Friction Source Terms Research Articles

A novel efficient and robust treatment of the friction source term in 2D shallow water inundation models

The technological advancements of the last few decades have fostered the use of two-dimensional (2D) Shallow water Equations (SWE) flood simulations not only in academic research but also in practical real-world applications and territorial planning. The evaluation of flow resistance due to friction is crucial as it plays a relevant role in a variety of flow conditions. However, problems arise in dam-break and overland flow applications involving very small water depth because the time scale connected to the friction term may be much smaller than the hydrodynamic time scale. To cope with this issue, known as source term stiffness, an implicit treatment of the friction term is frequently adopted in Finite Volume (FV) numerical schemes, but there are cases (null roughness coefficient or null flow velocity) where a division by zero may occur during calculations, leading to a crash of the algorithm. Herein, we propose a reformulation of the implicit friction term that is general, more stable and computationally efficient (with a speed-up of approximately 3 times for the routine running the friction computation) than currently available approaches. The novel implicit method allows to manage special conditions with null roughness or null discharge, without resorting to ad-hoc thresholds, and can be easily implemented in existing schemes with pointwise friction treatment. The novel friction approach is implemented in a purposely simple FV numerical scheme (hydrostatic reconstruction to account for the bed slope terms, first-order accuracy in time and space) - to better focus on the friction term treatment - and it is validated against analytical, synthetic, laboratory and real-world case studies, showing promising capabilities.

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

A well-balanced and positivity-preserving numerical model for overland flow under vegetation effects

In this study, we used the depth-averaged shallow water equations for modeling flows through vegetation field. The vegetation effects on flow are modeled using Morison’s equation taking into account drag and inertia forces which depend on both vegetation and flow properties. We compute and compare different formulations for the stem drag coefficient based on the Froude number or the vegetation volume fraction. Vegetation-induced turbulence is taken into account by adding diffusion terms in the momentum equations. The resulting system of equations is solved using a well-balanced and positivity preserving finite volume method to guarantee the balance between the flux and bed topography source terms, and the positivity of the computed water depth. In our approach, the drag force and bed friction source terms are combined into a unified form. We propose to discretize the obtained term using an implicit temporal method where an analytical technique is used. Special discretization techniques are used for the inertia force and turbulent diffusion terms. Numerical simulations are performed to validate the accuracy of the proposed numerical model. We investigate and compare different formulations for the stem drag coefficient in the vegetation model. Our results confirm the capability of the proposed numerical model for simulating overland flows under vegetation effects.

Numerical modelling of dense snow avalanches with a well-balanced scheme based on the 2D shallow water equations

Abstract A common technique for simulating non–Newtonian fluid dynamics, such as snow avalanches, is to solve the Shallow Water Equations (SWE), together with a rheological model describing the momentum dissipation by shear stresses. Friction and cohesion terms are commonly modelled using the Voellmy friction model and, recently, the Bartelt cohesion model. Here, an adaptation of the Roe scheme that ensures the balance between the flux and pressure gradients and the friction source term is presented. An upwind scheme was used for the discretisation of the SWE numerical fluxes and the non–velocity-dependent terms of the friction–cohesion model, whereas a centred scheme was used for the velocity-dependent source terms. The model was tested in analytically solvable settings, laboratory experiments and real cases. In all cases, the model performed well, avoiding numerical instabilities and achieving stable and consistent solution even for an avalanche stopping on a sloping terrain.

An asymptotic preserving scheme for the shallow‐water equations with Manning friction using viscous correction of the HLL scheme

AbstractThe aim of this article is to propose a numerical scheme for the shallow‐water equations with a Manning's friction source term able to preserve the diffusive regime, namely the behavior of the solutions in long time and stiff friction limit. Because of the non‐usual rescaling of the friction parameter, a strongly nonlinear derivative operator is involved in the diffusive limit and the development of a numerical scheme possessing this property is not straightforward. The scheme presented here is based on a perturbed HLL scheme with additional viscous terms.

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.

