Mathematical modeling of a river stream based on a shallow water approach

Flood control problems as well as problems connected with wastewater discharge into rivers are issues of current importance. Depth averaged shallow water equations are used to model flows where water depth is much less than the horizontal dimension of the computational area and the free surface greatly influences the flow

An unstructured finite volume model for unsteady turbulent shallow water flow with wet-dry fronts: numerical solver and experimental validation

[Resumen] El principal objetivo de esta tesis es aplicar las ecuaciones de aguas someras bidimensionales a diferentes flujos en lámina libre, centrándose especialmente en aquellos problemas en los que o bien el tratamiento de la turbulencia o bien el tratamiento del frente seco-mojado son de especial relevancia. Para ello se ha desarrollado un código de volúmenes finitos que resuelve las ecuaciones de aguas someras acopladas con diferentes modelos de turbulencia. El código incluye un modelo parabólico de viscosidad turbulenta, un modelo algebraico de longitud de mezcla y 3 versiones del modelo k-ε para aguas someras. Al mismo tiempo se ha propuesto e incluido un modelo de tensiones algebraicas para aguas someras. El código desarrollado se ha utilizado para simular el flujo en 4 aplicaciones prácticas diferentes que incluyen el oleaje de onda larga, el flujo inducido por la marea en un estuario, el flujo en canal con un codo de 90º, y el flujo en escalas de peces de hendidura vertical. En todos los casos los resultados numéricos se han comparado con datos experimentales.[Resumo] O principal obxetivo de esta tese é aplicar as ecuacións de augas someras bidimensionais a diferentes fluxos en lámina libre, centrándose especialmente naqueles problemas nos que ou ben o tratamento da turbulencia ou ben o tratamento do frente seco-mollado son de especial relevancia. Desenvolveuse un código en volúmenes finitos que resolve as ecuacións de augas someras acopladas a diferentes modelos de turbulencia. O código inclúe un modelo parabólico de viscosidade turbulenta, un modelo alxebraico de lonxitude de mezcla e tres versións do modelo k-ε para augas someras. Ó mesmo tempo, proponse un modelo de tensións alxebraicas para augas someras. O código realizado utilizouse en 4 aplicacións prácticas diferentes que inclúen a oleaxe de onda larga, o fluxo de marea nun estuario, o fluxo en canal cun codo de 90º, e o fluxo en escalas de peixes de fendidura vertical. En todolos casos os resultados numéricos compáranse con datos experimentais.[Abstract]The main goal of this thesis is to apply the depth averaged shallow water equations to several free surface flows in which the turbulence modelling and the treatment of wet-dry fronts are of special interest. In this context, four different flows have been studied: the generation, propagation and reflection of shallow waves in a 1D flume, the tidal flow in a coastal estuary, the flow in an open channel with a 90º bend, and the flow in vertical slot fishways. The aim of this work is to investigate, for the considered applications, which flow features can be resolved by a depth averaged model and which features are beyond the capabilities of the equations. In order to do so, a finite volume solver for the 2D-SWE, coupled with several depth averaged RANS turbulence models, has been developed. A depth averaged mixing length model and a k-ε model for shallow waters have been included in the code, and a depth averaged algebraic stress model has been proposed. At the same time experimental data has been used to analyse the characteristics of the flow as well as to compare with the numerical results

A simple and efficient unstructured finite volume scheme for solving the shallow water equations in overland flow applications

This paper presents the Decoupled Hydrological Discretisation (DHD) scheme for solving the shallow water equations in hydrological applications involving surface runoff in rural and urban basins. The name of the scheme is motivated by the fact that the three equations which form the two-dimensional shallow water system are discretised independently from each other and thus, the numerical scheme is decoupled in a mathematical sense. Its main advantages compared to other classic finite volume schemes for the shallow water equations are its simplicity to code and the lower computational cost per time step. The validation of the scheme is presented in five test cases involving overland flow and rainfall-runoff transformation over topographies of different complexity. The scheme is compared to the finite volume scheme ofRoe [1986], to the simple inertia formulation [Bates et al., 2010], and to the diffusive wave model. The test cases show that the DHD scheme is able to compute subcritical and supercritical flows in rural and urban environments, and that in overland flow applications it gives similar results to the second order scheme of Roe with a lower computational cost. The results obtained with the simple inertia and diffusive wave models are very similar to those obtained with the DHD scheme in rural basins in which the bed friction and topography dominate the flow hydrodynamics but they deteriorate in typical urban configurations in which the presence of supercritical flow conditions and small scale patterns boost the relevance of the inertial terms in the momentum equation

