is that Kelvin-Helmholtz type instabilities arise at the layers' interface, if the velocity difference between the layers becomes sufficiently large. A way to avoid the hyperbolicity loss is to locally introduce an extra momentum exchange between the layers, assessable from the system eigenstructure and aimed at roughly mimicking the dynamical effects of such instabilities. To better understand the hyperbolicity conditions, the eigenstructure of the two-layer model is methodically studied by an asymptotic analysis. The analysis for the limiting cases, where the layers' thicknesses are either comparable or very different from each other, reveals new stability criteria. These analytical criteria are, then, exploited to design a new family of approximate criteria, valid for any flow condition. Numerical investigations demonstrate the reliability of this approach, which can be easily implemented in numerical schemes for preserving the hyperbolicity.

A simplified depth-averaged debris flow model with Herschel-Bulkley rheology for tracking density evolution: a finite volume formulation

A simplified depth-averaged debris flow model with Herschel-Bulkley rheology for tracking density evolution: a finite volume formulation

Tridimensional nonhydrostatic transient rip currents in a wave-resolving model

Flash rips and surf eddies are transient horizontal structures of the order of 10 to 100 m, which can be generated in the surfzone in the absence of bathymetric irregularities. They are traditionally evaluated in a depth averaged setting which involves intrinsic horizontal shear instabilities and the direct generation of vorticity by short-crested waves. In this article, we revisit the processes of surf eddy generation with a new three-dimensional wave resolution model (CROCO) and provide a plausible demonstration of new 3D non-hydrostatic instability and turbulent cascade. We first present a quick overview of a compressible free surface approach suitable for nearshore dynamics. Its ability to simulate the propagation of surface gravity waves and nearshore wave-driven circulation is validated by two laboratory experiments. Next, we present a real world application from Grand Popo Beach, Benin, forced by waves with frequency and directional spreading. The generation of surf eddies by the 3D model differs from depth-averaged models, due to the vertical shear associated with shallow breaking waves. In this case, the generation of eddies from both horizontal shear instability and the breaking of short-crested waves is hampered, the former by stretching the alongshore current and the latter by inhibiting the inverse energy cascade. Instead, the vertical shear flow is subjected to forced wave group variability and Kelvin–Helmholtz type instability at an inflection point. Primary and secondary instabilities generate spanwise and streamwise vorticity connecting small-scale eddies to larger horizontal surfzone structures. Streamwise filaments, appearing as 5 m wide ribs or mini-rips, can extend beyond the surfzone but with moderate energy. These results appear consistent with the velocity spectra and observed patterns of tracers and suspended sediments at Grand Popo Beach. The timescale associated with the mean shear-induced turbulence is several times the wave period and suggests an intermediate range between breaker-induced turbulence and large-scale surf eddies.

A Moment-Based Chezy Formula for Bed Shear Stress in Varied Flow

Despite its limitations, the Chezy bed shear stress formula is commonly used in depth-averaged flow numerical models as closure for estimating mutual tractive stresses with underneath boundaries. This paper proposes a novel moment-based formula that could be considered a revised version of the Chezy formula and can be used to estimate local variations of the bed shear stress under more complex and varied flow conditions with accelerating–decelerating flow fields. The formula depends on two velocity scales: the depth-averaged velocity, Uo, and a new moment-based velocity scale, u1. The new formula is calibrated using 10 experiments for flow over fixed bedforms, and the calibration coefficient is found to linearly correlate with h/Δ and h/zo ratios. The formula is also applied for the case of air flow across a negative step, jet water flow downstream a gate, and 2D water flow downstream an oblique negative step, and reasonably satisfactory agreement with the measured data is found. The new formula could be used in vertically averaged and moment models to disclose part of the information already lost by the vertical integration procedure.

Simplified Spectral Model of 3D Meander Flow

Most 2D (two-dimensional) models either take vertical velocity profiles as uniform, or consider secondary flow in momentum equations with presupposed velocity profiles, which weakly reflect the spatio-temporal characteristics of meander flow. To tackle meander flow in a more accurate 3D (three-dimensional) way while avoiding low computational efficiency, a new 3D model based on spectral methods is established and verified in this paper. In the present model, the vertical water flow field is expanded into polynomials. Governing equations are transformed by the Galerkin method and then advection terms are tackled with a semi-Lagrangian method. The simulated flow structures of an open channel bend are then compared with experimental results. Although a zero-equation turbulence model is used in this new 3D model, it shows reasonable flow structures, and calculation efficiency is comparable to a depth-averaged 2D model.

