Shallow Water Equations Research Articles

Development of a Semi-Analytical Dynamic Force Model

A moving water mass generates force which is exerted on its moving path. Cyclone generated storm surge or earthquake generated tsunami are specific examples of moving water mass generated force along the coasts. In addition to human lives, these moving water masses cause severe damages to the coastal infrastructure due to tremendous force exerted on these structures. To assess the damage on these infrastructures, an essential parameter is the resultant force exerted on these structures. To evaluate the damages, there is hardly any quantitative method available to compute this force. In this paper we have developed a semi-analytical model, named as Dynamic Force Model (DFM), by using Variational Iteration Method to compute this force. We have derived the governing equation on the basis of Saint-Venant equations which are basically 1D shallow water equations derived from the Navier-Stokes equations. DFM is verified, calibrated, validated, and applied in Bangladesh coastal zone to compute dynamic thrust force due to tropical cyclone SIDR.

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

Potential Sunda Strait tsunami hazard due to the current deformation of Anak Krakatau

Abstract The tsunami Sunda Strait event at 2018 is a prove that volcanic activity may trigger a devastating and unpredictable tsunami. Several findings indicate that this catasrophe generated by flank collapse caused by the instability of Gunung Anak Krakatau (AK). Other processes of volcanic activity that cause tsunamis include pyroclastic flows, subaerial and submarine landslides, underwater explosions, blasts, and caldera collapse. In the 1883, Krakatau Mount produced the largest tsunami event, and it is apparent that AK is one of threats in Sunda Strait that may potentially produce tsunami in the future since it is still active until now. Some latest research shows there are two submarine landslide threats near Gunung Anak Krakatau; in the northeast part of AK and an elliptical landslide source in the west part of AK with the estimated landslide volume about 0.014 km3 and 0.6 km3, respectively. Those threats will be simulated by shallow water equation model to obtain the tsunami wave height near the source and at crucial areas along Sunda Strait as well as its tsunami time arrival. The first scenario is sourced from a new deposit from the 2018 eruption, and the latter is a change of slopes bathymetry due to volcanic or seismic activity. It is observed that the highest elevation of tsunami from the first scenario reached only 2.5 cm at Tanjung Lesung, Banten and 80 cm at the nearest island, Panjang Island. The second scenario has the height of 50 cm at Labuan, Banten and 70 cm at Panjang Island. The tsunami time travel at the surrounding islands is ranging between 0.6 - 5.8 minutes (scenario 1) and 0.7-4.9 minutes (scenario 2). Both scenario reached the inexpensive device of sea level monitoring at the east side of Rakata after 5 minutes and the height is 30 cm and 1.6 cm respectively for scenario 1 and 2.

Zeitlin Truncation of a Shallow Water Quasi‐Geostrophic Model for Planetary Flow

AbstractIn this work, we consider a Shallow‐Water Quasi Geostrophic equation on the sphere, as a model for global large‐scale atmospheric dynamics. This equation, previously studied by Verkley (2009, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), possesses a rich geometric structure, called Lie‐Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long‐time simulations of this equation. The method develops in two steps: first, we construct an N‐dimensional Lie‐Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite‐dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation‐induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section

This work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

Estimation of soil infiltration coefficient in the furrow irrigation using the combination of the hydrodynamics model and Richard's equation

Present study optimizes the coefficients of the Kostiakov-Lewis infiltration Eq. and estimates water infiltration in the soil with the help of simulation of the advancing phase using the numerical solution of the Saint-Venant equations. In order to validate the numerical model, two series of field data measured in two regions of Iran were used and the results of the simulation were compared with actual field results. Also, using the Richards' Eq., the HYDRUS-1D software, the Hydrus-2D software, and a combination of the Hydrodynamics model with each of the mentioned methods the infiltration simulation and prediction of coefficients of the Kostiakov-Lewis infiltration Eq. were provided. The criteria for the accuracy of the methods were considered the comparison between the infiltration volumes of the water (calculated from each method) and the actual infiltration volume of the water in Furrow irrigation. The results illustrated that the combination of the Hydrodynamics model with each of the HYDRUS-1D, HYDRUS-2D software and the numerical solution of the Richards' Eq. increased the accuracy of the estimation process of the coefficients of the Kostiakov-Lewis's infiltration equation and the infiltration volume. So, the best and worst methods for simulating water infiltration through the soil has been introduced based on the less error between numerical and observed values of the infiltration volume as the combined method of the Hydrodynamics- HYDRUS 2D with NRMSE = 17.62% and numerical solution of the Richard's Eq. with NRMSE = 36.02%, respectively. The results illustrated that the error of the Hydrodynamics- HYDRUS-2D compared with the numerical solution of the Richard's Eq. (t(10) = −3.018) was significantly less value.

