Shallow 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.

Accelerated numerical modeling of shallow water flows with MPI, OpenACC, and GPUs

In this paper, a time-explicit Finite-Volume method is adopted to solve the 2-D shallow water equations on an unstructured triangular mesh, using a two-stage Runge-Kutta integrator and a monotone MUSCL model to achieve second-order accuracy in time and space, respectively. A multi-GPU model is presented that uses the Message Passing Interface (MPI) with OpenACC and uses the METIS library to produce the domain decomposition. A CUDA-aware MPI library (GPUDirect) and overlapped MPI communication with computation are used to improve parallel performance. Two benchmark tests with wet and dry downstream beds are used to test the code's accuracy. Good results were achieved compared to the numerical simulations of published studies. Compared with the multi-CPU version of a 6-core CPU, maximum speedups of 56.18 and 331.51 were obtained using a single GPU and 8 GPUs, respectively. Higher mesh resolution enhances acceleration performance, and the model is applicable to other environmental modeling activities.

Cross-Scale Modeling of Shallow Water Flows in Coastal Areas with an Improved Local Time-Stepping Method

A shallow water equations-based model with an improved local time-stepping (LTS) scheme is developed for modeling coastal hydrodynamics across multiple scales, from large areas to detailed local regions. To enhance the stability of the shallow water model for long-duration simulations and at larger LTS gradings, a prediction-correction method using a single-layer interface that couples coarse and fine time discretizations is adopted. The proposed scheme improves computational efficiency with an acceptable additional computational burden and ensures accurate conservation of time truncation errors in a discrete sense. The model performance is verified with respect to conservation and computational efficiency through two idealized tests: the spreading of a drop of shallow water and a tidal flat/channel system. The results of both tests demonstrate that the improved LTS scheme maintains precision as the LTS grading increases, preserves conservation properties, and significantly improves computational efficiency with a speedup ratio of up to 2.615. Furthermore, we applied the LTS scheme to simulate tides at grid scales of 40,000 m to 200 m for a portion of the Northwest Pacific. The proposed model shows promise for modeling cross-scale hydrodynamics in complex coastal and ocean engineering problems.

DOT-type schemes for hybrid hyperbolic problems arising from free-surface, mobile-bed, shallow-flow models

Free-surface, mobile-bed, shallow-flow models may present Hybrid hyperbolic systems of partial differential equations characterised by conservative and non-conservative fluxes that can only be expressed in primitive variables. This paper presents the effort we made to derive DOT-type schemes (Osher-type schemes derived by Dumbser and Toro, 2011 [1]) for these kinds of systems formulated for the one-dimensional case. Firstly, for a Hybrid system, we managed to write a quasi-linear form characterised by the presence of a matrix, expressed as a function of the primitive variables, that multiplies the spatial derivative of the conserved variables. Next, we derived the first numerical flux by adapting the approach of Leibinger et al., 2016 [2] to this quasi-linear form. We called this result DOTHCP flux. To achieve a faster algorithm, instead of using an integration path in the space of conserved variables, as in the previous case, we employed a path in the space of primitive variables. We called this second formulation DOTHPP flux. Subsequently, we managed to account for certain physical constraints arising from the generalised Rankine-Hugoniot relations in the expression of one term of the previous flux formulation, thus obtaining the DOTHZR flux. Finally, we showed that these methods can also be applied to Combined systems characterised by conservative and non-conservative fluxes expressed in conserved variables. Several tests show the characteristics and good performances of the proposed methods when applied to Riemann problems of Hybrid and Combined systems deriving from free-surface models. Finally, thanks to the general formulation of the proposed DOT-type fluxes, these can also be applied to Hybrid and Combined hyperbolic systems deriving from different physical problems.

A consistent three-equation shallow-flow model for Bingham fluids

A consistent three-equation shallow-flow model for Bingham fluids

A central-upwind scheme for two-phase shallow granular flow model

In this article, a central-upwind scheme (CUP) is extended to solve two-phase shallow granular flow (TPSF) model. The flow in this case is considered as incompressible and it is regarded to be a shallow layer of granular and liquid material. The model consists of continuity and momentum equations for both solid and liquid phases. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model, the complexity of which raises numerical challenges. The proposed method is non-oscillating upwind biased that does not require a Riemann solver at each step. A few test problems are approximated numerically to check the performance of suggested scheme. For validation the outcomes are compared with the conservation element solution element (CE/SE) and kinetic flux vector splitting (KFVS) schemes.