Modelling finger propagation in elasto-rigid channels

Abstract

Equilibrium approach for modeling erosional failure of granular dams

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.

NUMERICAL SIMULATION OF MUDFLOW WITH A TWO-DIMENSIONAL DEPTH-AVERAGED MODEL

This study is motivated by the need to mitigate damages caused by mudflow disaster, which is occurring frequently due to intensive forest clearing and uncontrolled land use at highland. Numerical modelling of mudflow is challenging as the propagation speed and rheological behavior of the flow relies on the accuracy of the numerical scheme and choice of rheological model. In this study, a two-dimensional depth-averaged model was developed to simulate a simplified mudflow event. In order to capture shock in mudflow, the finite volume method (FVM) with third order accurate Monotonic Upstream-centered Scheme of Conservation Laws (MUSCL) scheme was adopted in the numerical model and the model was verified against a benchmark partial dam-break problem. By assuming that the mudflow has high amount of fine mud suspension. the rheological model was represented by a simplified Herschel-Bulkley model. Numerical results showed that the Herschel-Bulkley model could reproduce the viscoplastic behavior of mudflow well, especially in the estimation of final longitudinal flow spreading (2% difference compared to experimental value) and depth of peak wave (8.8% maximum difference). The robustness and stability of the model was demonstrated by simulating a simplified mudflow event with obstacles.

Modeling shallow geological flows on steep terrains using a specific differential transformation

Shallow geological flows such as landslides, debris flows, and floods are frequently modeled mathematically in the depth-averaged framework. As the bed topography becomes steep and irregular, the classical shallow-water model loses accuracy. How to handle bed topography and gravity effects in depth-averaged models becomes important, and the validity of the model needs verifying. In this study, a differential transformation called the G2L transformation linking the global height to the local depth of the surface material is proposed. This transformation helps to reconstruct the landslide/avalanche surface for one-dimensional and two-dimensional terrains that are obviously curved as well as real landslide examples such as the Huangtian and Rongsong landslides with shallowness ratios as high as 1/4. With this G2L transformation and closures of the lateral and bed stress, two versions of a model governing the geological flow dynamics are developed to avoid difficulties in terrain treatment when using differential geometry. Version 1 of the model is a coupled hyperbolic–differential system; version 2 is a modified hyperbolic system. The two proposed models are verified by a simple one-dimensional granular flow problem on a steep slope. Both mitigate problems associated with computational differential geometry of geological flows and the singularity problem of shallow-water models. In particular, the reduced gravity of several current models in landslide dynamics is a special instance of version 2. The two proposed models are well able to mathematically model the gravity geological hazards and may provide the tools required for prediction and mitigation of natural geological hazards.

Erosion-deposition dynamics and long distance propagation of granular avalanches

Abstract

Effects of interactions between transient granular flows and macroscopically rough beds and their implications for bulk flow dynamics

Steep-creek beds are macroscopically rough. This roughness causes channelized flow material to decelerate and dissipate energy, which are accounted for by depth-averaged mobility models (DMM). However, practical DMM implementations do not explicitly account for grain-scale basal interactions that influence macroscopic flow dynamics. In this study, we model flows using physical tests with smooth and macroscopically rough bases, and hence evaluate discrete element method (DEM) and DMM models. A scaling effect is identified relating to roughened beds: increasing the number of grains per unit depth tends to suppress dispersion, such that small-scale flows on smooth beds resemble large-scale flows on roughened beds, at least in terms of bulk density. Furthermore, the DEM shows that rougher beds reduce the peak bulk density by up to 15% compared to a smooth bed. Rough beds increase the vertical momentum transfer tenfold, compared to smooth ones. The DMM cannot account for density change or vertical momentum, so DMM flow depths are underestimated by 90% at the flow front and 20% in the body. The Voellmy model implicitly captures internal energy dissipation for flows on rough beds. The parameter ξ can allow velocity reductions due to rough beds observed in the DEM to be captured.

