Modelling rollers for shallow water flows

Hydraulic jumps, roll waves or bores in open channel flows are often treated as singularities by hydraulicians while slowly varying shallow water flows are described by continuous solutions of the Saint-Venant equations. Richard & Gavrilyuk (J. Fluid Mech., vol. 725, 2013, pp. 492–521) have enriched this model by introducing an equation for roller vorticity in a very elegant manner. This new model matches several experimental results that have resisted theoretical approaches for decades. This is the case of the roller of a stationary hydraulic jump as well as the oscillatory instability that the jump encounters when the Froude number is increased. The universality of their approach as well as its convincing comparisons with experimental results open the way for significant progress in the modelling of open channel flows

Fluid boundary of a viscoplastic Bingham flow for finite solid deformations

The modelling of viscoplastic Bingham fluids often relies on a rheological constitutive law based on a "plastic rule function" often identical to the yield criterion of the solid state. It is also often assumed that this plastic rule function vanishes at the boundary between the solid and fluid states, based on the fact that it is true in the limit of small deformations of the solid state or for simple yield criteria. We show that this is not the case for finite deformations by considering the example of a two state flow on a tilted plane where the solid state is described by a Neo-Hookean model with a Von Mises yield criterion. This opens new approaches for the modelling and the computation of the fluid state boundaries

Hydrodynamique et mécanique des fluides

Qu’il s’agisse du milliard de km3 de l’océan, du million de km3 dans le sous-sol ou des milliers de km3 dans les rivières, canaux ou canalisations, l’eau est en perpétuel mouvement. L’hydrodynamique étudie ces mouvements des échelles les plus grandes jusqu’à celles permet- tant encore de décrire la matière de manière continue (c'est à-dire pour des échelles encore très largement supérieures à la taille des molécules)

High-Reynolds shallow flow over an inclined sinusoidal bottom

An experimental study of a turbulent free-surface shallow flow over an inclined sinusoidal bottom with a fixed corrugation amplitude is presented. A parametric analysis is performed by varying both the inclination angle and the Reynolds number. We show that a “Pulse-Waves” regime, dominant for Reynolds smaller than 4 000, coexists with a “Roll-Waves” regime, which becomes dominant above this value. The relative energy of the waves is quantified in the parameter space. At Reynolds numbers larger than 8 000, these wave instabilities disappear

Linear stability of the 1D Saint-Venant equations and drag parameterizations

The stability of the homogeneous and steady flow based on the one-dimensional Saint-Venant equations for free surface and shallow water flows of constant slope is derived and displayed through graphs. With a suitable choice of units, the small and large drag limits, respectively, correspond to the small and large spatio-temporal scales of a linear system only controlled by the Froude number and two other dimensionless numbers associated with the bottom drag parameterization. Between the small drag limit, with the two families of marginal and non-dispersive shallow water waves, and the large drag limit, with the marginal and non-dispersive waves of the kinematic wave approximation, dispersive roll waves are detailed. These waves are damped or amplified, depending on the value of the three control parameters. The spatial generalized dispersion relations are also derived indicating that the roll-wave instability is of the convective type for all drag parameterizations

Contrail microphysics in the near wake of a realistic wing through RANS simulations

This paper focuses on Steady Reynolds Average Navier Stokes simulations (RANS) of ice particles growth in the near field of a wing-injector configuration. The multiphysics multiphase flow solver CEDRE, enriched with a microphysical model, has been developed in order to simulate the impact of a more real aircraft geometry in contrail formation studies. As a first evaluation case, a simplified aircraft description, i.e. a NACA0012 2D wing with two injectors, has been used. Ice formation has been simulated by assuming water condensation and instantaneous freezing on activated soot particles, initially emitted by aircraft engines. Our investigation focuses on the near field, extending from the nozzle exit until eight wing spans. Although the main goal is to address the question of ice formation, the aerodynamic flow field has been investigated and numerical results compared with existing experimental data. The first results indicate that the exhaust jet is correctly wrapped around the vortex and that the pattern of dilution qualitatively matches observations in the near field. Sensitivity studies to humidity and to the initial soot particle radius have also been performed