Debris Flow Run-Out Prediction Based on the Shallow-Water Flow Numerical Model—A Case Study of Xulong Gully

Here we present a method for predicting debris flow run-out based on a numerical model for shallow water flows, using a case study conducted on Xulong Gully, a proposed dam site for a hydropower station in the upper reaches of the Jinsha River. A field investigation and remote sensing interpretation methods were used to develop a comprehensive evaluation of debris flow zones and calculate the potential provenance volume in the Xulong Gully. Particle-size analysis was conducted on the early debris flow fan in the Xulong Gully to determine the rheological properties of the debris flow materials. A numerical model for shallow flows was constructed using the finite volume method to verify fluid motion across complex terrain and explore the debris flow run-out range with various provenance volumes. The model showed that for a total debris flow volume of less than two million m3, the debris flow impact area would remain within the Xulong Gully. However, if the total debris flow volume is more than two million m3, the debris flow would flush out into the Jinsha River, blocking a portion of the river. If all the provenance in the Xulong Gully were flushed out, the maximum flow velocity of the generated debris flow would be 11 m/s and the thickness of the debris flow at the Xulong Gully estuary would be about 28.8 m. The debris flow would completely block 470 m of the Jinsha River.

Research on the characteristics of fluid escape near the seabed in Bohai Bay

AbstractSeabed fluid flow happens due to the complex interactionamong the accumulation, migration and escape of hydrocarbon and/or fluids. As shallow gas accumulations are anomalously developed in Bohai Bay, studying fluid flow features provides insights for understanding environment fluid in the Bohai Sea. Therefore, in our study, we aimed to make practical and effective use of single‐channel seismic reflection data to delineate and interpret subsurface fluid flow features by analysing the variance, sweetness, instantaneous frequency, and root mean square amplitude of seismic attributes. Four types of submarine fluid escape features, including pockmarks, mounds, fluid vents, dimmed or chaotic seafloor reflections, and two shallow fluid migration pathways (gas chimneys and faults) are distinguished on the single‐channel seismic reflection profiles. Gas chimneys are divided into strict columnar and irregular columnar chimneys, fluid migration along faults is critically dependent on the activity of faults. Enhanced reflections and bright spots may be good indicators of lateral fluid migration and accumulation of fluids in porous formations. We propose a shallow fluid flow model in the study area that illustrates the dynamic fluid migration process in terms of possible fluid sources, shallow migration pathways, and submarine fluid escape features.

A GPU-Accelerated Two-Dimensional Hydrodynamic Model for Unstructured Grids

The precision of numerical overland flow models is limited by their computational cost. A GPU-accelerated 2D shallow flow model is developed to overcome this challenge in this study. The model employs a Godunov-type finite volume method (FVM) to solve shallow water equations (SWEs) with unstructured grids, while also considering rainfall, infiltration, bottom slope, and friction source terms. The numerical simulation demonstrates that this model has well-balanced and robust properties. In an experiment of urban rain-runoff and flood, the accuracy and stability of the model are further demonstrated. The model is programmed with CUDA, and each numerical computation term is processed in parallel to adopt multi-thread GPU acceleration technology. With the GPU computation framework, this model can achieve a speeding up ration around 75 to single-thread CPU in the dam-break flow for a large-scale application.

Simulating Debris Flow and Levee Formation in the 2D Shallow Flow Model D‐Claw: Channelized and Unconfined Flow

AbstractDebris flow runout poses a hazard to life and infrastructure. The expansion of human population into mountainous areas and onto alluvial fans increases the need to predict and mitigate debris flow runout hazards. Debris flows on unconfined alluvial fans can exhibit spontaneous self‐channelization through levee formation that reduces lateral spreading and extends runout distances compared to unchannelized flows. Here we modify the D‐Claw shallow flow model in two ways that are hypothesized to generate levees. We evaluate these modifications with observations from a large‐scale flume experiment. We investigate model performance when including the effect of two different friction sub‐models, as well as the inclusion of segregation effects on granular permeability. Results show that, for a wide range of plausible model input parameters, simulations including the effects of segregation promoted modeled levee formation, whereas simulations without the effects of segregation did not create levees. Further, using a forward predictive framework, simulations with the effects of segregation were more likely to better model the magnitude of debris flow depth and runout distance, whereas simulation timing of the debris flow was affected by the choice of friction sub‐model. Our results indicate that including the effects of segregation on granular permeability can improve the likelihood of better predictions of debris flow depth and runout prior to an event occurring.

