Depth-averaged Flow Model Research Articles

Combined dam-break modes between upper and lower reservoirs of pumped storage hydropower stations and associated flood risk assessment

ABSTRACT With the increasing construction of pumped storage hydropower projects in China, dam-break risk has become a matter of great concern. Typical pumped storage hydropower stations have relatively small storage capacity but are vulnerable to complicated combinations of dam breaks between their upper and lower reservoirs. This paper provides basic principles for determining different dam-break classes that could initiate at a pumped storage hydropower station and summarizes these in terms of risk source. After idealizing the breach development process, a simple dam-break model and a planar two-dimensional depth-averaged shallow flow model are used to calculate the discharge hydrograph of the dam-break flood and simulate the temporal and spatial characteristics of downstream inundation. Taking a montane rockfill-pumped storage project as an example, 12 dam-break schemes are used to examine different types of combined flow patterns driven by upper and lower dam breaks with modes that are gradual and/or instantaneous. It is found that predicted dam-break discharge hydrographs and downstream inundation indexes under adverse schemes fit common features of dam-break floods in mountain rivers. Downstream disaster losses are evaluated and flood evacuation routes are recommended. The findings of this study should be useful in ensuring the safe management of pumped storage hydropower stations.

A vertically non-uniform temperature approach for the friction term computation in depth-averaged viscoplastic lava flows

Recently, depth-averaged shallow flow models have been adapted to modelling liquefied lava flows, generally characterized by a marked temperature-dependent non-Newtonian rheology. Modelling these complex flows requires to include the effects of the depth-averaged temperature gradients in the governing equations. The most significant term to correctly predict the lava mobility is the flow resistance term, which is widely estimated using the linear viscoplastic Bingham model. This non-Newtonian model allows to relate the bed shear stress to the depth-averaged lava flow features by means of a cubic equation with analytical solution when assuming a uniform temperature distribution along the vertical profile. Nevertheless, the lava temperature is non-uniform along the vertical due to the heat transfer at the bottom and the free surface, and hence the classical cubic Bingham model is not valid anymore. In this work, a depth-averaged shallow flow model is adapted for realistic lava flows considering influence of the non-uniform vertical temperature profile in the non-Newtonian resistance. This requires to modify the rheological viscoplastic models for ensuring the coupling between flow dynamics and temperature evolution. Three non-uniform temperature vertical distributions are considered: linear, piece-wise and diffusion profiles. Synthetic tests are used to show the influence of the temperature vertical profile on the numerical results. Furthermore, laboratory experimental data are used to validate this novel viscoplastic resistance formulation and to show that the calibration of its parameters is possible.

Shallow Moments to Capture Vertical Structure in Open Curved Shallow Flow

Shallow water models are successfully used for simulating geophysical flows like river floods, tsunamis, sediment transport, or debris flows. Depth-averaged models are in general attractive due to their low computational cost. However, information on the vertical velocity is required a priori, typically by assuming a parametrized profile using a time-independent function, e.g., a constant. A systematic generalization of depth-averaged flow models is given by the shallow moment method. This method retains transient information about the vertical flow profile by using a finite Legendre expansion for said profile with time and space-dependent coefficients. The shallow moment approach allows to include vertical information and generates a hierarchy of equations that in the limit, recover the reference system before depth-averaging. Even a low number of basis functions significantly increases the model’s capability to resolve vertical information and, whenever this is relevant for the modeling task at stake, significantly improves the predictive power compared to classical shallow water systems. In the paper, we show a two-dimensional simulation result based on the shallow moment system to predict processes unobservable in classical shallow flow, namely secondary flow and the redistribution of velocity profiles, which are of relevance in curved channelized flow as experimentally demonstrated by Steffler.

IMEX_SfloW2D v2: a depth-averaged numerical flow model for volcanic gas–particle flows over complex topographies and water

