scholarly journals Comparison Between the Exact Solutions of Three Distinct Shallow Water Equations Using the Painlevé Approach and Its Numerical Solutions

2020 ◽  
Vol 16 (3) ◽  
pp. 463-477
Author(s):  
A. Bekir ◽  
◽  
M. Shehata ◽  
E. Zahran ◽  
◽  
...  
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions

scholarly journals Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

Models of non-Boussinesq lock-exchange flow

2011 ◽  
Vol 675 ◽  
pp. 1-26 ◽  
Author(s):  
R. ROTUNNO ◽  
J. B. KLEMP ◽  
G. H. BRYAN ◽  
D. J. MURAKI
Keyword(s):  
Free Boundary ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Water System ◽  
Analytic Solutions ◽  
Analytical Models ◽  
Navier Stokes ◽  
Exchange Flow

Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

scholarly journals Symmetries and exact solutions of the rotating shallow-water equations

2009 ◽  
Vol 20 (5) ◽  
pp. 461-477 ◽  
Author(s):  
A. A. CHESNOKOV
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
New Class ◽  
New Exact Solutions ◽  
Time Periodic Solutions ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations ◽  
Time Periodic

Lie symmetry analysis is applied to study the non-linear rotating shallow-water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow-water equations can be transformed to the classical shallow-water model. The derived symmetries are used to generate new exact solutions of the rotating shallow-water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

Computational study of some nonlinear shallow water equations

Open Physics ◽  
2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Ali Bhrawy ◽  
Mohamed Abdelkawy
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Periodic Solutions ◽  
Shallow Water Equations ◽  
Computational Study ◽  
Expansion Method ◽  
Coastal Areas ◽  
Atmospheric Modeling ◽  
Hydraulic Engineering ◽  
Nonlinear Shallow Water Equations

AbstractThe shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using $\frac{{G'}} {G} $-expansion method. Hyperbolic and triangular periodic solutions can be obtained from the $\frac{{G'}} {G} $-expansion method.

scholarly journals Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 345
Author(s):  
Sudi Mungkasi ◽  
Stephen Gwyn Roberts
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Adaptive Mesh ◽  
Finite Volume Methods ◽  
Water Type ◽  
Two Dimensional ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Smoothness Indicators

This paper proposes some formulations of weak local residuals of shallow-water-type equations, namely, one-, one-and-a-half-, and two-dimensional shallow water equations. Smooth parts of numerical solutions have small absolute values of weak local residuals. Rougher parts of numerical solutions have larger absolute values of weak local residuals. This behaviour enables the weak local residuals to detect parts of numerical solutions which are smooth and rough (non-smooth). Weak local residuals that we formulate are implemented successfully as refinement or coarsening indicators for adaptive mesh finite volume methods used to solve shallow water equations.

Maximum power from a turbine farm in shallow water

2013 ◽  
Vol 714 ◽  
pp. 634-643 ◽  
Author(s):  
Chris Garrett ◽  
Patrick Cummins
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Momentum Equation ◽  
Basic Flow ◽  
Maximum Power ◽  
Two Dimensional ◽  
Dominant Term ◽  
Tidal Flows ◽  
Linear Friction

AbstractThe maximum power that can be obtained from a confined array of turbines in steady or tidal flows is considered using the two-dimensional shallow-water equations and representing the turbine farm by a uniform local increase in friction within a circle. Analytical results supported by dimensional reasoning and numerical solutions show that the maximum power depends on the dominant term in the momentum equation for flows perturbed on the scale of the farm. If friction dominates in the basic flow, the maximum power is a fraction (half for linear friction and 0.75 for quadratic friction) of the dissipation within the circle in the undisturbed state; if the advective terms dominate, the maximum power is a fraction of the undisturbed kinetic energy flux into the front of the turbine farm; if the acceleration dominates, the maximum power is similar to that for the linear frictional case, but with the friction coefficient replaced by twice the tidal frequency.

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

2014 ◽  
Vol 178 (3) ◽  
pp. 278-298 ◽  
Author(s):  
Yu. A. Chirkunov ◽  
S. Yu. Dobrokhotov ◽  
S. B. Medvedev ◽  
D. S. Minenkov
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
One Dimensional ◽  
Nonlinear Shallow Water Equations
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