Depth-integrated Model Research Articles

Stokes drift and particle trajectories induced by surface waves atop a shear flow

Surface waves and currents are crucial to the mass transfer in the air-sea interaction as they can drive a variety of dynamical processes. How mass can be transported by surface waves and current coupling is addressed through a study of their induced motions of fluid parcels. To this end, a weakly nonlinear wavetrain is imposed on the background flow whose direction and magnitude are permitted to vary with water depth and second-order features of this configuration are investigated. A leading-order approximation to the Stokes drift is derived, correct to the second order in wave steepness, and applicable to an arbitrarily depth-dependent background flow. The reduced forms of the approximate Stokes drift are provided in a few limiting cases such as a current with an exponential profile or propagating in an orthogonal direction to the wave propagation. Novel features related to the Stokes drift and particle trajectories have been reported for the first time as a result of the rotation induced by the wave and current coupling. A non-vanishing component of the Stokes drift velocity and net-mean displacement of fluid parcels in the span-wise direction to the wave propagation are observed in the cases where a shear current propagates obliquely to the waves direction. A non-monotonic dependence on water depth of the stream-wise component of the Stokes drift is shown, and thereby the largest mass transport induced no longer occurs on the still water surface but some depth beneath. The non-monotonic behavior occurs beyond the regime of the near-irrotational assumption of wave-induced motions. It can also lead to the change of the signs for the stream-wise Stokes drift throughout the water column, and thus an overall cancellation of the integrated mass transport by waves over the water column, indicating that the depth-integrated models can likely lead to underestimated effects of the mass transport which is non-trivial at a local depth. The results from this study have far-reaching impact. The Stokes drift profile is a direct input to the parametrization of the surface waves forcing in ocean circulations and the obliquely propagating Stokes drift can be plausibly responsible for the formation of oblique Langmuir rolls to wave propagation in the open ocean.

Harnessing the heat below: Efficacy of closed-loop systems in the cooper basin, Australia

Transitioning to renewable energy is vital for combating climate change and reducing CO2 emissions. Geothermal energy, sourced from the earth's crust, presents a promising alternative. While shallow geothermal energy extraction grows steadily, tapping into deep Hot Dry Rock reservoirs poses challenges. Sustainable utilization of deep geothermal energy is crucial for large-scale electricity generation. Australia's Habanero enhanced geothermal system (EGS) projects faced obstacles due to complex fracture networks. Borehole heat exchangers (BHEs) offer a solution, circulating a working fluid without direct contact with geofluids. This study assesses coaxial BHEs' feasibility in the Habanero reservoir using a 2D axisymmetric model (2D-AM), comparing it with a simpler 1D depth-integrated model (1D-DIAM). Both models yield similar outcomes. Additionally, results indicated that the production temperature of the coaxial BHE exponentially decreased from 250 °C to 100 °C within 10 h, rendering energy recovery for electricity generation unsuitable after 10 h of operation. The study proposes an energy recovery cycle of 10 h followed by a 110-h shutdown. Injection temperatures and flow rates significantly impact production efficiency, while steel casing's thermal conductivity has minimal influence on heat exchanger performance.

OpenFOAM-avalanche 2312: depth-integrated models beyond dense-flow avalanches

Abstract. Numerical simulations have become an important tool for the estimation and mitigation of gravitational mass flows, such as avalanches, landslides, pyroclastic flows, and turbidity currents. Depth integration stands as a pivotal concept in rendering numerical models applicable to real-world scenarios, as it provides the required efficiency and a streamlined workflow for geographic information systems. In recent years, a large number of flow models were developed following the idea of depth integration, thereby enlarging the applicability and reliability of this family of process models substantially. It has been previously shown that the finite area method of OpenFOAM® can be utilized to express and solve the basic depth-integrated models representing incompressible dense flows. In this article, previous work (Rauter et al., 2018) is extended beyond the dense-flow regime to account for suspended particle flows, such as turbidity currents and powder snow avalanches. A novel coupling mechanism is introduced to enhance the simulation capabilities for mixed-snow avalanches. Further, we will give an updated description of the revised computational framework, its integration into OpenFOAM, and interfaces to geographic information systems. This work aims to provide practitioners and scientists with an open-source tool that facilitates transparency and reproducibility and that can be easily applied to real-world scenarios. The tool can be used as a baseline for further developments and in particular allows for modular integration of customized process models.

