## Weighted Average Flux

### BibTeX

@MISC{_weightedaverage,

author = {},

title = {Weighted Average Flux},

year = {}

}

### OpenURL

### Abstract

Roll waves simulation using shallow water equations and Weighted Average Flux method Simulation de train de rouleaux avec les équations des ondes longues et la méthode

### Citations

329 |
Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction
- Toro
- 1997
(Show Context)
Citation Context ...n the same conditions tested by Brock [4,5], in order to have a direct comparison with available experimental data. 1. Numerical implementation and source terms The Weighted Average Flux (WAF) method =-=[22,23]-=-, on which our numerical model is based, is a development of currently favoured methods such as Godunov’s and Roe’s; it is ‘shock-capturing’ in the sense that discontinuities are automatically treated... |

69 |
Water Waves: The mathematical theory with applications. Interscience
- Stoker
- 1957
(Show Context)
Citation Context ...y 28, 2003. free-surface, determined that the critical Froude number Fr cr was about 0.5; a similar result was obtained by Chen [7] through 1D SW equations. For turbulent flows, Jeffreys [14], Stoker =-=[19]-=- and Liggett [17] considered a wide rectangular channel, uniform velocity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], ... |

22 |
Wave Formation in Laminar Flow down an Inclined
- Benjamin
- 1957
(Show Context)
Citation Context ...de as time increases, the flow is said to be unstable; these small amplifying perturbations would eventually result in the visible large amplitude roll waves. For laminar flows, Yih [26] and Benjamin =-=[2]-=-, using the NavierStokes equation and assuming sinusoidal perturbations of the Revision received June 21, 2001. Open for discussion till February 28, 2003. free-surface, determined that the critical F... |

17 |
Riemann Problems and the WAF Method for Solving the Two-Dimensional Shallow Water Equations
- Toro
- 1992
(Show Context)
Citation Context ...n the same conditions tested by Brock [4,5], in order to have a direct comparison with available experimental data. 1. Numerical implementation and source terms The Weighted Average Flux (WAF) method =-=[22,23]-=-, on which our numerical model is based, is a development of currently favoured methods such as Godunov’s and Roe’s; it is ‘shock-capturing’ in the sense that discontinuities are automatically treated... |

15 |
Stability of liquid flow down an inclined plane, Phys
- Yih
- 1963
(Show Context)
Citation Context ...ncrease in amplitude as time increases, the flow is said to be unstable; these small amplifying perturbations would eventually result in the visible large amplitude roll waves. For laminar flows, Yih =-=[26]-=- and Benjamin [2], using the NavierStokes equation and assuming sinusoidal perturbations of the Revision received June 21, 2001. Open for discussion till February 28, 2003. free-surface, determined th... |

13 |
An Introduction to the Method of Characteristic
- Abbott
- 1966
(Show Context)
Citation Context ...ificant than effects produced by non-linearity. Moreover, vertical accelerations are not negligible for ondular bore, whereas for wave instability they play a secondary role, as pointed out by Abbott =-=[1]-=-. The propagation of bores in our model is solved exactly by the Riemann solver (as checked by Toro, [22,23]), but the assumption of steep front with negligible length does not correspond to reality, ... |

8 |
The flow of water in an inclined channel of rectangular
- Jeffreys
- 1925
(Show Context)
Citation Context ... till February 28, 2003. free-surface, determined that the critical Froude number Fr cr was about 0.5; a similar result was obtained by Chen [7] through 1D SW equations. For turbulent flows, Jeffreys =-=[14]-=-, Stoker [19] and Liggett [17] considered a wide rectangular channel, uniform velocity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler an... |

5 |
Development of roll-wave trains in open channels
- Brock
- 1969
(Show Context)
Citation Context ...ce profiles versus time show a gaussian symmetric structure during instability and an evident non-symmetric form during the final overtaking phase, in agreement with laboratory experiments (Fig. 6 in =-=[5]-=-) as reported, for instance, for Test 2 in Fig. 7. Fig. 7a shows the flow depth profile in a section at small JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 5 615 a) y [m] b) y [m] t [s] t [s] Fig.... |

