Solution Of Shallow Water Equations Research Articles

On the existence of flux as a function of the surface elevation for long wave solution of shallow water equations

The shallow water equations in mechanics of fluids, govern the motion of a shallow layer of water over a fixed impervious bed. In this paper, the bed form is assumed to be rough and horizontal, and the motion of water is assumed to be of the long wave type (Lamb [1], pp. 254-256) such that the free surface has a gradually varying propagating profile. Gravitation permits such motion but is resisted by the turbulence generated by the bed friction. A model of the governing equations based on the Reynolds averaged Navier-Stokes equations has recently been given by Bose [2], which is highly nonlinear. A heuristic approach of numerically solving the equations for the modified long waves is also presented in that article, by assuming that the horizontal flux across a section of flow is some function of the free surface elevation alone. This key assertion is analysed in this article and proved to hold provided some boundedness criteria are satisfied by the flux gradients. The theory is apparently applicable to find appropriate boundedness conditions on the flux of flow for numerically solving long wave equations in the case of other models for long wave propagation as well.

2-D Modeling of hyperconcentrated fluid flow in curvilinear coordinates: dam break study

The hyperconcentrated fluid flow occurs as a result of heavy rainfall, during which a large amount of sediments from the upstream basin is washed away and suddenly increases the flow concentration of the alluvial channels. The stresses exerted by this type of fluid on the bed and body of the stream/river and related structures such as dam lead to the failure them and cause many human and financial losses. One of the important topics in the simulation of dam break caused by non-Newtonian fluid flow is the modeling of frictional stresses. In this research, after collecting several relationships to model the coefficient of friction loss of non-Newtonian fluid, a two-dimensional model was developed based on the numerical solution of shallow water equations in curvilinear coordinates to simulate hyperconcentrated flow. The results of the validation of the model were presented by comparing the measurement data of the suddenly complete dam break caused by the non-Newtonian fluid flow in the form of graphs, which all emphasize the accuracy of the developed model. It was also shown that for a suddenly complete dam break, with an increase in fluid volume concentration from 13.8 to 36.4%, the flow depth at the failure site increases by 18.8%. Next, asymmetric two-dimensional partial dam break of non-Newtonian fluid was simulated and compared with the results of Newtonian fluid. The results showed that the maximum flow velocity in the center of the fracture wall for the non-Newtonian fluid with a concentration of 32.2% is less than half of the maximum velocity of the Newtonian fluid.

Parallelization of numerical solutions of shallow water equations by the finite volume method for implementation on multiprocessor systems and graphics processors

An overview of approaches to parallelization of grid-based numerical methods for solving shallow water equations for multiprocessor systems and graphics processors is presented. A multithreaded approach for shared-memory computing systems implemented on the basis of the OpenMP programming interface and a geometric decomposition approach with message-passing using the MPI library for distributed-memory computers are described. Multithreading for programming GPUs based on the OpenACC software interface is considered. For the COASTOX-UN system of two-dimensional modeling of hydrodynamics, sediment and radionuclide transport in river systems and coastal areas of the seas, the parallelization of its hydrodynamic model COASTOX-HD was carried out. In the developed numerical model, the shallow water equations are solved by finite-volume numerical methods on unstructured computational grids with triangular cells of variable size. The parallelization is implemented using a hybrid MPI+OpenACC approach targeting multiprocessor systems and GPUs. For multiprocessor computers, geometric decomposition and MPI-based messaging are used, and for GPUs, multithreading is implemented using OpenACC directives. The performance of the developed parallel hydrodynamic model was evaluated during the calculation of typical problems of hydrodynamics of shallow water bodies, river flood, and tsunami wave run-up on the coast on a Dell Precision Workstation 7920 multi-core workstation with two 20-core Intel Xeon Gold 6230 processors and NVIDIA Quadro RTX 5000 and NVIDIA GeForce RTX 3080 graphics cards. It is shown that the developed model has significantly accelerated the simulation on the considered multiprocessor system and the considered GPUs. The acceleration on GPUs depends on the size of the computational grid, increasing to saturation with an increase in the number of grid cells. It is established that for the developed parallel model, whose numerical schemes are related to algorithms with low computational intensity, the memory bandwidth of the NVIDIA architecture GPUs is a more important limiting factor of acceleration than their performance.