An Asymptotic Expansion for the Recharge-Discharge Model of ENSO

International audienceThe dynamics of El Niño-Southern Oscillation (ENSO) in the equatorial Pacific Ocean are largely associated with the slow thermocline adjustment at interannual and basin scales. This adjustment involves, among other things, the fast propagation and reflection of equatorial waves by wind stress forcing. A simple and straightforward asymptotic expansion of the long-wave equations is proposed using the low-frequency approximation. The asymptotic expansion is performed in Fourier space, retaining only the gravest equatorial long waves and baroclinic modes with the largest scale, and considering small dissipation by friction and boundary reflections. This leads to an asymptotic model for the thermocline response to wind stress forcing, which is in essence the ocean component of the recharge-discharge model of ENSO. The asymptotic model is nonheuristic and in broad agreement with some essential results scattered in previous studies. Thermocline variability is divided into a sloping "Tilt mode" that adjusts instantly to wind stress forcing and a zonal-mean "Warm Water Volume mode" that adjusts as a time integrator to wind stress curl. The model has a plausible energy budget and its solutions are in good agreement with observations. Results suggest that the net adjustment rather than the explicit delays of equatorial waves is essential for the slow thermocline adjustment, and this is best described by the recharge-discharge model

Correction of upstream flow and hydraulic state with data assimilation in the context of flood forecasting

The present study describes the assimilation of river water level observations and the resulting improvement in flood forecasting. The Kalman Filter algorithm was built on top of a one-dimensional hydraulic model which describes the Saint-Venant equations. The assimilation algorithm folds in two steps: the first one was based on the assumption that the upstream flow can be adjusted using a three-parameter correction; the second one consisted of directly correcting the hydraulic state. This procedure was applied using a four- day sliding window over the flood event. The background error covariances for water level and discharge were repre- sented with anisotropic correlation functions where the cor- relation length upstream of the observation points is larger than the correlation length downstream of the observation points. This approach was motivated by the implementation of a Kalman Filter algorithm on top of a diffusive flood wave propagation model. The study was carried out on the Adour and the Marne Vallage (France) catchments. The correction of the upstream flow as well as the control of the hydraulic state during the flood event leads to a significant improve- ment in the water level and discharge in both analysis and forecast modes

DATA ASSIMILATION ON A FLOOD WAVE PROPAGATION MODEL : EMULATION OF A KALMAN FILTER ALGORITHM

International audienceThis study describes the assimilation of synthetically-generated river water level observations in a flood wave propagation model. For this approach to be applied in the framework of real-time flood forecasting, the cost of the data assimilation procedure, mostly related to the estimation of the background error covariance matrix, should be bound. An Ensemble Kalman Filter (EnKF) algorithm is applied, with a steady observation network, to demonstrate how the assimilation modifies the background correlation function at the observation point. It is shown that an initially Gaussian correlation function turns into an anisotropic function at the observation point, with a shorter correlation length-scale downstream of the observation point than upstream, and that the variance of the error in the water level state is significantly reduced downstream of the observation point. The covariance matrix resulting from the EnKF is then used as an invariant background error covariance matrix for a series of successive Best Linear Unbiased Estimation (BLUE) algorithms which emulate an EnKF at a lower cost. This study shows how the background error covariance matrix can be computed off-line, with an advanced algorithm, and then used with a cheaper algorithm for real-time application

Steady and unsteady shear flows of a viscoplastic fluid in a cylindrical Couette cell

Abstract Yield stress fluid flows in Couette cells have been widely studied in the last decades for their intriguingly exhibiting phenomena. In this paper, we use a {PIV} technique to investigate the axisymmetric flow and rheological properties of a Carbopol gel in a relatively wide cylindrical Couette device. Carbopol gel is known to exhibit viscoplastic behavior and is often described using a Herschel–Bulkley law, which is characterized by a plastic yield stress τ y and a shear-dependent nonlinear viscosity. In some cases, the elasticity of the material has to be accounted for to understand the whole dynamics of the system, in particular for unsteady flows as observed in the present study. Two set of experiments are conducted here in order to highlight these different rheological behaviors and the resulting dynamics: (i) a steady shear configuration and (ii) an unsteady shear configuration, in which the angular velocity of the inner cylinder is either constant or time dependent ( sin profile), respectively. In the steady configuration, a simple optimization model, based on the Herschel–Bulkley law, is developed to extract the rheological parameters of the viscoplastic contribution of the gel from the steady velocity fields. Results are shown to be in good agreements with rheological parameters obtained from a standard rheometer. On the other hand, the elastic contribution of the material is highlighted in the unsteady shear configuration, for which a spatio-temporal transition between solid-elastic and fluid behaviors is observed. Different models are proposed to describe the dynamics of the unsteady flow. First, quasi-steady state models allow to predict both the fluid shear zone close to the inner cylinder and the elastic deformation of the material as long as their contributions can be decoupled in space and in time. For more complex dynamics, i.e. when the flow becomes strongly unsteady, an elasto-viscoplastic model is developed to describe the flow dynamics. It is shown to quantitatively reproduce the experimental measurements. Finally, an elastic wave model is derived to describe an elastic front propagating from the inner cylinder to the outer one, and observed at every half forcing period. The front velocity is thus shown to scale on the phase velocity of an elastic wave in a deformable solid