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![Elementary fluxes, ζ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta$$\end{document}, leaving a point and fluxes between two nodes, ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}](profile/Riadh-Ata/publication/283031773/figure/fig1/AS:941223338594333@1601416539378/Elementary-fluxes-zdocumentclass12ptminimal-usepackageamsmath.gif)
Elementary fluxes, ζ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta$$\end{document}, leaving a point and fluxes between two nodes, ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}
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In this work, a recent residual distribution scheme and a second-order finite volume method are compared to model the transport of a pollutant in free surface flows. The phenomenon is described in two dimensions using the shallow water (SW) system augmented by a scalar conservation law for the pollutant transport. The two numerical methods are deve...
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Experimental investigation of a mechanically generated regular surface wave over turbulent current is carried out to understand the role of surface wave in combined flow. The recorded data sets enable obtaining reliable statistical parameters characterizing wavefield with depth. Variation in velocity power spectra along the flow depth in the mixed...
Citations
... For these methods either structured or unstructured grid can be used (Hou et al. 2013;Hou et al. Chapter 2 18 2014; Pavan et al. 2015), while time marching schemes can be explicit, implicit or semi-implicit. There also exists an ongoing development of robust and highresolution numerical methods Simons 2020). ...
This thesis studies the numerical modelling of a range of hydrodynamic, solute transport and biochemical reaction processes in shallow water environment, combining both Euler and Lagrangian techniques. The research consists of two parts: the rapid flow modelling and the solute transport modelling. In the rapid flow modelling, the research is focused on demonstrating the impact of modifying the Boussinesq coefficient when simulating rapid flows in various situations. The traditional Alternating Direction Implicit (ADI) scheme has been proven to be incapable of modelling trans-critical flows. Its inherent lack of shock-capturing capability results in spurious oscillations and computational instabilities. However, the ADI scheme is still widely adopted in flood modelling software, and various special treatments have been designed to stabilise the computation. Modification of the Boussinesq coefficient to adjust the amount of fluid inertia is a numerical treatment that allows the ADI scheme to be applicable to rapid flows. A comprehensive study has been undertaken to examine the impact of this numerical treatment over a range of flow conditions. A shock-capturing TVD-MacCormack model is used to provide reference results. In the solute transport modelling, a mesh-free, depth-averaged and highly robust random walk model has been developed for simulating the two-dimensional solute transport processes. The development of the random model consists of two stages. In the first stage, extensive parametric studies have been undertaken to investigate the influences of the number of particles, the size of time steps and the technique of sampling processes used in the random walk model. The model is then applied to investigate the solute oscillation along a tidal estuary subject to extensive wetting and drying during tidal oscillations. The flow velocities are interpolated from the grid-based TVD-MacCormack flow solver. Finally, the model is applied to investigate wind-induced chaotic mixing in a circular shallow basin. The effect of diffusive processes on the chaotic mixing is investigated. The results of the first stage research provide a guideline for properly presenting the outputs of the random walk method for scientific analyses. In the second stage, the random-walk model is further developed to investigate the advection, diffusion and reaction processes of non-conservative materials. First, several classical test cases are presented to showcase the capability of the random walk model in addressing the problem of continuous release of non-conservative substances. Then, the method is applied to model the BOD-DO balance along a hypothetical river. The numerical results are in good agreement with the analytical solution. Finally, the developed scheme is used to study the pollutant transport in Thames Estuary. The current model is illustrated to be able to accurately predict the interaction between multiple pollutants in real-world situations with uneven bathymetry and extensive intertidal floodplains.
... En la literatura existe mucha investigación numérica en el área del transporte de contaminantes en aguas poco profundas, aplicando métodos de diferencias finitas [6], métodos de volúmenes finitos [7,8,11,12,13], métodos de elementos finitos [4,9,10], métodos tipo-Godunov [18] y métodos de Boltzmann [14,15,16]. ...