Physical information-fused deep learning model ensembled with a subregion-specific sampling method for predicting flood dynamics

The computational prediction of flood dynamics is a long-standing problem in hydrodynamics and hydrology. Until now, conventional numerical models based on shallow water equations (SWEs) have been the dominant approaches. Here, we show that by combining a data-driven methodology with some general physical governing equations, the resulting deep learning model can enable the prediction of the long-term dynamics of dam-break floods with satisfactory accuracy. For this purpose, we propose a deep learning model using a physics-informed neural network (PINN) embedded in SWEs. Training data are reconstructed from prior-known partial numerical solutions and a novel subregion-specific sampling method that adapts well to the violent variations in flood surfaces. Various traditional cases of 2D dam-break floods are used to illustrate and validate the effectiveness of the proposed model. The results show that the proposed PINN-embedded model predicts the dam-break flood dynamics well, as observed in the numerical simulations based on SWEs, and outperforms the pure data-driven multilayer perceptron (MLP) model and a similar model with the traditional random sampling method. We also demonstrate that the PINN-embedded model improves the computational efficiency by over 3 ∼ 4 orders of magnitude compared to the traditional 2D numerical solution of SWEs. Our study showcases how numerical simulation and deep learning techniques can be coupled to investigate the details of dam-break floods.

Numerical extensions to incorporate subgrid corrections in an established storm surge model

ABSTRACT Inundation models represent coastal regions with a grid of computational points, often with varying resolution of flow pathways and barriers. Models based on coarse grid solutions of shallow water equations have been improved recently via the use of subgrid corrections, which account for information (ground surface elevations, roughness characteristics) at smaller scales. In this work, numerical approaches of an established storm surge model are extended to include subgrid corrections. In an attempt to maintain continuity with existing users and results, model extensions were limited to those needed to provide basic subgrid capabilities, and included two major additions. First, a finite volume method is used to incorporate corrections to the mass and momentum equations using high-resolution ground surface elevations. Second, the no-slip condition imposed on the B-grid wet/dry interface in the model is modified to a slip condition to enable flows in channels with widths comparable to cell size. Numerical results demonstrate these numerical extensions can significantly enhance the accuracy of the model’s predictions of coastal flooding, with low additional computational cost.

Self-similar solutions of shallow water equations with porosity

Simulated free surface transients in periodic urban layouts have been reported to be self-similar in the space-time domain when averaged on the scale of the building period. Such self-similarity is incompatible with the head loss model formulae used in most porosity-based shallow water models. Verifying it experimentally is thus of salient importance. New dam-break flow laboratory experiments are reported, where two different configurations of idealized periodic buildings layouts are explored. A space-time analysis of the experimental water level fields validates the self-similar character of the flow. Simulating the experiment using the two-dimensional shallow water model also yields self-similar period-averaged flow solutions. Then, the Single Porosity (SP), Integral Porosity (IP) and Dual Integral Porosity (DIP) models are applied. Although all three models behave in a similar fashion when the storage and connectivity porosities are close to each other, the DIP model is the one that upscales best the refined 2D solution.

Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method

A multi-layer hydrostatic shallow-water model was developed in the present study. The layer-integrated hydrostatic nonlinear shallow-water was solved with θ time integration and the least-squares finite element method. Since the least-squares formulation was employed, the resulting system of equations was symmetric and positive–definite; therefore, it could be solved efficiently by the preconditioned conjugate gradient method. The model was first applied to simulate the von Karman vortex shedding. A well-organized von Karman vortex street was reproduced. The model was then applied to simulate the Kuroshio current-induced Green Island vortex street. A swirling recirculation was formed and followed by several pairs of alternating counter-rotating vortices. The size of the recirculation, as well as the temporal and spatial scale of the vortex shedding, were found to be consistent with ADCP-CDT measurements, X-band radar measurements, and analysis of the satellite images. It was also revealed that Green Island vortices were affected by the upstream Orchid Island vortices.