An efficient semi-implicit friction source term treatment for modeling overland flow

In this study, a new numerical treatment of the friction source term is proposed for modeling shallow water flows over complex bed topography. We design a novel semi-implicit temporal discretization technique based on Taylor series expansion of friction source terms. A well-balanced positivity-preserving finite volume scheme on unstructured triangular grids is used to solve the shallow water model. We present a series of numerical tests to demonstrate the robustness and effectiveness of the proposed numerical model where we used rapidly varying flows over complex topography as well as flows involving small values of water depth and wet/dry zones. The simulation results of the numerical model based on the proposed approach for friction source terms are compared to experimental data and to those obtained by using other existing techniques. The proposed numerical model is stable and second-order accurate and our predictions are in good agreement with experimental observations.

A fully well-balanced and asymptotic preserving scheme for the shallow-water equations with a generalized Manning friction source term

A fully well-balanced and asymptotic preserving scheme for the shallow-water equations with a generalized Manning friction source term

A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment

Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological processes was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the framework of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution of erosion and deposition over time. The research results provide guidance and a reference for the in-depth study of debris flow processes and disaster prevention and mitigation.

Simulation of Mosul Dam Break Using Finite Volume Method

Mosul dam is an earth-fill embankment located north of Iraq on the Tigris River forming a reservoir with 11.11 km3 water storage capacity which is the largest dam in the country. The dam is built on a rock bed foundation, in which the dissolution process is dynamic in the zone where gypsum and anhydrite layers present. During the construction development seepage locations were found in the dam foundation and the grouting process is in progress until now to control this problem. Therefore, the possibility of the Mosul dam break is highlighted by previous studies. In this research, a FORTRAN code based on the finite volume method is modified to solve the two-dimensional shallow water equations and simulating the Mosul dam break. The computational domain discretized using unstructured triangular mesh. The solver applied Harten lax van leer with contact (HLLC) wave approximate Riemann solver to calculate the cell interface fluxes, and the semi-implicit scheme employed to solve the friction source term. The numerical scheme applied to two benchmark test cases, and the results showed that the presented model was robust and accurate especially in handling wet/dry beds, mixed flow regimes, discontinuities, negative water depths, and complex topography. The results of this study demonstrate that flood waves may reach the center of Mosul city in < 6 h and water depth may rise to 34 m after 7 h of Mosul dam breaking. Finally, the simulation results of the Mosul dam break were used to prepare an emergency action plan.

A fully well-balanced scheme for the 1D blood flow equations with friction source term

We consider the 1D blood flow equations with friction source term. The main purpose of this work is the derivation of a positivity preserving and fully well-balanced scheme, which correctly approximates the solutions of this system, exactly preserves all the associated steady states, including the moving ones, and ensures the positivity of the cross-sectional area. To address such issues, a study of the moving steady states related to the friction source term is first performed. Afterwards, a Godunov-type scheme is constructed, by deriving a two-state approximate Riemann solver and by introducing a relevant discretization of the source term, to enforce the well-balanced property. The scheme is then adapted to the generalized model including a space-variable viscous resistance; a second-order well-balanced MUSCL extension of the scheme is also proposed. Numerical experiments are finally carried out in order to validate the scheme and highlight its properties.

A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction

A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate bottom level and the intermediate water height at the cell interface. We propose an explicit-implicit method to address the friction source term. The new second-order SR scheme together with the explicit-implicit method can preserve a special steady-state solution of the system and can maintain the positivity of the water depth. We also extend the new scheme to two-dimensional shallow water flows. To demonstrate the robustness and effectiveness of the new scheme, we use several classical numerical experiments for the shallow water flows over a complex bottom topography.

