Micro-fracture instabilities in granular solids

We study the aspect of unstable behavior (like strain localization bands) in elastic solids as a consequence of micro-fracturing. A two-scale approach of computational homogenization is considered. The macroscopic behavior is investigated by finite element computations on a unit cell. At the micro-level, we consider a granular structure with elastic grains. The inter-granular boundaries are modeled with cohesive laws, friction and unilateral contact. We show that decohesion between grains gives rise to macro-instabilities, indicated by the loss of ellipticity, typical for deformation localization bands. The relation between the microscopic softening on inter-granular boundaries and the onset of macro-instabilities is studied through numerical examples. The influence of the cohesive law and friction parameters is analyzed. For periodic distributions of granular structures, we prove the loss of periodicity by failure and the corresponding size dependence effect in the homogenized response. We present numerical examples of bifurcation of solutions for granular cell structures and of particular solutions specific to elementary volumes with periodic cell distribution. Size dependence appears in the unstable regime and is strongly influenced by cohesion and friction parameters.     

On the Formulation of Lagrangian Stochastic Models of Scalar Dispersion Within Plant Canopies

Lagrangian stochastic models, quadratic in velocity and satisfying the well-mixed condition for two-dimensional Gaussian turbulence, are used to make predictions of scalar dispersion within a model plant canopy. The non-uniqueness associated with satisfaction of the well-mixed condition is shown to be non-trivial (i.e. different models produce different predictions for scalar dispersion). The best agreement between measured and predicted mean concentrations of scalars is shown to be obtained with a small sub-class of optimal models. This sub-class of optimal models includes Thomson's model (J. Fluid Mech. 180, 529–556, 1987), the simplest model that satisfies the well-mixed condition for Gaussian turbulence, but does not include two other models identified recently as being in optimal agreement with the measured spread of tracers in a neutral boundary layer. It is therefore demonstrated that such models are not universal, i.e. applicable to a wide range of flows without readjustment of model parameters. Predictions for scalar dispersion in the model plant canopy are also obtained using the model of Flesch and Wilson (Boundary-Layer Meteorol. 61, 349–374, 1992). It is shown that, when used with a Gaussian velocity distribution or a maximum-missing-information velocity distribution, which accounts for the measured skewness and kurtosis of velocity statistics, the agreement between predictions obtained using the model of Flesch and Wilson and measurements is as good as that obtained using Thomson's model.     

On The Dynamical Content Of Lagrangian Stochastic Models In The Well-Mixed Class

It is shown how the correspondence between Lagrangian stochasticmodels and second-moment closures of the scalar-flux equation can be exploited to distinguishbetween Lagrangian stochastic models in the well-mixed class. It is found that physically realisticclosures of the scalar-flux equation correspond to Lagrangian stochastic models that have non-zero`spin' and so produce spiralling tracer-particle trajectories, whilst `zero-spin'models correspond to the isotropic-production model of scalar-fluxes.Lagrangian stochastic models consistent with rapid distortion theory and Speziale's transformation rule for the Reynolds stressequations in the extreme limit of two-dimensional turbulence are also shown to have non-zero spin.The residual non-uniqueness associated with satisfaction of thewell-mixed condition and the specification of mean spin is shown to be related to the helicity oftracer-particle trajectories. Investigations are also made of the influence upon turbulent dispersion oftime-dependent spin and of mean rotations of the fluctuating Lagrangian acceleration vector(i.e., second-order spin).     

A Two-Dimensional Lagrangian Stochastic Dispersion Model for Convective Boundary Layers with Wind Shear

Rotach, Gryning and Tassone constructed a two-dimensional Lagrangian stochastic model to describe the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral (w_* = 0) to fully-convective (u_* = 0). They found that the value of the Kolmogorov constant, C₀, as determined by optimizing model agreement with the measured spread of passive tracers, was dependent upon stability. Here, it is shown that the non-uniqueness, associated with satisfaction of the well-mixed condition, can be exploited to construct an alternative version of the model of Rotach et al. for which C₀ = 3 is universally applicable over the entire range of stabilities under consideration. This alternative model is shown to be in very close agreement with predictions, obtained in large-eddy simulations, for the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral to fully-convective.     

On Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models

Recently Wilson and Flesch (Boundary-Layer Meteorology, 84, 411-426, 1997) suggested that the average increment d z to the orientation = arctan(w/u) of the Lagrangian velocity-fluctuation vector can be used to distinguish the better Lagrangian stochastic models within the well-mixed class. Here it is demonstrated that the specification of d z constitutes neither a sufficient or universally applicable criterion to distinguish the better Lagrangian stochastic models within the well-mixed class. The hypothesis made by Wilson and Flesch that Lagrangian stochastic models with /P_E irrotational are zero-spin models, having d z=0, is proven