Role of three-dimensional vortex motions on horizontal eddies in an open-channel cavity

AbstractLateral cavities are a popular object of study in hydraulic research as they are widely found in rivers and hydraulic facilities and significantly impact flow patterns, sediment transport, and water quality in aquatic ecosystems. While the effects of open-channel cavities on various aspects including characteristics of three-dimensional structures have been extensively studied, the role of vertical flow structures in the cavity has not been focused on. This study examines the relationship between the shallowness parameter and the three-dimensional flow effects on the dynamics of an open-channel lateral cavity flow with horizontal vortex motion, comparing the conventional two-dimensional model (2DC), advanced depth-integrated models including general bottom velocity calculation method (GBVC) with the ability to consider vertical flow structures, simplified bottom velocity calculation method with the shallow water assumption (SBVC), a three-dimensional model (3DC), and experimental data. The comparison results demonstrate that the three-dimensional flow effect significantly impacts the velocity distribution and vortex evolution in the cavity. The GBVC model demonstrates a good agreement with the flow patterns by the 3DC model and experimental results in the cavity, whereas the 2DC and SBVC models are unsuitable for deep-water conditions.

Numerical Calculations for Curved Open Channel Flows with Advanced Depth-Integrated Models

This paper examines the effect of coupling several equations for vertical velocity profiles in a depth-integrated model to clarify the roles of three-dimensional flow structures minimizing the errors from the two-dimensional calculation model. In addition, this paper develops a numerical discretization method for the dispersion terms in the horizontal momentum equations. It is revealed that the use of the 2DC model underestimates the water surface elevation through the comparisons with the experimental datasets, advanced 2DC and 3DC results. The prediction of water surface elevation is considerably improved by taking into account of secondary flow effect with vorticity equations. It is clarified that the accuracy in predicting the water surface elevation is increased with increasing the degree of the function for vertical velocity profiles in advanced 2DC models or the number of vertical grids in 3DC model, while non-hydrostatic pressure and variation in vertical velocity have a second importance.

Depth-integrated models for three-dimensional flow over topography

Considered in this investigation is the three-dimensional, gravity-driven flow of a thin viscous fluid layer down an incline, and spreading over topography. Three depth-integrated models are presented and contrasted. These include an integral-boundary-layer model, a weighted-residual model and a hybrid model. A numerical solution procedure suited for solving three-dimensional flows is also proposed. Numerous simulations have been conducted using the models for various steady subcritical, and unsteady supercritical flows over several topographies. Good agreement among the three models was found. In addition, the models were also validated using experimental results, and, again, good agreement between the three models and with experiments was obtained.

Two-layer gravity currents of generalized Newtonian fluids

We examine gravity-driven two-layer flow of generalized Newtonian fluids. The two layers have different densities and constitutive laws, and the flow is assumed to be shallow and inertia-less. A depth-integrated model for flow over one-dimensional topography is derived with the volume fluxes written in terms of functions that depend only on the fluids’ rheology. The model enables two-layer flows of any combination of generalized Newtonian fluids to be computed without explicit knowledge of the velocity profile. For viscoplastic layers, the formulation provides a convenient way to determine the layer evolution without having to analyse the multiple yield surfaces that may occur. Motivated by the lubricated flow of ice sheets, we analyse the case in which the lower layer is relatively thin. The model reduces to a one-layer flow with an effective slip law that encapsulates the thickness and generalized Newtonian rheology of the lower layer. For flow over two-dimensional topography, a depth-integrated two-layer model cannot generally be derived because the direction of the shear stress varies across the lower layer. Progress is possible in the special cases that the lower layer is Newtonian or is relatively thin.

MULTI-LINEAR-ELEMENT DEPTH-INTEGRATED MODELS FOR FLOWS WITH A FREE SURFACE

In this study, we derive a new set of depth-integrated models for solving unsteady flows that satisfy the Euler equations and nonlinear free surface boundary conditions in the sigma -coordinates. One of the obvious differences between the present approach and the ones directly solving the 3D Euler equations using Finite Element Method (FEM) is that in the present model the vertical velocity and the pressure field are eliminated by integrating the continuity equation and vertical momentum equation, respectively. Therefore, the present models only solve the horizontal velocity components and the free surface displacement in the two-dimensional horizontal (2DH) space. The new models are also more advantageous in using fewer elements in the vertical direction while achieving better performance. At this stage, the capability and application of the new models in dealing with free surface wave propagation problems are addressed.