A GPU-based numerical model coupling hydrodynamical and morphological processes

Sediment transport simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model coupling hydrodynamical and morphological processes was developed to simulate water flow, sediment transport, and morphological changes. Aiming at accurately predicting the sediment transport and sediment scouring processes, the model resolved the realistic features of sediment transport and used a GPU-based parallel computing technique to the accelerate calculation. This model was created in the framework of a Godunov-type finite volume scheme to solve the shallow water equations (SWEs). The SWEs were discretized into algebraic equations by the finite volume method. The fluxes of mass and momentum were computed by the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated by the proposed a splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which could run on GPUs to substantially accelerate the computation. The aim of the work was to develop a GPU-based numerical model to simulate hydrodynamical and morphological processes. The novelty is the application of the GPU techniques in the numerical model, making it possible to simulate the sediment transport and bed evolution in a high-resolution but efficient manner. The model was applied to two cases to evaluate bed evolution and the effects of the morphological changes on the flood patterns with high resolution. This indicated that the GPU-based high-resolution hydro-geomorphological model was capable of reproducing morphological processes. The computational times for this test case on the GPU and CPU were 298.1 and 4531.2 s, respectively, indicating that the GPU could accelerate the computation 15.2 times. Compared with the traditional CPU high-grid resolution, the proposed GPU-based high-resolution numerical model improved the reconstruction speed more than 2.0–12.83 times for different grid resolutions while remaining computationally efficient.

A GPU-Accelerated Shallow-Water Scheme for Surface Runoff Simulations

The capability of a GPU-parallelized numerical scheme to perform accurate and fast simulations of surface runoff in watersheds, exploiting high-resolution digital elevation models (DEMs), was investigated. The numerical computations were carried out by using an explicit finite volume numerical scheme and adopting a recent type of grid called Block-Uniform Quadtree (BUQ), capable of exploiting the computational power of GPUs with negligible overhead. Moreover, stability and zero mass error were ensured, even in the presence of very shallow water depth, by introducing a proper reconstruction of conserved variables at cell interfaces, a specific formulation of the slope source term and an explicit discretization of the friction source term. The 2D shallow water model was tested against two different literature tests and a real event that recently occurred in Italy for which field data is available. The influence of the spatial resolution adopted in different portions of the domain was also investigated for the last test. The achieved low ratio of simulation to physical times, in some cases less than 1:20, opens new perspectives for flood management strategies. Based on the result of such models, emergency plans can be designed in order to achieve a significant reduction in the economic losses generated by flood events.

Finite element solution of a new formulation for gas flow in a pipe with source terms

A new formulation for isothermal gas flow in a pipeline with gravity and friction source terms is introduced and solved by finite element methods using FEniCS open-source computing platform. The proposed formulation can be easily adapted to simulate different types of real boundary conditions and involves only one scalar unknown instead of two ones (typically mass flux and pressure or density). The main motivation for presenting the work is to find a compact formulation which is able to be solved with a high order accuracy scheme just by changing the type of elements. Also, it can be extended to gas network easily. In order to check the performance of the numerical method, it has been applied to several test problems. At first, with the purpose of finding the order of accuracy, two manufactured tests for different types of boundary conditions are solved. The results show that, for nodal finite elements of degree one, the order of accuracy in space is more than two and for time is between two and three. Then, two test cases with step boundary conditions are solved. The corresponding numerical results show that, even with large time-step and coarse mesh, they are in good agreement with those obtained with a fine mesh. Finally, a real pipe from the Spanish gas network is solved and the results compared with on-line measurements.

An implicit friction source term treatment for overland flow simulation using shallow water flow model

Aiming at resolving the numerical problems caused by the improper friction source term treatment when simulating overland flow using 2D shallow water flow models, a proposed implicit method computing the friction source term is developed in this work. The method is able to not only accurately evaluate the tricky thin overland flow by considering the flow velocity varying in a single time step, but also eliminates the redundant iterations for regular implicit approach through converting the implicit scheme to an explicit equation. It is therefore an improved one in terms of accuracy and efficiency comparing to the existing methods. Furthermore, it is an independent part and can be straightforwardly and universally incorporated into any FVM based shallow water flow model. Such performances are validated against three test cases involving theoretical and practical overland flow problems. The results prove the proposed method treating friction source term could simulate overland flow problems in a relatively more accurate and efficient way, and therefore can effectively help extend the applicability of the shallow water models reliably computing both the open channel and overland flows.