The life and fate of a bubble in a geometrically perturbed Hele-Shaw channel

Abstract

Numerical modeling of bed deposition in rapid flow-like landslides: a case study of the Gaolou landslide in Shaanxi Province, China

Some flow-like landslides tend to lose materials while moving on a relatively dry sliding surface. This phenomenon is called bed deposition. In contrast to the bed entrainment phenomenon, bed deposition is relatively poorly understood. Therefore, an improved depth-averaged model is proposed to quantify this phenomenon. The deposition depth is calculated according to the momentum conservation of the deposited mass, and the rheological property of the sliding mass on the bottom is modified considering an abrupt increase in the depth of the sliding surface after deposition. Utilizing the proposed model, the Gaolou landslide, a typical flowlike landside occurring on October 6th, 2006 with an obvious bed deposition phenomenon in Shaanxi Province of China, is simulated to investigate the influence of bed deposition on its propagation process. The results indicate that the proposed model can effectively depict the bed deposition phenomenon in the Gaolou landslide. Bed deposition dissipates part of the kinetic energy of this landslide; thus, the simulated debris inundation area would likely be overestimated when this effect is neglected. On the other hand, the thin liquefied layer formed by the bed deposition process reduces the friction energy dissipation, contributing to the high mobility of this landslide.

Two-Dimensional Numerical Modeling of Flow in Physical Models of Rock Vane and Bendway Weir Configurations

While there are a wide range of design recommendations for using rock vanes and bendway weirs as streambank protection measures, no comprehensive, standard approach is currently available for design engineers to evaluate their hydraulic performance before construction. This study investigates using 2D numerical modeling as an option for predicting the hydraulic performance of rock vane and bendway weir structure designs for streambank protection. We used the Sedimentation and River Hydraulics (SRH)-2D depth-averaged numerical model to simulate flows around rock vane and bendway weir installations that were previously examined as part of a physical model study and that had water surface elevation and velocity observations. Overall, SRH-2D predicted the same general flow patterns as the physical model, but over- and underpredicted the flow velocity in some areas. These over- and underpredictions could be primarily attributed to the assumption of negligible vertical velocities. Nonetheless, the point differences between the predicted and observed velocities generally ranged from 15 to 25%, with some exceptions. The results showed that 2D numerical models could provide adequate insight into the hydraulic performance of rock vanes and bendway weirs. Accordingly, design guidance and implications of the study results are presented for design engineers.

Baffled clarification basin hydrodynamics and elution in a continuous time domain

Clarification basins have been implemented for millennia as a unit operation (UO). Modern basins are intended to manage hydrologic/hydraulic phenomena while sequestering particulate matter (PM) and PM-bound constituent loads. Water treatment systems, subject to steady flows, have used baffles to modulate dead zones, residence time and hydrodynamics. Yet, permeable baffling impacts to larger-scale basin hydrodynamics subject to highly unsteady continuous time domain flows is a novel investigation. In this study, in situ monitoring and numerical simulations are conducted for a full-scale prototype urban drainage basin, while geometrically oversized with respect to the relatively coarse particle size distribution and hydraulic loading, retrofitted with gabion baffles composed of crushed carbonated recycled concrete. A 40-day tracer monitoring of hydrodynamics within the retrofit basin is tested against a novel application of a depth-averaged unsteady Reynolds-averaged Navier-Stokes equations (URANS) as a computational fluid dynamics (CFD) model in the OpenFOAM framework. A dual-peak elution pattern is observed in response to tracer injections in the physical and simulated (fully transient and quasi-steady) results. Short-circuiting around the baffles in the as-built retrofit is elucidated with monitoring and modeling. Simulations of retrofit configurations (pre-retrofit, as-designed, as-built and impervious baffles) indicate, in comparison to the pre-retrofit, a well-baffled system reduces short-circuiting, delays elution, and improves PM separation. Permeable baffles provide hydrodynamic benefits as compared to impervious baffles. The novel application of this depth-averaged URANS CFD model accommodates complex geometry and physics; and elucidates contrasting Reynolds number distributions of inter- and intra-baffle flows. The CFD model with continuous time domain flows is a viable tool for design, analysis, permitting and management of basins with or without baffle retrofits. The model extends insights beyond analytical tools and complements tools such as the SWMM (Stormwater Management Model).