Bounded dam-break flows with tailwaters

2011 ◽  
Vol 686 ◽  
pp. 160-186 ◽  
Author(s):  
Alexander J. N. Goater ◽  
Andrew J. Hogg
Keyword(s):  
Analytical Solutions ◽  
Shallow Water Equations ◽  
Analytical Techniques ◽  
Rear Wall ◽  
Continuous Solutions ◽  
Nonlinear Shallow Water Equations ◽  
Stationary Layer ◽  
Hodograph Transformation ◽  
Finite Extent ◽  
Quiescent Fluid

AbstractThe gravitationally driven collapse of a reservoir into an initially stationary layer of fluid, termed the tailwater, is studied using the nonlinear shallow water equations. The motion is tackled using the hodograph transformation of the governing equation which allows the solutions for velocity and depth of the shallow flowing layer to be constructed by analytical techniques. The front of the flow emerges as a bore across which the depth of the fluid jumps discontinuously to the tailwater depth. The speed of the front is initially constant, but progressively slows once the finite extent of the reservoir begins to influence the motion. There then emerges a variety of phenomena depending upon the depth of the tailwater relative to the initial depth of the reservoir. Provided that the tailwater is sufficiently deep, a region of quiescent fluid emerges adjacent to the rear wall of the reservoir, followed by a region within which the velocity is negative. Also it is shown that for non-vanishing tailwater depths, continuous solutions for the velocity and height of the flowing layer breakdown after a sufficient period and develop an interior bore, the location and time of inception of which are calculated directly from quasi-analytical solutions.

Solving the nonlinear shallow-water equations in physical space

2009 ◽  
Vol 643 ◽  
pp. 207-232 ◽  
Author(s):  
M. ANTUONO ◽  
M. BROCCHINI
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Physical Space ◽  
Wave Interaction ◽  
Transformation Method ◽  
Perturbation Approach ◽  
Order Condition ◽  
First Order ◽  
Nonlinear Shallow Water Equations ◽  
Hodograph Transformation

The boundary value problem for the nonlinear shallow-water equations with a beach source term is solved by direct use of physical variables, so that solutions are more easily inspected than those obtained by means of hodograph transformations. Beyond an overall description of the near-shoreline flows in terms of the nonlinear shallow-water equations, significant results are provided by means of a perturbation approach which enables much of the information on the flow to be retained. For sample waves of interest (periodic and solitary), first-order solutions of the shoreline motion and of the near-shoreline flows are computed, illustrated and successfully compared with the equivalent ones obtained through a hodograph transformation method previously developed by the authors. Wave–wave interaction, both at the seaward boundary and within the domain, is also accurately described. Analytical conditions for wave breaking within the domain are provided. These, compared with the authors' hodograph model, show that the first-order condition of the present model is comparable to the second-order condition of that model.

Run-up and backwash bore formation from dam-break flow on an inclined plane

2009 ◽  
Vol 640 ◽  
pp. 151-164 ◽  
Author(s):  
MATTEO ANTUONO ◽  
ANDREW J. HOGG
Keyword(s):  
Analytical Solution ◽  
Shallow Water Equations ◽  
Inclined Plane ◽  
Nonlinear Shallow Water Equations ◽  
Dam Break Flow ◽  
Height Fields ◽  
Hodograph Transformation ◽  
Run Up ◽  
Initial Distributions ◽  
Moving Fluid

Nonlinear shallow water equations are employed to model the inviscid slumping of fluid along an inclined plane and analytical solutions for the motion are derived using the hodograph transformation to reveal the run-up and the inception of a bore on the backwash. Starting from rest, the fluid slumps along the inclined plane, attaining a maximum run-up, before receding and forming a relatively thin and fast moving backwash. This interacts with the less rapidly moving fluid within the interior to form a bore. The evolution of the bore and the velocity and height fields either side of it are also calculated to reveal that it initially grows in magnitude before diminishing and intersecting with the shoreline. This analytical solution reveals features of the solution, such as the onset of the bore and its growth and decline, previously known only through numerical computation and the method presented here may be applied quite widely to the run-up of other initial distributions of fluid.

