Shallow Flow Model Research Articles

A 2D shallow flow model for practical dam-break simulation

By YUELING WANG, QUIHUA LIANG, GEORGES KESSERWANI and JIM W. HALL, J. Hydraulic Res. 49(3), 2011, 307–316. On p. 312, the authors state that “The classical analytical test of oscillatory flow in a ...

Closure to “A 2D shallow flow model for practical dam-break simulation”

"Closure to “A 2D shallow flow model for practical dam-break simulation”." Journal of Hydraulic Research, 50(5), pp. 544–545

A 2D well-balanced shallow flow model for unstructured grids with novel slope source term treatment

Within the framework of the Godunov-type cell-centered finite volume (CCFV) scheme, this paper proposes a 2D well-balanced shallow water model for unstructured grids. In this model, the face-based van Albada limiting scheme is employed in conjunction with a directional correction to reconstruct second order spatial values at the midpoint of the considered face. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) is applied to compute the fluxes of mass and momentum, while the splitting implicit method is utilized to solve the friction source terms. The novel aspects of the model include the new limited directional correction with which the new local extrema caused by the unlimited correction are prevented efficiently, the simplified non-negative water depth reconstruction used to get rid of numerical instabilities and in turn to preserve mass conservation at wet–dry interfaces and the novel slope source term treatment which suits complex unstructured grids well by transforming the slope source of a cell into fluxes at its faces. This model is able to preserve the C-property and mass conservation, to achieve good convergence to steady state, to capture discontinuous flows and to handle complex flows involving wetting and drying over uneven beds on unstructured grids with poor connectivity in an accurate, efficient and robust way. These capabilities are verified against analytical solutions, numerical results of alternative models and experimental and field data.

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.

Simulations of granular flow along an inclined plane using the Savage–Hutter model

In this work a velocity-dependent friction is introduced into a depth-averaged Savage–Hutter dynamical model for shallow granular flows. The process of granular material flowing along an inclined plane and then depositing on a horizontal plane is simulated. The surface profiles and evolution of various types of energy are investigated and compared when using the standard Coulomb-type friction versus velocity-dependent friction. Interestingly, there is a small difference between the two different types of friction.

Analysis of secondary current effect in the modeling of shallow flow in open channels

The 3D information of a secondary flow evolved in curved or confluent channels, which is usually lost in conventional shallow water models due to the depth-averaging process, can be partially retrieved by incorporating the secondary current velocity profile in the vertical direction. In the present study, the depth-averaged shallow water equations with dispersion stresses were solved by the SU/PG scheme among the various numerical methods of the FEM. In the proposed model, to quantify the strength of the secondary flow, the comprehensive factors such as the deformation of the vertical profile of the streamwise velocity, the roughness coefficient, and the curvature ratio were embedded into the dispersion stress terms. Two sets of experimental data, one with a sharply curved channel by Rozovskii (1961) [40], and the other with a confluent channel by Shumate (1998) [51] were used to validate the proposed model. The computed values of the water surface profile and the depth-averaged velocity across the channel showed good agreement with experimental data, which indicated that secondary velocity profiles were preserved properly. The proposed model was applied to a natural stream with moderate curvature to test the field applicability. The simulation results obtained by the proposed model with dispersion terms matched quite well with the ADCP data, whereas the model without dispersion terms produced excessive velocities at both banks, and the commercial RMA-2 model yielded uniform span-wise velocity distributions at all sections. The analysis of the momentum balance and relative dominance of the secondary current demonstrated that the convection and pressure gradient terms played major roles in achieving equilibrium in momentum equations, and the bottom friction ranked next, followed by the remaining dispersion stresses and viscous term of a similar order. It was also found that the pressure gradient term was the primary factor that triggered velocity redistribution. As the Fr increased, the convective acceleration that formed along the channel curvature was activated as a secondary factor in velocity redistribution, whereas the viscous stress term lost its influence. The bottom friction term had minor exertion on the redistribution mechanism.

