scholarly journals Improving the stability of a simple formulation of the shallow water equations for 2-D flood modeling

2012 ◽  
Vol 48 (5) ◽  
Author(s):  
Gustavo A. M. de Almeida ◽  
Paul Bates ◽  
Jim E. Freer ◽  
Maxime Souvignet
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Flood Modeling ◽  
Simple Formulation ◽  
The Stability

scholarly journals A Finite-Volume Grid Using Multimoments for Geostrophic Adjustment

2006 ◽  
Vol 134 (9) ◽  
pp. 2515-2526 ◽  
Author(s):  
F. Xiao ◽  
X. D. Peng ◽  
X. S. Shen
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Control Volume ◽  
Numerical Dispersion ◽  
Geostrophic Adjustment ◽  
Simple Formulation ◽  
Volume Method

Abstract This paper presents a novel finite-volume grid that uses not only the volume-integrated average (VIA) like the traditional finite-volume method, but also the surface-integrated average (SIA) as the model variables. The VIA and SIA are generically called “moments” in the context used here and are carried forward in time separately as the prognostic quantities. With the VIA defined in the control volume while the SIA is on the surface of the control volume, the discretization based on VIA and SIA leads to some new features in the numerical dispersions. A simple formulation using both VIA and SIA for shallow-water equations is presented. The numerical dispersion of the resulting grid, which is denoted as the “M grid,” is discussed with comparisons to the existing ones.

scholarly journals Urban flood modeling with porous shallow-water equations: A case study of model errors in the presence of anisotropic porosity

2015 ◽  
Vol 523 ◽  
pp. 680-692 ◽  
Author(s):  
Byunghyun Kim ◽  
Brett F. Sanders ◽  
James S. Famiglietti ◽  
Vincent Guinot
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Model Errors ◽  
Flood Modeling ◽  
Urban Flood

scholarly journals Urban Stormwater Modeling with Local Inertial Approximation Form of Shallow Water Equations: A Comparative Study

2021 ◽  
Author(s):  
Weiqi Wang ◽  
Wenjie Chen ◽  
Guoru Huang
Keyword(s):  
Shallow Water ◽  
Computational Efficiency ◽  
Environmental Protection Agency ◽  
Shallow Water Equations ◽  
Drainage System ◽  
Flood Modeling ◽  
Urban Stormwater ◽  
Urban Flood ◽  
Storm Water Management Model ◽  
Form Model

AbstractThis study focused on the performance and limitations of the local inertial approximation form model (LIM) of the shallow water equations (SWEs) when applied in urban flood modeling. A numerical scheme of the LIM equations was created using finite volume method with a first-order spatiotemporal Roe Riemann solver. A simplified urban stormwater model (SUSM) considering surface and underground dual drainage system was constructed based on LIM and the US Environmental Protection Agency Storm Water Management Model. Moreover, a complete urban stormwater model (USM) based on the SWEs with the same solution algorithm was used as the evaluation benchmark. Numerical results of the SUSM and USM in a highly urbanized area under four rainfall return periods were analyzed and compared. The results reveal that the performance of the SUSM is highly consistent with that of the USM but with an improvement in computational efficiency of approximately 140%. In terms of the accuracy of the model, the SUSM slightly underestimates the water depth and velocity and is less accurate when dealing with supercritical flow in urban stormwater flood modeling. Overall, the SUSM can produce comparable results to USM with higher computational efficiency, which provides a simplified and alternative method for urban flood modeling.

scholarly journals A well-balanced spectral volume scheme with the wetting–drying property for the shallow-water equations

2012 ◽  
Vol 14 (3) ◽  
pp. 745-760 ◽  
Author(s):  
Luca Cozzolino ◽  
Renata Della Morte ◽  
Giuseppe Del Giudice ◽  
Anna Palumbo ◽  
Domenico Pianese
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Step Length ◽  
Swash Zone ◽  
Time Step ◽  
Third Order ◽  
Spectral Volume ◽  
The Stability ◽  
The One ◽  
Volume Method

The shallow-water equations are widely used to model surface water bodies, such as lakes, rivers and the swash zone in coastal flows. Physically congruent solutions are characterized by non-negative water depth, and many numerical methods may fail to preserve this property at the discrete level when moving wet–dry transitions are present in the physical domain. In this paper, we present a spectral-volume method for the approximate solution of the one-dimensional shallow-water equations, which is third-order accurate in wet regions, far from discontinuities, and which is well balanced for water at rest states: the stability of the solution is ensured if reconstruction and limitation of variables preserves non-negativity of the depth and a suitable constraint for the time step length is satisfied. A number of numerical experiments are reported, showing the promising capabilities of the model to solve problems with non-trivial topographies and friction.

On the stability of elliptical vortex solutions of the shallow-water equations

1987 ◽  
Vol 183 ◽  
pp. 343-363 ◽  
Author(s):  
P. Ripa
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Rossby Number ◽  
Coriolis Parameter ◽  
Vortex Solutions ◽  
The Stability ◽  
The One ◽  
Two Parameters

The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.

scholarly journals Urban flood modeling using shallow water equations with depth-dependent anisotropic porosity

2016 ◽  
Vol 541 ◽  
pp. 1165-1184 ◽  
Author(s):  
Ilhan Özgen ◽  
Jiaheng Zhao ◽  
Dongfang Liang ◽  
Reinhard Hinkelmann
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Flood Modeling ◽  
Urban Flood

scholarly journals Two-Dimensional Shallow-Water Model with Porosity for Urban Flood Modeling

Proceedings ◽  
2018 ◽  
Vol 2 (20) ◽  
pp. 1307
Author(s):  
Malika Benslimane ◽  
Saâdia Benmamar ◽  
André Paquier
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Urban Areas ◽  
Large Scale ◽  
Momentum Equation ◽  
Water Model ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Flood Modeling ◽  
Urban Flood

In the world, floods are at the forefront of natural hazard. Urban areas are often at risk of flooding and just as often unprepared for management. Flood modeling is nowadays a very important topic in the theme of water, it inevitably involves the numerical resolution of the shallow water equations derived from the Navier Stocks equations governing flows. Two-dimensional shallow water models with porosity appear as an interesting path for the large-scale modeling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two-dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. The developed solution method consists in solving the two-dimensional shallow water equations with porosity via a finite volume scheme solving the conservative form of the equations which can be reduced to a calculation of flux through an edge, a problem that can be approached by a one-dimensional problem in the normal direction at the edge (Riemann problem).

Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling

2008 ◽  
Vol 362 (1-2) ◽  
pp. 19-38 ◽  
Author(s):  
Brett F. Sanders ◽  
Jochen E. Schubert ◽  
Humberto A. Gallegos
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Integral Formulation ◽  
Flood Modeling ◽  
Urban Flood

MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

2017 ◽  
Author(s):  
Xiao-Hua Zhu ◽  
Xiao-Hua Zhu ◽  
Ze-Nan Zhu ◽  
Ze-Nan Zhu ◽  
Xinyu Guo ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Mean Amplitude ◽  
Momentum Equation ◽  
Tidal Currents ◽  
Acoustic Tomography ◽  
Mean Values ◽  
Residual Currents ◽  
Water Depths ◽  
Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

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