Bathymetry reconstruction from experimental data using PDE-constrained optimisation

Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming, expensive and inaccurate. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flow as constraint. In many cases, the shallow water equations can be used to describe the flow. While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements. This paper shows that the approach can, at least qualitatively, reconstruct a Gaussian-shaped bathymetry in a wave flume from measurements of the free surface level at up to three points. Achieved normalised root mean square errors (NRMSE) are in line with other approaches.

Parametric analysis of the performance of the SPH solutions of unsteady free-surface flow

ABSTRACT In this article, the performances of the SPH method to solve Shallow Water Equations SWEs with three investigation parameters were studied, such as the type of kernel functions, namely: cubic spline, Gaussian and quintic spline kernels, the number of particles used and the stabilization terms injected, specifically: Lax Friedrichs flux, artificial viscosity and two shocks Riemann solver. Three benchmarking tests make the subject of unsteady free surface flow in this study. It is 1D typical dam-break on wet and dry bottom; 2D partial dam-break on a dry floodplain and 2D partial dam-break on channel with 90° bend. The analysis of the different errors between the numerical and analytical solutions and/or the experimental data shows that the SPH method gives reliable values with the selected optimal parameters which are the cubic kernel function and the artificial viscosity term. The increase in the number of particles increases the precision but also the calculation time.

Numerical investigation for water flow in an irregular channel using Saint-Venant equations

Given recent flood events in Indonesia, understanding the hydraulic dynamics in open channels, mirroring real-world drainage and river scenarios, has become paramount. To address this, our research employs a numerical model to simulate water flow in both regular and irregular open channels. Specifically, we utilize the Saint-Venant Equations to explore fluid flow evolution in irregular channels. This model is solved numerically using a staggered finite volume method. We complement our research with laboratory experiments and validate our numerical model against the data collected. Furthermore, we cross-reference our numerical results with those from HEC-RAS to examine the robustness and accuracy of our model in predicting water levels and velocities under varying conditions. To deepen our understanding, we conduct sensitivity analyses that offer valuable insights into the core factors influencing water levels and velocity. This knowledge provides a foundation for practitioners to normalize rivers.

Nonlinear processes in tsunami simulations for the Peruvian coast with focus on Lima and Callao

Abstract. This investigation addresses the tsunami inundation in Lima and Callao caused by the massive 1746 earthquake (Mw 9.0) along the Peruvian coast. Numerical modeling of the tsunami inundation processes in the nearshore includes strong nonlinear numerical terms. In a comparative analysis of the calculation of the tsunami wave effect, two numerical codes are used, Tsunami-HySEA and TsunAWI, which both solve the shallow water (SW) equations but with different spatial approximations. The comparison primarily evaluates the flow velocity fields in inundated areas. The relative importance of the various parts of the SW equations is determined, focusing on the nonlinear terms. Particular attention is paid to the contribution of momentum advection, bottom friction, and volume conservation. The influence of the nonlinearity on the degree and volume of inundation, flow velocity, and small-scale fluctuations is determined. The sensitivity of the solution concerning the bottom friction parameter is also investigated, showing the effects of nonlinearity processes in the inundated areas, wave heights, current velocity, and the spatial structure variations shown in tsunami inundation maps.

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Numerical simulations of dam-break flows of viscoplastic fluids via shallow water equations

This paper presents simulations of dam-break flows of Herschel–Bulkley viscoplastic fluids over complex topographies using the shallow water equations (SWE). In particular, this study aims to assess the effects of rheological parameters: power-law index (n), consistency index (K), and yield stress (τc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _{c}$$\\end{document}), on flow height and velocity over different topographies. Three practical examples of dam-break flow cases are considered: a dam-break on an inclined flat surface, a dam-break over a non-flat topography, and a dam-break over a wet bed (downstream containing an initial fluid level). The effects of bed slope and depth ratios (the ratio between upstream and downstream fluid levels) on flow behaviour are also analyzed. The numerical results are compared with experimental data from the literature and are found to be in good agreement. Results show that for both dry and wet bed conditions, the fluid front position, peak height, and mean velocity decrease when any of the three rheological parameters are increased. However, based on a parametric sensitivity analysis, the power-law index appears to be the dominant factor in dictating fluid behaviour. Moreover, by increasing the bed slope and/or depth ratio, the wave-frontal position moves further downstream. Furthermore, the presence of an obstacle is observed to cause the formation of an upsurge that moves in the upstream direction, which increases by increasing any of the three rheological parameters. This study is useful for an in-depth understanding of the effects of rheology on catastrophic gravity-driven flows of non-Newtonian fluids (like lava or mud flows) for risk assessment and mitigation.Graphical abstract

Numerical Implementation of Various Boundary Conditions on the Shallow Water Equation using a Novel Conformal and Non-conformal Finite Element Methods

Numerical Implementation of Various Boundary Conditions on the Shallow Water Equation using a Novel Conformal and Non-conformal Finite Element Methods

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Physics-informed neural network for diffusive wave model