A new coupled depth-integrated model incorporating 3D DDA on debris flow with large boulders

Debris flow frequently occurs in mountainous regions and can cause serious damages to residents and structures in susceptible areas. Conventional debris flow simulation (DFS) with a depth-integrated model has been widely used to study debris flow behavior. In DFS, two-phase fluid of water and solid is considered. However, large boulders are often entrained in debris flows in practice. Obviously, the behaviors of debris flow with large boulders cannot be correctly described by DFS with two-phase fluid model. Therefore, a new coupling method for simulating debris flow with large boulders is developed by combining three-dimensional discontinuous deformation analysis (3D DDA) with conventional DFS. The dynamic behavior of those entrained boulders in debris flow can be solved by 3D DDA while the kinematic and mechanic behaviors of two-phase fluid are predicted by conventional DFS. An interactive module is introduced to address the interaction between solid motion in 3D DDA module and fluid field in DFS module. The imposed interaction forces from fluid on solid block include buoyancy, drag force and static pressure. Drag effect between adjacent blocks in debris flow is evaluated empirically to make the calculated drag force more realistic. For the block resistance to the debris flow, an equivalent solution is proposed by time-varying elevation of terrain node. Subsequently, the developed 3D DDA-DFS method is validated by several simple examples. Further, a case study is carried out to discuss the kinematic and mechanic behaviors of debris flow with large boulders based on a real catchment. The dam safety under different dam material properties is investigated when debris flow with large boulders happens. The simulation results suggest debris flow with large boulders can be more destructive than a pure debris flow. The developed 3D DDA-DFS method is a promising approach to study debris flow with large boulders.

A preliminary depth-integrated model for tsunamis propagation including water compressibility and seafloor elasticity

A model for tsunamis propagation is derived by coupling a weakly compressible liquid layer, standing for the ocean, to a viscoelastic solid layer of constant depth, representing the upper layer of the Earth. Below this layer, the Earth is assumed to be perfectly rigid. The resulting equations are averaged over the depth of the water layer for the liquid part and are integrated over the thickness of the solid layer for the solid part. The obtained system of equations is hyperbolic and admits an exact equation of energy conservation. The model is dispersive and includes an elastic branch, an acoustic branch and a gravity branch. The elasticity of the solid layer entails: (1) a reverse dispersion effect i.e. the phase velocity of the gravity branch decreases for small wave numbers if the wave number decreases; (2) an arrival time delay of the tsunami and (3) a leading negative phase i.e. a negative water elevation before the arrival of the main wave. All these effects are absent if the sea floor is assumed to be perfectly rigid. The arrival time delay due to elasticity comes in addition to the delay due to seawater compressibility. These effects are in agreement with previous works on tsunamis propagation. This preliminary model is a promising alternative approach for tsunamis propagation due to its ability to capture the main physical effects with a relatively simple mathematical structure of equations, which is easy to solve numerically.

Keynote lecture. A general two phase depth integrated model considering pore water pressure and dewatering

Debris flows can be considered a type of landslide with large velocities and long run-out distances. There are many types of debris flows, depending on the properties of the solid and fluid components of the mixture. The triggering and propagation of debris flows can be studied using a single 3D mathematical model. The computational cost can be very high because of their length, and depth-integrated models provide a good combination of accuracy and cost. Both types of models can be combined in the analysis, using 3D models for initiation and at singular points where more accuracy is wanted. As in a chain where the strength is never higher than that of the weaker link, we have to ensure that all the models are accurate enough in a joint model. This paper deals with a new depth-integrated model which can take into account the changes caused by dewatering in a debris flow.

A Numerical Method of Granular Flow for Hazard Prediction Based on Depth-Integrated Model and High-Resolution Algorithm

A Numerical Method of Granular Flow for Hazard Prediction Based on Depth-Integrated Model and High-Resolution Algorithm

A new two-layer two-phase depth-integrated SPH model implementing dewatering: Application to debris flows

Debris flows can be considered a type of landslide with large velocities and long run-out distances. There are many types of debris flows, depending on the properties of the solid and fluid components of the mixture. The triggering and propagation of debris flows can be studied using a single 3D mathematical model. The computational cost can be very high because of their length, and depth-integrated models provide a good combination of accuracy and cost. Both types of models can be combined in the analysis, using 3D models for initiation and at singular points where more accuracy is wanted. As in a chain where the strength is never higher than that of the weaker link, we have to ensure that all the models are accurate enough in a joint model. This paper deals with a new depth-integrated model which can take into account the changes caused by dewatering in a debris flow. An important limitation of existing two-phase models allowing different velocities of solid and water particles is that when water abandons the mixture, porosity decreases and tends to zero. Here, a two-layer model is introduced, including an unsaturated upper layer on top of a saturated layer.