A new efficient implicit scheme for discretising the stiff friction terms in the shallow water equations

Abstract Discretisation of the friction terms to ensure numerical stability and accuracy remains to be challenging for the development of robust numerical schemes to solve the shallow water equations (SWEs), particularly for applications involving very shallow flows (e.g. overland flows and wet/dry fronts) over complex domain topography. The key challenge is to ensure relaxation of the flow towards an equilibrium state characterised by the balance between friction and gravity in a computationally efficient way. To overcome this numerical challenge, this paper proposes a novel approach for discretising the friction source terms in the SWEs in the context of an explicit finite volume method. The overall numerical scheme adopts the HLLC Riemann solver and surface reconstruction method (SRM) to explicitly discretise the flux and bed slope source terms. Whilst a fully implicit scheme is used to handle the friction source terms, solution to the implicit formulation is analytically derived to explicitly update the flow variables. Compared with the existing approaches, the proposed scheme effectively resolves the issue associated with stiff relaxation without necessity to use an iteration method and it supports efficient simulation using time steps controlled only by the Courant–Friedrichs–Levy (CFL) condition. The current friction term discretisation scheme is not coupled with flux and bed slope calculation and therefore may be readily implemented in any other explicit finite volume SWE models. After being successfully validated against two benchmark tests with analytical solutions, the resulting new SWE model is applied to reproduce a rainfall-flooding event in the Upper Lee catchment in the UK.

A new depth-averaged model for flow-like landslides over complex terrains with curvatures and steep slopes

Flow-like landslides are one of the most catastrophic types of natural hazards due to their high velocity and long travel distance. They travel like fluid after initiation and mainly fall into the ‘flow’ movement type in the updated Varnes classification (Hungr et al., 2014). In recent years, depth-averaged models have been widely reported to predict the velocity and run-out distance of flow-like landslides. However, most of the existing depth-averaged models present different shortcomings for application to real-world simulations. This paper presents a novel depth-averaged model featured with a set of new governing equations derived in a mathematically rigorous way based on the shallow flow assumption and Mohr-Coulomb rheology. Particularly, the new mathematical formulation takes into account the effects of vertical acceleration and curvature effects caused by complex terrain topographies. The model is derived on a global Cartesian coordinate system so that it is easy to apply in real-world applications. A Godunov-type finite volume method is implemented to numerically solve these new governing equations, together with a novel scheme proposed to discretise the friction source terms. The hydrostatic reconstruction approach is implemented and improved in the context of the new governing equations, providing well-balanced and non-negative numerical solutions for mass flows over complex domain topographies. The model is validated against several test cases, including a field-scale flow-like landslide. Satisfactory results are obtained, demonstrating the model's improved simulation capability and potential for wider applications.

A well-balanced scheme for the shallow-water equations with topography or Manning friction

We consider the shallow-water equations with Manning friction or topography, as well as a combination of both these source terms. The main purpose of this work concerns the derivation of a non-negativity preserving and well-balanced scheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, the scheme has to deal with vanishing water heights and transitions between wet and dry areas. To address such issues, a particular attention is paid to the study of the steady states related to the friction source term. Then, a Godunov-type scheme is obtained by using a relevant average of the source terms in order to enforce the required well-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights. A second-order well-balanced MUSCL extension is designed, as well as an extension for the two-dimensional case. Numerical experiments are performed in order to highlight the properties of the scheme.

Shallow-water simulations by a well-balanced WAF finite volume method: a case study to the great flood in 2011, Thailand

A well-balanced finite volume method for solving two-dimensional shallow water equations with weighted average flux (WAF) is developed in this work to simulate flooding. Friction source terms are estimated with a semi-implicit scheme resulting in an efficient numerical method for simulating shallow water flows over irregular domains, for both wet and dry beds. A wet/dry cell tracking technique is also presented for reducing computational time. The accuracy of these methods are investigated by application to well-studied cases. For practical purposes, the developed scheme is applied to simulate the flooding of the Chao Phraya river from Chai Nat to Sing Buri provinces in Thailand during October 13–17, 2011. The numerical simulations yield results that agree with the existing data obtained from the satellite images.