Shallow Water Flow Research Articles

Numerical Simulation of Two-Layer Shallow Water Flows. Exchange Flow in the Lombok Strait

Numerical Simulation of Two-Layer Shallow Water Flows. Exchange Flow in the Lombok Strait

Manning Coefficient Effect on the Shallow Water Flow Equation

This work will seek to solve shallow water equations, using numerical integration methods, to solve partial differential equations (PDEs) and subsequently present a brief physical interpretation of the solutions. The derivation of the equations is demonstrated, along with the explanation of each physical quantity present. The discretization of non-conservative shallow water equations is carried out using the method of finite differences advanced in time and central in space (FTCS), in explicit form. With the deductions, different Manning coefficients will be addressed in the solutions, verifying their effects.

Randomly Supported Variations of Deterministic Models and Their Application to One-Dimensional Shallow Water Flows

Randomly Supported Variations of Deterministic Models and Their Application to One-Dimensional Shallow Water Flows

NUMERICAL COMPUTATION OF ONE- AND TWO-LAYER SHALLOW FLOW MODEL

In this research, we study a proficient computational model designed to simulate shallow flows involving one- and two-layer shallow flow. This numerical model is built upon the Saint Venant equations, which are widely used in hydraulics to depict the behavior of shallow water flow. The numerical scheme used here is constructed based on the conventional leapfrog technique implemented on a staggered grid framework, referred to as MCS. The primary objective of this research is to re-examine and implement the MCS in accurately modelling the free surface and interface waves produced by different flows passing through irregular geometries. Unlike the conventional MCS, we modify the momentum conservation principle to be more general, accommodating a non-negative wet cross-sectional area due to irregular geometry. We successfully conduct numerous numerical simulations by examining various scenarios involving one-layer and two-layer flow through irregularly shaped channels or structures. Our results show that the correct surface wave profile generated by a one-dimensional dam break through the triangular obstacle in the open channel can be simulated very well. Comparison with the existing experimental data seems promising although some disparities are being found due to dispersive phenomena with RMSE less than 5%. Furthermore, our scheme is successfully extended to simulate the steady sub-maximal exchange in two-layer flows using specific boundary conditions. The alignment between the submaximal numerical results with exchange flow theory is noticeable in the interface profile, characteristics of flow conditions and the flux values achieved when the steady situation occurs. These satisfying results indicate that our proposed numerical model can be used for practical needs involving various flow situations both one and two-layer cases

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

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

Weakly Compressible Two-Layer Shallow-Water Flows Along Channels

In this paper, we formulate a model for weakly compressible two-layer shallow water flows with friction in general channels. The formulated model is non-conservative, and in contrast to the incompressible limit, our system is strictly hyperbolic. The generalized Rankine–Hugoniot conditions are provided for the present system with non-conservative products to define weak solutions. We write the Riemann invariants along each characteristic field for channels with constant width in an appendix. A robust well-balanced path-conservative semi-discrete central-upwind scheme is proposed and implemented to validate exact solutions to the Riemann problem. We also present numerical tests in general channels to show the merits of the scheme.

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

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

Experiments on flow velocity profiles in mountain river channels

ABSTRACT Research on the vertical profiles of flow velocity in mountainous river channels is limited, particularly in scenarios where complex bed geometries are absent. Due to the coarse roughness and seepage flow on streambeds composed of gravel, the conventional formulae for flow velocity profiles derived from fluvial river channels do not apply to mountainous river channels. Based on flume experiments with a bed packed with natural gravel and a slope ranging from 0.006 to 0.16, we derived a theoretical formula for flow velocity profiles. This new formula integrates the influence of the subsurface flow and velocity reduction near the water surface, demonstrating a strong alignment with measurements. Our findings indicate that for shallow water flow over rough bed surfaces, the turbulence intensity diminishes along the vertical direction in the near-bed region while remaining relatively constant in the upper water body. Contrary to conventional theories which attribute the increase in flow resistance and the decrease in sediment transport rates in mountainous river channels to form drag, our study emphasizes that the subsurface flow plays a significant role in the overall flow resistance of mountainous river channels and should not be overlooked.

