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Researchers using Darcy's law to model flow in porous media must satisfy the requirement for sufficient scale separation between the pore scale and the model scale. This requirement is analogous to that for any continuum model, where application is restricted to scales larger than the underlying discrete structure. In the case of Darcy's law when the model scale becomes too small, the measurement of the permeability - the material property required to close the relationship - becomes polluted by the boundary conditions, either physical or numerical. The requirements for adequate scale separation to obtain permeability measurements (also known as satisfying the conditions for a representative elementary volume, or REV, for permeability) have not been previously reported. Likewise, the behavior of Darcy models when applied at sub-REV length scales has not been reported. Here, the results of Stokes simulations of transverse flow in 90,000 sequential random packings of monodisperse cylinders at a variety of liquid fractions and averaging-volume sizes show that approximately 200 cylinders must be present in an averaging volume before the effects of periodic boundary conditions on the Stokes simulations (the conventional choice for permeability measurements using Stokes flow) are no longer evident in the measured permeability. Direct comparisons between flow predictions from a two-dimensional, tensor-based Darcy model and a Stokes model for additional 10,000 domains show that the Darcy model is an unbiased predictor of the flow distribution in the system, even when the permeability is expected to contain boundary-condition artifacts. Though unbiased, the Darcy models do show considerable reduction in accuracy as the model scale shrinks toward the pore scale, with significant declines observed after the side length of a square averaging volume reaches 10 times the cylinder diameter. Finally, a novel approach for visualizing flows using the linear properties of the Stokes equations shows how the periodic boundary conditions affect the flow, and motivates the development of a generalized approach for obtaining permeability that does not require periodic boundary conditions. Modest improvements in the Darcy model relative to the actual Stokes flow result when the new approach is used to obtain permeability at small averaging volumes.

Thermosolutal fluid flow has a strong influence on the evolution of solidification microstructures. While porous media theory and volume-averaged permeability relations give a basis to quantify these phenomena, traditional methods of permeability estimation used for random porous media fail to adequately characterize the full relation of microstructural morphology to volume-average permeability. Most significantly, the link between microstructural parameters and permeability is treated as a deterministic function at all scales, ignoring the variability inherent in porous media.The variation in permeability inherent to random porous media is investigated by the numerical solution of Stokes equations on an ensemble of porous media, which represent of many scales of sampling and morphological character. Based on volume-averaging and statistical treatment, the stochastic character of tensoral permeability in porous media is numerically investigated. Quantification of permeability variation and autocorrelation structure are presented as conditions, which future realistic stochastic permeability fields must respect.

A floating object's drift is governed by its buoyancy, shape, and the wind, waves and currents it experiences. Here, I develop a drift modelling framework for marine mammal carcases in coastal waters. The resulting models were run forwards and backwards in time to provide insights into strategies for environmental monitoring under two scenarios. The first explored the beach search options for carcases resulting from potentially fatal collisions between tidal-stream turbines and marine mammals. The second applied the reverse problem for known-location mass strandings to highlight potential at-sea mortality sites. The drift properties of carcase-like objects were assessed in at-sea experiments. Wave transport was found to be greater than Stokes drift alone and in a complex coastal area could not be represented by a downwind multiplier as many previous models have assumed. A high resolution unstructured grid wave model was set up to complement existing wind and current models for the West Coast of Scotland, and these components were combined to build a carcase drift model. In the forward case, from tidal turbine locations, the drift model showed a wide spread of potential stranding sites, suggesting monitoring a limited number of beaches is unlikely to be fruitful. However, selecting beaches in response to immediate wind direction would improve efficiency. Stranding locations alone can only provide evidence of turbine interactions if the number of animals affected is large. In the reverse case, when applied to a mass stranding in Chile, the drift model showed the ability to exclude areas of origin, even though it could not pinpoint an exact mortality site. This work advances understanding of wave transport of surface floating objects, of carcase drift modelling, and of the feasibility of strandings monitoring. The decomposition rate of carcases is a source of uncertainty in the model where further work should be undertaken.

