Nombre de Froude

Résumé

Le nombre de Froude, de l'hydrodynamicien anglais William Froude, est un nombre sans dimension qui caractérise dans un fluide l'importance relative de l'énergie cinétique de ses particules par rapport à son énergie potentielle gravitationnelle. Il s'exprime donc par un rapport entre la vitesse d'une particule et la force de pesanteur qui s'exerce sur celle-ci. Ce nombre apparaît essentiellement dans les phénomènes à surface libre, en particulier dans les études de cours d'eau, de barrages, de ports et de navires (architecture navale). Il est également important en météorologie pour le calcul de l'écoulement de l'air en montagne. En dynamique des fluides il fait partie des trois nombres sans dimension les plus utilisés : il caractérise l'importance de la pesanteur alors que le nombre de Reynolds prend en compte la viscosité et le nombre de Mach la compressibilité. Le nombre de Froude peut être indicatif de la vitesse du fluide normée par une distance caractéristique. Ainsi les rapid

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Roll waves are known to occur in the frictional flow of a thin layer of water down an inclined solid surface. For a layer of constant depth, the formation of these waves on a solid plane with small slope angle have been explained as a hydrodynamic instability occurring above a critical Froude number by analyzing the temporal stability of the constant velocity flow. Here we analyze the linear, spatial stability of the shallow-water flow down an inclined plane of arbitrary slope including the first order effects of the gradients of both the velocity and the water depth. For constant water depth (parallel case) we reproduce previous results of the temporal stability analysis. Then we apply the nonparallel stability formulation to the kinematic wave approximation of the shallow-water flow down an inclined plane of arbitrary slope and characterize, analytically, the frequency and the wavelength of the most unstable waves as a function of the Froude number, slope angle, velocity and velocity gradient. We find important qualitative differences with respect to the parallel (constant depth) case. These stability results are used to discuss the numerical solution to the nonlinear shallow-water flow equations for the dam-break problem on an inclined surface of arbitrary slope. A good agreement between the waves resulting from the numerical simulations and the predictions of the stability analysis is found.

Taylor and Francis Group2006

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The flow after the rupture of a dam on an inclined plane of arbitrary slope, and the induced transport of non-cohesive sediment, is analysed using the shallow-water approximation. An asymptotic (analytical) solution is presented for the flow hydrodynamics, and compared with the numerical simulation of the dam-break flood. Differences arise due to the appearance of hydrodynamic instabilities (hereafter called roll-waves) in the numerical solution. These roll-waves point out the unstable behaviour of the dam-break wave. It is found that roll-waves enhance the transport of suspended sediment. The limitations of this simple model to predict the transport of sediment in dam-break floods on steep inclines are discussed. Then, a numerical experiment is designed to analyse the unstable character of the dam-break wave, that constitutes a non-parallel and unsteady base flow. By analysing the linear and non-linear numerical evolution of small perturbations, it is possible to reveal how the nature of the ensuing flow depends not only on the Froude number (as it happens in the classical problem of roll-waves over a uniform and steady flow) but also on the non-parallel and time-varying characteristics of the background flow. Consequently, it is also shown that these effects stabilise turbulent roll-waves and raise the critical Froude number required to achieve an unstable flow. This stability result differs with that obtained with a non-parallel spatial stability analysis, pointing out the strong influence of the base-flow time-dependence in the stability criteria. A novel Continuum Mechanics model is presented to study the transport of sediment in a laminar/turbulent free-surface flow. The mixture equations for non-cohesive sediment transport in turbulent free-surface flow are derived from the ensemble averaged Navier-Stokes equations of the three-phases (water, sediment and air). This model avoids the limitations of traditional shallow-water models, and is suitable to study, for instance, the transport of sediment in non-hydrostatic shallow-water flows over bed of arbitrary bottom slopes. The model developed in this work reveals a mathematical equivalence between the propagation of the volumetric concentration of the sediment and the phase function used to capture the free surface. We take advantage of this fact to formulate an explicit Finite Volume Method (FVM) with the exact conservation property, that is implemented in the open source software OpenFOAM. Finally, this model is applied to solve the problem of local scour around a pipeline and the transport of sediment after the rupture of a horizontal dam. It is demonstrated that one-dimensional models based on depth-averaged variables (e.g. generalisations of the one-dimensional Saint-Venant equations to predict mor- phological changes) are superseded by more sophisticated and accurate procedures valid for hyperconcentrated shallow-water flows over bed of arbitrary bottom slopes (e.g. the model described herein).

University of Malaga, Spain2008

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Competition between kinematic and dynamic waves in floods on steep slopes

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values or the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude number Fr-p and a measure of the local slope of the free-surface height phi by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, phi > 0, introduce substantial differences with respect to the plane-parallel flow with phi = 0. In particular, we find that instabilities do not occur at Froude numbers Fr-cr much larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill & Whitham (Proc. R. Soc. A, vol. 229, 1955, pp. 281-345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tall of the darn-break flood.

Cambridge University Press2010

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University of Malaga, Spain2008

Competition between kinematic and dynamic waves in floods on steep slopes

Cambridge University Press2010