Exact solution for Riemann problems of the shear shallow water model

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

Analytical study of atmospheric internal waves model with fractional approach

This study looks at the mathematical model of internal atmospheric waves, often known as gravity waves, occurring inside a fluid rather than on the surface. Under the shallow-fluid assumption, internal atmospheric waves may be described by a nonlinear partial differential equation system. The shallow flow model’s primary concept is that the waves are spread out across a large horizontal area before rising vertically. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for any given model. This aids in the modelling of the global atmosphere, which has applications in weather and climate forecasting. For the integer-order value (α=1), the FRDTM solution is compared to the precise solution, EADM, and HAM to assess the correctness and efficacy of the proposed technique.

MPS modeling of cross-sectional averaged shallow water flows with open boundaries using TVD-MacCormack predictor-corrector

The present study utilizes a moving particle simulation (MPS) method to solve cross-sectional averaged shallow water equations applicable for non-prismatic open channels. The well-known MacCormack predictor–corrector scheme is extended to the Lagrangian framework, ensuring the second accuracy in time and space. For this purpose, upwind and downwind MPS gradient operators are introduced. The total variation diminishing (TVD) flux-limiter scheme is developed to the MPS formulation to preserve monotonicity, which does not require characteristic speeds of the system. In mesh-based methods, the non-conservative form of the momentum equation is problematic in scenarios with discontinuities. However, the non-conservative convective flux term does not exist under the Lagrangian framework in preserving the momentum conservation. Moreover, the cross-sectional area is computed by the algebraic density-ratio equation in the present work since the mass is automatically conserved. This advantage is beneficial for open boundary treatment resulting in less computational effort than mesh-based methods because the partial differential continuity equation is omitted as a boundary value problem. The present high-order solver has positivity-preserving and well-balancing properties without the need for any threshold value for treating the dry bed and friction term. Additionally, it can model free-surface flows in prismatic and non-prismatic open channels with various open and/or closed boundary conditions.

Unsteady Linearisation of Bed Shear Stress for Idealised Storm Surge Modelling

The modelling of time-varying shallow flows, such as tides and storm surges, is complicated by the nonlinear dependency of bed shear stress on flow speed. For tidal flows, Lorentz’s linearisation circumvents nonlinearity by specifying a (steady) friction coefficient r based on a tide-averaged criterion of energy equivalence. However, this approach is not suitable for phenomena with episodic and irregular forcings such as storm surges. Here, we studied the implications of applying Lorentz’s energy criterion in an instantaneous sense, so that an unsteady friction coefficient r(t) adjusts to the temporal development of natural wind-driven flows. This new bed-stress parametrisation was implemented in an idealised model of a single channel, forced by time-varying signals of wind stress (acting over the entire domain) and surface elevation (at the channel mouth). The solution method combines analytical solutions of the cross-sectionally averaged linearised shallow-water equations, obtained in the frequency domain, with an iterative procedure to determine r(t). Model results, compared with a reference finite-difference solution retaining the quadratic bed shear stress, show that this new approach accurately captures the qualitative and quantitative aspects of the surge dynamics (height and timing of surge peaks, sloshing, friction-induced tide-surge interaction) for both synthetic and realistic wind forcings.

A fifth order WENO scheme for numerical simulation of shallow granular two-phase flow model

In this article, a weighted essentially non-oscillatory (WENO) scheme is implemented to simulate two-phase shallow granular flow (TPSF) model. The flow is assumed to be incompressible and it is regarded as shallow layer of granular and liquid material. The mathematical model consists of two phases, that is, solid and liquid. Each phase has its continuity and momentum equation. The presence of the equations are coupled together involving the derivatives of unknowns which make it more challenging to solve. An efficient numerical technique is needed to tackle the numerical complexities. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model. The weighted essentially non-oscillatory (WENO) scheme of order 5 is utilized to handle the shock waves and contact discontinuities appear in the solution. The results are compared with the results already available in the literature by conservation element and solution element (CESE) scheme. It is observed the WENO scheme produces less errors as compared to CESE scheme and also effectively handle the shocks.

Mathematical modeling of atmospheric internal waves phenomenon and its solution by Elzaki Adomian decomposition method

This article involves the study of atmospheric internal waves phenomenon, also referred to as gravity waves. This phenomenon occurs inside the fluid, not on the surface. The model is based on a shallow fluid hypothesis represented by a system of nonlinear partial differential equations. The basic assumption of the shallow flow model is that the horizontal size is much larger than the vertical size. Atmospheric internal waves can be perfectly represented by this model as the waves are spread over a large horizontal area. Here we used the Elzaki Adomian Decomposition Method (EADM) to obtain the solution for the considered model along with its convergence analysis. The Adomian decomposition method together with the Elzaki transform gives the solution in a convergent series without any linearization or perturbation. Comparisons are built between the results obtained by EADM and HAM to examine the accuracy of the proposed method.