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method

Solution of fully-coupled shallow water equations and contaminant transport using a primitive variable Riemann solver and a semi-discrete SUPG method

In the present dissertation, a finite volume and a finite element model are developed and tuned for the solution of the fully-coupled two-dimensional Shallow Water and Contaminant transport Equations with arbitrary bed topography and wetting-drying fronts. A Riemann-solver finite volume scheme, using primitive variables rather than conserved variables, and a semi-discrete Streamline Upwind Petrov-Galerkin (SUPG) method in finite element context are applied to compare the performance of these two numerical models. The Riemann-solver scheme is based on the unstructured finite volume discretization using primitive-variable Roe-flux approximation with an entropy fix. Second-order accuracy in space and time, an implicit scheme based on Newton-iterative algorithm, and an Euler explicit scheme are applied for the finite volume model. For the SUPG finite element model, a new exact source-term balancing method is introduced in this study. This new balancing method satisfies the C-property for both still water and dry regions on a non-flat bed. Two different stabilization terms are applied to compare their performance for wet-bed problems and a shock-capturing scheme is implemented to accommodate shock wave fronts. Linear triangular elements are used to decompose the computational domain and a second-order backward differentiation (BDF2) implicit method is used for the time integration. The resulting nonlinear system is solved using a Newton-type method where the linear system is solved at each step using the Generalized Minimal Residual (GMRES) algorithm. Both finite volume and finite element formulations are applied to moving-boundary problems on fixed numerical meshes. In order to examine the accuracy and robustness of the present scheme to predict the flow variables and contaminant transport, numerical results are verified by several test cases. These cases include wet and dry dam break problems, evolution of a dam break wave with an obstacle downstream of the dam, oscillation of a bead of water in a parabolically-shaped basin, supercritical flow in a constricted channel, as well as advection and diffusion of contaminant with the flow. The scenario of contaminant transport in a notional river is also simulated to demonstrate that the present work can be implemented on practical applications involving flooding and contaminant transport

Development and Validation of Turbulence Closures for Three-Dimensional Reynolds-Averaged and Partially-Averaged Navier-Sotkes Application to Open-Channel Flow in Bends and Meanders

