Modified Shallow Water Equations With Application for Horizontal Centrifugal Casting of Rolls

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
Abdellah Kharicha ◽  
Jan Bohacek ◽  
Andreas Ludwig ◽  
Menghuai Wu
Keyword(s):  
Liquid Metal ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Centrifugal Casting ◽  
Axial Direction ◽  
Mean Amplitude ◽  
Angular Frequency ◽  
Average Height ◽  
Mold Deformation

A numerical model based on the shallow water equations (SWE) was proposed to simulate the two-dimensional (2D) average flow dynamics of the liquid metal spreading inside a horizontally rotating mold. The SWE were modified to account for the forces, such as the centrifugal force, Coriolis force, shear force with the mold wall, and gravity. In addition, inherent vibrations caused by a poor roundness of the mold and the mold deformation due to temperature gradients were applied explicitly by perturbing the gravity and the axis bending, respectively. Several cases were studied with the following initial conditions: a constant average height of the liquid metal (5, 10, 20, 30, and 40 mm) patched as a flat or a perturbed surface. The angular frequency Ω of the mold (∅1150–3200) was 71.2 (or 30) rad/s. Results showed various wave patterns propagating on the free surface. In early stages, a single longitudinal wave moved around the circumference. As the time proceeded, it slowly diminished and waves traveled mainly in the axial direction. It was found that the mean amplitude of the oscillations grows with the increasing liquid height.

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.

Solving shallow water equations with crisp and uncertain initial conditions

2018 ◽  
Vol 28 (12) ◽  
pp. 2801-2815 ◽  
Author(s):  
Perumandla Karunakar ◽  
Snehashish Chakraverty
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Triangular Fuzzy Number ◽  
Governing Equations ◽  
Content Type ◽  
Variational Iteration

Purpose This paper aims to deal with the application of variational iteration method and homotopy perturbation method (HPM) for solving one dimensional shallow water equations with crisp and fuzzy uncertain initial conditions. Design/methodology/approach Firstly, the study solved shallow water equations using variational iteration method and HPM with constant basin depth and crisp initial conditions. Further, the study considered uncertain initial conditions in terms of fuzzy numbers, which leads the governing equations to fuzzy shallow water equations. Then using cut and parametric concepts the study converts fuzzy shallow water equations to crisp form. Then, HPM has been used to solve the fuzzy shallow water equations. Findings Results obtained by both methods HPM and variational iteration method are compared graphically in crisp case. Solution of fuzzy shallow water equations by HPM are presented in the form triangular fuzzy number plots. Originality/value Shallow water equations with crisp and fuzzy initial conditions have been solved.

scholarly journals Multiple Solutions for the Riemann Problem in the Porous Shallow Water Equations

10.29007/31n4 ◽  
2018 ◽  
Author(s):  
Luca Cozzolino ◽  
Raffaele Castaldo ◽  
Luigi Cimorelli ◽  
Renata Della Morte ◽  
Veronica Pepe ◽  
...  
Keyword(s):  
Shallow Water ◽  
Riemann Problem ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Numerical Models ◽  
Control Volume ◽  
Fluid Phase ◽  
Problem Solution ◽  
The One ◽  
Solid Obstacles

The Porous Shallow water Equations are widely used in the context of urban flooding simulation. In these equations, the solid obstacles are implicitly taken into account by averaging the classic Shallow water Equations on a control volume containing the fluid phase and the obstacles. Numerical models for the approximate solution of these equations are usually based on the approximate calculation of the Riemann fluxes at the interface between cells. In the present paper, it is presented the exact solution of the one-dimensional Riemann problem over the dry bed, and it is shown that the solution always exists, but there are initial conditions for which it is not unique. The non-uniqueness of the Riemann problem solution opens interesting questions about which is the physically congruent wave configuration in the case of solution multiplicity.

On axisymmetric intrusive gravity currents: the approach to self-similarity solutions of the shallow-water equations in a linearly stratified ambient

2007 ◽  
Vol 463 (2085) ◽  
pp. 2165-2183 ◽  
Author(s):  
Tamar Zemach ◽  
Marius Ungarish
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Fixed Ratio ◽  
Analytical Prediction ◽  
Ambient Fluid ◽  
Time Motion ◽  
Ring Shape ◽  
Long Time ◽  
Self Similar

The axisymmetric intrusion of a fixed volume of fluid, which is released from rest and then propagates radially at the neutral buoyancy level in a deep linearly stratified ambient fluid, is investigated. Attention is focused on the development of self-similar propagation. The shallow-water equations representing the high-Reynolds-number motion are used. For the long-time motion, an analytical similarity solution indicates propagation with t 1/3 , but the shape is peculiar: the intrusion propagates like a ring with a fixed ratio of inner to outer radii; the inner domain contains clear ambient fluid. To verify the similarity analytical prediction, a long-time finite-difference solution with realistic initial conditions was performed. To avoid accumulation of numerical errors, the problem was reformulated in terms of new variables. It is shown that the numerical solution has a ‘tail-ring’ shape. The ‘tail’ decays like t −2 and the ‘ring’ tends to the analytical similarity prediction. The initial geometry of the lock does not influence this result. Comparison with the non-stratified case is also presented. It was found that for the non-stratified case, there is a stage of propagation in which the intrusion has a similar ‘tail-ring’ form; however, this stage is only a transient to a self-similar shape which is different from that obtained for the stratified ambient.

RSW Modons and their Surprising Properties: RSW Turbulence

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Energy Spectra ◽  
2D Turbulence ◽  
Theoretical Predictions ◽  
Scatter Plots ◽  
Collision Processes ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

By using quasi-geostrophic modons constructed in Chapter 6 as initial conditions, rotating-shallow-water modons are obtained through the process of ageostrophic adjustment, both in one- and in two-layer configurations. Scatter plots show that they are solutions of the rotating shallow-water equations. A special class of modons with an internal front (shock) is shown to exist. A panorama of collision processes of the modons, leading to formation of tripoles, nonlinear modons, or elastic scattering is presented. The modon solutions are then used for initialisations of numerical simulations of decaying rotating shallow-water turbulence. The results are analysed and compared to those obtained with standard in 2D turbulence initializations, and differences are detected, showing non-universality of decaying 2D turbulence. The obtained energy spectra are steeper than theoretical predictions for ‘pure’ 2D turbulence, and pronounced cyclone–anticyclone asymmetry and dynamical separation of waves and vortices are observed.

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.

Filtering inertia-gravity waves from the initial conditions of the linear shallow water equations

2007 ◽  
Vol 19 (3-4) ◽  
pp. 204-218 ◽  
Author(s):  
Alexander Barth ◽  
Jean-Marie Beckers ◽  
Aida Alvera-Azcárate ◽  
Robert H. Weisberg
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Inertia Gravity Waves
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