Inverse grading in granular flow
The current work concerns with the parametrical dependencies of inverse grading in granular flows from material properties of individual particles involved in the flow. The investigations were initiated and motivated by the search for an explanation of the function principle of the avalanche airbag, which prevents a skier from going under in a flowing avalanche. The functionality of the avalanche airbag cannot be understood in the frame of the classical hydrodynamical avalanche models: the hydrostatic buoyancy caused by the avalanche airbag is not sufficient to keep a skier on the surface of a flowing avalanche. Starting from this motivation, a short phenomenological and historical overview over snow avalanches and the development of the avalanche airbag is given. Existing field observations and laboratory experiments are described to outline the occurrence and main features of granular flows. It is discussed, how far these observations and experiments indicate specific properties of granular flows such as inverse grading. Mathematical modelling of avalanches or granular flows can easily be adjusted to the specific investigated problems and does not consume as much property as physical modelling in laboratory experiments does. Different classes of mathematical and numerical models of avalanches and granular flows are reported with respect to the question how far they allow qualitative or quantitative insight into the nature of inverse grading. Comparing continuum dynamic avalanche models, analytic models of granular flow and discrete element models (DEM), it can be seen that DEMs are well suited for the investigation of inverse grading. According to the type of the granular flow which has to be investigated, either the hard sphere or the soft sphere type of DEM is the appropriate description for the flow. Quasistatic granular flows, which are dominated by multiparticle interactions over longer durations are described by soft sphere models. For rapid granular flows, hard sphere models are the appropriate approach. In this case, it is assumed, that the flow is dominated by instantaneous binary contacts which extend over time intervals that can be neglected compared to the time intervals of free particle motion. In the current work, snow avalanches are considered as rapid flows. According to this, a hard sphere DEM was used for the investigation of inverse grading in granular flow. 1n the used model, binary collisions of particles are parametrized by the particle elasticities e and b in longitudinal and transversal direction and by the Coulomb coefficient μ of the colliding particles. The model can reproduce the effect of inverse grading in a defined range of particle material parameters. Therefore, the global flow and segregational behaviour of the granular flow can be considered to depend from the interaction behaviour of individual particles. The longitudinal restitution coefficient e, which describes the particle elasticity in longitudinal direction, seems to be the parameter, which influences the flow– and segregational behaviour most strongly. It is tried to establish a link between the model results and continuum dynamic granular flow models by searching for a constitutive relation between stress and strain in the model flow. However, a unique relation could not be found, probably because the model flow fails to meet some principle requirements for the derivation of stress tensors in granular flows. On the other hand, the method of stochastic analysis could successfully be applied to the model flow data: the time series of the vertical position of individual particles are apparently noisy and have Markov properties. This allows to derive deterministic, particle size dependent segregational dynamics of one single particle from its time series of vertical position. The fixed points of these dynamics for large particles lie in upper flow layers, where the fixed points of smaller particles are situated near the bottom. The results of the investigations in the frame of the numerical DEMs are compared with field observations and with results of laboratory measurements of the longitudinal restitution coefficient e of snow. Field measurements of mean snow lump size profiles correspond qualitatively to the particle size profiles obtained from the numerical simulations. The measured value of e ≈ 0.1...0.3 for snow differs significantly from the value of e ≈ 0.8, which is necessary to obtain granular model flows which are exhibiting inverse grading. This difference can be explained by the fact that snow flow avalanche dynamics also incorporate particle interactions extended in time, for which the numerical hard sphere model cannot account. Conceptually, this work spans an arc from the application of the avalanche airbag to an explanation of the typical granular flow behaviour or inverse grading in the frame of a numerical hard sphere approach.