St Venant Equations Research Articles

A finite volume scheme for solving an alternative morphodynamic model with porosity

AbstractThis work proposes an extended numerical scheme based on the finite volume method that is intended to solve a new morphodynamic model with porosity. The St-Venant equations and the Exner equation are coupled in this model, accounting for the variation of the porosity function . In the morphodynamic system, the Jacobian matrix's sign is taken into account when reconstructing the numerical fluxes using an improved Roe's approach to assure correct numerical computations.

The drainage of glacier and ice sheet surface lakes

Supraglacial lakes play a central role in storing melt water, enhancing surface melt and ultimately in driving ice flow and ice shelf melt through injecting water into the subglacial environment and through facilitating fracturing. Here, we develop a model for the drainage of supraglacial lakes through the dissipation-driven incision of a surface channel. The model consists of the St Venant equations for flow in the channel, fed by an upstream lake reservoir, coupled with an equation for the evolution of channel elevation due to advection, uplift and downward melting. After reduction to a ‘stream power’-type hyperbolic model, we show that lake drainage occurs above a critical rate of water supply to the lake due to the backward migration of a shock that incises the lake seal. The critical water supply rate depends on advection velocity and uplift (or more precisely, drawdown downstream of the lake) as well as model parameters such as channel wall roughness and the parameters defining the relationship between channel cross-section and wetted perimeter. Once lake drainage does occur, it can either continue until the lake is empty, or terminate early, leading to oscillatory cycles of lake filling and draining, with the latter favoured by large lake volumes and relatively small water supply rates.

A new one-dimensional numerical model for unsteady hydraulics of sediments in rivers

This paper presents the model UMHYSER-1D (Unsteady Model for the HYdraulics of SEdiments in Rivers 1-D), a one-dimensional hydromorphodynamic model capable of representing water surface profiles in a single river or a multiriver network, with different flow regimes considering cohesive or non-cohesive sediment transport. It has both steady and unsteady flow and sediment modules. For steady gradually varied flows, UMHYSER-1D uses the standard step method to solve the energy equation and the “NewC” scheme for the de St Venant equations. For sediment transport, UMHYSER-1D uses two methods: for long-term simulation, the unsteady terms of the sediment transport continuity equation are ignored, and a non-equilibrium sediment transport method is used. For short-term simulation, the convection–diffusion equation, with a source term arising from sediment erosion/deposition is solved using the fractional step method. The equation without the source term is solved with an implicit finite-volume method, then the equation with source term is solved. Internal boundary conditions, such as time-stage tables, rating curves, weirs, bridges, and gates are simulated. Incorporated is the active layer concept, which allows selective erosion, enabling the simulation of bed armoring. Non-cohesive sediment transport equations and cohesive sediment physical processes are applied to calculate the sediment deposition and erosion. Finally, UMHYSER-1D empirically accounts for bed geometry adjustments by using a relationship between erosion width and flow rate, an angle of repose condition for bank stability and three minimization theories. The presented validation and application cases show UMHYSER-1D’s capabilities and predicts its promising role in solving complex, real engineering cases.

Dynamic flood wave routing in prismatic channels with hydraulic and hydrologic methods

AbstractTime-dependent, unsteady flow has been studied in prismatic open channels with symmetric trapezoidal and triangular cross sections and small bottom slope. The St Venant equations without lateral inflow have been discretized in explicit as well as in implicit form and solved numerically, for unsteady, subcritical flow. The inflow hydrograph used can be applied for different flood events by adjusting its parameters accordingly. The results presented are derived from the explicit schemes Lax-Diffusive, MacCormack, Lambda as well as the implicit Preissmann scheme, and are compared to those from the Muskingum-Cunge method and the widely used commercial software HEC-RAS. The peak flow computed by the Lax-Diffusive scheme was reduced at the downstream end of the channel and the arrival time of the peak increased if compared to the other methods. The Muskingun-Cunge method forecasted the shortest peak flow arrival time at the downstream end cross section. Mass conservation computed from inflow and outflow hydrographs has been confirmed, since the maximum error did not exceed 2.60%. All codes were implemented in house using Matlab®.

