Two fluid numerical model for coastal applications

Wave breaking is an important coastal process, influencing hydro-morphodynamic processes such as turbulence generation and wave energy dissipation, run-up on the beach and overtopping of coastal defence structures. During breaking, waves are complex mixtures of air and water (“white water”) whose properties affect velocity and pressure fields in the vicinity of the free surface and, depending on the breaker characteristics, different mechanisms for air entrainment are usually observed. Several laboratory experiments have been performed to investigate the role of air bubbles in the wave breaking process (Chanson & Cummings, 1994, among others) and in wave loading on vertical wall (Oumeraci et al., 2001; Peregrine et al., 2006, among others), showing that the air phase is not negligible since the turbulent energy dissipation involves air-water mixture. The recent advancement of numerical models has given valuable insights in the knowledge of wave transformation and interaction with coastal structures. Among these models, some solve the RANS equations coupled with a free-surface tracking algorithm and describe velocity, pressure, turbulence and vorticity fields (Lara et al. 2006 a-b, Clementi et al., 2007). The single-phase numerical model, in which the constitutive equations are solved only for the liquid phase, neglects effects induced by air movement and trapped air bubbles in water. Numerical approximations at the free surface may induce errors in predicting breaking point and wave height and moreover, entrapped air bubbles and water splash in air are not properly represented. The aim of the present thesis is to develop a new two-phase model called COBRAS2 (stands for Cornell Breaking waves And Structures 2 phases), that is the enhancement of the single-phase code COBRAS0, originally developed at Cornell University (Lin & Liu, 1998). In the first part of the work, both fluids are considered as incompressible, while the second part will treat air compressibility modelling. The mathematical formulation and the numerical resolution of the governing equations of COBRAS2 are derived and some model-experiment comparisons are shown. In particular, validation tests are performed in order to prove model stability and accuracy. The simulation of the rising of a large air bubble in an otherwise quiescent water pool reveals the model capability to reproduce the process physics in a realistic way. Analytical solutions for stationary and internal waves are compared with corresponding numerical results, in order to test processes involving wide range of density difference. Waves induced by dam-break in different scenarios (on dry and wet beds, as well as on a ramp) are studied, focusing on the role of air as the medium in which the water wave propagates and on the numerical representation of bubble dynamics. Simulations of solitary and regular waves, characterized by both spilling and plunging breakers, are analyzed with comparisons with experimental data and other numerical model in order to investigate air influence on wave breaking mechanisms and underline model capability and accuracy. Finally, modelling of air compressibility is included in the new developed model and is validated, revealing an accurate reproduction of processes. Some preliminary tests on wave impact on vertical walls are performed: since air flow modelling allows to have a more realistic reproduction of breaking wave propagation, the dependence of wave breaker shapes and aeration characteristics on impact pressure values is studied and, on the basis of a qualitative comparison with experimental observations, the numerical simulations achieve good results.

Stability, viscous dissipation and local thermal non-equilibrium in fluid saturated porous media

In this thesis, the field of study related to the stability analysis of fluid saturated porous media is investigated. In particular the contribution of the viscous heating to the onset of convective instability in the flow through ducts is analysed. In order to evaluate the contribution of the viscous dissipation, different geometries, different models describing the balance equations and different boundary conditions are used. Moreover, the local thermal non-equilibrium model is used to study the evolution of the temperature differences between the fluid and the solid matrix in a thermal boundary layer problem. On studying the onset of instability, different techniques for eigenvalue problems has been used. Analytical solutions, asymptotic analyses and numerical solutions by means of original and commercial codes are carried out.

Magma injections and destabilization of basaltic volcanoes: a numerical study. Application to La Reunion (Indian ocean, France) and Stromboli (Tyrrhenian sea, Italy)

