The early stages of shallow flows in an inclined flume

2009 ◽  
Vol 633 ◽  
pp. 285-309 ◽  
Author(s):  
MATTEO ANTUONO ◽  
ANDREW J. HOGG ◽  
MAURIZIO BROCCHINI
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Algorithms ◽  
Test Case ◽  
Discontinuous Solutions ◽  
Shallow Flows ◽  
Moving Interface ◽  
Nonlinear Shallow Water Equations ◽  
Two Dimensional Flow ◽  
Height Fields

The motion of an initially quiescent shallow layer of fluid within an impulsively tilted flume is modelled using the nonlinear shallow water equations. Analytical solutions for the two-dimensional flow are constructed using the method of characteristics and, in regions where neither of the characteristic variables is constant, by adopting hodograph variables and using the Riemann construction for the solution. These solutions reveal that the motion is strongly influenced by the impermeable endwalls of the flume. They show that discontinuous solutions emerge after some period following the initiation of the flow and that for sufficiently long flumes there is a moving interface between wetted and dry regions. Using the hodograph variables we are able to track the evolution of the flow analytically. After the discontinuities develop, we also calculate the velocity and height fields by using jump conditions to express conservation of mass and momentum across the shock and thus we show how the hydraulic jump moves within the domain and how its magnitude grows. In addition to providing the behaviour of the flow in this physical scenario, this unsteady solution also provides an important test case for numerical algorithms designed to integrate the shallow water equations.

A shock solution for the nonlinear shallow water equations

2010 ◽  
Vol 658 ◽  
pp. 166-187 ◽  
Author(s):  
MATTEO ANTUONO
Keyword(s):  
Shock Wave ◽  
General Solution ◽  
Theoretical Analysis ◽  
Asymptotic Behaviour ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Boundary Data ◽  
Depth Analysis ◽  
Nonlinear Shallow Water Equations ◽  
Shock Solution

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.

scholarly journals A test case for the inviscid shallow-water equations on the sphere

2015 ◽  
Vol 142 (694) ◽  
pp. 488-495 ◽  
Author(s):  
R. K. Scott ◽  
L. M. Harris ◽  
L. M. Polvani
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Test Case

Predicting mesh density for adaptive modelling of the global atmosphere

2009 ◽  
Vol 367 (1907) ◽  
pp. 4523-4542 ◽  
Author(s):  
Hilary Weller
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Large Scale ◽  
Mesh Adaptation ◽  
Similar Proportion ◽  
Test Case ◽  
Uniform Mesh ◽  
Coarse Mesh ◽  
Mesh Density ◽  
The Cost

The shallow water equations are solved using a mesh of polygons on the sphere, which adapts infrequently to the predicted future solution. Infrequent mesh adaptation reduces the cost of adaptation and load-balancing and will thus allow for more accurate mapping on adaptation. We simulate the growth of a barotropically unstable jet adapting the mesh every 12 h. Using an adaptation criterion based largely on the gradient of the vorticity leads to a mesh with around 20 per cent of the cells of a uniform mesh that gives equivalent results. This is a similar proportion to previous studies of the same test case with mesh adaptation every 1–20 min. The prediction of the mesh density involves solving the shallow water equations on a coarse mesh in advance of the locally refined mesh in order to estimate where features requiring higher resolution will grow, decay or move to. The adaptation criterion consists of two parts: that resolved on the coarse mesh, and that which is not resolved and so is passively advected on the coarse mesh. This combination leads to a balance between resolving features controlled by the large-scale dynamics and maintaining fine-scale features.

scholarly journals A Finite Volume Method for Modeling Shallow Flows with Wet-Dry Fronts on Adaptive Cartesian Grids

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
Sheng Bi ◽  
Jianzhong Zhou ◽  
Yi Liu ◽  
Lixiang Song
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Second Order ◽  
Topographic Features ◽  
Shallow Flows ◽  
Local Complex ◽  
Proposed Model ◽  
Volume Method

A second-order accurate, Godunov-type upwind finite volume method on dynamic refinement grids is developed in this paper for solving shallow-water equations. The advantage of this grid system is that no data structure is needed to store the neighbor information, since neighbors are directly specified by simple algebraic relationships. The key ingredient of the scheme is the use of the prebalanced shallow-water equations together with a simple but effective method to track the wet/dry fronts. In addition, a second-order spatial accuracy in space and time is achieved using a two-step unsplit MUSCL-Hancock method and a weighted surface-depth gradient method (WSDM) which considers the local Froude number is proposed for water depths reconstruction. The friction terms are solved by a semi-implicit scheme that can effectively prevent computational instability from small depths and does not invert the direction of velocity components. Several benchmark tests and a dam-break flooding simulation over real topography cases are used for model testing and validation. Results show that the proposed model is accurate and robust and has advantages when it is applied to simulate flow with local complex topographic features or flow conditions and thus has bright prospects of field-scale application.

Dispersive Nonlinear Shallow-Water Equations

2009 ◽  
Vol 122 (1) ◽  
pp. 1-28 ◽  
Author(s):  
M. Antuono ◽  
V. Liapidevskii ◽  
M. Brocchini
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Nonlinear Shallow Water Equations

The Diabatic Contour-Advective Semi-Lagrangian Algorithms for the Spherical Shallow Water Equations

2009 ◽  
Vol 137 (9) ◽  
pp. 2979-2994 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
David G. Dritschel
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Relative Strength ◽  
Test Case ◽  
Diagnostic Measures ◽  
Contour Advection ◽  
Rapid Generation ◽  
Quasigeostrophic Model ◽  
Conventional Algorithm ◽  
Grid Based

Abstract The diabatic contour-advective semi-Lagrangian (DCASL) algorithm is extended to the thermally forced shallow water equations on the sphere. DCASL rests on the partitioning of potential vorticity (PV) to adiabatic and diabatic parts solved, respectively, by contour advection and a grid-based conventional algorithm. The presence of PV in the source term for diabatic PV makes the shallow water equations distinct from the quasigeostrophic model previously studied. To address the more rapid generation of finescale structures in diabatic PV, two new features are added to DCASL: (i) the use of multiple sets of contours with successively finer contour intervals and (ii) the application of the underlying method of DCASL at a higher level to diabatic PV. That is, the diabatic PV is allowed to have both contour and grid parts. The added features make it possible to make the grid part of diabatic PV arbitrarily small and thus pave the way for a fully Lagrangian DCASL in the presence of forcing. The DCASL algorithms are constructed using a standard semi-Lagrangian (SL) algorithm to solve for the grid-based part of diabatic PV. The 25-day time evolution of an unstable midlatitude jet triggered by the action of thermal forcing is used as a test case to examine and compare the properties of the DCASL algorithms with a pure SL algorithm for PV. Diagnostic measures of vortical and unbalanced activity as well as of the relative strength of the grid and contour parts of the solution for PV indicate that the superiority of contour advection can be maintained even in the presence of strong, nonsmooth forcing.

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.

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