Depth-integrated Shallow Water Equations Research Articles

Trying to choose the less bad route: Individual migratory behaviour of Atlantic salmon smolts (Salmo salar L.) approaching a bifurcation between a hydropower station and a navigation canal

Contrary to small- and medium-sized rivers, little attention has been paid to the downstream migration of Atlantic salmon smolts in large-sized rivers and the size-related impact of hydropower stations. From 2014 to 2016, we investigated the downstream migration of n = 72 acoustic-tagged smolts in the Meuse river at a bifurcation zone between a hydropower station equipped with three Kaplan turbines and a navigation canal. A hydrodynamic model that solves the depth-integrated shallow water equations on a Cartesian grid using a finite volume technique was used to infer the influence of water discharge and flow velocity on the smolts' behaviour upstream of the hydroelectric complex. Of the migrating smolts, 41.5% performed back and forth movements before approaching the complex for the first time, sometimes over long distances and at a slow pace, leading to significant delays (3–298 h). Beyond about 250 m3 s−1, the water flow direction changes towards the hydropower station with a gradual acceleration. A median water discharge of 161 m3 s−1 and associated median flow velocity of 0.14 m s−1 tended to favour a more direct and downstream movement towards the hydropower station. On the other hand, the navigation canal was mainly approached at low water discharge (median 132 m3 s−1), due to a higher flow velocity (median 0.11 m s−1) at the entrance. Of the released smolts, only 38.6% passed through the complex, of which 36.4% migrated by the navigation canal and 63.6% by the hydropower station, with a median research time of 04:44. Among all the released individuals, the escapement rate at the end of the study site was 2.9% by the canal and 8.3% by the Meuse river. This site, which offers two non-optimal, unattractive and unsafe migration routes, turns out to be problematic for successful downstream smolt migration.

Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework

Lee, H. and Lee, N., 2018. Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 106–110. Coconut Creek (Florida), ISSN 0749-0208.The characteristic Galerkin finite element model for the discrete Boltzmann equation is developed for the depth-integrated shallow water equations. The lattice Boltzmann method (LBM) simulates microscopic collision and streaming of fluid particles and, then, evaluates the macroscopic velocity, pressure, etc. from the computed particle distribution. Due to the inherent geometric flexibility of the unstructured mesh in finite element method, the mesh clustering near large gradients and the numerical treatments of complex geometries has become possible. Some benchmark problems of practical importance (subcritical flow over hump, flow around cylinder, curved channel, etc.) are tested and the numerical results are in good agreement with previous experimental or analytic solutions.

On weak solutions to the 2D Savage–Hutter model of the motion of a gravity-driven avalanche flow

ABSTRACTWe consider the Savage–Hutter system consisting of two-dimensional depth-integrated shallow water equations for the incompressible fluid with the Coulomb-type friction term. Using the method of convex integration we show that the associated initial value problem possesses infinitely many weak solutions for any finite energy initial data. On the other hand, the problem enjoys the weak-strong uniqueness property provided the system of equations is supplemented with the energy inequality.

Non-oscillatory continuous FEM for transport and shallow water flows

The paper presents a sign-preserving continuous finite element model (FEM) for coupled transport equations, and in particular for the depth-integrated shallow water equations. The model incorporates an enhanced flux corrected transport method (FCT) with a high order solution given by a generalization of the characteristic based split finite element method (CBS), and a predictor positive algorithm founded in a finite element upwind methodology. Numerical experiments explore the efficacy of the procedure, with emphasis in the simulation of flows with evolutionary boundaries determined by null values of any flow property, and in severe conditions of propagation such as dam-breaking waves.

Euler’s predictor–corrector technique for solving the depth-integrated shallow water equations

Euler’s predictor–corrector technique combined with finite analysis method is applied to solve 2D advection–diffusion shallow water equations. In this algorithm the momentum equations are calculated by the finite analysis method based on a single mesh, while the continuity equation is solved by Euler’s predictor–corrector technique. To verify the performance of this approach, the simulation of tidal flow in the Huangpu estuary is carried out. The numerical results are found to be consistent with the field results, implying that this proposed method is effective and applicable.

A discontinuous Galerkin algorithm for the two-dimensional shallow water equations

A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.

A second-order analytic solution for oscillatory wind-induced flow in an idealized shallow lake

We present a second-order analytic solution to the nonlinear depth-integrated shallow water equations for free-surface oscillatory wind-driven flow in an idealized lake. Expressing the solution as an asymptotic expansion in the dimensionless wave amplitude ( ζ/ h), which is considered to be a small parameter, enables simplification of the governing equations and permits the use of a perturbation approach to solve them. This analytic solution provides a benchmark for testing numerical models. In particular, the main merit of this solution is that it accounts for advective effects, which are typically omitted from analytic solutions of two-dimensional free surface flow. In order to retain these effects in an analytic solution, we restrict our attention to forcing from a monochromatic wind stress, consider a constant depth rectangular lake, and simplify the governing equations by omitting the Coriolis and eddy viscosity terms and using a linearised friction factor. As such, the analytic solution is of limited use for considering real world problems. Due to the complexity of the analytic solution computer code for this solution is available online. Our solution is valid for cases where changes in the water surface level are small compared with the depth of the lake, and the advective terms in the momentum equations are small compared with acceleration terms. We examine the validity of these assumptions for three test cases, and compare the second-order analytic solution to numerical results to verify an existing hydrodynamic model.