Most basaltic volcanoes are affected by recurrent lateral instabilities during their evolution. Numerous factors have been shown to be involved in the process of flank destabilization occurring over long periods of time or by instantaneous failures. However, the role of these factors on the mechanical behaviour and stability of volcanic edifices is poorly-constrained as lateral failure usually results from the combined effects of several parameters. Our study focuses on the morphological and structural comparison of two end-member basaltic systems, La Reunion (Indian ocean, France) and Stromboli (southern Tyrrhenian sea, Italy). We showed that despite major differences on their volumes and geodynamic settings, both systems present some similarities as they are characterized by an intense intrusive activity along well-developed rift zones and recurrent phenomena of flank collapse during their evolution. Among the factors of instability, the examples of la Reunion and Stromboli evidence the major contribution of intrusive complexes to volcano growth and destruction as attested by field observations and the monitoring of these active volcanoes. Classical models consider the relationship between vertical intrusions of magma and flank movements along a preexisting sliding surface. A set of published and new field data from Piton des Neiges volcano (La Reunion) allowed us to recognize the role of subhorizontal intrusions in the process of flank instability and to characterize the geometry of both subvertical and subhorizontal intrusions within basaltic edifices. This study compares the results of numerical modelling of the displacements associated with high-angle and low-angle intrusions within basaltic volcanoes. We use a Mixed Boundary Element Method to investigate the mechanical response of an edifice to the injection of magmatic intrusions in different stress fields. Our results indicate that the anisotropy of the stress field favours the slip along the intrusions due to cointrusive shear stress, generating flank-scale displacements of the edifice, especially in the case of subhorizontal intrusions, capable of triggering large-scale flank collapses on basaltic volcanoes. Applications of our theoretical results to real cases of flank displacements on basaltic volcanoes (such as the 2007 eruptive crisis at La Reunion and Stromboli) revealed that the previous model of subvertical intrusions-related collapse is a likely mechanism affecting small-scale steeply-sloping basaltic volcanoes like Stromboli. Furthermore, our field study combined to modelling results confirms the importance of shallow-dipping intrusions in the morpho-structural evolution of large gently-sloping basaltic volcanoes like Piton de la Fournaise, Etna and Kilauea, with particular regards to flank instability, which can cause catastrophic tsunamis.

Aeroelastic stability of structures: flutter analysis using Computational Fluid Dynamics

Thanks to the increasing slenderness and lightness allowed by new construction techniques and materials, the effects of wind on structures became in the last decades a research field of great importance in Civil Engineering. Thanks to the advances in computers power, the numerical simulation of wind tunnel tests has became a valid complementary activity and an attractive alternative for the future. Due to its flexibility, during the last years, the computational approach gained importance with respect to the traditional experimental investigation. However, still today, the computational approach to fluid-structure interaction problems is not as widely adopted as it could be expected. The main reason for this lies in the difficulties encountered in the numerical simulation of the turbulent, unsteady flow conditions generally encountered around bluff bodies. This thesis aims at providing a guide to the numerical simulation of bridge deck aerodynamic and aeroelastic behaviour describing in detail the simulation strategies and setting guidelines useful for the interpretation of the results.

Numerical modelling of fluid-structure interaction with application in hemodynamics

The thesis deals with numerical algorithms for fluid-structure interaction problems with application in blood flow modelling. It starts with a short introduction on the mathematical description of incompressible viscous flow with non-Newtonian viscosity and a moving linear viscoelastic structure. The mathematical model consists of the generalized Navier-Stokes equation used for the description of fluid flow and the generalized string model for structure movement. The arbitrary Lagrangian-Eulerian approach is used in order to take into account moving computational domain. A part of the thesis is devoted to the discussion on the non-Newtonian behaviour of shear-thinning fluids, which is in our case blood, and derivation of two non-Newtonian models frequently used in the blood flow modelling. Further we give a brief overview on recent fluid-structure interaction schemes with discussion about the difficulties arising in numerical modelling of blood flow. Our main contribution lies in numerical and experimental study of a new loosely-coupled partitioned scheme called the kinematic splitting fluid-structure interaction algorithm. We present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Here, we assume both, the nonlinearity in convective as well is diffusive term. We analyse the convergence of proposed numerical scheme for a simplified fluid model of the Oseen type. Moreover, we present series of experiments including numerical error analysis, comparison of hemodynamic parameters for the Newtonian and non-Newtonian fluids and comparison of several physiologically relevant computational geometries in terms of wall displacement and wall shear stress. Numerical analysis and extensive experimental study for several standard geometries confirm reliability and accuracy of the proposed kinematic splitting scheme in order to approximate fluid-structure interaction problems.

Numerical simulation of some viscoelastic fluids

Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.

