scholarly journals Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

2018 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Md. Masum Murshed ◽  
◽  
Kouta Futai ◽  
Masato Kimura ◽  
Hirofumi Notsu ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Energy Estimates ◽  
Numerical Studies

scholarly journals Seiche in the Gulf of Tonkin

1996 ◽  
Vol 18 (1) ◽  
pp. 27-33
Author(s):  
Pham Van Ninh ◽  
Tran Thi Ngoc Duyet
Keyword(s):  
Boundary Condition ◽  
Numerical Modelling ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Forced Oscillation ◽  
Two Dimensional ◽  
Nonlinear Shallow Water Equations ◽  
Liquid Boundary ◽  
Gulf Of Tonkin

Steichen in the Gulf of Tonkin has been studied by numerical modelling based on the two-dimensional nonlinear shallow water equations system with liquid boundary condition given in the form of forced oscillation. The main proper periods have been defined as follows: 23-25 hours, 1-12 hours, 5-7 hours, 2-4 hours. Among them the 23 hours period is the most evident. The obtained results coincide with observed ones at the long shore hydrometeological stations of the Gulf.

scholarly journals Numerical approximation of the shallow water equations with coriolis source term

2021 ◽  
Vol 70 ◽  
pp. 31-44
Author(s):  
E. Audusse ◽  
V. Dubos ◽  
A. Duran ◽  
N. Gaveau ◽  
Y. Nasseri ◽  
...  
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Numerical Approximation ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Geostrophic Balance ◽  
Low Froude Number ◽  
Numerical Fluxes

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

scholarly journals Long Wave Run-Up Resonance in a Multi-Reflection System

2020 ◽  
Vol 10 (18) ◽  
pp. 6172
Author(s):  
Shanshan Xu ◽  
Frédéric Dias
Keyword(s):  
Boundary Condition ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Incoming Wave ◽  
Wave Trapping ◽  
Relaxation Zone ◽  
Long Wave ◽  
Nonlinear Shallow Water Equations ◽  
Run Up ◽  
Sloping Beach

Wave reflection and wave trapping can lead to long wave run-up resonance. After reviewing the theory of run-up resonance in the framework of the linear shallow water equations, we perform numerical simulations of periodic waves incident on a linearly sloping beach in the framework of the nonlinear shallow water equations. Three different types of boundary conditions are tested: fully reflective boundary, relaxation zone, and influx transparent boundary. The effect of the boundary condition on wave run-up is investigated. For the fully reflective boundary condition, it is found that resonant regimes do exist for certain values of the frequency of the incoming wave, which is consistent with theoretical results. The influx transparent boundary condition does not lead to run-up resonance. Finally, by decomposing the left- and right-going waves into a multi-reflection system, we find that the relaxation zone can lead to run-up resonance depending on the length of the relaxation zone.

Integral flow properties of the swash zone and averaging

1996 ◽  
Vol 317 ◽  
pp. 241-273 ◽  
Author(s):  
M. Brocchini ◽  
D. H. Peregrine
Keyword(s):  
Boundary Condition ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Lower Boundary ◽  
Stokes Drift ◽  
Swash Zone ◽  
Dimensional Solution ◽  
Short Waves ◽  
Averaged Model

The swash zone is that part of a beach over which the instantaneous shoreline moves back and forth as waves meet the shore. This zone is discussed using the nonlinear shallow water equations which are appropriate for gently sloping beaches. A weakly three-dimensional extension of the two-dimensional solution by Carrier & Greenspan (1958) of the shallow water equations for a wave reflecting on an inclined plane beach is developed and used to illustrate the ideas. Thereafter attention is given to integrated and averaged quantities. The mean shoreline might be defined in several ways, but for modelling purposes we find the lower boundary of the swash zone to be more useful. A set of equations obtained by integrating across the swash zone is investigated as a model for use as an alternative boundary condition for wave-resolving studies. Comparison with sample numerical computations illustrates that they are effective in modelling the dynamics of the swash zone and that a reasonable representation of swash zone flows may be obtained from the integrated variables. The longshore flow of water in the swash zone is in many ways similar to the Stokes’ drift of propagating water waves. Further averaging is made over short waves to obtain results suitable as boundary conditions for longer period motions including the effect of incident short waves. In order to clearly present the work a few simplifications are made. The main result is that in addition to the kinematic type of boundary condition that occurs on a simple, e.g. rigid, boundary two further conditions are found in order that both the changing position of the swash zone boundary and the longshore flow in the swash zone may be determined. Models of the short waves both outside and inside the swash zone are needed to complete a full wave-averaged model; only brief indication is given of such modelling.

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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