Abstract. We present developments to the physical model and the open-source numerical code IMEX_SfloW2D (de' Michieli Vitturi et al., 2019). These developments consist of a generalization of the depth-averaged (shallow-water) fluid equations to describe a polydisperse fluid–solid mixture, including terms for sedimentation and entrainment, transport equations for solid particles of different sizes, transport equations for different components of the carrier phase, and an equation for temperature/energy. Of relevance for the simulation of volcanic mass flows, vaporization and entrainment of water are implemented in the new model. The model can be easily adapted to simulate a wide range of volcanic mass flows (pyroclastic avalanches, lahars, pyroclastic surges), and here we present its application to transient dilute pyroclastic density currents (PDCs). The numerical algorithm and the code have been improved to allow for simulation of sub- to supercritical regimes and to simplify the setting of initial and boundary conditions. The code is open-source. The results of synthetic numerical benchmarks demonstrate the robustness of the numerical code in simulating transcritical flows interacting with the topography. Moreover, they highlight the importance of simulating transient in comparison to steady-state flows and flows in 2D versus 1D. Finally, we demonstrate the model capabilities to simulate a complex natural case involving the propagation of PDCs over the sea surface and across topographic obstacles, through application to Krakatau volcano, showing the relevance, at a large scale, of non-linear fluid dynamic features, such as hydraulic jumps and von Kármán vortices, to flow conditions such as velocity and runout.

An integrated hydrodynamic model for runoff-generated debris flows with novel formulation of bed erosion and deposition

Debris flow is one of the most common geohazards in mountainous regions, posing significant threats to people, property and infrastructure. Among different types of debris flows, runoff-generated debris flows are attributed to rain storms, which provide abundant runoff that entrain large quantities of bed material, resulting in the formation of a solid-liquid current known as a debris flow. One of the keys to effectively simulating runoff-generated debris flows is modelling the erosion-deposition process. The commonly used approach for formulating erosion and deposition, although constrained by physics, suffers from a singularity in the presence of vanishing velocity, which poses a major challenge for practical applications. It is also argued that the deposition rate cannot be represented by simply reversing the sign of the erosion rate. To address these two issues, we have developed a depth-averaged debris flow model with a novel method of calculating the erosion-deposition rate. We have demonstrated that the singularity is due to the non-linear erosion-deposition term but quickly disappears while the flow converges to the equilibrium that is defined by the classic Takahashi's formula. To resolve the non-linearity and avoid the singularity, an implicit method within a Godunov-type finite volume framework has been proposed. An additional parameter is introduced to differentiate the erosion rate from the deposition rate. The model is validated against several test cases, including a real-world debris flow event. Satisfactory results are obtained, demonstrating the model's simulation capability and potential for wider applications such as risk assessment and impact-based early warning.

Complex dam break simulation using the 2-D depth-averaged SPH flow model: a validation for tsunami application

Dam break flow has a similar characteristic to the tsunami surge after the breaking wave. Therefore, it is often used in laboratory-scale experiments to study tsunamis and can be used as a benchmark for validating a numerical model. This paper presents a simulation of a complex dam break flow to validate the two-dimensional (2-D) depth-averaged Smoothed Particle Hydrodynamics (SPH) flow model which involves a non-submersible obstacle in the domain. The method can also be referred to as shallow-water equations SPH (SWE-SPH). With SPH, the wetting and drying interfaces are automatically solved without any additional trick or handling. Therefore, a dry bed dam break is specifically tested here. The numerical results are validated using experimental data from the literature. The model shows the ability to reproduce the flow fields, shocks, vortices and flow region transitions with reasonable accuracy compared to the experimental data, despite the vertical averaging process in the formulation. This model can be a robust tool for predicting the hazards caused by extreme floodings, such as dam breaks, flash floods and tsunamis in development planning.

Development of depth-averaged nonlinear flow model for fractures considering heterogeneity and significant inertial effects

Fluid flow through fractures is central to many engineering applications and geophysical processes. Although the intertwined effects of aperture heterogeneity and inertial forces on fluid flow have been interrogated in last few decades, it remains challenging to accurately predict fracture flow field at high pressure gradient conditions when inertial effects are significant. To address this challenge, we rigorously derived a depth-averaged nonlinear flow model by introducing a quadratic velocity term into the linear flow model (i.e., Local Cubic Law that describes a linear relationship between pressure gradient and fluid flux) at the mechanistic level. The supporting laboratory experiments and direct numerical simulations through six rough-walled fractures with varying roughness further confirmed the robustness of the proposed model. We found that the predictive model outperforms existing alternatives with acceptable relative errors in mean velocity (<7%) considering a broad range of Reynolds number (0.01–167), suggesting its superior performance of handling pronounced inertial effects. Lastly, we parameterized the new model by establishing an empirical function that related parameter (ω) in the model to measurable fracture properties (i.e., aperture); this makes the predictive model viable and straightforward for predicting local velocity fields in pressurized engineered environments.