En este artículo presentamos la aproximación del modelo acoplado de las ecuaciones del movimiento de un fluido en aguas poco profundas con la ecuación convección-difusión-reacción (CDR) del transporte de contaminantes. Dicha aproximación se realiza mediante elementos finitos de alto orden y usando métodos variacionales estabilizados de subescalas. El sistema acoplado de ecuaciones, previamente discretizado en el tiempo y linealizado, lo escribimos como una ecuación vectorial transitoria de CDR. Los métodos estabilizados de elementos finitos utilizados son los conocidos métodos de subescalas ASGS y OSS, los mismos que nos permiten usar igual interpolación para todas las incógnitas, así como tratar con flujos de convección y reacción dominantes. Consideramos la posibilidad de no linealidad tanto en el término convectivo como en el de reacción. No consideraremos el posible desarrollo de choques en la solución. Con el fin de examinar la precisión y robustez de los métodos ASGS y OSS, presentamos cuatro casos de prueba: convergencia en malla, transporte de un contaminante en una cavidad cuadrada, transporte de un contaminante en el golfo de Roses y en la desembocadura del río Guadalquivir, y el modelo depredador-presa, que puede escribirse como una ecuación vectorial de CDR transitoria con no linealidad en el término de reacción. 1 Introducción Un contaminante es aquel componente que está presente en el agua a niveles perjudiciales para la vida de los seres humanos, plantas y animales ([1], véase también los informes de la U.S. Environmental Protecction Agency, https://www.epa.gov/). La simulación del transporte de contaminantes para la predicción de la concentración de contaminantes en ríos, lagos, lagunas y regiones costeras tiene importancia estratégica en el análisis y diseño de soluciones de los problemas de contaminación ambiental del planeta. En el fenómeno físico del movimiento de contaminantes se deben distinguir tres procesos: la difusión, la convección y la reacción. La difusión es el proceso físico debido al cual el contaminante se mueve como resultado del movimiento intermolecular de las partículas de ambas sustancias, el fluido que la transporta (agua) y el contaminante. La convección es el movimiento del soluto (contaminante) debido al movimiento del agua, por lo cual si el agua permanece en reposo no hay convección. Finalmente, la reacción tiene en cuenta el posible efecto de crecimiento o decrecimiento de un contaminante por factores externos (calor, especies con las que puede combinarse); en el caso de haber distintas especies de contaminantes, la reacción puede modelar la interacción entre ellas. La formulación del problema del transporte de contaminantes se fundamenta, como la de todos los fenó-menos físicos, en las ecuaciones de equilibrio y las ecuaciones constitutivas. Las ecuaciones del transporte de sustancias disueltas o en suspensión en el flujo se basan en el principio de conservación de la masa de dichas sustancias en caso de que no haya reacción o en el modelo de crecimiento o decrecimiento en caso de que la haya. En el problema que queremos abordar, la dinámica del transporte de contaminantes involucra los modelos de flujos del campo de velocidades en aguas someras y del transporte, difusión y reacción de contaminantes, los cuales tienen características de la ecuación transitoria de convección, difusión y reacción (CDR).
... Moreover, the discussion allows to clarify the equivalence between the formulation of the schemes in the shallow water context and the classical RD formulation for a scalar equation (see [4,6]). ...
... The aim of this section is to recall the RD formulation for steady problems, using a FE technique. Major details about this part can be found in part in [5], then in [4,6]. ...
... where + and − are chosen in order to store a positive value for a flux going from node i to node j. As explained in [6], the same fluxes can be obtained through a different method, which will be useful to relate our scheme to the RD classical schemes. This method consists in computing from ζ i an intermediary flux, called λ N ij and then the final flux φ el ij . ...
We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge-Kutta schemes for scalar equations [1]. The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in [1,2], we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor-corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns.
... Inundation can be simulated correctly and the results of the coupled 1D-2D model are compared with those of the complete 2D model also considering CPU time. A residual distribution and a finite-volume scheme have been developed in the framework of the Telemac2D system by Pavan et al. (2015). The two schemes are compared in several test cases simulating pollutant transport on unstructured grids considering numerical diffusion, mass conservation, and monotonicity. ...
The developments of linearity preserving (LP) residual distribution (RD) schemes for unsteady hyperbolic and hyperbolic-parabolic equations are presented. This is achieved by utilizing a classical aerodynamics philosophy on the standard finite-element technique to modify classic RD schemes such as LDA to be truly LP even for unsteady advection-diffusion cases using an explicit time integration. The new approach herein would also highlight what previous analytical approaches lacked. Abstract The developments of linearity preserving (LP) residual distribution (RD) schemes for unsteady hyperbolic and hyperbolic-parabolic equations are presented. This is achieved by utilizing a classical aerodynamics philosophy on the standard finite-element technique to modify classic RD schemes such as LDA to be truly LP even for unsteady advection-diffusion cases using an explicit time integration. The new approach herein would also highlight what previous analytical approaches lacked.
