Development of an algorithm for calculating stable solutions of the Saint-Venant equation using an upwind implicit difference scheme

The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that: 1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region; 2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC; 3) the proposed computational algorithm is stable; 4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable); 5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time. Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.

Urban hydrological model (UHM) developed for urban flash flood simulation and analysis of sensitivity of flood intensity to urbanization

A two-dimensional raster gridded urban hydrological model has been developed to simulate the hydrologic response to urban land surfaces with consideration of the hydraulic characteristics of urban areas, and to produce mappings of urban inundation associated with rainstorms. The model is forced using radar-observed QPEs, in conjunction with parameter sets of land use and land cover (LULC) derived from satellite multispectral images and high spatial resolution GIS datasets relating to urban hydrology and land surface hydrodynamic properties. Urban drainage flow capacity is derived from a GIS road-network dataset using a generalization method. Submodels deduce runoffs of both the impervious and the pervious. Methodologically, the D8 method (eight slope directions) is used to derive the channel paths for gravity-driven nondispersive streamflow, which its hydrodynamics can be described by the hydraulic model based on simplified 1D − 2D St. Venant equation. A case study was undertaken to reproduce the urban flash flooding that occurred in Beijing following thunderstorms on 21 July 2012. The model results were verified qualitatively using media reports of the flooding. Through manipulation of model parameters, the test on the sensitivity of flash flood intensity to urban LULC variability and drainage network settings revealed the following: 1) flood intensity is enhanced slightly if the current urban LULC is substituted with a pure impervious, 2) increasing the pervious surface area (PSA) attenuates flood intensity considerably, and 3) flash flood intensity will increase by 30–60% in the absence of an underground drainage system.

Automated Detection of Instability-Inducing Channel Geometry Transitions in Saint-Venant Simulation of Large-Scale River Networks

A new sweep-search algorithm (SSA) is developed and tested to identify the channel geometry transitions responsible for numerical convergence failure in a Saint-Venant equation (SVE) simulation of a large-scale open-channel network. Numerical instabilities are known to occur at “sharp” transitions in discrete geometry, but the identification of problem locations has been a matter of modeler’s art and a roadblock to implementing large-scale SVE simulations. The new method implements techniques from graph theory applied to a steady-state 1D shallow-water equation solver to recursively examine the numerical stability of each flowpath through the channel network. The SSA is validated with a short river reach and tested by the simulation of ten complete river systems of the Texas–Gulf Coast region by using the extreme hydrological conditions recorded during hurricane Harvey. The SSA successfully identified the problematic channel sections in all tested river systems. Subsequent modification of the problem sections allowed stable solution by an unsteady SVE numerical solver. The new SSA approach permits automated and consistent identification of problem channel geometry in large open-channel network data sets, which is necessary to effectively apply the fully dynamic Saint-Venant equations to large-scale river networks or for city-wide stormwater networks.

ON TWO MODELS RELATED TO THE FLOODING AND DEPOSITION OF THE DON DELTA

The results of modeling changes in the water surface level in the eastern part of the Taganrog Bay and the main Don branches in its delta area are presented. A numerical study of the process of saltwater inflow from the Taganrog Bay to the Don delta has been carried out. The hydrodynamics in the Taganrog Bay, as well as the saltwater transport process, are specified using the corresponding two-layer models. A system of Don arms is presented in the form of a graph, the edges of which correspond to open channels, and the vertices correspond to branching points and end nodes. The flow in the main Don branches is described by the Saint-Venant equation. It is assumed that there is no distributed lateral inflow, and the channel cross-section has a parabolic profile. Saltwater inflow into the arms is described by a one-dimensional transport equation. Boundary conditions are specified for each sleeve. At the branching nodes, conditions are set for the equality of the water levels, as well as the equality of the inflowing and outflowing discharges. The description of the algorithm of the process of flooding/drainage of the Don delta area is given. Considering the values of the depths at the nodes of the flat grid, the cells located in water or on land are determined. A logical array characterizing the type of cells (“water”, “land”) sets the configuration of the entire computational domain. Comparison of the calculation results with the observed values of salinity and water level is carried out.