Programa Oficial de Doutoramento en Enxeñaría Civil . 5011V01[Resumo] Desenvolvemento e validación de modelos de turbulencia para Reynolds-Averaged e Partially-Averaged Navier-Stokes en tres dimensións. Aplicación a fluxo en canais abertos curvos e meandriformes Entender e ser quen de predicir o comportamento do fluxo en canles abertos curvos e meandriformes é un elemento crucial para a enxeñería fluvial. O presente documento analiza este tipo de casos mediante a utilización de modelos computacionais tridimensionais e non hidrostáticos baseados nas ecuacións de conservación da masa e o momentum de Navier-Stokes. Dado que a turbulencia é omnipresente en fluxos ambientais e que a sua influencia é extremadamente relevante, este traballo analiza o efecto dunha variedade de modelos de peche para o termo que encapsula a aportación das fluctuacións turbulentas en tres escenarios: un canal curvo de 270°, un canal meandriforme consistente nunha sucesión de dúas curvas contrapostas e un meandro infinito. A análise céntrase específicamente na descripción do fluxo secundario, os mecanismos de xeración e regulación das estruturas coherentes e a súa influencia na distribución das tensións tanxenciais. Os modelos empregados nesta investigación pódense encadrar dentro de tres familias fundamentais que se diferenzan no xeito en que resolven ou aproximan as tensións turbulentas: URANS, PANS e LES. As prediccións obtidas nestas simulacións foron comparadas e validadas numérica e experimentalmente. A influenza nos resultados de parámetros numéricos como a condición de contorno de entrada e a discretización do termo convectivo da ecuación de momentum recibiu particular atención. Os resultados mostran que determinadas configuracións de PANS predín particularmente ben os fluxos primario e secundario e a estrutura da turbulencia con respecto aos datos experimentais e os resultados obtidos con LES. URANS combinado co modelo de turbulencia k-ε produce simulacións robustas e fiábeis para os escenarios considerados, pero manifesta deficiencias na predicción dalgúns mecanismos do fluxo secundario e a cuantificación da enerxía cinética turbulenta debido ao exceso de disipación. Implementáronse modelos non lineais baseados no concepto de viscosidade turbulenta en cobinación con URANS; os resultados foron irregulares e, en xeral, non melloraron a capacidade predictiva de k-ε. Os resultados sinalan que o desenvolvemento da turbulencia e a ‘memoria’ previa do fluxo tras percorrer sucesivas curvas alternas en canais meandriformes son cruciais para definir a estrutura e magnitude do fluxo secundario. Esta investigación amosa como estruturas coherentes formadas en curvas consecutivas interaccionan entre si e son recicladas entre un meandro e o seguinte, o cal ten importantes repercusións para o transporte de sedimentos e contaminantes en fluxos ambientais. As fluctuacións turbulentas identificadas nos canais en curva son intensamente anisotrópicas e non poden ser descritas con rigor e exclusivamente por modelos baseados en hipóteses de turbulencia isotrópica. Este traballo servirá como base a novas liñas de investigación para o desenvolvemento de modelos dinámicos inspirados en PANS que produzan ferramentas predictivas tridimensionais rápidas, fiables e precisas para enxeñería fluvial.[Resumen] Desarrollo y validación de modelos de turbulencia para Reynolds-Averaged y Partially-Averaged Navier-Stokes en tres dimensiones. Aplicación a flujo en canales abiertos curvos y meandriformes Entender y ser capaz de predecir el comportamiento del flujo en canales abiertos curvos y meandriformes es un elemento crucial en ingeniería fluvial. El presente documento analiza este tipo de casos mediante la aplicación de modelos computacionales tridimensionales y no hidrostáticos basados en las ecuaciones de conservación de masa y momentum de Navier-Stokes. Dado que la turbulencia es omnipresente en flujos ambientales y que su influencia es extremadamente relevante, se analizó el efecto producido por diferentes modelos de cierre para el término turbulento en tres escenarios: un canal curvo de 270°, un canal meandriforme consistente en una sucesión de dos curvas alternas y un meandro infinito. El análisis se centra específicamente en la descripción del flujo secundario, los mecanismos de generación y la evolución de las estructuras coherentes y la interdependencia de lo anterior con las tensiones tangenciales. Los modelos empleados en este trabajo se pueden encuadrar en tres familias principales atendiendo al modo en que resuelven o aproximan las tensiones turbulentas: URANS, PANS y LES. Las predicciones obtenidas en dichas en estas simulaciones han sido comparadas y validadas numérica y experimentalmente. La influencia de parámetros como la condición de contorno de entrada del flujo y la discretización del término convectivo de la ecuación de momentum recibió particular atención. Los resultados revelan que determinadas configuraciones de PANS predicen con acierto los flujos primario y secundario, así como la estructura de la turbulencia, con respecto a los datos experimentales y simulaciones hechas con LES. URANS combinado con el modelo de turbulencia k-ε produce simulaciones robustas y fiables para los escenarios considerados, en especial del flujo primario, pero manifiesta deficiencias en la predicción de algunos mecanismos del flujo secundario y la cuantificación de la energía cinética turbulenta debido al exceso de disipación. Se han aplicado modelos no-lineales para la predicción de la viscosidad turbulenta en combinación con URANS; los resultados son irregulares y, en general, no mejoraron la capacidad predictiva de k-ε. Los resultados señalan que el desarrollo de la turbulencia y la ‘memoria’ previa del flujo tras recorrer sucesivas curvas alternas en canales meandriformes son clave para definir la estructura y magnitud del flujo secundario. Esta investigación muestra como estructuras coherentes formadas en curvas consecutivas interactúan entre sí y son recicladas entre un meandro y el siguiente, lo cual tiene importantes repercusiones para el transporte de sedimentos y contaminantes en flujos ambientales. Las fluctuaciones turbulentas identificadas en los canales en curva son intensamente anisotrópicas y no pueden ser descritas con rigor exclusivamente por modelos basados en hipótesis de turbulencia isotrópica. Este trabajo servirá de base a nuevas líneas de investigación sobre el desarrollo de modelos dinámicos inspirados en PANS capaces de producir herramientas predictivas en tres dimensiones rápidas, fiables y precisas para la ingeniería fluvial.[Abstract] Development and validation of turbulence closures for three-dimensional Reynolds-Averaged and Partially-Averaged Navier-Stokes. Application to open-channel flow in bends and meanders Understanding and being able to predict curved and meandering flow behaviour is key to river engineering. This work analyses this kind of flows using three-dimensional, non-hydrostatic computational models. Given the ubiquitous presence of turbulence in environmental flows and its crucial importance, different turbulence closures are applied to three curved and meandering open-channel flow scenarios: a single 270° bend, a two-bends meandering channel and an infinite meander. The analysis particularly focuses on the structure of the secondary flow, the mechanisms of generation and modulation of coherent structures, and their influence on the shear stresses. The modelling approaches utilised during this research fall within three fundamental families, URANS, PANS and LES. The fundamental difference among them is their approach to solve or model the turbulence stresses. The predictions provided by these models were tested, compared and validated. The influence of several modelling parameters – besides the turbulence closure itself - on their performance is also analysed, with a special emphasis on the discretisation scheme for the convective term and the inflow conditions. The results show that some configurations of PANS remarkably match the available experimental and LES datasets regarding the prediction of primary and secondary flow and turbulence structure. URANS combined with a k-ε turbulence closure provides a very robust and consistent forecasting of the scenarios under investigation, particularly the primary flow, while lacking on the prediction of some of the mechanisms driving the secondary motion and the turbulence structure. Non-linear eddy viscosity models were tested with irregular results, and overall failing to improve k-ε performance. Turbulence development and the memory of prior bends in meandering channels seem to be key to the secondary flow structure. It was found that coherent structures of consecutive bends interact with each other, which has important repercussions to sediment and pollution transport in environmental flows. It was also found that the turbulent fluctuations within the bends are strongly anisotropic and cannot be well described by models reliant on isotropic assumptions. Future lines of work based on this research could provide dynamic PANS models and new turbulence closures that will generate quick, reliable, and accurate three-dimensional tools for river engineering modelling