Exploring Advanced Analysis Technique for Shallow Water Flow Models with Diverse Applications

The present article focuses on the analytical approach using fractional orders and its applicationin the dynamics of physical processes. Fractional order models align better with experimental data compared tonon-fractional ones. This study primarily focuses on employing the new approximate analytical method to solveshallow water models with fractional orders. Numerical examples within the Caputo fractional derivative showcasethe method’s application. Results for both integer and fractional orders are graphically depicted, demonstrating thefractional solutions’ closeness to actual data. Analysis of 3D and 2D fractional order graphs reveals convergencetoward integer order graphs as fractional derivatives approach non-fractional ones. This method shows promise fordirect application in solving targeted problems and can be easily adapted for other fractional nature problems.

Flow resistance of emergent rigid vegetation in steady flow

Abstract Enhanced understanding of flow resistance in open channels with emergent vegetation is essential for flood management and river ecosystem restoration. The presence of vegetation can significantly alter bed resistance, leading to a challenge in accurately predicting flow discharge, water levels, sediment transport, and bed deformation. Previous studies on vegetated flows have focused on vegetation resistance, on which the impact of vegetation has been ignored or poorly estimated. This study proposes a new analytical model, built upon the momentum conservation law, to predict flow resistance to vegetated zones in a plain bed without bed forms, explicitly quantifying bed resistance and vegetation resistance in a corollary manner. The proposed model is benchmarked against five typical sets of laboratory experiments. It is demonstrated that the present model using a modified logarithmic velocity distribution performs best, whereas that assuming a uniform velocity profile considerably overestimates the vegetation resistance and neglects the effect of vegetation on bed resistance. The ratio of bed resistance to the total resistance is shown to range between 5% and 40%, and it decreases with increasing vegetation density and decreases with water depth. Therefore, bed resistance cannot be ignored when modelling shallow water flow with sparsely distributed vegetation. It is also revealed that vegetation arrangements significantly affect flow resistance, and therefore a model incorporating the effect of vegetation arrangement performs better. Overall, the present model facilitates a viable and promising tool for quantifying flow resistance in emergent vegetated channels.

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.

A modified Rusanov method for simulating two-layer shallow water flows with irregular topography

A modified Rusanov method for simulating two-layer shallow water flows with irregular topography

Development of a cascaded lattice Boltzmann model for two‐layer shallow water flows

SummaryMany environmental phenomena, such as flows in rivers or in coastal region can be characterised by means of the ‘shallow approach’. A multi‐layer scheme allows to extend it to density layered shallow water flows (e.g., gravity currents). Although a variety of models allowing numerical investigation of single and multi‐layer shallow water flows, based on continuum and particle approaches, have been widely discussed, there are still some computational aspects that need further investigation. Focusing on the Lattice Boltzmann models (LBM), available multi‐layer models generally use the standard linear collision operator (CO). In this work we adopt a multi relaxation time (MRT) cascaded collision operator to develop a two‐layered liquid Lattice‐Boltzmann model (CaLB‐2). Specifically, the model solves the shallow water equations, taking into account two separate sets of particle distribution function (PDF), one for each layer, solved separately. Layers are connected through coupling terms, defined as external forces that model the mutual actions between the two layers. The model is validated through comparisons with experimental and numerical results from test cases available in literature. First results are very promising, highlighting a good correspondence between simulation results and literature benchmarks.

Energy stable and structure-preserving schemes for the stochastic Galerkin shallow water equations

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

An implicit 1D-2D deeply coupled hydrodynamic model for shallow water flows

The 1D-2D coupled hydrodynamic models have been widely developed, to simulate free-surface flows in large shallow water systems. A deeply coupled solution of 1D and 2D submodels refers to the involvement of a solution of flow equations across the 1D-2D interfaces, which will ensure the ability, stability and accuracy of the 1D-2D coupled model in simulating various flow regimes and scenarios. A deeply coupled solution is relatively easy for explicit submodels, but difficult for implicit submodels because of their complex iterative solutions. Techniques for coupling the solution of implicit 1D and 2D submodels are systematically studied, which includes a dimension reduction pretreatment for linking 1D and 2D grids, an upwind scaling method for solving the flow advection across a 1D-2D interface, a method for the drying-wetting simulation across 1D-2D interfaces, etc. The domain-decomposition prediction-correction method is extended to solve velocity–pressure coupling problems of the mixed 1D and 2D subdomains synchronously. Based on these techniques, a 1D-2D deeply coupled model is developed, which allows large time steps, requires only minor data exchange at 1D-2D interfaces, and can be well parallelized. Model test is first done using a real dam-break flow and a hypothetical periodic flow in an experimental flume, to demonstrate the ability of the new model in simulating unsteady supercritical and subcritical flows. The simulation results are shown to agree with the experimental measurements, the simulation results of an existing 1D-2D coupled model and the HEC-RAS. The second test uses a real large river–lake system, and the efficiency of the proposed model is compared with those of a pure 1D model, a pure 2D model and a newly reported 1D-2D coupled model (the proposed model is shown to run about 2 orders of magnitude faster than the newly reported model). Moreover, four 1D-2D coupled models, characterized by different levels of coupling degrees, are tested and compared. With the key physical terms of the governing equations being solved across the 1D-2D interfaces, the proposed model achieves essential improvements in accuracy compared to the loosely coupled model.