Effectiveness of BIOECODS for peak flow attenuation: an appraisal using InfoWorks SD

ABSTRACTThe Bio-Ecological Drainage System (BIOECODS) is a sustainable drainage system, which adopts a “control at source” approach for urban storm water management in Malaysia. This study attempts to model a small-scale BIOECODS using InfoWorks SD. New modelling techniques are used to fully integrate the surface and on-line subsurface conveyance system, in which overland flow routing is described by a storm water management model that uses a nonlinear reservoir method and the kinematic wave approximation of the St Venant equation, and subsurface flow is described by the Horton method in conjunction with the Soil Conservation Service (SCS) curve number (CN) method. The observed water levels at primary outlets are compared with those obtained from model simulation. The modelling approach has been proven successful as the hydrographs (predicted and observed) match each other closely, with a mean error in the range of 4.58–7.32%. Results from the model showed that the BIOECODS is able to attenuate peak flow by 60–75%, and increase the lag time by 20 min within an area of <28 300 m2 when compared with a traditional drainage system.

SURCOS: A software tool to simulate irrigation and fertigation in isolated furrows and furrow networks

A software tool useful for the numerical computation of surface irrigation and fertigation in furrows and furrow networks was developed. The model solves the complete one-dimensional St-Venant equations together with the transport equation of a passive solute. The flow equations and the solute advection are solved with a high resolution TVD explicit Eulerian scheme. The solute dispersion is solved with a centered implicit Eulerian scheme to avoid further restriction in the allowable time step. The computational speed of the model is high in isolated furrows. In cases of large furrow networks over extended irrigation times the model is slower but affordable computational speed is achieved. The computational model has been designed to be robust, intuitive and able to supply useful visual results. Both the executable and the source code, as well as the examples presented can be downloaded, edited and distributed under a BSD type license.

Cyclic steps and roll waves in shallow water flow over an erodible bed

AbstractThe St Venant equations in conjunction with a phenomenological law for erosion are used to explore the nonlinear dynamics of cyclic steps – linearly unstable bedform patterns which emerge when uniform flow over an erodible bed becomes supercritical. The instability saturates by blocking the overlying flow and creating hydraulic jumps just downstream of the steepest part of the steps. Near onset, steadily migrating, nonlinear step patterns are constructed and shown to suffer a short-wavelength secondary instability that ‘roughens’ the bed and renders the staircase patterns less regular and time-dependent. An eddy viscosity is needed to regularize both the onset of the primary steps and the secondary instabilities. Further beyond the critical Froude number, the steps block the flow sufficiently to arrest erosion significantly, creating complicated patterns mixing migrating steps and stationary bedforms. The reduction in flux also stabilizes roll waves – a second, hydrodynamic instability of uniform supercritical flow. It is further shown that roll waves are purely convective instabilities, whereas cyclic steps can be absolute. Thus, in the finite geometries of the laboratory or field, it may be difficult to excite roll waves. On the other hand, the complicated spatiotemporal patterns associated with the cyclic-step instability should develop naturally. The complicated patterns resulting from the secondary instability do not appear to have been observed experimentally, calling into question the validity of the model.

Temporal variability of the selected flood parameters in the Biebrza River valley

Temporal variability of the selected flood parameters in the Biebrza River valley The paper presents the application of the hydrodynamic one-dimensional model of river flow based on St Venant equations for calculation of annual floods characteristics in the Lower Biebrza River Basin situated in the Northeast of Poland. This model was combined with the Digital Elevation Model (DEM) for determining of area of flooding, its mean depth and flood volume. The model was calibrated using measurements of flood extent and verified comparing calculated flood extent to satellite images. The water level values calculated with the numerical model of flood-flow for cross-sections were next used to determine the digital model of the floodwater table in the valley. Then, inundation extent maps and water depth maps were calculated for whole area of the valley by overlaying the DEM and water table layers. This procedure, flood simulations with the hydrodynamic model and GIS analysis for determinations of inundation extent, was repeated for each day of the analyzed period of 1966-2000. A set of each year maximum calculated areas of flooding with related mean water depths and flood volumes was created and it was submitted to the trend analysis in order to determine possible change tendencies. Statistical analysis of three above described time series of annual flood characteristics was conducted with the use of moving average technique and linear regression method. Moving averages were calculate in two variants: with time step of 5 years and of 10 years. Next, linear trends were calculated by linear regression method. The results indicate that the developed hydrodynamic model of river flow combined with the DEM was useful for the calculation of annual maximum floods in the Lower Biebrza Valley and their parameters. Analysis of calculated flood parameters variability within the analyzed time period of 1961-2000 show decreasing trends for investigated area. This means that results of climate changes are perceptible in investigated area and they form serious danger to riparian wetlands.