Computation of shallow wakes with the fractional step finite element method

The depth-integrated shallow-water equations are used to simulate shallow wake flow around a solid plate-like obstacle. Influences of different terms in the equations are analyzed by normalizing the momentum equation. A fractional step finite element method is used to discretize the shallow-water equations with third-order accuracy and uniform Courant-Friedrichs-Lewy (CFL) property. The established numerical model is used to simulate the shallow wake in three typical cases and the computational flow fields are compared with digital Particle Image Velocimetry (PIV) measurements and flow visualization pictures. Three turbulence models are considered: The results of the standard k–ε model agree with the experimental results better than those of zero equation and sub-depth scale turbulence models. Furthermore, the results of the simulation also show that it is the wake stability parameter that mainly controls the flow patterns of shallow wakes.

A Discontinuous Galerkin Method for Three-Dimensional Shallow Water Equations

We describe the application of a local discontinuous Galerkin method to the numerical solution of the three-dimensional shallow water equations. The shallow water equations are used to model surface water flows where the hydrostatic pressure assumption is valid. The authors recently developed a DG\linebreak method for the depth-integrated shallow water equations. The method described here is an extension of these ideas to non-depth-integrated models. The method and its implementation are discussed, followed by numerical examples on several test problems.

Coupled discontinuous and continuous Galerkin finite element methods for the depth-integrated shallow water equations

In this paper, we investigate a new approach for the two-dimensional fully non-linear depth-averaged shallow water equations by coupling continuous and discontinuous Galerkin methods. We apply a discontinuous Galerkin method to the primitive continuity and momentum equations and couple this to a continuous method discretizing the so-called “wave continuity” and momentum equation. Herein we present the formulation and derive an a priori error estimate, and demonstrate some preliminary numerical results. This work represents the culmination of a series of papers exploring coupled methods for the shallow water equations.

Discontinuous/continuous Galerkin methods for coupling the primitive and wave continuity equations of shallow water

In this paper, we investigate a new approach for the numerical solution of the two-dimensional depth-integrated shallow water equations, based on coupling discontinuous and continuous Galerkin methods. In this approach, we couple a discontinuous Galerkin method applied to the primitive continuity equation, coupled to a continuous Galerkin method applied to the so-called “wave continuity equation.” The formulation is derived and a priori error estimates are given. Preliminary numerical results for a simple model of flow in a channel are presented.

Tides of the Ross Sea and Ross Ice Shelf cavity

Two new ocean tide models for the Ross Sea including the ocean cavity under the Ross Ice Shelf, are described. The optimum model for predicting ice shelf surface height variability is based on assimilation of gravimetry-derived tidal constituents from the Ross Ice Shelf. Synthetic aperture radar interferograms provide an independent test of model performance. The standard deviation of tide height variability is largest under the eastern ice shelf along the Shirase and Siple Coasts, where it can exceed 0.8 m. The maximum peak-to-peak tidal range in this region is ∼3 m. The best predictor for ocean tidal currents north of the ice front is a dynamics-based model that solves the depth-integrated shallow water equations with a linear representation of benthic friction rather than the more usual quadratic form. Tidal currents over the open Ross Sea are dominated by diurnal, topographically trapped vorticity waves. The strongest modelled currents exceed 1 m s−1 at spring tide in a narrow band along the upper continental slope in the north-western Ross Sea. Typical tidal currents in the central continental shelf area of the Ross Sea are 10–20 cm s−1. Under the ice shelf the typical currents are ∼5 cm s−1.

A discontinuous Galerkin method for two-dimensional flow and transport in shallow water

A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species.

A Numerical Study of the Storm Surge Generated by Tropical Cyclone Jane

Abstract In 1983 Tropical Cyclone Jane crossed the North West Coast of Australia generating a storm surge. Currents associated with this storm surge were recorded at two offshore moorings south of the cyclone track. The data from these moorings are suggestive of the propagation of a continental shelf wave between the two stations. This hypothesis is tested by carrying out a numerical simulation of this storm surge based on the depth-integrated shallow-water equations, with wind-wave-enhanced bottom friction. Analysis of the numerical results shows that the storm surge can be interpreted as due to continental shelf waves.

Application of Depth-Integrated Shallow Water Equations with Boundary-Fitted Mesh Generation to Hydrodynamics in Meandering Mangrove Estuaries

Physical features of tropical mangrove estuaries are meandering and asymmetric variations in tidal-induced flow at the estuary mouth. In the present study we use two-dimensional depth-integrated shallow water equations with boundaryfitted mesh generation for clarifying the interaction between creeks and swamps. Curvilinear coordinated scheme is introduced. At first, the results of numerical experiments are compared with those of hydraulic experiments. After evaluating the model's potential to simulate the movements in sinuous open channel flow, the hydrodynamics in the Buanje Estuary is discussed by the model.

Finite Element Analysis of Tidal Flow in the Osaka Bay with Subgrid Scale Eddy Coefficient

The finite element Galerkin procedure is applied to space domain of the depth-integrated shallow water equations using an isoparametric element for accurately representing the complicatedly curved boundaries. Smagorinsky's assumption is used for determining the an eddy viscosity coefficient ν T which can be made for the SGS Reynolds stresses. Since the value of ν T is evaluated from the tidal flow field of each time step, it varies in time and in space. The SGS eddy viscosity coefficients in the Osaka Bay are found to be an order of 103- 105 cm2/s heavily depending upon the flow intensity.