A numerical formulation for fully resolved simulations of freely moving rigid particles in turbulent flows is presented. This work builds upon the fictitious-domain based approach for fast computation of fluid-rigid particle motion by Sharma & Patankar ([1] Ref. J. Compt. Phys., (205), 2005). The approach avoids explicit calculation of distributed Lagrange multipliers to impose rigid body motion and reduces the computational overhead due to the particle-phase. Implementation of the numerical algorithm in co-located, finite-volume-based, energy conserving fractional-step schemes on structured, Cartesian grids is presented. The numerical approach is first validated for flow over a fixed sphere at various Reynolds numbers and flow generated by a freely falling sphere under gravity. Grid and time-step convergence studies are performed to evaluate the accuracy of the approach. Finally, simulation of 125 cubical particles in a decaying isotropic turbulent flow is performed to study the feasibility of simulations of turbulent flows in the presence of freely moving, arbitrary-shaped rigid particles.

Very complex flow structures occur during separation that appear in a wide variety of applications involving flow over a bluff body. This study examines the ability to detect the dynamic interactions of vortical structures generated from a Helmholtz instability caused by separation over bluff bodies at large Reynolds number of approximately $10^4$ based on cross stream characteristic length of the geometry. Accordingly, two configurations, a thin airfoil with flow at an angle of attack of $20^0$ and a square cylinder with normally incident flow are examined. A time-resolved, three-component PIV data set is collected in a symmetry plane for the airfoil, whereas direct numerical simulations are used to obtain flow over the square cylinder. The experimental data consists of the velocity field, whereas simulations provide both velocity and pressure-gradient fields. Two different approaches analyzing vector field and tensor field topologies are considered to identify vortical structures and local, swirl regions. The vector field topology uses (1) the $\Gamma$ function that maps the degree of rotation rate (or pressure-gradients) to identify local swirl regions, and (2) Entity Connection Graph (ECG) that combines the Conley theory and Morse decomposition to identify vector field topology consisting of fixed points (sources, sinks, saddles) and periodic orbits, together with separatrices (links connecting them). The tensor field feature uses (1) the $\lambda_2$ method that examines the gradient fields of velocity or pressure-gradient to identify local regions of pressure minima, and (2) tensor field feature that decomposes the velocity-gradient or pressure Hessian tensor into isotropic scaling, rotation, and anisotropic stretching parts to identify regions of high swirl. The vector-field topology requires spatial integration of the velocity or pressure-gradient fields and represents a global descriptor of vortical structures. The tensor field feature, on the other hand, is based on gradients of the velocity of pressure-gradient vectors and represents a local descriptor. A detailed comparison of these techniques is performed by applying them to velocity or pressure-based data and using spatial filtered data sets to identify the multiscale features of the flow. It is shown that various techniques provide useful information about the flow field at different scales that can be used for further analysis of many fluid engineering problems of practical interest.

Many situations of physical and biological interest involve diffusions on manifolds. It is usually assumed that irregularities in the geometry of these manifolds do not influence diffusions. The validity of this assumption is put to the test by studying Brownian motions on nearly flat 2D surfaces. It is found by perturbative calculations that irregularities in the geometry have a cumulative and drastic influence on diffusions, and that this influence typically grows exponentially with time. The corresponding characteristic times are computed and discussed. Conditional entropies and their growth rates are considered too.

We derive error estimates for fully discrete scheme using primal discontinuous Galerkin discretization in space and backward Euler discretization in time. The estimates in the energy norm are optimal with respect to the mesh size and suboptimal with respect to the polynomial degree. The proposed scheme is of high order as polynomial approximations of pressure and concentration can take any degree. In addition, the method can handle different types of boundary conditions and is well-suited for unstructured meshes.

Multiresolution methods, such as the wavelet decompositions, are increasingly used in physical applications where multiscale phenomena occur. We present in this paper two applications illustrating two different aspects of the wavelet theory.
In the first part of this paper, we recall the bases of the wavelets theory. We describe how to use the continuous wavelet decomposition for analyzing multifractal patterns. We also summarize some results about orthogonal wavelets and wavelet packets decompositions.
In the second part, we show that the wavelet packet filtering can be successfully used for analyzing two-dimensional turbulent flows. This technique allows the separation of two structures: the solid rotation part of the vortices and the remaining mainly composed of vorticity filaments. These two structures are multiscale and cannot be obtained through usual filtering methods like Fourier decompositions. The first structures are responsible for the inverse transfer of energy while the second ones are responsible for the forward transfer of enstrophy. This decomposition is performed on numerical simulations of a two dimensional channel in which an array of cylinders perturb the flow.
In the third part, we use a wavelet-based multifractal approach to describe qualitatively and quantitatively the complex temporal patterns of atmospheric data. Time series of geopotential height are used in this study. The results obtained for the stratosphere and the troposphere show that the series display two different multifractal behaviors. For large time scales (several years), the main Hölder exponent for the stratosphere and the troposphere data are negative indicating the absence of correlation. For short time scales (from few days to one year), the stratosphere series present some correlations with Hölder exponents larger than 0.5, whereas the troposphere data are much less correlated.