Superposition of local zoom models and simultaneous calibration for 1D–2D shallow water flows

We address the problem of coupling 2D shallow water equations with 1D shallow water equations (St-Venant equations), as applied to river-floodplain flows. Mathematical coupling conditions are derived classicaly from the 3D Navier–Stokes equations by integrating over the vertical wet section, when overflowing occurs. It leads to extra source terms in the 1D equations. Next we assume to be in a variational data assimilation context, then the optimal control process allows to couple both models and assimilate data simultaneously (Joint Assimilation Coupling algorithms). Two different versions of JAC algorithms are presented and compared. In a numerical test case, we superimpose the local 2D model on the 1D global model. The results show the efficiency of the present simultaneous superposition–assimilation approach.

Identification of compound channel flow parameters

Identification of compound channel flow parametersOptimization methods are used to estimate parameters required for routing floods through open compound channels. Besides initial and boundary flow conditions, data required especially include, cross-sectional area (A) of flow and conveyance (K) as functions of flow depth (y) for a representative cross-section of the study reach. Thus, instead of optimizing upon channel's geometric and hydraulic parameters, optimization is performed upon non-physical parameters in assumedA(y) andK(y) relationships. The optimization method selected for this application is the Nelder and Mead Simplex Algorithm. The objective function is expressed in terms of the relative differences between observed and simulated stages and discharges, which are evaluated based on the complete numerical solution of St Venant equations. This approach to formulating the optimization problem was applied to unsteady flow data sets for an experimental reach of the River Main in Northern Ireland. Based on statistical analysis, simulated and observed stages were found to be in good agreement.

The natural volume method (NVM): Presentation and application to shallow water inviscid flows

AbstractIn this paper a fully Lagrangian formulation is used to simulate 2D shallow water inviscid flows. The natural element method (NEM), which has been used successfully with several solid and fluid mechanics applications, is used to approximate the fluxes over Voronoi cells. This particle‐based method has shown huge potential in terms of handling problems involving large deformations. Its main advantage lies in the interpolant character of its shape function and consequently the ease it allows with respect to the imposition of Dirichlet boundary conditions.In this paper, we use the NEM collocationally, and in a Lagrangian kinematic description, in order to simulate shallow water flows that are boundary moving problems. This formulation is ultimately shown to constitute a finite‐volume methodology requiring a flux computation on Voronoi cells rather than the standard elements, in a triangular or quadrilateral mesh. St Venant equations are used as the mathematical model. These equations have discontinuous solutions that physically represent the existence of shock waves, meaning that stabilization issues have thus been considered. An artificial viscosity deduced from an analogy with Riemann solvers is introduced to upwind the scheme and therefore stabilize the method. Some inviscid bidimensional flows were used as preliminary benchmark tests, which produced decent results, leading to well‐founded hopes for the future of this method in real applications. Copyright © 2008 John Wiley &amp; Sons, Ltd.

The residual formulation of the method of manufactured solutions for computationally efficient solution verification*

The method of manufactured solutions (MMS) is a solution verification methodology for determining whether the implementation of a discretization method is achieving its theoretical order of accuracy. This methodology combines the advantages of grid refinement studies and comparison with exact solution, by modifying the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and comparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be investigated. However, in its current form, converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, by comparing the MMS to the method for determining the theoretical order of accuracy of a discretization method, the residual formulation of the MMS is developed. This new formulation only requires that the residual of the discretized governing equations to be calculated and not the solution to the discretized equations, thus avoiding the computational cost and difficulties inherent in obtaining converged solutions. Furthermore, since only the residuals are interrogated, individual components of the flow solver can be tested, in isolation, allowing the MMS to be used more effectively in locating errors within the code. This new approach is demonstrated to yield the same order of accuracy as the original MMS using three different cases—one-dimensional porous media equation, one-dimensional St Venant equations and two-dimensional unstructured Navier–Stokes simulations.