A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms

Turbulence models are critical for depth-averaged flow models in at least two ways: (i) as closures for momentum equations and (ii) as indicators of the spatial variability in the turbulence intensity field, which is crucial for sediment transport and bedform evolutions. This paper introduces a novel moment-based depth-averaged k-ε turbulence (MDAKE) model that could be considered as a revised version for the standard k-ε Rastogi–Rodi (SDAKE) model and can be used to estimate the true values for the depth-averaged turbulence kinetic energy in more complex and varied flow conditions with accelerating–decelerating flow fields. The study in hand shows that the SDAKE model tends to overestimate the true depth-averaged turbulent kinetic energy (k¯u) by 50 to 130% in the benchmark case of uniform flow over a flatbed. Further, the SDAKE model assumes that the bed shear velocity is an appropriate scale for the generation terms of both turbulent kinetic energy and dissipation. When bed topographic features vary, a shear flow zone is formed and the assumption is invalid. Since most of the turbulence is generated by shear flow zones away from the bed, the SDAKE model’s estimates for the depth-averaged turbulent kinetic energy field are out of phase with measurements for the flow over a train of bedforms. Therefore, a newly developed depth-averaged KE model based on the moment concept (MDAKE) is presented here. The model replaces bed shear velocity with the integral moment velocity scale (u1). The calibrated MDAKE model is used to predict turbulent kinetic energy over a train of bedforms. The results of the MDAKE model are in phase and generally in reasonable agreement with the measurements.

Balanced Meshless Method for Numerical Simulation of Pollutant Transport by Shallow Water Flow over Irregular Bed: Application in the Strait of Gibraltar

This paper focuses on the implementation of an accurate meshless scheme for the simulation of the advection–diffusion of non-active pollutant in a two-dimensional depth-averaged flow. The depth-averaged flow model includes the shallow water system and the pollutant propagation described by the advection–diffusion equation with diffusion tensor. The mathematical model was implemented by using a meshless method based on local radial basis functions. This method was used to numerically evaluate the spatial derivatives and complemented with the second-order Runge–Kutta method for the time evolution. To remove the non-physical oscillations, which appear at the discontinuity, a filter based on a hyperviscosity operator was applied. To investigate the effectiveness and accuracy of the proposed scheme, a number of tests are presented, including the dam break problem, and the pure transport of a pollutant in a long channel. Finally, a hypothetical example of a pollutant transport in the Strait of Gibraltar is modeled. The results obtained are compared both with analytical solutions and with simulation results obtained by a finite volume method based on the Roe-MUSCL scheme. The main advantages of the proposed method are: (i) the simplicity of implementation, (ii) the ability to handle calculations of slowly varying flows or concentrations, as well as rapidly varying flows containing shocks or discontinuities, and (iii) the ability to satisfy the C-property and guarantee positive values of both water level and pollutant concentration.

A new two-phase shallow water hydro-sediment-morphodynamic model based on the HLLC solver and the hybrid LTS/GMaTS approach

In this paper, we present a new computationally-efficient and high-resolution depth-averaged two-phase flow model for hydro-sediment-morphydynamic processes, featuring an advance over existing models in terms of accuracy and efficiency of numerical solution. Under the framework of finite volume method (FVM) on unstructured grids, the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver is proposed to compute inter-cell fluxes by applying the classical upwind HLLC approach to the water-sediment mixture and the sediment phases separately, in contrast to previous two-phase flow models using centered schemes. Moreover, to improve computational efficiency, the local-time-stepping (LTS) approach is implemented, the first attempt in the field of two-phase flow modelling. After a convergence rate study, the model is tested against a series of flow-sediment-bed evolutions induced by (1) two refilling processes of dredged trenches, (2) two instantaneous dam-break flooding flows, and (3) one levee breaching process by overtopping flows. It features encouraging performance when compared to a two-phase flow model based on a centered scheme and global time bound, characterized by more accurate results and much less computational cost. The present modelling framework shows promise in practical shallow water hydro-sediment-morphodynamic modelling applications.

Modeling Tephra Fall and Sediment-Water Flows to Assess Their Impacts on a Vulnerable Building Stock in the City of Arequipa, Peru