In the vadose zone, fluids, which can transport contaminants, move within unsaturated rock fractures. Surface roughness has not been adequately accounted for in modeling movement of fluid in these complex systems. Many applications would benefit from an understanding of the physical mechanism behind fluid movement on rough surfaces. Presented are the results of theoretical investigation of the effect of surface roughness on fluid spreading. The model presented classifies the regimes of spreading that occur when fluid encounters a rough surface: i) microscopic precursor film, ii) mesoscopic invasion of roughness and iii) macroscopic reaction to external forces. Theoretical diffusion-type laws based on capillarity and fluid and surface frictional resistive forces developed using different roughness shape approximations are compared to available fluid rise on roughness experiments. The theoretical diffusion-type laws are found to be the same apparent functional dependence on time; methods that account for roughness shape better explain the data as they account for more surface friction.

Modeling and analysis of texture evolution in polycrystalline materials is a major challenge in materials science. It requires understanding grain boundary or interface evolution at the network level, where topological reconfigurations (critical events) play an important role. In this paper, we investigate grain boundary evolution in a simplified one-dimensional system designed specifically to target microstructural critical event, evolution. We suggest a stochastic framework that may be used to model this system and compare predictions of the model with simulations. We discuss limitations and possible extensions of this approach to higher-dimensional cases.

We construct Lebesgue and Sobolev spaces of functions defined on a continuous distribution of domains {$Y_x \subset \mathbb{R}^m$ : $x \in \Omega$}. The resulting spaces can be viewed as a generalisation of the Bochner spaces $L_p(\Omega;W_q^l(Y))$ for the case that $Y$ depends on $x \in \Omega$. Furthermore, we introduce a Lebesgue space of functions defined on the boundaries {$∂Y_x : x \in \Omega$}. The latter construction relies on a uniform Lipschitz parametrisation of the above collection of boundaries, interpreted as a higher-dimensional manifold. The results are applied to prove existence, uniqueness and upper and lower bounds for a distributed-microstructure model of reactive transport in a heterogeneous porous medium.

Traditional unsaturated flow models use a capillary pressure-saturation relationship determined under static conditions. Recently it was proposed to extend this relationship to include dynamic effects and in particular flow rates. In this paper, we consider numerical modeling of unsaturated flow models incorporating dynamic capillary pressure terms. The resulting model equations are of nonlinear degenerate pseudo-parabolic type with or without convection terms, and follow either Richards' equation or the full two-phase flow model. We systematically study the difficulties associated with numerical approximation of such equations using two classes of methods, a cell-centered finite difference method (FD) and a locally conservative Eulerian-Lagrangian method (LCELM) based on the finite difference method. We discuss convergence of the methods and extensions to heterogeneous porous media with different rock types. In convection-dominated cases and for large dynamic effects instabilities may arise for some of the methods while those are absent in other cases.

Fully-saturated and partially fissured media, in which supplementary flow and transport arise from direct cell-to-cell diffusion paths, have been described accurately over wide range of scales by discrete secondary-flux models. These models were constructed as an extension of classical double-porosity models for totally fissured media by two-scale modeling considerations. There is some substantial literature on the analysis of continuously distributed secondary-flux models, and the corresponding discrete models have been proven to give efficient and accurate simulations when compared to recently available experimental data. These are particularly effective in the presence of advection. In this note, a summary description is given for the two-scale convergence of the discrete secondary-flux model to the corresponding continuous double-porosity secondary-flux model.

A groundwater flow model based on a specified hydraulic conductivity field in the modeling domain has a unique solution only if either the head or the normal flux component is specified on the boundary. On the other hand, specification of both head and flux as boundary conditions may be used to determine the conductivity field, or at least improve an initial estimate of it. The specified head and flux data may be obtained from measurements on the boundary, including the wells. We have presented a relatively simple, but instructive approach: the Double Constraint (DC) method. The method is exemplified in the context of upscaling and its inverse: downscaling. The DC method is not only instructive, but also easy to implement because it is based on existing groundwater modeling software. The exemplifications shown in this paper related to downscaling and demonstrate that the DC method has practical relevance.