Investigation of Flood Routing Using Variable Parameter Kinematic Wave Model (VPKWM) for Non-Prismatic Natural Channel in an Ungauged Basin

This research concerns about the development and application of Variable Parameter Kinematic Wave Numerical model (VPKWM) based on 1-D Saint-Venant equation, to study the behaviour of the propagation of a flood wave in Non-prismatic natural waterways in an ungauged basin. The channel slope and wetted perimeter are considered as variable because of the irregularity of the boundary of the channel and the change in magnitude of discharge. The scarcity of reliable inflow data at upstream is a serious problem for the flood routing process in an ungauged basin. In this study the inflow hydrograph and lateral inflow hydrographs are obtained using SCS-CN method as rainfall runoff model. The performance of the model assessed considering four parameters such as root mean square error (RMSE), peak discharge, peak time and total volume. The results indicated that the VPKWM for non-prismatic channel provided reasonable output compared with the observed data.

Experimental and Numerical Analysis of the Gas Flow in the Axisymmetric Radial Clearance

This paper focuses on the analysis of the gas flow in the axisymmetric mini gap bounded by the surface of the top of the labyrinth seal tooth and the surface of the body. It includes the results of experimental research and numerical calculations. Experimental research focused on the analysis of gas flow for six clearance heights in a wide range of pressure drops. Based on this research, we determined the mass flow in the clearance. Using the Saint-Venant equation, we determined the flow coefficient versus the pressure ratio upstream and downstream from the seal. Flow coefficients for various clearance heights obtained from the experiment can be divided into two data groups, the values of which differ significantly. To explain changes in the value of the gas flow coefficient for selected clearance heights, numerical analysis of the said gas flow was performed using the Fluent software. This analysis allowed us to explain the reason for the variability of the flow coefficient. This research can be the basis for determining the change of seal integrity during operation for staggered and stepped seals.

Applicability of Kinematic Wave Model for Flood Routing under Unsteady Inflow

This study implemented kinematic wave and dynamic wave approximation of flood routing for a prismatic rectangular channel. The results of the two methods were compared by differences in maximum flow depth, and the applicability of kinematic wave equation was discussed. The influences of hydraulic and geometrical factors on the applicability of kinematic wave equation were considered. It was found that a portion of the numerical results violated existing criteria used to indicate the applicability of kinematic wave equation, particularly when geometrical and hydraulic factors were considered together. This is because the characteristics of upstream inflow were rarely or incompletely considered in these criteria. Therefore, the present study proposed a new criterion. The theoretical influence of all factors was considered using three parameters, namely, KF02, ηts/T0′ and Qbottom/Qpeak (K, F0, ηts, T0′, Qbottom, and Qpeak represent the kinematic wave number, Froude number, the time span of discharge exceeding 90% of maximum discharge in hydrograph, wave travel time in the channel, base flow discharge, and peak discharge, respectively, while the subscript 0 represent the value of reference discharge). The influences of these three parameters were illustrated by the momentum equation of one-dimensional Saint-Venant equation. The numerical results showed that the value of ηts/T0′ (KF02)D could be used to determine the relative error ξh of kinematic wave equation. In addition, for each Qbottom/Qpeak the value of ηts/T0′ (KF02)D used to depict the same relative error ξh was different. This new criterion was validated using two real case studies, and it showed a good performance.

Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.

1D and 2D Debris Flow Modeling with HEC-RAS

<p>The Hydrologic Engineering Center River Analysis System (HEC-RAS) is a free software developed by the United States Army Corps of Engineers for simulating hydraulics, sediment transport, and water quality.  We present on the recent and ongoing developments of non-Newtonian flow and mobile bed modeling within HEC-RAS. The numerical models solve the in one-dimensional (1D) St. Venant equation, and the two-dimensional (2D) Diffusion Wave and Shallow Water Equations with corrections and modifications for non-Newtonian flows and steep slopes. The equations are solved using a combination of Finite-Difference and Finite-Volume methods on unstructured grids (for 2D). Several flow resistance laws are implemented including the Bingham, Coulomb, Herschel-Bulkley, and Voellmy models. Sediment transport is simulated in 2D with a total-load advection-diffusion model with corrections for steep slopes and high concentrations. A subgrid modeling approach is utilized for hydraulics and sediment transport, which allows for larger computational cells while maintaining accuracy. The numerical models have been verified with analytical test cases, and validated with small and large scale physical experiments and field applications. The results demonstrate the applicability of HEC-RAS as a tool for natural hazard studies involving non-Newtonian flows.</p>