4 |
Numerical solution of the shallowwater equations on a beach using the weighted average flux method.” Computational Fluid Dynamics ’92
- Watson, Peregrine, et al.
- 1992
(Show Context)
Citation Context ...s near bores (Toro [23]). To avoid systematic errors at second order, caused by treating S with simple time operator splitting, S may be incorporated into the Riemann problem, following Watson et al. =-=[25]-=-. This is achieved by transforming the problem into a reference frame with the horizontal acceleration gβ that makes the momentum equation homogeneous. After some calculations, not reported here for s... |

3 |
Unstable turbulent flow in open channels
- Berlamont, Vanderstappen
- 1981
(Show Context)
Citation Context ...locity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], Craya [9], Koloseus and Davidian [15], Berlamont and Vanderstappen =-=[3]-=-, Iwasa [13]) have derived stability criteria for varying degrees of generality, showing that Fr cr depends on the channel shape, the frictional resistance law and the velocity distribution. A flow wi... |

3 |
1970a Relative validities of water wave theories
- DEAN
(Show Context)
Citation Context ...etisation on computations. 3.2.1 Shallow water approximation The possibility of applying the SW equations to the conditions tested in laboratory by Brock was checked on the basis of the diagram (Dean =-=[10]-=-) that represents the domain of applicability of different wave theories as functions of wave period (in a reference system fixed to the water body), wave height and flow depth. For sea wave theory, t... |

3 |
Instability of concentrated free surface flows
- Trowbridge
- 1987
(Show Context)
Citation Context ...IC RESEARCH, VOL. 40, 2002, NO. 5sheight is reached. For generalized Newtonian flows, Chun [8] performed a linear stability analysis determining the curve of stability; for Bingham fluids, Trowbridge =-=[24]-=- carried out the same analysis using SW equations. The first attempt to describe analytically large amplitude roll waves was by Thomas [21], who considered a periodic train of waves of constant shape ... |

2 |
Development of roll waves in open channels
- Brock
- 1967
(Show Context)
Citation Context ...n unvarying friction factor. On the basis of Dressler’s [11] work, Brock [6] developed a theory for periodic permanent roll-waves using the SW equations and comparing it with his experimental results =-=[4, 5]-=-. The aim of this paper is to prove how and how far a numerical code based on SW equations is able to represent roll-waves development. The paper is composed of three main parts: the first one reviews... |

2 |
Momentum and energy coefficients based on power-law velocity profile
- Chen
- 1992
(Show Context)
Citation Context ...of the Revision received June 21, 2001. Open for discussion till February 28, 2003. free-surface, determined that the critical Froude number Fr cr was about 0.5; a similar result was obtained by Chen =-=[7]-=- through 1D SW equations. For turbulent flows, Jeffreys [14], Stoker [19] and Liggett [17] considered a wide rectangular channel, uniform velocity distribution and a constant friction factor, finding ... |

2 |
The criterion for possibility of roll-wave formation’’, Gravity waves
- Craya
- 1952
(Show Context)
Citation Context ...iggett [17] considered a wide rectangular channel, uniform velocity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], Craya =-=[9]-=-, Koloseus and Davidian [15], Berlamont and Vanderstappen [3], Iwasa [13]) have derived stability criteria for varying degrees of generality, showing that Fr cr depends on the channel shape, the frict... |

2 |
The criterion for instability of steady uniform flows in open channels
- Iwasa
- 1954
(Show Context)
Citation Context ...ribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], Craya [9], Koloseus and Davidian [15], Berlamont and Vanderstappen [3], Iwasa =-=[13]-=-) have derived stability criteria for varying degrees of generality, showing that Fr cr depends on the channel shape, the frictional resistance law and the velocity distribution. A flow with a value o... |

2 |
La formazione spontanea dei treni d’onde su canali a pendenza molto forte’’, L’Energia Elettrica
- Montuori
- 1961
(Show Context)
Citation Context ... the channel shape, the frictional resistance law and the velocity distribution. A flow with a value of Fr even considerably higher than Fr cr may not exhibit visible roll waves, as noted by Montuori =-=[18]-=- in his investigation of field data collected in Europe and Russia. This led him to develop a criterion for predicting formation of roll waves (but not their dimensions), involving not only the value ... |