Arequipa, Peru’s second economic center hosting c. 1,110,000 inhabitants, is the largest South American city exposed to a large variety of natural hazards. At least 200,000 live in areas likely to be affected by hazards from El Misti volcano, located 17 km to the NE. A multidisciplinary project aims to address the impacts of tephra fall and frequent mass flows on the vulnerable building stock and roofs along two ravines that cross the city, enabling decision-makers to undertake retrofitting projects and improve urban risk planning. Two recent eruptions, that is, the 1440–1470 CE Vulcanian event and c. 2070 years BP Plinian eruption, were chosen as references for probable scenarios of potential tephra fall impacts from El Misti on the building roofs. Tephra fall impacts on the city depend on the eruptive style, column height, and patterns of wind directions and velocities over south Peru and roof mechanical resistance. Estimates of potential damage levels and cost range values rely on nine structural types and four classes of vulnerable roofs. Simulation runs of hyperconcentrated flows (HCF) and debris flows (DF), using three depth-averaged flow models (Titan2F, VolcFlow, and Flo-2D) along two drainage basins on the SW flank of El Misti and across Arequipa, examined three scenarios from a database of 39 recent events and other historical lahars. Simulation results showcase the extent toward the city, inundation depths ≤4.6 m, flow velocities ≤9 m/s, and dynamic pressure up to 100 kPa from three different magnitude HCFs and DFs. In both ravines, overbank flows occurred in key urban areas due to channel sinuosity and constrictions near bridges. Potential impacts on habitat stem from ranges of flow dynamic pressure and measurements of construction material. We estimated the monetary loss of buildings according to hyperconcentrated flows and debris flows scenarios to contribute to retrofitting procedure, implementation of defense work, and relocation policy.

Devising a procedure to calculate and analyze parameters for passing the flood and breakthrough wave taking into consideration the topographical and hydraulic riverbed irregularities

It has been established that the most likely period of breakthrough wave occurrence is the time of spring flooding or heavy rain when water-head facilities are subjected to significant loads that lead to the collapse of their individual elements or the entire structure. In addition, the possibility of man-made accidents that can occur at any time cannot be ruled out. It has been proven that breakthrough wave formation depends on the nature of the destruction or the overflow through a water-head facility. For the study reported in this paper, a model of the kinematics of riverbed and breakthrough flows was used, which is based on the equations of flow, washout, and transport of sediments that are averaged for the depths of the stream. The differential equations describing the nonstationary flow averaged for depth are solved using the numerical grid system FST2DH (2D Depth-averaged Flow and Sediment Transport Model), which implements a finite-element method on the plan of a riverbed's topographic region. These tools are publicly available, which allows their wide application to specific loads and boundary conditions of mathematical models. The construction of an estimation grid involving the setting of boundary conditions and the use of geoinformation system tools makes it possible to simulate the destruction of a culvert of the pressure circuit and obtain results for a specific case of an actual riverbed and a water-head facility. It has been established that there is a decrease in the speed of wave propagation along the profile, from 3 m/s to 1 m/s. The impact of bottom irregularities, the effect of floodplains, and the variety of bottom roughness have also been assessed, compared to the results of their calculation based on one-dimensional models given in the regulatory documents. Hydraulic calculations were carried out taking into consideration the related properties of the main layer of the floodplain, which consists of peat accumulations, and the heterogeneity of the depths and roughness of floodplain surfaces of soils. It has been established that there is almost no erosion of supports in the floodplain zone in this case. It was found that as the distance between the flow and breakthrough intersection increases, there is a decrease in the height of the head from 2.1 m to 1.25 m.

A bed pressure correction of the friction term for depth-averaged granular flow models

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen friction laws, do not in some cases preserve the physical threshold of motion. In particular, the simulated granular mass can start to flow (or stay at rest) even if the slope angle of its free surface is lower (or higher) than the repose angle of the granular material involved. The problem is related to the hydrostatic pressure assumption, associated with the direction of integration, which is orthogonal to a reference plane or a reference bottom. We propose here an initial method to correct this misleading behavior. Firstly, we define a correction of the friction term that accounts for the Jacobian of a change of coordinates, making it possible to reproduce the physical threshold of motion and thus the solutions at rest. Secondly, we observe that the 3D model presented in [F. Bouchut, I. Ionescu, and A. Mangeney. An analytic approach for the evolution of the static-flowing interface in viscoplastic granular flows. Commun, Math. Sci., 14(8):2101–2126, 2016] verifies the physical thresholds of motion because it is based on a second order correction of the pressure valid for slow granular flows. The correction proposed here ensures that the model preserves, up to the second order, the physical threshold of motion defined by the repose angle of the material. Several numerical tests are presented to illustrate certain problems related to classical depth averaged models and the remedial effect of the proposed correction, in particular through comparisons with experimental data. We finally show that this correction is not exact far from the starting and stopping phases of the granular avalanche and should be improved by adding other second order terms in the pressure approximation.

Mega riverbed-patterns: linear and weakly nonlinear perspectives.

In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre–manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart–Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.