Numerical models for the simulation of longitudinal and transverse Stokes flow in cylindrical periodic porous media are presented. The models, which are based on a finite-volume formulation in primitive variables, utilize digital image representations of the geometries to simulate, making them particularly well-suited for the rapid automated analysis of creeping flow in porous media with complex morphologies. Complete details of the model formulations are given, including extensive treatment of the pressure boundary conditions at the solid-liquid interface needed to guarantee convergence with all possible geometries. The convergence behavior of both models is tested, and the models are shown to be second-order accurate.The models are used to simulate flow over the whole range of volume fractions of liquid in several regular geometries. The longitudinal model is used to simulate flow in square arrays of circular and square ducts, and both models are used to simulate flow in square and hexagonal arrays of circular cylinders and square arrays of square cylinders rotated by varying amounts. For each of the geometries, accurate empirical expressions for the Darcy permeability as a function of volume fraction solid are presented. Where applicable, model predictions of permeability are compared to existing analytical results.Subsequently, the models are used to simulate Stokes flow in random domains over a wide range of fractions liquid. The sequential random packing algorithm is used to generate 1,000 random packings of circular cylinders at each of 14 fractions of liquid, and longitudinal and transverse flow simulations are performed for each geometry. Histograms and summary statistics are computed for the permeability for each fraction liquid, and empirical expressions for mean permeability as a function of fraction liquid are given. The autocorrelation structure of the geometry and of the fluid velocity is analyzed, and an analysis of the scaling of longitudinal permeability variance is presented. In transverse flow at high packing densities, it is found that lightning-like patterns emerge in the fluid velocity. It is also found that the details of flows in such geometries are strongly sensitive to the placement of individual solid obstacles.

To increase our understanding of how ice sheets and glaciers interact with the climate system, numerical models have become an indispensable tool. However, the complexity of these systems and the natural limitation in computational power is reflected in the simplifications of the represented processes and the spatial and temporal resolution of the models. Whether the effect of these limitations is acceptable or not, can be assessed by theoretical considerations and by validating the output of the models against real world data. Equally important is to verify if the numerical implementation and computational method accurately represent the mathematical description of the processes intended to be simulated. This thesis concerns a set of numerical models used in the field of glaciology, how these are applied and how they relate to other study areas in the same field. The dynamical flow of glaciers, which can be described by a set of non-linear partial differential equations called the Full Stokes equations, is simulated using the finite element method. To reduce the computational cost of the method significantly, it is common to lower the order of the used elements. This results in a loss of stability of the method, but can be remedied by the use of stabilization methods. By numerically studying different stabilization methods and evaluating their suitability, this work contributes to constraining the values of stabilization parameters to be used in ice sheet simulations. Erroneous choices of parameters can lead to oscillations of surface velocities, which affects the long term behavior of the free-surface ice and as a result can have a negative impact on the accuracy of the simulated mass balance of ice sheets. The amount of basal sliding is an important component that affects the overall dynamics of the ice. A part of this thesis considers different implementations of the basal impenetrability condition that accompanies basal sliding, and shows that methods used in literature can lead to a difference in velocity of 1% to 5% between the considered methods. The subglacial hydrological system directly influences the glacier's ability to slide and therefore affects the velocity distribution of the ice. The topology and dominant mode of the hydrological system on the ice sheet scale is, however, ill constrained. A third contribution of this thesis is, using the theory of R-channels to implement a simple numerical model of subglacial water flow, to show the sensitivity of subglacial channels to transient processes and that this limits their possible extent. This insight adds to a cross-disciplinary discussion between the different sub-fields of theoretical, field and paleo-glaciology regarding the characteristics of ice sheet subglacial hydrological systems. In the study, we conclude by emphasizing areas of importance where the sub-fields have yet to unify: the spatial extent of channelized subglacial drainage, to what degree specific processes are connected to geomorphic activity and the differences in spatial and temporal scales. As a whole, the thesis emphasizes the importance of verification of numerical models but also acknowledges the natural limitations of these to represent complex systems. Focusing on keeping numerical ice sheet and glacier models as transparent as possible will benefit end users and facilitate accurate interpretations of the numerical output so it confidently can be used for scientific purposes. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p> / Greenland Analogue Project