Fully-Coupled Modeling of Shallow Water Flow and Pollutant Transport on Unstructured Grids

Understanding the space-time dynamics of pollutant transport remains an essential impediment to accurate prediction of impacts on the ecology of rivers and coastal areas and also for establishing efficient strategies for pollution control and environmental protection. Numerical models are a powerful tool to study the water flows and pollutant transport, and recently a new generation of models is being developed to simulate the coupled flow and pollutant transport in shallow water. In this paper, a two-dimensional fully-coupled model of shallow water flows and pollutant transport was developed using a triangular unstructured grid (TIN: triangular irregular network), which is also an important module of the PIHM-Hydro modeling system. The model is based on a cell-centered upwind finite volume method using the HLL approximate Riemann solver. A multidimensional linear reconstruction technique and multi-dimensional slope limiter was implemented to achieve a second-order spatial accuracy. In order to make the model efficient and stable, an explicit-implicit method was used in temporal discretization by an operator splitting technique. A test case of the pollutant transport in a square cavity is used to validate the model. Then the model was further applied to two pollutant transport scenarios: microscale pollutant transport following dam break and mesoscale pollutant transport driven by storm surge in Galveston Bay. The numerical results show that the model could accurately predict the flow dynamics and pollutant transport in extreme events such as a dam break and a storm surge. According to the prediction of the model, the storm surge caused by the Hurricane Ike significantly extended the polluted area.

Effect of magnetic forces on bubble transport and MHD stability of aluminium electrolysis cells

The effect of electro-magnetophoretic force on gas bubbles due to the additional magnetic pressure distribution in the electrolyte is analyzed and a mathematical model is derived. According to the results, the integral force on an individual bubble is of similar order as the typical drag force associated with the electrolyte flow, opposing the motion of bubbles along the slightly inclined base of the anode. This could explain certain features of the anode effect onset. It is demonstrated that the presence of electrolyte channels, where the bubbles escape, has a crucial effect on the shape and size of the metal-bath interface deformation. A shallow layer flow model accounting for the channels is derived. Results on the interface stationary shape and MHD wave stability are presented.

SPH Modeling of Shallow Flow with Open Boundaries for Practical Flood Simulation

A smoothed particle hydrodynamics (SPH) numerical model for shallow water equations (SWEs) is presented for simulating flood inundation owing to rapidly varying flow, such as dam breaks, tsunamis, and levee breaches. Important theoretical and numerical developments have recently been made, and the model in this paper incorporates these developments and implements open boundary conditions, resulting in a general, accurate computational tool suitable for practical application. The method is attractive for flood simulation over large domains in which the extent of inundation is unknown because computation is carried out only in wet areas and is dynamically adaptive. The open boundary algorithm is very general, on the basis of a simplified version of the characteristics method, handling both supercritical and subcritical inflow and outflow. This is tested against reference solutions for flows over a hump involving shocks. The model is then applied to two very different flood inundations resulting from the Okushiri tsunami in Japan and from a hypothetical dyke breach at Thamesmead in the United Kingdom. The SPH-SWE model compares well with established commercial and state-of-the-art finite-volume codes.

On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm

AbstractIn cities, flood waves may propagate over street surfaces below which lie complicated pipe networks used for storm drainage and sewage. The flood and pipe flows can interact at connections between the underground pipes and the street surface. The present paper examines this interaction, using the shallow water equations to model the flood wave hydrodynamics. Sources and sinks in the mass conservation equation are used to model the pipe inflow and outflow conditions at bed connections. We consider the problem reduced to one dimension. The shallow water equations are solved using a Godunov‐type wave propagation scheme. Wave speeds are modified in the wave propagation algorithm to enable flows to be simulated over nearly dry beds and dry states. First, the model is used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. Comparisons are made with numerical predictions from STAR‐CD, a commercial Navier–Stokes solver that models the free‐surface motions, and a parameter study is undertaken to investigate the effect of the one‐dimensional approximation of the present model, for a range of non‐dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all time provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered. Copyright © 2010 John Wiley & Sons, Ltd.