Strong oblique shock waves in granular free-surface flows

Strong oblique shock waves of granular flow are a steady-state solution formed when a granular free-surface flow deflects around a wedge-shaped obstacle at a supercritical speed, but they do not usually occur because their formation requires specific conditions to be satisfied downstream of the shock wave. This paper discusses the method of generating the strong oblique shock wave in a laboratory experiment and numerical simulation. The experiment is conducted on a plexiglass chute inclined at an angle to the horizontal, in which a dry granular material is released from a hopper at the top of the chute to form a channelized flow that passes a wedge at a downslope location. In order to generate a strong oblique shock wave, a second gate is established at the downstream of the wedge to control the material to flow out only at the designed time and height. Such a granular flowing process is simulated with a depth-averaged granular flow model, where the above two-gate system is mirrored into the inlet and outlet boundaries, respectively. The formation of the strong oblique shock is investigated through the transient solution of the flow field, and a good agreement is observed between the experiment and the simulation. Then, the steady-state solution of the interaction between the weak and strong oblique shocks is analyzed in the experiment and simulation. This result can be regarded as the third solution of granular shock because it can be formed by just changing the opening time of the second gate. With the dramatic change in flow thickness and velocity across the strong oblique shock, the bulk inertial number, used to quantify the rheological relation of granular materials, becomes extremely small, but it does not seem to affect the behavior of the flow discussed in this paper.

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.

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

A depth-averaged model for Newtonian fluid flows in shallow microchannels

Pressure-driven flow has been widely used in microfluidic devices to pump fluids and particles through planar microchannels for various applications. The variation in channel geometry (e.g., contraction or expansion) may lead to complex flow phenomena (e.g., recirculations) useful for microfluidic sampling, such as fluid mixing and particle focusing. In this work, we develop a depth-averaged inertial flow model for Newtonian fluids in shallow microchannels based on an asymptotic analysis of the continuity and momentum equations. The validity and accuracy of this two-dimensional model are assessed through comparisons with the experimental measurements and three-dimensional numerical simulations for water flow through contraction–expansion microchannels of varying depths. Our proposed depth-averaged model provides the accuracy of three-dimensional modeling if the channel depth-to-width ratio remains small (specifically, at ∼0.1 or less).

Instability of large-scale riverbed patterns

In this paper, we explore the instability of large-scale riverbed patterns, performing a linear stability analysis of a zero-pressure gradient free-surface flow in a wide straight channel with an erodible bed. The local depth-averaged turbulence state is governed by two key parameters: namely, the turbulent kinetic energy (TKE) and its dissipation rate. A depth-averaged flow model coupled with the transport equations of the TKE and its dissipation rate and the particle transport model are developed to examine the formation of large-scale patterns. Both the modes of particle transport as bedload and suspended load are considered herein, allowing for the extension of the conventional theories to cover from hydraulically smooth to transitional flow regimes. The classical Exner equation of the bed evolution is modified in the presence of suspended particles, whose concentration is coupled with the steady-state advection–diffusion equation. Applying a standard linearization technique, the periodic perturbations in both streamwise and spanwise directions are imposed on the bed to find the dispersion relationship. The stability maps for the growth rate of large-scale patterns are obtained as a function of streamwise and spanwise wavenumbers and of key parameters associated with the flow and particles.

Numerical Analysis of the Potential Effect of Wetlands on Reducing Tidal Currents in the Liao River Estuary, China

In this study, an explicit depth-averaged 2D flow model that considers vegetation effects was established to investigate the interactions between tidal currents and vegetation in coastal wetland waters. Roe’s approximate Riemann solver, coupled with the drying-wetting boundary technique, was proposed to evaluate the interface fluxes and track the moving shoreline. In addition, the vegetation-induced drag force was added to the momentum equations as an internal source. Model validation based on two laboratory experiments demonstrated that the model results were in good agreement with the measured results. The model was then applied to evaluate the interactions between vegetation and tidal currents in the Liao River Estuary, China. The simulation results showed that vegetation played a critical role in attenuating flow velocities in vegetated waters. In the Liao River wetland, the reductions in the amplitudes of the peak depth-averaged velocities in Phragmites austral (P. australis) wetlands were significantly higher than those of the Suaeda heteroptera (S. heteroptera) wetlands. Moreover, the flow velocity changes in the wetland waters also impacted the kinetic energy, creating a new flow field structure. The simulation results also indicated that higher vegetation densities resulted in a larger flow velocity attenuation rate in the vegetated areas.