Shallow Water Flow Research Articles

Space–time residual distribution on moving meshes

This article investigates the potential for an r-adaptation algorithm to improve the efficiency of space–time residual distribution schemes in the approximation of time-dependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually “relocated” as the flow evolves so that features of interest remain resolved as they move within the domain.Adaptive strategies of this type are well suited to the space–time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of time-step, even on the distorted space–time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the time-step, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixed-mesh scheme is retained on the moving mesh in the majority of applications tested.Space–time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the so-called C-property (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved.The r-adaptation is applied within each time-step by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error.

Response of GPS-Tracked Drifters to Wind and Water Currents in a Tidal Estuary

Lagrangian field data in tidal shallow waters are rare, but valuable for the understanding of the spatiotemporal structure of flow and particle transport. The response of drifters to the wind and water flow in tidal shallow water was examined using correlation, spectral, and coherence analyses. Under moderate wind conditions (0-4 m/s), floating drifter motions in bounded sheltered water are affected by wind through low-frequency induced wind current rather than direct wind drag, when only a small portion of the drifter is unsubmerged. The field validation of both high- and low-resolution drifters with surface measured velocity from a fixed acoustic Doppler current profiler is good in the streamwise direction. The correlation between the drifter and fixed instrument velocities is low in the cross-stream direction due to strong spatial variability of the flow field. The evaluation shows that drifters are applicable to studying the flow dynamics of tidal water bodies in relation to small-scale processes.

Assessing Adaptive Mesh Refinement (AMR) in a Forced Shallow-Water Model with Moisture

AbstractTwo forced shallow-water flow scenarios are explored in a 2D fourth-order finite-volume dynamical core with adaptive mesh refinement (AMR) to investigate AMR’s ability to track and resolve complex evolving features. Traditional shallow-water test cases are mainly characterized by large-scale smooth flows that do not effectively test the multiscale abilities of variable-resolution and AMR models to resolve sharp gradients and small-scale flow filaments. Therefore, adding forcing mechanisms to the shallow-water system to model key atmospheric processes adds complexity and creates small-scale phenomena. These can serve as foci for dynamic grid refinement while remaining simple enough to study the numerical design of a model’s dynamical core. The first shallow-water flow scenario represents a strengthening, tropical cyclone–like, vortex that is driven by a Betts–Miller-like convection scheme. The second shallow-water test is built upon a barotropically unstable jet with an added Kessler-like warm rain scheme that leads to precipitating frontal zones. The key feature of both tests is that there is significant sensitivity to the model grid while converging (structurally) at high resolution. Both test cases are investigated for a series of uniform resolutions and a variety of AMR tagging criteria. The AMR simulations demonstrate that grid refinement can resolve local features without requiring global high-resolution meshes. However, the results are sensitive to the refinement criteria. Criteria that trigger refinement early in a simulation reproduce the uniform-resolution reference solutions most reliably. In contrast, AMR criteria that delay refinement for several days require careful tuning of the AMR thresholds to improve results compared with uniform-resolution simulations.

Hybrid Artificial Viscosity–Central-Upwind Scheme for Recirculating Turbulent Shallow Water Flows

AbstractIn this paper, a hybrid artificial viscosity–central-upwind (AV-CU) scheme is proposed for simulating recirculating turbulent shallow water flows by combining the artificial viscosity (AV) ...

Sheared free-surface flow over three-dimensional obstructions of finite amplitude

When shallow water flows over uneven bathymetry, the water surface is modulated. This type of problem has been revisited numerous times since it was first studied by Lord Kelvin in 1886. Our study analytically examines currents whose unperturbed velocity profile$U(z)$follows a power law $z^{q}$, flowing over a three-dimensional uneven bed. This particular form of $U$, which can model a miscellany of realistic flows, allows explicit analytical solutions. Arbitrary bed shapes can readily be imposed via Fourier’s theorem provided their steepness is moderate. Three-dimensional vorticity–bathymetry interaction effects are evident when the flow makes an oblique angle with a sinusoidally corrugated bed. Streamlines are found to twist and the fluid particle drift is redirected away from the direction of the unperturbed current. Furthermore, a perturbation technique is developed which satisfies the bottom boundary condition to arbitrary order also for large-amplitude obstructions which penetrate well into the current profile. This introduces higher-order harmonics of the bathymetry amplitude. States of resonance for first- and higher-order harmonics are readily calculated. Although the method is theoretically restricted to bathymetries of moderate inclination, a wide variety of steeper obstructions are satisfactorily represented by the method, even provoking occurrences of recirculation. All expressions are analytically explicit and sequential fast Fourier transformations ensure quick and easy computation for arbitrary three-dimensional bathymetries. A method for separating near and far fields ensures computational convergence under the appropriate radiation condition.