Linear analysis on the non-hydrostatic pressure field of depth-integrated models

In the development of depth-integrated models, certain assumptions are made on the vertical profile of the horizontal velocity, and the pressure field is eliminated through vertical integration of the vertical momentum equation. In this study, analytical forms of the approximated non-hydrostatic pressure field are derived from five most representative depth-integrated models, which include three Boussinesq-type models, a weighted residual type model and a non-hydrostatic model. A Stokes wave-type Fourier analysis is then conducted to examine their capability in reproducing the vertical profile of the non-hydrostatic pressure field, which is compared with linear wave theory. We found the Boussinesq-type models are able to reproduce the non-hydrostatic pressure field from shallower water up to the kd values similar to that of concerning linear wave phase velocity. Specifically, the two-layer Boussinesq model generally outperforms the higher-order (μ4) Boussinesq model, especially for higher kd values. The weighted residual type model can reasonably capture the non-hydrostatic pressure near to the free surface even at high kd values, but undulations appear in the lower part of the water column where non-hydrostatic pressure is close to zero. The non-hydrostatic model shows rapid convergence of linear wave phase velocity by using few vertical layers, however, its capability in reproducing the non-hydrostatic pressure field is limited, especially at the highest layer interface, where nonphysical phenomena of reversed pressure could appear. This suggests that it is necessary to employ more layers when detailed information on the pressure field is needed, e.g. wave–structure interaction problems.

Two-Phase Two-Layer Depth-Integrated SPH-FD Model: Application to Lahars and Debris Flows

The complex nature of debris flows suggests that the pore-water pressure evolution and dewatering of a flowing mass caused by the high permeability of soil or terrain could play an essential role in the dynamics behavior of fast landslides. Dewatering causes desaturation, reducing the pore-water pressure and improving the shear strength of liquefied soils. A new approach to landslide propagation modeling considering the dewatering of a mass debris flow has drawn research attention. The problem is characterized by a transition from saturated to unsaturated soil. This paper aims to address this scientific gap. A depth-integrated model was developed to analyze the dewatering of landslides, in which, desaturation plays an important role in the dynamics behavior of the propagation. This study adopted an SPH numerical method to model landslide propagation consisting of pore-water and a soil skeleton in fully or partially saturated soils. In a two-phase model, the soil–water mixture was discretized and represented by two sets of SPH nodes carrying all field variables, such as velocity, displacement, and basal pore-water pressure. The pore-water was described by an additional set of balance equations to take into account its velocity. In the developed two-layer model, an upper desaturated layer and a lower saturated layer were considered to enhance the description of dewatering. This is the so-called two-phase two-layer formulation, which is capable of simulating the entire process of landslides propagation, including the large deformation of soils and corresponding pore-water pressure evolutions, where the effect of the dewatering in saturated soils is also taken into account. A dam-break problem was analyzed through the new and previously developed model. A flume test performed at Trondheim was also used to validate the proposed model by comparing the numerical results with measurements obtained from the experiment. Finally, the model was applied to simulate a real case lahar, which is an appropriate benchmark case used to examine the applicability of the developed model. The simulation results demonstrated that taking into account the effects of dewatering and the vital parameter of relative height is essential for the landslide propagation modeling of a desaturated flowing mass.

Study on numerical calculation of flow structures in a curved open channel with advanced depth-integrated models

A better understanding of flow structures distribution in rivers is crucial to determine the safety degree of rivers. A practical and reliable model is required to overcome the issue of the long computational time of a three-dimensional calculation model and the lack of computation detail of a two-dimensional calculation model for flow structures distribution simulation in rivers. This paper presented an advanced depth-integrated numerical calculation method called the bottom velocity calculation (BVC) to reproduce the flow structures in a curved open channel. BVC method is an integrated multiscale simulation of flows in rivers that can evaluate vertical distributions of velocities and bottom velocity distributions by introducing depth-averaged horizontal vorticity and horizontal momentum equations on a water surface. It has several models, such as simplified bottom velocity calculation (SBVC) with shallow water assumption and general bottom velocity calculation (GBVC) method without the assumptions. The advantages of the BVC method, including SBVC and GBVC models, are validated in this paper using experimental datasets of a curved open channel and compared to two-dimensional and three-dimensional models. The results show that the BVC method has good reproducibility to simulate the flow structures distribution in the channel.