4-Point FDF of Muskingum method based on the complete St Venant equations

Using the Froude number, a nonlinear convection–diffusion equation was derived from the St Venant equations of continuity and momentum. It was made applicable to discrete space using a mixing-cell method, resulting in a first-order nonlinear ordinary differential equation. A 4-point finite difference scheme was used to solve the first-order nonlinear ordinary differential equation to achieve a nonlinear algebraic form that is very similar in structure to the traditional Muskingum form thus its naming as the ‘4-point FDF of Muskingum Method (4-point FDF M-M)’. In this method, the space interval Δ x is a characteristic channel length and a variable that is dependent on flow discharge, water depth, and flow velocity. An iterative search method was applied to simultaneously obtain the flow discharge and the optimal space interval Δ x. The method developed was tested with both synthetic numerical examples and observed events and the results were compared with those of the Lambda scheme and the method of characteristics. The outflow hydrographs produced by this new 4-point FDF M-M were of comparable accuracy. The parameters used in the new method are based on the physical attributes of the channel and thus do not need calibration as required for the Muskingum method.

Pressure transient control: Part II—simulation and design of a positive surge protection device for building drainage networks

Air pressure transient propagation within building drainage and vent systems may feature either, or both, positive and negative pressure changes in response to decelerating or accelerating flow conditions. Local venting, or the use of inwards relief air admittance valves, has traditionally suppressed negative transients. Positive transients have only been dealt with by the use of open roof terminations. This paper introduces an approach to positive air pressure transient suppression based upon the use of a flexible, variable volume containment vessel capable of reducing the rate of change of the entrained air following a system surcharge. The experimental programme to provide proof of concept, together with the simulation of the device utilizing the method of characteristics solution of the St Venant equations is detailed. Recommendations as to the suitability of bag materials and installation choices are presented, along with comparisons of surge relief efficiency. Practical application: The control of air pressure transients in building drainage systems has been limited due to the need to attenuate positive pressure propagation via an open termination at roof level - a poor solution as the transient will have affected all system trap seals before reaching the relief vent. The Positive Air Pressure Attenuator - an expandable bag that controls the rate of change of entrained airflow within the system - reduces the possibility of trap-seal loss due to positive transient propagation. Potentially this is a major contribution to vent system design that could revolutionize 150 years of design methodology.

Simulation of integrated surface-water/ground-water flow and salinity for a coastal wetland and adjacent estuary

The SWIFT2D surface-water flow and transport code, which solves the St Venant equations in two dimensions, was coupled with the SEAWAT variable-density ground-water code to represent hydrologic processes in coastal wetlands and adjacent estuaries. A sequentially coupled time-lagged approach was implemented, based on a variable-density form of Darcy's Law, to couple the surface and subsurface systems. The integrated code also represents the advective transport of salt mass between the surface and subsurface. The integrated code was applied to the southern Everglades of Florida to quantify flow and salinity patterns and to evaluate effects of hydrologic processes. Model results confirm several important observations about the coastal wetland: (1) the coastal embankment separating the wetland from the estuary is overtopped only during tropical storms, (2) leakage between the surface and subsurface is locally important in the wetland, but submarine ground-water discharge does not contribute large quantities of freshwater to the estuary, and (3) coastal wetland salinities increase to near seawater values during the dry season, and the wetland flushes each year with the onset of the wet season.

Transient identification of defective trap seals

The prevention of the ingress of contaminated air into habitable spaces has been a central concern in the design of building drainage and vent systems since the 1850s. Failure to provide the necessary protection, primarily through appliance water trap seals, was shown to have been a significant contributor to the Severe Acute Respiratory Syndrome (SARS) epidemic spread in Hong Kong, China, in 2003. Prevention depends upon both good design, to limit the air pressure transients propagated within the system, and good maintenance. Good practice has evolved to include wherever practical long radius bends to minimize the local air pressure changes. Similarly, modern usage of drainage networks includes the random discharge of disposable sanitary and hygiene products that may interact with undesirable rapid changes in flow direction and thereby contribute to local surcharge conditions. However, a maintenance regime within a large complex building network also requires a degree of prior knowledge of possible defective appliance trap seal locations. This paper proposes the use of air pressure transient simulation and transient response measurement to identify, during periods of system non-use, the location of depleted trap seals. The simulation of air pressure transient propagation within building drainage and vent systems is detailed, based on the proven capabilities of the method of characteristics solution of the St Venant equations. Validation of system transient response simulation is presented together with indications of the impact of a defective appliance trap seal on the system response to an applied low-amplitude, short-duration, pressure surge. Comparison of the defect free and defective system response is shown to be sufficiently discriminatory to allow the identification of an appliance trap seal in need of replenishment. Identification of persistent trap seal depletion would allow local installation of active control devices to prevent further failures – including the addition of either air admittance valves to limit trap seal loss to negative transients or variable volume containment positive air pressure attenuators to limit trap seal loss due to positive pressure transients.