On Study of Generalized Novikov Equation by Reduced Differential Transform Method

Objectives: The object of the work is essentially to examine the generalization of NovikovPartial Differential Equations through differential transform algorithm. This work also shows that the method can allow us to construct explicit solutions highly nonlinear equations. We have also plotted the constructed solutions. Methods: We have constructed the approximate solutions of mentioned equation using a relatively new algorithm, known as reduced differential transform algorithm. Findings: It turns out that our solutions agree with the abstract findings known in key papers that we followed. Applications: Generalization of Novikov Partial Differential Equations models several physical phenomena such as shallow water flow, dynamics of enzymes in the human cells etc.Keywords: Nonlinear Equations, Novikov Partial Differential Equations, Partial Differential Equations (PDE)

Efficient computational models for shallow water flows over multilayer erodible beds

Purpose The purpose of this paper is to present a new numerical model for shallow water flows over heterogeneous sedimentary layers. It is already several years since the single-layered models have been used to model shallow water flows over erodible beds. Although such models present a real opportunity for shallow water flows over movable beds, this paper is the first to propose a multilayered solver for this class of flow problems. Design/methodology/approach Multilayered beds formed with different erodible soils are considered in this study. The governing equations consist of the well-established shallow water equations for the flow, a transport equation for the suspended sediments, an Exner-type equation for the bed load and a set of empirical equations for erosion and deposition terms. For the numerical solution of the coupled system, the authors consider a non-homogeneous Riemann solver equipped with interface-tracking tools to resolve discontinuous soil properties in the multilayered bed. The solver consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. Findings This paper reveals that modeling shallow water flows over multilayered sedimentary topography can be achieved by using a coupled system of partial differential equations governing sediment transport. The obtained results demonstrate that the proposed numerical model preserves the conservation property, and it provides accurate results, avoiding numerical oscillations and numerical dissipation in the approximated solutions. Originality/value A novel implementation of sediment handling is presented where both averaged and separate values for sediment species are used to ensure speed and precision in the simulations.

The role of the Rankine-Hugoniot relations in staggered finite difference schemes for the shallow water equations

The purpose of this work is to point out the relevance of the Rankine-Hugoniot jump relations regarding the numerical solution of the inviscid shallow water equations. To arrive at physically relevant solutions in rapidly varied flow, it is of crucial importance that continuity of mass flux and momentum flux across a steady discontinuity is fulfilled at the discrete level. By adopting this viewpoint, finite difference schemes can be studied that may be well suited to solve shallow water flow problems involving discontinuities, while they are not based on a characteristic decomposition of the governed hyperbolic equations. Three schemes on staggered grids with either the water level or the water depth at the cell centre and the flow velocity or the depth-integrated velocity at the cell interface are examined. They differ in (1) the character of the transport velocity to bias the discretization of the advective acceleration term in the upwind direction, and (2) the determination of the water depth at the cell face with which the depth-integrated velocity must be linked to the flow velocity. A detailed analysis is provided and aimed at highlighting the necessity of fulfilling the Rankine-Hugoniot jump conditions for preventing the odd-even decoupling problem. The accuracy and robustness of three selected schemes is assessed by means of convergence tests, three idealized 1D test problems with exact solutions and a 1D laboratory experiment of the breaking, runup and rundown of a solitary wave on a sloping beach. Numerical results reveal that schemes satisfying exactly the jump conditions display improved performance over schemes which do not share this property. Also, these results support strong evidence on the link between not fulfilling the jump conditions and the appearance of odd-even oscillations.