SPH numerical modelling of landslide movements as coupled two-phase flows with a new solution for the interaction term

In this paper, the theoretical framework is a depth-integrated two-phase model capable of considering many essential physical aspects such as reproducing the propagation of debris flows with soil permeability ranging from high to low and considering the pore-water pressure evolution. In this model, the pore fluid is described by an additional set of depth-integrated balance equations in order to take into account the velocity of pore fluid. The model employs a frictional rheological law for the granular material, and the interstitial fluid is treated as a Newtonian fluid. A drag law describes the interaction between interstitial fluid and grains. The variables of permeability, porosity, and drag force are included in the governing equations to consider the interaction between the phases. This paper aims to extend a generalized two-phase depth-integrated model to enhance the description of the interaction between the two phases and their respective movements. It allows us to increase our understanding of the mechanism behind natural rapid landslides. To evaluate the developed approach, a set of dam-break problems has been performed. These simulations provide interesting information in simple and controlled situations on the landslide propagations with different degrees of soil permeability and the interaction between solid and fluid phases. The extended model has also been applied to simulate the dynamics of the Acheron rock avalanche, which is an appropriate benchmark to examine the applicability of the model to real cases.

Two-phase SPH–FD depth-integrated model for debris flows: application to basal grid brakes

In some cases, debris flow consists of low-permeability soils in which run-out distances, velocities and lateral spreading are highly influenced by pore-water pressures. Basal screens are energy dissipation structures designed to reduce these velocities and run-out distances. These structures consist of grids built on horizontal decks where basal pore pressures of debris flows are made equal to the atmospheric pressure, which increases basal friction. Once the debris flows exit the structure, basal pore pressure increases as the basal surface is impermeable again, but the values have been reduced. In order to model these phenomena, it is necessary to use debris flow simulation models considering two phases and pore-water pressures. Here, a depth-integrated two-phase smooth particle hydrodynamics (SPH) model, previously developed by the authors, is improved by including the vertical structure of pore-water pressure. This is done by incorporating finite-differences meshes associated to each SPH node that represents a solid particle. In this way, a higher precision is obtained in describing excess pore pressure along depth, including the influence of vertical consolidation, height variation and changes in basal permeability. The proposed model is applied to describe debris flows with two different ranges of soil permeability propagating over grid structures. The authors have used laboratory tests run at the Norwegian University of Science and Technology and studied the efficiency of the basal grids by comparing cases equipped by them and cases without any obstacle. The results obtained from the proposed numerical experiments suggest that the proposed two-phase coupled model is able to properly reproduce the behaviour of debris flows, describing more accurately the time–space evolution of pore-water pressures during the whole propagation stage, considering both propagation impermeable beds and pore pressure dissipation structures such as grids.

Comparison between the Lagrangian and Eulerian Approach for Simulating Regular and Solitary Waves Propagation, Breaking and Run-Up

The present paper places emphasis on the most widely used Computational Fluid Dynamics (CFD) approaches, namely the Eulerian and Lagrangian methods each of which is characterized by specific advantages and disadvantages. In particular, a weakly compressible smoothed particle (WCSPH) model, coupled with a sub-particle scale (SPS) approach for turbulent stresses and a new depth-integrated non-hydrostatic finite element model were employed for the simulation of regular breaking waves on a plane slope and solitary waves transformation, breaking and run-up. The validation of the numerical schemes was performed through the comparison between numerical and experimental data. The aim of this study is to compare the two modeling methods with an emphasis on their performance in the simulation of hydraulic engineering problems.

Characteristics and Dynamic Process Modeling of the Rainfall-Induced Landslide on August 21, 2020 in Hanyuan County, China

On August 21, 2020, a landslide occurred in Zhonghai village of Hanyuan County, Sichuan Province, China. The landslide is triggered by two successive rounds of heavy rainfall. This landslide can be clearly divided into an initial landslide and a main landslide. The main landslide is activated by the intense impact and overloading of the initial landslide. The depth-integrated continuum method is adopted to simulate the dynamic process. Due to the complicated failure process, it is found that there is no proper unified parameters in Coulomb model which could well reproduce the two successive landslides. It implies that the dynamic process of landslides is highly associated with the characteristics of sliding bodies. Here, an implement of variable frictional coefficients for different parts is proposed and the parameters are calibrated. It is demonstrated that results from numerical modeling match well with the field investigation. The complicated landslides in two different stage can both be efficiently revealed by depth-integrated continuum modeling.