Reduced Order Models for Optimal Flow Control

AbstractData assimilation models allow to fill the gap between numerical simulations and experimental data. Optimal control problems governed by parametrized partial differential equations PDE(µ) is suited for this kind of application, where you want to track problem solutions towards known quantities, given by data collections or previous knowledge. Still, the computational effort increases when one has to deal with nonlinear time‐dependent governing equations. Reduced order methods are an effective approach to solve data assimilation problems in a reliable and faster way. We apply the POD‐Galerkin methodology in environmental marine sciences where different parameters describe several physical configurations. We present a nonlinear time‐dependent tracking problem for velocity‐height solutions of shallow water equations.

A note on the Riemann problem for shallow water equations with discontinuous topography

In a recent paper LeFloch and Thanh (2011) constructed the Riemann solutions for shallow water equations with discontinuous topography, where the stationary wave is determined by solving steady state equations. Here we consider the case when the original steady equations do not have a solution automatically. We thus construct the supplement Riemann solutions and prove that a backward shock wave emits to decelerate the flow which makes the steady equations solvable. Global entropy condition is imposed subsequently to select the physical relevant solution which states that the defined energy function should not only increase but increase to the maximum rate. Numerical results are given to verify our analysis.

Numerical Assessment of Criteria for Mesh Adaptation in the Finite Volume Solution of Shallow Water Equations

Numerical Assessment of Criteria for Mesh Adaptation in the Finite Volume Solution of Shallow Water Equations

Analytical Solution of Shallow Water Equations for Ideal Dam-Break Flood along a Wet-Bed Slope

AbstractThe existing analytical solutions of dam-break flow do not consider simultaneously the effects of wet downstream bottom and bed slope on the dam-break wave propagation. In this study, a new...

An exact source-term balancing scheme on the finite element solution of shallow water equations

In this paper, a numerical model is presented based on a semi-discrete Streamline Upwind Petrov–Galerkin scheme for solving the two-dimensional shallow water equations with arbitrary bed topography and wetting–drying fronts. The present finite element method is implemented for moving-boundary problems on fixed numerical meshes. In order to satisfy the C-property for both still water and dry regions, a new exact source-term balancing method is presented within the finite element framework. The formulation is stabilized using two different stabilization terms, and a discontinuity-capturing scheme is used to simulate steep wave fronts. The second-order scheme is applied for both temporal and spatial discretization. The proposed mathematical balancing method is verified through well-known test cases including the traditional dam break problem, evolution of a dam break wave over an obstacle, and oscillation of a bead of water in a parabolically-shaped basin.

The problem of lifting a symmetric convex body from shallow water

Abstract A problem of a plane-parallel flow induced by vertical lifting of a symmetric convex body partly immersed in water filling a rectangular prismatic channel with a horizontal bottom is solved within the framework of the shallow water theory. The body width coincides with the channel width, its flat side surfaces are perpendicular to the channel bottom, and its lower downward-convex surface has sufficiently small curvature and is completely immersed in water at the initial time. The liquid flow is obtained analytically in the region adjacent to the lower surface of the body and by means of the numerical solution of shallow water equations by the second-order CABARET (compact accurately boundary-adjusting high-resolution technique) scheme outside this region. Equations that define the motion of the boundary line between the liquid and the lower surface of the body are derived. It is shown that the form of these equations is determined by the sign of the spatial derivative of pressure on this boundary line. Numerical results demonstrating liquid lifting behind the body leaving the water medium are presented.

Implicit discontinuous Galerkin scheme for shallow water equations

The numerical solutions of shallow water equations are presented with implicit discontinuous Galerkin (IDG) method. The motivations for the development of an implicit scheme is stated. Details of the developed model is described including the spatial and temporal discretization, the approximate Riemann solver, and the slope limiter. For the validation of the model, the channel transition flow of contraction and expansion are carried out. As the last case study, the classical dam-break flow is simulated. In all cases, linear triangular meshes are employed and the implicit backward Euler time integration scheme is used. As an approximate Riemann solver, the Roe numerical flux is employed and a van Albada type gradient-reconstruction type slope limiter was applied. Good agreement was observed with experimental observations and exact solutions in all case studies.

