Sharp transition in modes of dynamic crack propagation in dry-snow slab avalanche

Dry-snow slab avalanche release can be separated in four distinct phases : (i) failure initiation in a weak snow layer, (ii) onset, (iii) dynamics of crack propagation in the weak layer and eventually (iv)slab release. While a lot has been done to study the first two phases, less is known about dynamic crack propagation, especially at slope scale.In this study, we used the Material Point Method and elastoplasticity to simulate the dynamics of20 m long Centered Propagation Saw Tests (PST). We improved the recent constitutive snow model of Gaume et al. (2018) by developing a new softening law based on the total plastic deformation (volumetric and deviatoric parts) to account for damages caused by shearing the weak layer. Interestingly, several regimes of propagations are observed depending on slope angle. For slope angles smaller than the friction angle (θφ), a sharp transition is observed once the crack reaches a critical length (lf). We interpret this transition as a change from slab bending to slab tension due to the increasing load in the downstream direction. An estimation of this critical length (lf) is proposed using a basic analytical shear model with residual friction similar to the one developped by McClung in 1979. The crack speed seems only bounded by the elastic wave speed in the slab. By this study, we explain the gap between propagation speeds based on 2 m PSTs and slope scale observations. Finally, our results bring back to light the pioneer work of McClung which appears sufficient to model slab avalanche release on steep terrain.

Dynamics of supercooled liquids: Density fluctuations and mode coupling theory

Giuseppe Foffi, Francesco Sciortino

We write equations of motion for density variables that are equivalent to Newton's equations. We then propose a set of trial equations parametrized by two unknown functions to describe the exact equations. These are chosen to best fit the exact Newtonian equations. Following established ideas, we choose to separate these trial functions into a set representing integrable motions of density waves, and a set containing all effects of non-integrability. The density waves are found to have the dispersion of sound waves, and this ensures that the interactions between the independent waves are minimized. Furthermore, it transpires that the static structure factor is fixed by this minimum condition to be the solution of the Yvon-Born-Green equation. The residual interactions between density waves are explicitly isolated in their Newtonian representation and expanded by choosing the dominant objects in the phase space of the system, that can be represented by a dissipative term with memory and a random noise. This provides a mapping between deterministic and stochastic dynamics. Imposing the fluctuation-dissipation theorem allows us to calculate the memory kernel. We write exactly the expression for it, following two different routes, i.e. using explicitly Newton's equations, or instead, their implicit form, that must be projected onto density pairs, as in the development of the well established mode coupling theory. We compare these two ways of proceeding, showing the necessity to enforce a new equation of constraint for the two schemes to be consistent. Thus, while in the first 'Newtonian' representation a simple Gaussian approximation for the random process leads easily to the mean spherical approximation for the statics and to MCT for the dynamics of the system, in the second case higher levels of approximation are required to have a fully consistent theory.

2002

Gaussian density fluctuations and mode coupling theory for supercooled liquids

Giuseppe Foffi, Francesco Sciortino

The equations of motion for the density modes of a fluid, derived from Newton's equations, are written as a linear generalized Langevin equation. The constraint imposed by the fluctuation-dissipation theorem is used to derive an exact form for the memory function. The resulting equations, solved under the assumption that the noise, and consequently density fluctuations, of the liquid are Gaussian distributed, are equivalent to the random phase approximation for the static structure factor and to the well-known ideal mode coupling theory (MCT) equations for the dynamics. This finding suggests that MCT is a theory of fluid dynamics that becomes exact in a mean-field limit.

2001

Turbulent Flows over Rough Permeable Beds in Mountain Rivers: Experimental Insights and Modeling

Gauthier Paul Daniel Marie Rousseau

Steep mountain streams exhibit shallow waters with roughness elements such as stones and pebbles that are comparable in size to flow depth. Owing to the difficulty in measuring fluid velocities at the interface, i.e., from the rough permeable bed to the free surface, experimental results are rare although they are essential to improve models. Using a novel experimental procedure, this thesis attempts to improve predictions of the vertical structure of turbulent flows over rough permeable beds. To explore flows at the bed interface, I devised an experimental set-up where a fluid flowed over glass spheres (8 mm < dp < 14 mm) in a narrow flume (W = 6 cm) with slopes varying from 0.5 % to 8 %. The Refractive Index Matching (RIM) technique has been employed. This involves matching the refractive index of the fluid with that of the glass spheres, thereby allowing the interior of the medium to be examined and velocities to be measured by Particle Image Velocimetry (PIV). Vertical profiles are retrieved by employing the spatiotemporal double averaging method. In the course of this manuscript, flow processes are studied at the mesoscopic scale, i.e., by averaging quantities over distances ranging from 5 to 10 grain diameters. For open-channel flows over rough permeable beds, the spatial averaging procedure yields a continuous porosity profile. When applied to the Navier-Stokes equations, it produces a momentum equation with several terms including drag forces and three stresses: the turbulent, dispersive, and viscous stresses. The momentum equation was employed to devise a one dimensional (1D) model describing the vertical structure of unidirectional turbulent flow. A turbulent boundary layer over the rough bed was observed while experiments were performed at intermediate Reynolds numbers, i.e., Re = O (1000). In such conditions, viscosity plays a critical role through the van Driest damping effect. To model vertical profiles, the Darcy-Ergün equation is well suited to the prediction of friction forces in the permeable bed, i.e., in roughness and subsurface layers. Based on the \textit{Prandtl mixing length theory}, turbulent stress is predicted from a mixing length distribution that considers dispersive effects and assumes a continuous porosity profile. This alternative contrasts with most existing boundary layer models which postulate a discontinuous porosity profile for permeable or impermeable walls. Finally, hydraulic conditions collected by an Unmanned Aerial Vehicle (UAV) and classical flow resistance equations (Chézy, Keulegan, ...) were compared with profile simulations and demonstrate a good agreement between predictions and observations. It reveals the crucial role of fluid depth definition in equations in small submergence conditions. Furthermore, incipient sediment motion conditions have been estimated and compared to empirical results showing the importance of turbulence and lift force for grain entrainment. With regard to fluid dynamics, mountain streams are a case study of the larger scientific family of turbulent flows interacting with porous structures. Insights and developments acquired in the course of this thesis are likely to be transferable to other domains working with these phenomena such as flows over buildings, vegetal canopies or rough wings.

EPFL2019

Dynamics of supercooled liquids: Density fluctuations and mode coupling theory

Giuseppe Foffi, Francesco Sciortino

2002

Turbulent Flows over Rough Permeable Beds in Mountain Rivers: Experimental Insights and Modeling

Gauthier Paul Daniel Marie Rousseau

EPFL2019