Numerical Investigation of Wave Period Influence on Long Wave Run-Up on Slope With Rigid Vegetation

2016 ◽  
Author(s):  
Jun Tang ◽  
Yongming Shen
Keyword(s):  
Shallow Water Equations ◽  
Water Wave ◽  
Wave Period ◽  
Coastal Vegetation ◽  
Coastal Structures ◽  
Rigid Vegetation ◽  
Comparison With Experiment ◽  
Long Wave ◽  
Nonlinear Shallow Water Equations ◽  
Run Up

Coastal vegetation can not only provide shade to coastal structures but also reduce wave run-up. Study of long water wave climb on vegetation beach is fundamental to understanding that how wave run-up may be reduced by planted vegetation along coastline. The present study investigates wave period influence on long wave run-up on a partially-vegetated plane slope via numerical simulation. The numerical model is based on an implementation of Morison’s formulation for rigid structures induced inertia and drag stresses in the nonlinear shallow water equations. The numerical scheme is validated by comparison with experiment results. The model is then applied to investigate long wave with diverse periods propagating and run-up on a partially-vegetated 1:20 plane slope, and the sensitivity of run-up to wave period is investigated based on the numerical results.

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 APPLICATION OF ABOODH TRANSFORM ON SOME FRACTIONAL ORDER MATHEMATICAL MODELS

2020 ◽  
Vol 20 (3) ◽  
pp. 661-672
Author(s):  
JAWARIA TARIQ ◽  
JAMSHAD AHMAD
Keyword(s):  
Mathematical Models ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Iteration Method ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Physical Phenomena ◽  
Convergent Solution

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo’s sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order.

scholarly journals Mandelbrot, Fama and the emergence of econophysics

2016 ◽  
Vol 35 (69) ◽  
pp. 637-662
Author(s):  
Boris Salazar Trujillo
Keyword(s):  
Analytical Solutions ◽  
Financial Economics ◽  
Analytical Techniques ◽  
Infinite Variance ◽  
Pareto Distributions

It is argued that Mandelbrot's stable Lévy-Pareto distributions were not accepted into the emerging field of financial economics due to their incompatibility with the analytical techniques and properties of equilibrium economics, and to the absence -both in physics and in economics- of analytical solutions to the infinite variance associated with those distributions. Whilst physicists made stable Lévy distributions plausible, creating Econophysics in the meantime, economists just forgot about them, suggesting their strong bias towards desirable properties and against established facts.

Identification of Nonlinear Systems Parameters Using Polyspectral Measurements and Analysis

1995 ◽  
Author(s):  
Muhammad R. Hajj ◽  
Ali H. Nayfeh ◽  
Pavol Popovic
Keyword(s):  
Experimental Study ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Analytical Techniques ◽  
Modal Interactions ◽  
Unknown Parameters ◽  
Nonlinear Parameters ◽  
The Subject ◽  
Beam Frame ◽  
Subharmonic Resonances

Abstract Experimental and analytical techniques that characterize nonlinear modal interactions in structures are used to quantify parameters in representative nonlinear systems. The subject of the experimental study is a three-beam frame. Subharmonic resonances and interaction between widely spaced modes are exploited to determine nonlinear parameters in models that represent these interactions. The phases of the auto-bispectra of the response of this structure appear in the analytical solutions of the representative models. Values of these phases could thus aid in determining other unknown parameters of nonlinear systems.

scholarly journals The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 987 ◽  
Author(s):  
A. A. Alderremy ◽  
Hassan Khan ◽  
Rasool Shah ◽  
Shaban Aly ◽  
Dumitru Baleanu
Keyword(s):  
Mathematical Modeling ◽  
Analytical Solution ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Analytical Techniques ◽  
Whitham Equations ◽  
Suggested Technique ◽  
Approximate Analytical Solutions ◽  
Physical Phenomena

This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

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