Physically-constrained data-driven inversions to infer the bed topography beneath glaciers flows. Application to East Antarctica

A method for infering bed topography beneath glaciers from surface measurements (elevation from altimetry and velocity from InSAR) and sparse thickness measurements is developed and evaluated. The method is based on an original non-isothermal Reduced Uncertainty (RU) version of the Shallow Ice Approximation (SIA) equation that natively incorporates the surface measurements. The flow model has a single dimensionless multi-physics parameter γ. This parameter takes into account the basal slipperiness and the variable vertical rate factor profiles, thus the vertical thermal variations. The inversions are based on three steps involving: an Artificial Neural Network (ANN) and two Variational Data Assimilation (VDA) processes. The ANN-based stage aims at estimating the multi-physics number γ from the thickness measurements; the resulting estimator is remarkably robust. The full inversion method is valid for half-sheared flows (presenting a moderate basal slipperiness): it can be applied to inland ice-sheets areas. Also these estimates connect continuously with estimates from mass conservation only, i.e. with areas of sliding flows. Numerical results are presented for areas of the East Antarctica Ice Sheet where bed elevation can be very uncertain (Bedmap2 values). Estimates are valid for wavelengths longer than \(\sim 10 \bar h\) (due to the long wave assumption, shallow flow model) with resolution at \(\sim \bar h\) (\(\bar h\) a characteristic thickness value).

TRENT2D❄: An accurate numerical approach to the simulation of two-dimensional dense snow avalanches in global coordinate systems

The paper presents a novel and accurate numerical approach to the simulation of two-dimensional, dense snow avalanches described as a single-phase, shallow fluids with a Voellmy friction law.Unlike the majority of the shallow-flow models present in the literature, using a local coordinate system (one axis normal to the bed and the other two lying in the tangent plane), our approach considers a global three-dimensional Cartesian coordinate system with an axis opposite to the gravity vector and the other two lying in a horizontal plane. The relevant flow equations, accounting for significant bed slope, have been derived in conservative form for the 1D and 2D cases. This choice allows overcoming an intrinsic limit of the local approach that ceases to be valid in case of vertical walls.From a numerical point of view, the model employs a finite volume scheme applied to a regular Cartesian grid where the fluxes are evaluated with a well-balanced Godunov method. The source term is computed with an implicit operator-splitting technique tailored to deal not only with dynamic conditions but also with stopping conditions due to the Coulomb-type term in the Voellmy friction law. Finally, an effective numerical strategy was developed to treat static conditions with an inclined free surface.Several tests, some with analytical solution and some without, were performed to evaluate the capabilities of the proposed approach. Results are satisfactory: all the analytical solutions are accurately reproduced while the other tests give reliable results. Finally, no instability arises in any situation. For these reasons, TRENT2D❄ is a candidate to become a good model also for practical applications such as hazard mapping and hazard assessments.

Numerical simulation and analysis of a geological disaster chain in the Peilong valley, SE Tibetan Plateau

The Yarlung Tsangpo river basin in the southeastern part of the Tibetan Plateau contains many maritime glaciers, where the massive debris flow formation caused by the glacial degradation leads to the issue of regional warming. It often forms a continuous geological disaster chain. In this study, the topographical features and debris flow deposits in the Peilong valley were investigated in November 2018. It was found that the geological conditions in the valley are complex and numerous unstable sources are existed. Field investigations, laboratory tests, and a geomorphological analysis were conducted to obtain relevant data for the valley. And comparison of Landsat images indicates that the rate of change of the glacial area is 51.62% from winter to summer. Discrete element software (PFC3D) is used to simulate glacial landslides on the left bank. The simulation results show that a dam with a height of 141 m was formed, creating a lake. The debris flow after the dam break is simulated for different peak discharge values and durations based on a free surface shallow flow (SFLOW) model, with a final accumulation depth of 21.6 m. The comprehensive results of numerical simulation show that the chain of geological disasters induced by glacial landslide poses a huge threat to residents and roads near Peilong valley. Therefore, it is recommended to monitor the glacier activities in real time to enable forecasting of debris flows and reduce the property losses and prevent loss of life.