A dynamic multilayer shallow water model for polydisperse sedimentation

2019 ◽  
Vol 53 (4) ◽  
pp. 1391-1432 ◽  
Author(s):  
Raimund Bürger ◽  
Enrique D. Fernández-Nieto ◽  
Víctor Osores
Keyword(s):  
Shallow Water ◽  
Bottom Topography ◽  
Mining Industry ◽  
Mass Conservation ◽  
Variable Density ◽  
Solid Particles ◽  
Water Model ◽  
Reduced Form ◽  
Momentum Balance ◽  
Shallow Water Model

A multilayer shallow water approach for the approximate description of polydisperse sedimentation in a viscous fluid is presented. The fluid is assumed to carry finely dispersed solid particles that belong to a finite number of species that differ in density and size. These species segregate and form areas of different composition. In addition, the settling of particles influences the motion of the ambient fluid. A distinct feature of the new approach is the particular definition of the average velocity of the mixture. It takes into account the densities of the solid particles and the fluid and allows us to recover the global mass conservation and linear momentum balance laws of the mixture. This definition motivates a modification of the Masliyah–Lockett–Bassoon (MLB) settling velocities of each species. The multilayer shallow water model allows one to determine the spatial distribution of the solid particles, the velocity field, and the evolution of the free surface of the mixture. The final model can be written as a multilayer model with variable density where the unknowns are the average velocities and concentrations in each layer, the transfer terms across each interface, and the total mass. An explicit formula of the transfer terms leads to a reduced form of the system. Finally, an explicit bound of the minimum and maximum eigenvalues of the transport matrix of the system is utilized to design a Harten–Lax–van Leer (HLL)-type path-conservative numerical method. Numerical simulations illustrate the coupled polydisperse sedimentation and flow fields in various scenarios, including sedimentation in a type of basin that is used in practice in mining industry and in a basin whose bottom topography gives rise to recirculations of the fluid and high solids concentrations.

scholarly journals Hybrid 4DVAR with a Local Ensemble Tangent Linear Model: Application to the Shallow-Water Model

2016 ◽  
Vol 145 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Douglas R. Allen ◽  
Craig H. Bishop ◽  
Sergey Frolov ◽  
Karl W. Hoppel ◽  
David D. Kuhl ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Shallow Water ◽  
Linear Model ◽  
Bottom Topography ◽  
Grid Point ◽  
Model Error ◽  
Water Model ◽  
State Variables ◽  
Shallow Water Model

Abstract An ensemble-based tangent linear model (TLM) is described and tested in data assimilation experiments using a global shallow-water model (SWM). A hybrid variational data assimilation system was developed with a 4D variational (4DVAR) solver that could be run either with a conventional TLM or a local ensemble TLM (LETLM) that propagates analysis corrections using only ensemble statistics. An offline ensemble Kalman filter (EnKF) is used to generate and maintain the ensemble. The LETLM uses data within a local influence volume, similar to the local ensemble transform Kalman filter, to linearly propagate the state variables at the central grid point. After tuning the LETLM with offline 6-h forecasts of analysis corrections, cycling experiments were performed that assimilated randomly located SWM height observations, based on a truth run with forced bottom topography. The performance using the LETLM is similar to that of the conventional TLM, suggesting that a well-constructed LETLM could free 4D variational methods from dependence on conventional TLMs. This is a first demonstration of the LETLM application within a context of a hybrid-4DVAR system applied to a complex two-dimensional fluid dynamics problem. Sensitivity tests are included that examine LETLM dependence on several factors including length of cycling window, size of analysis correction, spread of initial ensemble perturbations, ensemble size, and model error. LETLM errors are shown to increase linearly with correction size in the linear regime, while TLM errors increase quadratically. As nonlinearity (or forecast model error) increases, the two schemes asymptote to the same solution.

scholarly journals SHALLOW WATER EQUATION SOLUTION IN 2D USING FINITE DIFFERENCE METHOD WITH EXPLICIT SCHEME

2017 ◽  
Vol 17 (2) ◽  
pp. 105
Author(s):  
Nuraini Nuraini ◽  
Syamsul Rizal ◽  
Marwan Marwan
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Shallow Water Equation ◽  
Mass Conservation ◽  
Conservation Equation ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Three Dimensional Model

Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes.Keyword: shallow water equation, finite difference and schema explisit.REFERENSI 1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468.2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York.3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies.4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan  Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS  dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123.5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348.6. Triatmodjo, B. 2002. Metode Numerik  Beta Offset. Yogyakarta.

scholarly journals An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

2021 ◽  
Vol 90 (1) ◽  
Author(s):  
E. Guerrero Fernández ◽  
M. J. Castro Díaz ◽  
M. Dumbser ◽  
T. Morales de Luna
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Stationary Solutions ◽  
Variable Density ◽  
High Order ◽  
Water Model ◽  
Variable Pressure ◽  
Shallow Water Model ◽  
Hyperbolic Pde ◽  
Coarse Meshes

AbstractIn this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

scholarly journals A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 848
Author(s):  
Ernesto Guerrero Fernández ◽  
Manuel Jesús Castro-Díaz ◽  
Tomás Morales de Luna
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Hyperbolic Equations ◽  
Laboratory Data ◽  
Variable Density ◽  
Second Order ◽  
Water Model ◽  
Finite Volume Scheme ◽  
Reconstruction Technique ◽  
Shallow Water Model

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented.

The instability of vertically curved fronts in a two-layer shallow water model

2020 ◽  
Vol 32 (12) ◽  
pp. 124117
Author(s):  
M. W. Harris ◽  
F. J. Poulin ◽  
K. G. Lamb
Keyword(s):  
Shallow Water ◽  
Water Model ◽  
Shallow Water Model

scholarly journals Modelling Weirs in Two-Dimensional Shallow Water Models

Water ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2152
Author(s):  
Gonzalo García-Alén ◽  
Olalla García-Fonte ◽  
Luis Cea ◽  
Luís Pena ◽  
Jerónimo Puertas
Keyword(s):  
Experimental Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Experimental Dataset ◽  
River Hydraulics ◽  
Shallow Water Models ◽  
Water Models

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.

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