A Simple Modelling Framework for Shallow Subsurface Water Storage and Flow

Water storage and flow in shallow subsurface drives runoff generation, vegetation water use and nutrient cycling. Modelling these processes under non-steady state conditions is challenging, particularly in regions like the subtropics that experience extreme wet and dry periods. At the catchment-scale, physically-based equations (e.g., Richards equation) are impractical due to their complexity, while conceptual models typically rely on steady state assumptions not found in daily hydrological dynamics. We addressed this by developing a simple modelling framework for shallow subsurface water dynamics based on physical relationships and a proxy parameter for the fluxes induced by non-unit hydraulic gradients. We demonstrate its applicability for six generic soil textures and for an Acrisol in subtropical China. Results showed that our new approach represents top soil daily fluxes and storage better than, and as fast as, standard conceptual approaches. Moreover, it was less complex and up to two orders of magnitude faster than simulating Richards equation, making it easy to include in existing hydrological models.

MRT-Lattice Boltzmann Model for Multilayer Shallow Water Flow

The objectives of this study are to introduce a multiple-relaxation-time (MRT) lattice Boltzmann model (LBM) to simulate multilayer shallow water flows and to introduce graphics processing unit (GPU) computing to accelerate the lattice Boltzmann model. Using multiple relaxation times in the lattice Boltzmann model has an advantage of handling very low kinematic viscosity without causing a stability problem in the shallow water equations. This study develops a multilayer MRT-LBM to solve the multilayer Saint-Venant equations to obtain horizontal flow velocities in various depths. In the multilayer MRT-LBM, vertical kinematic viscosity forcing is the key term to couple adjacent layers. We implemented the multilayer MRT-LBM to a GPU-based high-performance computing (HPC) architecture. The multilayer MRT-LBM was verified by analytical solutions for cases of wind-driven, density-driven, and combined circulations with non-uniform bathymetry. The results show good speedup and scalability for large problems. Numerical solutions compared well to the analytical solutions. The multilayer MRT-LBM is promising for simulating lateral and vertical distributions of the horizontal velocities in shallow water flow.

A well-balanced adaptive Haar wavelet finite volume scheme for 1D free surface water flows

Abstract This paper studies a Haar-wavelet based finite volume method in terms of its capability of preserving a well-balanced property compared to the classical finite volume method, as well as with its application on the real case of the shallow water flow which has irregular bed elevation. The well-balanced property is considered as an important benchmark for validation of the numerical scheme, particularly when the system of the governing equations is inhomogeneous (i.e. the source term is not zero). Haar wavelet finite volume scheme is obtained via merging the theory of wavelets with the formulation of finite volume method. The Roe approximate Riemann solver is adopted to evaluate the numerical fluxes at cell boundaries. The source term is represented by the frictionless topography in the test case. Furthermore, the numerical results are compared to the classical finite volume scheme and the real case and it is in a good agreement with the exact and reference finite volume scheme in which the adaptive scheme preserved the equilibrium state of the system. The adaptive scheme inherits the features of the classical finite scheme but with less computational effort in which the adaptive scheme needs a maximum of 211 cells (8.8% of the finest uniform mesh refinement) to get the convergence of the scheme. While in the partially wet sub-case, it needs 223 cells (9.3% of the finest uniform mesh refinement). The numerical scheme was preserving the root mean square error of the computational results within the range of the machine precision.

Vertically-averaged and moment equations for flow and sediment transport

Simulation of river flow processes including sediment transport is usually conducted using the shallow water flow equations, which are based on a hydrostatic pressure distribution. To increase the accuracy of predictions in a variety of scenarios involving horizontal length scales of the order of vertical length scales, an improved representation of the vertical flow structure is necessary. The mathematical approximation to field variables like the velocity and fluid pressure must be enhanced during the depth-integrating process. Therefore, this paper presents a 1D non-hydrostatic flow and sediment transport model developed by using the method of the weighted residuals into the RANS equations. Using continuity, momentum, and moment equations, the fluid pressure distribution is modelled using a quadratic predictor with perturbation parameters to deviate the vertical momentum balance from the hydrostatic law. The flow equations are a generalized non-hydrostatic flow solver, where the fluid density variation due to suspension of sediments and the bed deformation due to erosion-sedimentation processes are accounted for. A hybrid semi-implicit finite volume-finite difference numerical scheme is developed to solve the system of conservation laws. Two different approaches are used to model the sediment transport processes: (i) Unified computation of the total-load transport and (ii) separate computation of suspended and bed loads. The accuracy of the non-hydrostatic model is demonstrated by comparison with experimental data, highlighting better results accounting for separate determinations of the suspended and bed loads in highly erosive flows.

Stability of Hydraulic Shock Profiles

We establish nonlinear $$H^2\cap L^1 \rightarrow H^2$$ stability with sharp rates of decay in $$L^p$$, $$p\ge 2$$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans–Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.