A simple and efficient unstructured finite volume scheme for solving the shallow water equations in overland flow applications

An edited version of this paper was published by AGU. Copyright (2015) American Geophysical Union.This paper presents the Decoupled Hydrological Discretisation (DHD) scheme for solving the shallow water equations in hydrological applications involving surface runoff in rural and urban basins. The name of the scheme is motivated by the fact that the three equations which form the two-dimensional shallow water system are discretised independently from each other and thus, the numerical scheme is decoupled in a mathematical sense. Its main advantages compared to other classic finite volume schemes for the shallow water equations are its simplicity to code and the lower computational cost per time step. The validation of the scheme is presented in five test cases involving overland flow and rainfall-runoff transformation over topographies of different complexity. The scheme is compared to the finite volume scheme ofRoe [1986], to the simple inertia formulation [Bates et al., 2010], and to the diffusive wave model. The test cases show that the DHD scheme is able to compute subcritical and supercritical flows in rural and urban environments, and that in overland flow applications it gives similar results to the second order scheme of Roe with a lower computational cost. The results obtained with the simple inertia and diffusive wave models are very similar to those obtained with the DHD scheme in rural basins in which the bed friction and topography dominate the flow hydrodynamics but they deteriorate in typical urban configurations in which the presence of supercritical flow conditions and small scale patterns boost the relevance of the inertial terms in the momentum equations.Peer ReviewedPostprint (published version