Nonsimilar solutions of the viscous shallow water equations governing weak hydraulic jumps

2010 ◽  
Vol 22 (11) ◽  
pp. 112108 ◽  
Author(s):  
Ratul Dasgupta ◽  
Rama Govindarajan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Hydraulic Jumps ◽  
Nonsimilar Solutions

scholarly journals Numerical shockwave anomalies in the resolu- tion of the Shallow Water Equations with bed variations

2018 ◽  
Vol 40 ◽  
pp. 05026
Author(s):  
Adrian Navas-Montilla ◽  
Javier Murillo
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Scientific Community ◽  
Hyperbolic Systems ◽  
Hydraulic Jumps ◽  
Riemann Solvers ◽  
Systems Of Conservation Laws ◽  
Spurious Oscillations ◽  
Transient Events ◽  
Variable Topography

The presence of numerical shockwave anomalies appearing in the resolution of hyperbolic systems of conservation laws is a well-known problem in the scientific community. The most common anomalies are the carbuncle and the slowly-moving shock anomaly. They have been studied for decades in the framework of Euler equations, but only a few authors have considered such problems for the Shallow Water Equations (SWE). In this work, the SWE are considered and the aforementioned anomalies are studied. They arise in presence of hydraulic jumps, which are transcritical shockwaves mathematically modelled as a pure discontinuity. When solving numerically such discontinuities, an unphysical intermediate state appears and gives rise to a spurious spike in the momentum. This is observed in the numerical solution as a spike in the discharge appearing in the cell containing the jump. The presence of the spurious spike in the discharge has been taken for granted by the scientific community and has even become a feature of the solution. Even though it does not disturb the rest of the solution in steady cases, it produces an undesirable shedding of spurious oscillations downstream when considering transient events. We show how it is possible to define a coherent spike reduction technique that reduces the magnitude of this anomaly and ensures convergence to the exact solution with mesh refinement. Concerning the carbuncle, which may also appear in presence of strong hydraulic jumps, a combination of Riemann solvers is proposed to circumvent it. Also, it will be shown how there is still room from improvement when treating anomalies in moving hydraulic jumps over variable topography.

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.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

2020 ◽  
Vol 35 (6) ◽  
pp. 355-366
Author(s):  
Vladimir V. Shashkin ◽  
Gordey S. Goyman
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Time Integration ◽  
Standard Test ◽  
Integration Scheme ◽  
Test Problems ◽  
Cubed Sphere ◽  
Lagrangian Scheme ◽  
Inertia Gravity Waves

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

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