A 2D shallow flow model for practical dam-break simulations

Dam-break flows usually occur in domains with complex geometric and topographic features and involve abrupt flow patterns. A dam-break model must therefore be able to effectively handle different flow types including transcritical flows or hydraulic jumps, deal with complex domain topography, capture repeating wet–dry interface and represent high roughness values in the floodplain. Herein, all of these objectives are achieved by extending a recent one-dimensional finite volume Godunov-type model into two dimensions for solving the shallow-water equations. While doing so, a much simplified condition to maintain well-balanced solutions around a wet–dry front is proposed and a two-dimensional friction source term discretization is derived under a suitable stability condition in relation to practical simulations. The two-dimensional model is successfully validated against three analytical benchmark tests and then assessed for predicting realistic dam-break flood events.

Testing a new adaptive grid‐based shallow flow model for different types of flood simulations

AbstractThis work aims to test a robust numerical tool for modelling different types of floods, e.g. rapid‐varying dam breaks or slow‐evolving inundations. Based on a novel adaptive grid system, the model solves the 2D shallow water equations using a first‐order accurate Godunov‐type scheme to approximate the flood hydrodynamics. The grid dynamically adapts to local topographic features, moving wet–dry interfaces and other flow features. High‐resolution mesh is only generated to those areas of interests. Therefore, it facilitates the accurate prediction of flood fronts and routs and meanwhile maintains efficient simulation. Results are presented for an analytical case of surge reflecting from a standing wall, a laboratory‐scale rapidly varying dam‐break flow and a prototypical case of slowly evolving flood inundation in the 36 km2 floodplain at Thamesmead. To produce results with a similar resolution, the adaptive grid‐based model is found to be considerably more efficient than its uniform grid‐based counterpart.

Solution properties and approximate Riemann solvers for two-layer shallow flow models

The 4 × 4 system of governing equations for two-layer shallow flow models is known to exhibit particular behaviours such as loss of hyperbolicity under certain flow configurations. An eigenvalue analysis of the conservation part of the equations shows that the loss of hyperbolicity is due only to the reaction exerted by each fluid onto the other at the interface between the fluids. Three Riemann solvers derived from the HLL formalism are presented. In the first solver, the pressure-induced terms are accounted for by the source term; in the second solver, they are incorporated into the fluxes; the third solver uses the same formulation as the first, except that the mass and momentum balance for the bottom layer are replaced with the balance equations for the system formed by the two layers as a whole. Numerical results using the three solvers are presented for (1) static conditions such as two fluids of identical densities at rest above each other, (2) dam-break flows involving the collapse of a body of light fluid over a uniform layer of a denser fluid, and (3) Liska and Wendroff’s ill-posed test cases [24] involving two-layer flows over a topographic bump. The three solvers produce quasi-undistinguishable results for the dam-break flows, and produce sharp solutions over the full range of density ratio, from 0 to 1. However, only the third solver allows a strict preservation of static configurations. Moreover, a method is proposed to assess the convergence of the numerical solutions in the configurations for which no analytical solution can be obtained.

SAFETY EVACUATION FROM UNDERGROUND MALL DURING URBAN FLOODING

Torrential rainfalls have been observed frequently in recent years and these rainfalls caused inundations in urban and rural areas. Intensity of such rainfalls were sometimes excess over the design rainfall of drainage systems and rain was running over the road. In these floods, some portion of rainwater intruded into underground spaces and the users faced the danger of underground flooding. In this paper, underground flooding in a mega-underground mall were investigated by using 2D shallow flow model with structural mesh and evacuation from there was also studied with the criteria of safety evacuation obtained by evacuation tests. The results show that flooding processes and safety of evacuation are quite different in the two cases with and without the consideration of drainage system. It means that the inflow conditions into underground space are depend on calculation results of rainwater flooding.

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe’s Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe’s method, while maintaining Roe’s low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions. As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [T. Gallouët, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77–84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.