A numerical method for computing three-dimensional Stokes flow driven by a doubly-periodic array of regularized forces is presented. In the non-periodic direction either a free boundary or a homogeneous Dirichlet condition is enforced. The method consists of finding a regularized Green's function in Fourier space analytically. Then only an inverse fast Fourier transform (inverse FFT) has to be computed. Accuracy is verified by comparing numerical results to a solution that is independent of the method. In an Ewald splitting, the FFT method can be used to compute the smooth component of the flow, which allows for a splitting parameter as small as a few grid cells. This selection makes the sum in physical space converge extremely fast. Numerical examples demonstrate that fact. Since the forces are regularized, in some cases splitting is not even needed, depending on the relative sizes of the numerical parameters. The method is applied to model the flow created by carpets of nodal cilia based on cilium shape. / acase@tulane.edu

Η περιγραφή των αλληλεπιδράσεων που αναπτύσσονται μεταξύ ενός συνόλου σωμάτων που είτε είναι ακίνητα είτε κινούνται και ενός ρευστού, ρέοντος ή ηρεμούντος, βρίσκεται στον πυρήνα ενός μεγάλου αριθμού επιστημονικών προσπαθειών. Η μοντελοποίηση των αλληλεπιδράσεων αυτών μπορεί να γίνει με πολλούς τρόπους, καθένας από τους οποίους εμφανίζει πλεονεκτήματα και μειονεκτήματα, ανάλογα με το είδος του συστήματος στο οποίο επιχειρείται να εφαρμοστεί. Σκοπός της εργασίας αυτής είναι η ανάδειξη ενός εξ αυτών που είναι η Μέθοδος των Προτύπων Ροών. / The description of interactions between a number of solid particles, which may be in motion or fixed in space and a fluid which in turn may either be flowing or resting, is placed at the core of a large number of scientific publications. Modelling such interactions may follow a variety of approaches. Each approach has its own pros and cons, regarding the nature and the specific features of the system which is aimed to describe. Current thesis goal is to describe one of these approaches, which is the approach of Singularities or the Method of Elementary Flows.

This thesis reports on measurements of drag on an unbounded cylinder in a viscoelastic Stokes flow, and compares these values with a Newtonian equivalent. Cylinders of diameter 0.5 to 3.34 mm were submerged 10 to 36 mm into slowly rotating annular tanks with channel widths between 133 to 152 mm. Theoretical formulas and computer simulations were used to correct for the effects of ends and walls, yielding estimates of the unbounded drag. The methodology was verified by testing Newtonian fluids and comparing the results to Kaplun’s (1957) prediction for unbounded drag. The test fluids used were a silicone oil, a polybutene, and two Boger fluids. By comparing the Boger fluid results to equally viscous Newtonian results, the contributions of elasticity to the drag were determined. The Deborah number (De) was used to represent the magnitude of flow elasticity, and an onset of elastic effects was measured between 0.5 and 0.7.

This thesis reports on measurements of drag on an unbounded cylinder in a viscoelastic Stokes flow, and compares these values with a Newtonian equivalent. Cylinders of diameter 0.5 to 3.34 mm were submerged 10 to 36 mm into slowly rotating annular tanks with channel widths between 133 to 152 mm. Theoretical formulas and computer simulations were used to correct for the effects of ends and walls, yielding estimates of the unbounded drag. The methodology was verified by testing Newtonian fluids and comparing the results to Kaplun’s (1957) prediction for unbounded drag. The test fluids used were a silicone oil, a polybutene, and two Boger fluids. By comparing the Boger fluid results to equally viscous Newtonian results, the contributions of elasticity to the drag were determined. The Deborah number (De) was used to represent the magnitude of flow elasticity, and an onset of elastic effects was measured between 0.5 and 0.7.