2 |
Sopra le onde di traslazione nei canali, Rendiconti Lincei XXIX
- Supino
- 1960
(Show Context)
Citation Context ...erturbation moves downstream the channel with decreasing amplitude, because of the bed resistance, and parameters as y 0 , ν, ω, are related by equations that the interested reader may find on Supino =-=[20]-=-. The rectangular channel assumed in computations is 1000.0 m long; the water flow is characterised by three different Fr=0.5, 2.0, 3.0, and by a constant uniform flow depth y 0 =1.0 m. The Chézy coef... |

1 |
Periodic permanent roll-waves
- Brock
- 1970
(Show Context)
Citation Context ...l waves. Dressler [11], using Thomas’s basic idea, was able to find closed form solutions for a wide rectangular channel with an unvarying friction factor. On the basis of Dressler’s [11] work, Brock =-=[6]-=- developed a theory for periodic permanent roll-waves using the SW equations and comparing it with his experimental results [4, 5]. The aim of this paper is to prove how and how far a numerical code b... |

1 |
Stability of flow of a generalized Newtonian fluid down an inclined plane
- Chun
- 1982
(Show Context)
Citation Context ...ust compensate the reduced energy dissipation in the regular flow when the regime flow 610 JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 5sheight is reached. For generalized Newtonian flows, Chun =-=[8]-=- performed a linear stability analysis determining the curve of stability; for Bingham fluids, Trowbridge [24] carried out the same analysis using SW equations. The first attempt to describe analytica... |

1 |
Mathematical solution of the probJOURNAL
- Dressler
- 1949
(Show Context)
Citation Context ...riodic train of waves of constant shape and velocity and, piecing together two gradually varied water-surface profiles for unsteady flow, built a wave profile similar to observed roll waves. Dressler =-=[11]-=-, using Thomas’s basic idea, was able to find closed form solutions for a wide rectangular channel with an unvarying friction factor. On the basis of Dressler’s [11] work, Brock [6] developed a theory... |

1 |
surface instabilities correlations’’, Geological Survey Water-Supply Paper 1592-C
- Koloseus
(Show Context)
Citation Context ...de rectangular channel, uniform velocity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], Craya [9], Koloseus and Davidian =-=[15]-=-, Berlamont and Vanderstappen [3], Iwasa [13]) have derived stability criteria for varying degrees of generality, showing that Fr cr depends on the channel shape, the frictional resistance law and the... |

1 |
2000: ‘‘Roll waves e dissipazione in correnti idriche e nei debris flows’’, Atti del XXVII Convegno Nazionale di Idraulica
- Lamberti, Longo
(Show Context)
Citation Context ...ia. This led him to develop a criterion for predicting formation of roll waves (but not their dimensions), involving not only the value of Fr cr but also the length of the channel. Lamberti and Longo =-=[16]-=- used an energy argument to evaluate a regime wave height, provided that wave period and length are known: energy dissipated in bores must compensate the reduced energy dissipation in the regular flow... |

1 |
Unsteady flow
- Liggett
- 1975
(Show Context)
Citation Context ...surface, determined that the critical Froude number Fr cr was about 0.5; a similar result was obtained by Chen [7] through 1D SW equations. For turbulent flows, Jeffreys [14], Stoker [19] and Liggett =-=[17]-=- considered a wide rectangular channel, uniform velocity distribution and a constant friction factor, finding the Fr cr equal to 2.0. Since then, many others (Dressler and Pole [12], Craya [9], Kolose... |

1 |
The propagation of waves in steep prismatic conduits
- Thomas
- 1939
(Show Context)
Citation Context ...ining the curve of stability; for Bingham fluids, Trowbridge [24] carried out the same analysis using SW equations. The first attempt to describe analytically large amplitude roll waves was by Thomas =-=[21]-=-, who considered a periodic train of waves of constant shape and velocity and, piecing together two gradually varied water-surface profiles for unsteady flow, built a wave profile similar to observed ... |