Computations of suspended sediment transport in a shallow side-cavity using depth-averaged 2D models with effects of secondary currents

Two-dimensional (2D) depth-averaged shallow flow models’ integrated effects of secondary currents are applied to open channel flows with a side cavity, and the accuracy of predictions of the flow structures as well as the behavior of suspended sediment transport is examined. The flow in a side-cavity is characterized by formations of a steady recirculation with a vertical axis and associated secondary current due to a centrifugal force (a secondary current of the first kind). The flow structures become completely three-dimensional and suspended sediment transports are affected by such complicated flow features. In this study, we applied depth-averaged shallow flow equations including effects of secondary currents, instead of 3D flow equations considering computational efficiency. We applied 4 different types of models for shallow flow equations and compared the performances of each model. First, these models were applied to a simple flow in a torus-shape channel with rigidly rotating flow, in order to consider the fundamental characteristics of suspended sediment transport. Then the models were applied to open channel flows with a rectangular side cavity, in order to consider both flow structures and sediment transport affected by the secondary current. The computations were performed under the same conditions of the laboratory tests and applicability of each model was discussed through the comparison between computational and experimental results. The model considering the lag of development of secondary current behind the streamline curvature, and the deformation of mean velocity profiles affected by secondary currents showed excellent performance for predicting flow phenomena with recirculation.

Numerical modeling of shallow water flow around arrays of emerged cylinders

The flow around arrays of emerged cylinders is investigated numerically in this paper. The resistant effects produced by the cylinders are modeled by adding momentum sink terms in the governing equations of fluid flow. A 3D LES (Large Eddy Simulation) model is employed with the sub-grid scale turbulence determined by the Smagorinsky model. The governing equations are solved by a split-operator finite difference scheme, in which the advection, diffusion and pressure propagation terms are solved separately. The numerical simulation reproduces the stagnant zone of flow behind the cylinders and the wake oscillation phenomenon further downstream. The length of the wake region and the Strouhal number are correctly predicted for cases with high number density of cylinders, but are over-predicted for the other cases in which the number density of cylinders is low. More accurate numerical predictions can be obtained by imposition of random velocity fluctuations in the zone occupied by the cylinders to trigger the earlier occurrence of wake oscillation, and/or by using an artificially higher value of the drag coefficient for the cylinders. The results indicate that the use of the spatially averaged governing equations with momentum sink terms for cylinder-array precludes the simulation of small eddies and requires a higher flow resistance or explicit velocity fluctuations to model the small scale turbulence effects.

Flood Simulation Using a Well-Balanced Shallow Flow Model

This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against...

Mathematical modelling of shallow flows: Closure models drawn from grain-scale mechanics of sediment transport and flow hydrodynamicsThis paper is one of a selection of papers in this Special Issue in honour of Professor M. Selim Yalin (1925–2007).

Mathematical modelling of river processes is, nowadays, a key element in river engineering and planning. River modelling tools should rest on conceptual models drawn from mechanics of sediment transport, river mechanics, and river hydrodynamics. The objectives of the present work are (i) to describe conceptual models of sediment transport, deduced from grain-scale mechanics of sediment transport and turbulent flow hydrodynamics, and (ii) to present solutions to specific river morphology problems. The conceptual models described are applicable to the morphologic evolution of rivers subjected to the transport of poorly sorted sediment mixtures at low shear stresses and to geomorphic flows featuring intense sediment transport at high shear stresses. In common, these applications share the fact that sediment transport and flow resistance depend, essentially, on grain-scale phenomena. The idealized flow structures are presented and discussed. Numerical solutions for equilibrium and nonequilibrium sediment transport are presented and compared with laboratory and field data.

Two-layer models for shallow avalanche flows over arbitrary variable topography

This paper presents a three-dimensional, two-layer model for shallow geophysical mass flows, such as debris flows, hydraulic sediment transport, or sub-aquatic turbidity currents down arbitrary natural topographic terrains. The bottom layer is a dense granular fluid which interacts with the stagnant basal topography through an erosion/deposition mechanism. Above this layer is a lighter fluid layer. There is no mass exchange at the layer interface and at the free upper surface, and the materials in both layers are treated as density preserving. The intrinsic modelling equations are written in non-dimensional form and then formulated relative to a topography-adjusted coordinate system. The mass balance equations and momentum balance equations parallel to the bottom topography are depth-averaged over the layers. The emerging governing system of equations is subsequently simplified on the basis of problem-adapted scales, in which a small parameter e, the shallowness parameter, plays a central role. The proposed ordering scheme is motivated by an earlier analysis, [1], and depends on the rheological complexities of the stress parameterizations of the two fluids. The ensuing equations are complemented by constitutive assumptions in each layer, at the bottom topography and at the layer interface.