High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry

In this paper, we present high order finite difference weighted essentially non-oscillatory (WENO) schemes for the shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady state solutions, namely preserve steady state at the discrete level, when there is a exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross section, the construction of the well-balanced schemes is not a trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then propose to construct the numerical fluxes by means of a flux modification technique, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.

Frictional effect on stability of discontinuity interface in tangential velocity of a shallow-water flow

We examine a frictional effect on the linear stability of an interface of discontinuity in tangential velocity. The fluid is moving with uniform velocity U in a region but is at rest in the other, and the bottom surface is assumed to exert drag force, quadratic in velocity, on the thin fluid layer. In the absence of the drag, the instability of the Kelvin-Helmholtz type is suppressed for U>8c, with c being the propagating speed of the gravity wave. We find by asymptotic analyses for both small and large values of the drag strength that the drag, regardless of its strength, makes the flow unstable for the whole range of the Froude number U/c.

R.sim.terrain 1.0: a landscape evolution model with dynamic hydrology

Abstract. While there are numerical landscape evolution models that simulate how steady-state flows of water and sediment reshape topography over long periods of time, r.sim.terrain is the first to simulate short-term topographic change for both steady-state and dynamic flow regimes across a range of spatial scales. This free and open-source Geographic Information Systems (GIS)-based topographic evolution model uses empirical models for soil erosion and a physics-based model for shallow overland water flow and soil erosion to compute short-term topographic change. This model uses either a steady-state or unsteady representation of overland flow to simulate how overland sediment mass flows reshape topography for a range of hydrologic soil erosion regimes based on topographic, land cover, soil, and rainfall parameters. As demonstrated by a case study for the Patterson Branch subwatershed on the Fort Bragg military installation in North Carolina, r.sim.terrain simulates the development of fine-scale morphological features including ephemeral gullies, rills, and hillslopes. Applications include land management, erosion control, landscape planning, and landscape restoration.

Cylinder wakes in shallow oscillatory flow: the coastal island wake problem

Topographic complexity on continental shelves is the catalyst that transforms the barotropic tide into the secondary and residual circulations that dominate vertical and cross-shelf mixing processes. Island wakes are one such example that are observed to significantly influence the transport and distribution of biological and physical scalars. Despite the importance of island wakes, to date, no sufficient, mechanistic description of the physical processes governing their development exists for the general case of unsteady tidal forcing. Controlled laboratory experiments are necessary for the understanding of this complex flow phenomenon. Here, three-dimensional velocity field measurements of cylinder wakes in shallow-water oscillatory flow are conducted across a parameter space that is typical of tidal flow around shallow islands. The wake form in steady flows is typically described in terms of the stability parameter $S=c_{f}D/h$ (where $D$ is the island diameter, $h$ is the water depth and $c_{f}$ is the bottom boundary friction coefficient); in tidal flows, there is an additional dependence on the Keulegan–Carpenter number $KC=U_{0}T/D$ (where $U_{0}$ is the tidal velocity amplitude and $T$ is the tidal period). In this study we demonstrate that when the influence of bottom friction is confined to a Stokes boundary layer the stability parameter is given by $S=\unicode[STIX]{x1D6FF}^{+}/KC$ where $\unicode[STIX]{x1D6FF}^{+}$ is the ratio of the wavelength of the Stokes bottom boundary layer to the depth. Three classes of wake form are observed with decreasing wake stability: (i) steady bubble for $S\gtrsim 0.1$; (ii) unsteady bubble for $0.06\lesssim S\lesssim 0.1$; and (iii) vortex shedding for $S\lesssim 0.06$. Transitions in wake form and wake stability are shown to depend on the magnitude and temporal evolution of the wake return flow. Scaling laws are developed to allow upscaling of the laboratory results to island wakes. Vertical and lateral transport depend on three parameters: (i) the flow aspect ratio $h/D$; (ii) the amplitude of tidal motion relative to the island size, given by $KC$; and (iii) the relative influence of bottom friction to the flow depth, given by $\unicode[STIX]{x1D6FF}^{+}$. A model of wake upwelling based on Ekman pumping from the bottom boundary layer demonstrates that upwelling in the near-wake region of an island scales with $U_{0}(h/D)KC^{1/6}$ and is independent of the wake form. Finally, we demonstrate an intrinsic link between the dynamical eddy scales, predicted by the Ekman pumping model, and the island wake form and stability.