DEM-based modelling of surface runoff using diffusion wave equation

A digital elevation model (DEM)-based overland flow routing model was developed for computation of surface runoff for isolated storm events. The model operates on a grid or cell basis and routes the rainfall excess generated over the cells, following the DEM-derived drainage paths, to the catchment outlet. The rainfall excess for each cell of the catchment was computed using the Philip two-term infiltration model utilizing the physical properties of the cell. The overland flow was described by a finite volume-based numerical solution of the diffusion wave approximation of the St Venant equations. The cell physical properties, such as topographic characteristics, land use, soil, etc., were extracted from published maps for discretized cells of the catchment using a Geographic Information System. The results of model application indicate that the model satisfactorily predicted the runoff hydrograph. The cell-based structure of the model allowed for generation of spatially distributed catchment information in terms of the model-computed variables, such as the depth of flow and discharge.

River networks and groundwater flow: a simultaneous solution of a coupled system

Coupled hydrologic modeling of surface and subsurface systems are necessary to obtain a better understanding of hydrologic pathways in a watershed. In early studies, the coupling is achieved by linking the two systems through an iterative process in which convergence of parameters linking the two domains are defined as the criteria for an acceptable solution. In this study, we propose a new approach based on the simultaneous solution of coupled channel and groundwater flow systems. The proposed method is based on the idea of solving a single global matrix at once rather than solving separate matrices for each flow domain while improving the solution iteratively. The proposed solution technique is implemented by coupling a one-dimensional channel network flow model that uses the complete form of the St Venant equation with a two-dimensional vertically averaged groundwater flow model. The proposed approach provides a more efficient solution strategy for the coupled flow problem and demonstrates an effective process of integrating the two hydrologic systems.

A GIS based distributed rainfall–runoff model

A grid or cell based process oriented distributed rainfall–runoff model capable of handling the catchment heterogeneity in terms of distributed information on landuse, slope, soil and rainfall is developed and applied to isolated storm events in several catchments. Model inputs such as slope, flow direction and overland flow sequencing (drainage path) are generated for each cell of the catchment using a digital elevation model and information about landuse, soil, etc. were derived through digital analysis of satellite data and published information. The input variables for each cell area are provided to the model through geographic information system. Infiltration in cell areas is computed by Philip two-term infiltration model, the parameters of which were determined mainly through the information on soil type in the cell. The mechanics of overland flow is described by the diffusion wave approximation of St Venant equations which are numerically solved for depth of flow and runoff by the method of finite volume. The model utilizes a relationship explaining the dependence of flow resistance on depth of flow and surface roughness. Results from several catchments indicate that the model can simulate reasonably well the runoff hydrograph at the catchment outlet. The model also realistically predicts temporal variation of the spatial distribution of flow depth and runoff over the catchment. Also the grid or cell based structure of the model allows studying the effect of catchment modifications in terms of soil and landuse changes on spatial and temporal distribution of the runoff. However, the proposed model has the limitation that the values of some parameters of the model need to be obtained through calibration.

Positive air pressure transient propagation in building drainage and vent systems

A major objective of drainage research over the past 100 years has been a reduction in the complexity of building drainage vent systems associated with the retention of appliance trap seals. Further simplification requires that the system operation is recognized as time dependent, where changes in water flow conditions result in the propagation of air pressure transients. Negative pressure transients that reduce trap seal levels by induced siphonage are well understood. More problematic is the propagation of positive air pressure transients, generated by stack or branch surcharge. The paper identifies the sources of positive air pressure transient propagation and demonstrates that such transients may be described by the St Venant equations of unsteady flow. Solutions are provided based on the proven finite difference methods and the method of characteristics and an understanding of the boundary conditions that represent the constituent components of a building drainage and vent system.