A fully coupled dynamic water-mooring line system: Numerical implementation and applications

Several numerical challenges exist in the analysis of water-mooring line systems which require robust, yet practical, methods to address this type of fully coupled nonlinear dynamic problems. The present study proposes a novel class of numerical techniques for the formulation and implementation of a fully coupled dynamic system which involves water flows and catenary mooring line system. In particular, the three-dimensional water flow model is replaced by a simplified multilayer shallow water system with mass exchange terms between the layers including frictional forces at the bed topography and wind-driven forces at the water free-surface. Coupling conditions between the multilayer shallow water model and the mooring line system are also investigated in the current work. As numerical solvers we implement a fast finite volume method for the multilayer shallow water equations and a nonlinear dynamic analysis for the mooring line based on elastic catenary cable elements. Efficient calculations of the interaction forces between the shallow water flow and the submerged mooring line system and associated numerical implementations are also discussed. The accuracy and computational advantages of the proposed fully coupled system are verified using a series of well-established benchmark problems and wind-driven flows over both flat and non-flat beds. The computational results obtained show high performance the developed model and demonstrate the ability of the method to simulate fully coupled dynamic water-mooring line systems.

A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering

<abstract><p>This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.</p></abstract>

Identification of rheological parameters for shallow water flows of viscoplastic fluids using elevation hydrographs

In this paper, rheological parameters, in particular yield stress and consistency index, for viscoplastic fluids are inferred from elevation hydrographs derived from experiments. The direct model consisting of shallow water equations with a Herschel–Bulkley rheology is used to simulate a fluid flowing down an inclined plane and past a cylindrical occlusion. Numerical simulations are validated with experimental and related results from the literature. The aim is to infer the unknown rheological parameters using hydrograph measurements in the contact line region between the fluid and the occlusion. The rheological identification problem is formulated to minimize an objective functional that measures the discrepancy between the elevation hydrographs from the model output and experimental data. The inverse solver is tested on both synthetic and laboratory data. The set of rheological parameters inferred is compared with the values measured on a rheometer for the fluid used in the experiments. Inference of the unknown flow quantities from the wetting free-surface data has direct applications not only in industrial settings, to predict the wetting dynamics, but also in geophysical ones for risk assessments and management plans.

Finite-volume two-step scheme for solving the shear shallow water model

<abstract><p>The shear shallow water (SSW) model introduces an approximation for shallow water flows by including the effect of vertical shear in the system. Six non-linear hyperbolic partial differential equations with non-conservative laws make up this system. Shear, contact, rarefaction, and shock waves are all admissible in this model. We developed the finite-volume two-step scheme, the so-called generalized Rusanov (G. Rusanov) scheme, for solving the SSW model. This method is split into two stages. The first one relies on a local parameter that permits control over the diffusion. In stage two, the conservation equation is recovered. Numerous numerical instances were taken into consideration. We clarified that the G. Rusanov scheme satisfied the C-property. We also compared the numerical solutions with those obtained from the Rusanov, Lax-Friedrichs, and reference solutions. Finally, the G. Rusanov technique may be applied for solving a wide range of additional models in developed physics and applied science.</p></abstract>

High-Order Well-Balanced Finite Difference AWENO Schemes for Shallow Water Flows along Channels with Irregular Geometry

High-Order Well-Balanced Finite Difference AWENO Schemes for Shallow Water Flows along Channels with Irregular Geometry