Numerical analysis of the relation between interactions and structure in a molecular fluid

Coarse graining is a popular technique used in physics to speed up the computer simulation of molecular fluids. An essential part of this technique is a method that solves the inverse problem of determining the interaction potential or its parameters from the given structural data. Due to discrepancies between model and reality, the potential is not unique, such that stability of such method and its convergence to a meaningful solution are issues.rnrnIn this work, we investigate empirically whether coarse graining can be improved by applying the theory of inverse problems from applied mathematics. In particular, we use the singular value analysis to reveal the weak interaction parameters, that have a negligible influence on the structure of the fluid and which cause non-uniqueness of the solution. Further, we apply a regularizing Levenberg-Marquardt method, which is stable against the mentioned discrepancies. Then, we compare it to the existing physical methods - the Iterative Boltzmann Inversion and the Inverse Monte Carlo method, which are fast and well adapted to the problem, but sometimes have convergence problems.rnrnFrom analysis of the Iterative Boltzmann Inversion, we elaborate a meaningful approximation of the structure and use it to derive a modification of the Levenberg-Marquardt method. We engage the latter for reconstruction of the interaction parameters from experimental data for liquid argon and nitrogen. We show that the modified method is stable, convergent and fast. Further, the singular value analysis of the structure and its approximation allows to determine the crucial interaction parameters, that is, to simplify the modeling of interactions. Therefore, our results build a rigorous bridge between the inverse problem from physics and the powerful solution tools from mathematics. rn

Novel magnetic iron oxide nanoparticles coated with poly(ethylene imine)-g-poly(ethylene glycol) for potential biomedical application: synthesis, stability, cytotoxicity and MR imaging

Magnetic iron oxide nanoparticles have found application as contrast agents for magnetic resonance imaging (MRI) and as switchable drug delivery vehicles. Their stabilization as colloidal carriers remains a challenge. The potential of poly(ethylene imine)-g-poly(ethylene glycol) (PEGPEI) as stabilizer for iron oxide (γ-Fe₂O₃) nanoparticles was studied in comparison to branched poly(ethylene imine) (PEI). Carrier systems consisting of γ-Fe₂O₃-PEI and γ-Fe₂O₃-PEGPEI were prepared and characterized regarding their physicochemical properties including magnetic resonance relaxometry. Colloidal stability of the formulations was tested in several media and cytotoxic effects in adenocarcinomic epithelial cells were investigated. Synthesized γ-Fe₂O₃ cores showed superparamagnetism and high degree of crystallinity. Diameters of polymer-coated nanoparticles γ-Fe₂O₃-PEI and γ-Fe₂O₃-PEGPEI were found to be 38.7 ± 1.0 nm and 40.4 ± 1.6 nm, respectively. No aggregation tendency was observable for γ-Fe₂O₃-PEGPEI over 12 h even in high ionic strength media. Furthermore, IC₅₀ values were significantly increased by more than 10-fold when compared to γ-Fe₂O₃-PEI. Formulations exhibited r₂ relaxivities of high numerical value, namely around 160 mM⁻¹ s⁻¹. In summary, novel carrier systems composed of γ-Fe₂O₃-PEGPEI meet key quality requirements rendering them promising for biomedical applications, e.g. as MRI contrast agents.

Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding.

Effect of high order interpolation in the stability and efficiency of the time-integration process in vorticity-velocity CFD algorithms

The numerical solution of the incompressible Navier-Stokes Equations offers an effective alternative to the experimental analysis of Fluid-Structure interaction i.e. dynamical coupling between a fluid and a solid which otherwise is very complex, time consuming and very expensive. To have a method which can accurately model these types of mechanical systems by numerical solutions becomes a great option, since these advantages are even more obvious when considering huge structures like bridges, high rise buildings, or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the KLE to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ODE time integration schemes and thus allows us to tackle the various multiphysics problems as separate modules. The current algorithm for KLE employs a structured or unstructured mesh for spatial discretization and it allows the use of a self-adaptive or fixed time step ODE solver while dealing with unsteady problems. This research deals with the analysis of the effects of the Courant-Friedrichs-Lewy (CFL) condition for KLE when applied to unsteady Stoke’s problem. The objective is to conduct a numerical analysis for stability and, hence, for convergence. Our results confirmthat the time step ∆t is constrained by the CFL-like condition ∆t ≤ const. hα, where h denotes the variable that represents spatial discretization.