The diffusive wave model (DWM), a nonlinear second-order simplified form of the shallow water equation, has been widely used in hydraulic, hydrologic and irrigation engineering. Solution of the forward problem of the DWM can be utilized to predict evolution in water levels and discharge. Solution of its inverse problem allows for the identification of crucial parameters (such as Manning coefficient, rainfall intensity, etc.) based on observations. This paper applied the physics-informed neural network (PINN) with novel improvements to solve the DWM for both forward and inverse problems. In the forward problem, compared to traditional numerical methods, PINN was able to predict the evolution at any location. In the inverse problem, PINN provided a simple and efficient solution process. In order to overcome the gradient explosion in the training process caused by the characteristics of the DWM, the stop-gradient technique was adopted to train the neural network. To improve the estimation of DWM parameters, the concept of time division was developed, and a new network structure was proposed. To verify the effectiveness of PINN and its improved algorithm for DWM, seven examples were simulated. The PINN solutions for forward problems were compared with the results obtained by classical numerical methods, while the correct rainfall pattern was identified for the inverse problem.

High order well-balanced asymptotic preserving IMEX RKDG schemes for the two-dimensional nonlinear shallow water equations

In this paper, a high order well-balanced asymptotic preserving scheme is presented for the two-dimensional nonlinear shallow water equations over variable bottom topography in all Froude number regimes. To obtain the well-balanced property, the system is first reformulated as a new form by introducing an auxiliary parameter. The flux is then split into a linear stiff part to be treated implicitly and a nonlinear non-stiff part to be treated explicitly, and the source term is treated explicitly. An implicit-explicit Runge-Kutta discontinuous Galerkin scheme is designed for solving the equations. The proposed scheme can be proved to be well-balanced, asymptotic preserving and asymptotically accurate. Finally, several numerical tests are carried out to validate the performance of our proposed scheme.

A 2D shallow water flow model with 1D internal boundary condition for subgrid-scale topography

In this work, a dynamic internal boundary condition is used as subgrid model in a two-dimensional (2D) model based on the shallow water equations in order to model narrow regions in the domain. In this way, computational savings are sought, since it is not necessary to discretize these regions with cells of reduced size. The new internal boundary condition simplifies other works where 1D–2D coupled models were presented, since the 1D model is a subgrid for the 2D mesh, so the coupling between both models is simple and direct. The coupling is performed using mass conservation, simplifying the calculation in the transfer between both models. Test cases are studied to validate the implemented boundary condition, and a mountain catchment as a realistic case. The results obtained with a fully 2D mesh and a 2D mesh with rills in narrow regions are very similar, with a large reduction in computational cost when using rills, both in test cases and in the realistic case. Thus, the use of the implemented internal boundary condition is an effective tool to study regions with narrow regions by reducing the computational cost with little loss of accuracy in the results.

Open‐source stand‐alone version of atmospheric model Aeolus 2.0 software

AbstractIn this discourse, we present the unveiling of an open‐source software package designed to facilitate engagement with the atmospheric model, Aeolus 2.0. This particular iteration stands as a self‐contained model of intermediate complexity. The model's dynamical core is underpinned by a multi‐layer pseudo‐spectral moist‐convective Thermal Rotating Shallow Water (mcTRSW) model. The pseudo‐spectral problem‐solving tasks are handled by the Dedalus algorithm, acknowledged for its spin‐weighted spherical harmonics. The model captures the temporal and spatial evolution of vertically integrated potential temperature, thickness, water vapour, precipitation, and the intricate influence of bottom topography. It comprehensively characterizes velocity fields in both the lower and upper troposphere, employing resolutions spanning a spectrum from the smooth to the coarse, enabling the exploration of a wide range of dynamic phenomena with varying levels of detail and precision.

Numerical Tools For Wave Overtopping At Rubble Mound Breakwaters With Submerged Berms

In the era of climate change, the adaptation of existing coastal structures as a response to potential increasing loads, has become a trending topic in coastal engineering. The combination of sea level rise, storm surge, tides and waves is going to dramatically increase the wave overtopping. In this context, this research is focused on the evaluation of wave overtopping for conventional rubble mound breakwaters, modified by the introduction of a submerged berm. Recent studies illustrated how the introduction of emerged berms on the seaward slope reduces the wave overtopping. The goal then becomes to turn the spotlight on the submerged berms, whose effectiveness has yet to be attested, although it has proven useful in reducing wave loads on the armour layer and seabed pressure under the structure. Therefore, the reliability of two different numerical models in detecting overtopping phenomenon have been assessed. IHFOAM and SWASH have been used (see Figure 1). The first one can solve both Reynolds Averaged Navier Stokes and the Volume Averaged Reynolds Averaged Navier–Stokes equations. The second one solves the non-linear shallow water equations with a non-hydrostatic pressure term, representing a simplified form of the Navier-Stokes equations, with associated limited computational cost. The numerical models have been validated based on experimental tests carried out in a wave flume at the University of L’Aquila.