One-Dimensional/Two-Dimensional Coupling Approach with Quadrilateral Confluence Region for Modeling River Systems

We study shallow water flows in river systems. An accurate description of such flows can be obtained using the two-dimensional (2-D) shallow water equations, which can be numerically solved by a shock-capturing finite-volume method. This approach can, however, be inefficient and computationally unaffordable when a large river system with many tributaries and complex geometry is to be modeled. A popular simplified approach is to model flow in each uninterrupted section of the river (called a reach) as one-dimensional (1-D) and connect the reaches at the river junctions. The flow in every reach can then be modeled using the 1-D shallow water equations, whose numerical solution is dramatically less computationally expensive compared with solving its 2-D counterpart. Even though several point-junction models are available, most of them prove to be sufficiently accurate only in the case of a smooth flow though the junction. We propose a new 1-D/2-D river junction model, in which each reach of the river is modeled by the 1-D shallow water equations, while the confluence region, where the mixing of flows from the different directions occurs, is modeled by the 2-D ones. We define the confluence region to be a trapezoid with parallel vertical sides. This allows us to take into account both the average width of each reach and the angle between the directions of flow of the tributary and the principal river at the confluence. We choose a trapezoidal confluence region as it is consistent with the 1-D model of the river. We implement well-balanced positivity preserving second-order semi-discrete central-upwind schemes developed in Kurganov and Petrova (Commun Math Sci 5:133–160, 2007) for the 1-D shallow water equations and in Shirkhani et al. (Comput Fluids 126: 25–40, 2016) for the 2-D shallow water equations using quadrilateral grids. For the 2-D junction simulations in the confluence region we choose a very coarse 2-D mesh as the goal of our model is not to resolve the fine details of complex 2-D vortices that form around the junction, but to efficiently compute average water depth and velocity in the connected 1-D reaches. A special ghost cell technique is developed for coupling the reaches to the confluence region, which is one of the most important parts of a good 1-D/2-D coupling method. The proposed approach leads to very significant computational savings compared to numerically solving the full 2-D problem. We perform several numerical experiments to demonstrate plausibility of the proposed 1-D/2-D coupling model.

Mathematical modeling of shallow-water flows on steep slopes

Abstract A 2D hydrodynamic (labeled as CAR) model has been proposed in a rectangular Cartesian coordinate system with two axes within the horizontal plane and one axis along the vertical direction (global coordinates), considering the effects of bed slope on both pressure distribution and bed shear stresses. The CAR model satisfactorily reproduces the analytical solutions of dam-break flow over a steep slope, while the traditional Saint-Venant Equations (labeled as SVE) significantly overestimate the flow velocity. For flood events with long duration and large mean slope, the CAR and the SVE models present distinguishable discrepancies. Therefore, the proposed CAR model is recommended for applications to real floods for its facility of extending from 1D to 2D version and ability to model shallow-water flows on steep slopes.

Non-linear shallow water flow modelling over topography with depth-averaged potential equations

Hunter Rouse was the father of the Fluid Mechanics of Open Channel Flow. In this review article the fundamental advances on the non-hydrostatic modeling of shallow open channel flow over topography, since the publication of his 1938 book, are discussed. Flows over uneven topography, like at open channel controls, occurs in a short length, thus rendering ideal fluid flow theory an adequate mathematical tool. This paper emphasis is on depth-averaged modeling of these flows using Boussinesq equations, given the significant advances basically since the 80’s. The one-dimensional steady flow model from Picard iteration is reviewed in detail, including energy and momentum formulations, flow at a weir crest, and on a slope. A new numerical scheme is developed for the solution of transcritical steady weir flow, showing excellent match with experiments. A new derivation for the two-dimensional depth-averaged unsteady ideal fluid model is presented based on a continuum mechanics formulation. The result is the Serre–Green–Naghdi equations (SGNE). It is demonstrated that the extended Bernoulli equation derived from Picard iteration is an integral form of the SGNE for 1D steady flow. A robust MUSCL-Hancock finite volume model is developed to solve the unsteady 1D SGNE, showing excellent agreement with the steady transcritical flow solutions previously constructed. The accuracy of the MUSCL-Hancock solver is critically assessed using a higher-order solver for the SGNE.