Exact solutions of shallow water equations for the water oscillation problem in a simulated basin and their implementation in verifying numerical algorithms

Exact solutions of shallow water equations for the water oscillation problem in a simulated basin and their implementation in verifying numerical algorithms

Inertial Oscillations and the Galilean Transformation

This paper presents a general solution of shallow-water equations on the f-plane. The solution describes the generation of inertial oscillations by wind-pulse forcing over the background of currents arbitrarily changing in time and space in a homogeneous fluid. It is shown that the existence of such a complete solution of shallow-water equations on the f-plane is related to their invariance with respect to the generalized Galilean transformations. Examples of velocity hodographs of inertial oscillations developing over the background of a narrow jet are presented which explain the diversity in their forms.

Dispersion Characteristics and Circulation Associated with Boreal Summer Westward-Traveling Mixed Rossby–Gravity Wave–Like Disturbances

Abstract An algorithm has been developed to track synoptic-scale, westward-traveling mixed Rossby–gravity (MRG) wave–like disturbances with a cross-equatorial component. Applied to space–time-filtered meridional wind data, this algorithm finds locations with Gaussian-shaped wind structures stated in the solutions of shallow-water equations (SWEs). Based on 850-hPa meridional wind from the global National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis (CFSR) dataset, local and instantaneous wave properties including the occurrence time, wavenumber, intrinsic frequency, and magnitude were examined. It was found that these low-level MRG wave–like disturbances can be classified into a longer-wavelength group and a shorter-wavelength group. While most waves identified in the eastern Pacific give longer wavelengths, disturbances in the western Pacific tend to have a wider range of wavenumbers. Composite analysis revealed that east of ~140°E, low-level disturbances are characterized by cross-equatorial wind anomalies with alternating signs, thus consistent with the MRG wave solution. West of ~140°E, they appear as northeast–southwest-tilted eddies that propagate northwestward. Examination of their energetics suggests that such a tilting structure is favorable to the maintenance of these transients because of the meridional shear of background zonal wind west of ~140°E in the off-equatorial Pacific. Farther east, the confluent nature of the low-level background flow plays a dominant role in maintaining the MRG wave–like disturbances because of barotropic conversion. Finally, there is evidence of downward energy dispersion in the mid- to upper levels, suggesting that the upper-level wave activity might be important in triggering these low-level waves in the Pacific basin.

Applications of Coupled Explicit–Implicit Solution of SWEs for Unsteady Flow in Yangtze River

In engineering practice, the unsteady flows generated from the operation of hydropower station in the upstream region could significantly change the navigation system of waterways located in the middle-lower reaches of the river. In order to study the complex propagation, convergence and superposition characteristics of unsteady flows in a long channel with flow confluence, a numerical model based on the coupling of implicit and explicit solution algorithms of Shallow Water Equations (SWEs) has been applied to two large rivers in the reach of Yangtze River, China, which covers the distance from Yibin to Chongqing located upstream side of the Three Gorges Dam. The accuracy of numerical model has been validated by both the steady and unsteady flows using the prototype hydrological data. It is found that the unsteady flows show much more complex water level and discharge behaviors than the steady ones. The studied unsteady flows arising from the water regulation of two upstream hydropower stations could influence the region as far as Zhutuo hydrologic station, which is close to the city of Chongqing. Meanwhile, the computed stage–discharge rating curves at all observation stations demonstrate multi-value loop patterns because of the presence of additional water surface gradient. The present numerical model proves to be robust for simulating complex flows in very long engineering rivers up to 400 km.

Traveling wave solutions for shallow water equations

An extended homogeneous balance method is suggested in this paper. Based on computerized symbolic computation and the homogeneous balance method, new exact traveling wave solutions of nonlinear partial differential equations (PDEs) are presented. The shallow-water equations represent a simple yet realistic set of equations typically found in atmospheric or ocean modeling applications, we consider the exact solutions of the nonlinear generalized shallow water equation and the fourth order Boussinesq equation. Applying this method, with the aid of Mathematica, many new exact traveling wave solutions are successfully obtained.

Lie Symmetries and Exact Solutions of Shallow Water Equations with Variable Bottom

Abstract In the present paper, Lie symmetries of nonlinear shallow water equations with variable shapes of the bottom that include horizontal, inclined plane and a parabolic bottom are obtained. Exact particular solutions of the governing system are then obtained using the invariance of the system under these symmetries using Lie’s method. The evolutionary behaviour of the C 1 $${C^1}$$ discontinuity wave, influenced by the amplitude of the discontinuity wave and the geometry of the bottom, is discussed in detail and some contrasting observations are made.