Computational study of internal and external condensing flows and experimental synthesis to investigate their attainability and stability in ground-based and space-based environments

This doctoral thesis presents the computational work and synthesis with experiments for internal (tube and channel geometries) as well as external (flow of a pure vapor over a horizontal plate) condensing flows. The computational work obtains accurate numerical simulations of the full two dimensional governing equations for steady and unsteady condensing flows in gravity/0g environments. This doctoral work investigates flow features, flow regimes, attainability issues, stability issues, and responses to boundary fluctuations for condensing flows in different flow situations. This research finds new features of unsteady solutions of condensing flows; reveals interesting differences in gravity and shear driven situations; and discovers novel boundary condition sensitivities of shear driven internal condensing flows. Synthesis of computational and experimental results presented here for gravity driven in-tube flows lays framework for the future two-phase component analysis in any thermal system. It is shown for both gravity and shear driven internal condensing flows that steady governing equations have unique solutions for given inlet pressure, given inlet vapor mass flow rate, and fixed cooling method for condensing surface. But unsteady equations of shear driven internal condensing flows can yield different “quasi-steady” solutions based on different specifications of exit pressure (equivalently exit mass flow rate) concurrent to the inlet pressure specification. This thesis presents a novel categorization of internal condensing flows based on their sensitivity to concurrently applied boundary (inlet and exit) conditions. The computational investigations of an external shear driven flow of vapor condensing over a horizontal plate show limits of applicability of the analytical solution. Simulations for this external condensing flow discuss its stability issues and throw light on flow regime transitions because of ever-present bottom wall vibrations. It is identified that laminar to turbulent transition for these flows can get affected by ever present bottom wall vibrations. Detailed investigations of dynamic stability analysis of this shear driven external condensing flow result in the introduction of a new variable, which characterizes the ratio of strength of the underlying stabilizing attractor to that of destabilizing vibrations. Besides development of CFD tools and computational algorithms, direct application of research done for this thesis is in effective prediction and design of two-phase components in thermal systems used in different applications. Some of the important internal condensing flow results about sensitivities to boundary fluctuations are also expected to be applicable to flow boiling phenomenon. Novel flow sensitivities discovered through this research, if employed effectively after system level analysis, will result in the development of better control strategies in ground and space based two-phase thermal systems.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.

Numerical simulation of the evolution of viscous rotating drops

In this contribution we simulate numerically the evolution of a viscous fluid drop rotating about a fixed axis at constant angular velocity ? or constant angular momentum L, surrounded by another viscous fluid. The problem is considered in the limit of large Ekman number and small Reynolds number. In the lecture we will describe the numerical method we have used to solve the PDE system that describes the evolution of the drop (3D boundary element method). We will also present the results we have obtained, paying special attention to the stability/instability of the equilibrium shapes.

A robust numerical framework for the analysis of material failure of fibre reinforced soft tissue

In this work, robustness and stability of continuum damage models applied to material failure in soft tissues are addressed. In the implicit damage models equipped with softening, the presence of negative eigenvalues in the tangent elemental matrix degrades the condition number of the global matrix, leading to a reduction of the computational performance of the numerical model. Two strategies have been adapted from literature to improve the aforementioned computational performance degradation: the IMPL-EX integration scheme [Oliver,2006], which renders the elemental matrix contribution definite positive, and arclength-type continuation methods [Carrera,1994], which allow to capture the unstable softening branch in brittle ruptures. The IMPL-EX integration scheme has as a major drawback the need to use small time steps to keep numerical error below an acceptable value. A convergence study, limiting the maximum allowed increment of internal variables in the damage model, is presented. Finally, numerical simulation of failure problems with fibre reinforced materials illustrates the performance of the adopted methodology.

Coupling time-stepping numerical methods and standard aerodynamics codes for instability analysis of flows in complex geometries

The development of a global instability analysis code coupling a time-stepping approach, as applied to the solution of BiGlobal and TriGlobal instability analysis 1, 2 and finite-volume-based spatial discretization, as used in standard aerodynamics codes is presented. The key advantage of the time-stepping method over matrix-formulation approaches is that the former provides a solution to the computer-storage issues associated with the latter methodology. To-date both approaches are successfully in use to analyze instability in complex geometries, although their relative advantages have never been quantified. The ultimate goal of the present work is to address this issue in the context of spatial discretization schemes typically used in industry. The time-stepping approach of Chiba 3 has been implemented in conjunction with two direct numerical simulation algorithms, one based on the typically-used in this context high-order method and another based on low-order methods representative of those in common use in industry. The two codes have been validated with solutions of the BiGlobal EVP and it has been showed that small errors in the base flow do not have affect significantly the results. As a result, a three-dimensional compressible unsteady second-order code for global linear stability has been successfully developed based on finite-volume spatial discretization and time-stepping method with the ability to study complex geometries by means of unstructured and hybrid meshes