Kernel Functions and New Applications of an Accurate Technique

In this article, some general reproducing kernel Sobolev spaces was constructed. We find the general functions in these reproducing kernel Sobolev spaces. Many higher order boundary value problems can be investigated by these special functions

Numerical Modelling of Turbulent Free Surface Flows over Rough and Porous Beds Using the Smoothed Particle Hydrodynamics Method

Understanding turbulent flow structure in open channel flows is an important issue for Civil Engineers who study the transport of water, sediments and contaminants in rivers. In the present study, turbulent flows over rough impermeable and porous beds are studied numerically using the Smoothed Particle Hydrodynamics (SPH) method. A comprehensive review is carried out on the methods of turbulence modelling and treatment of bed boundary in open channel flows in order to identify the limitations of the existing particle models developed in this area. 2D macroscopic SPH models are developed for simulating turbulent free surface flows over rough impermeable and porous beds under various flow conditions. For the case of impermeable beds, a drag force model is proposed to take the effect of bed roughness into account, while for the case of porous beds, macroscopic governing equations are developed based on the SPH formulation, incorporating the effects of drag and porosity. To simulate the effect of turbulence on the average flow field, a Macroscopic SPH-mixing-length (MSPH-ML) model is proposed based on the Large Eddy Simulation (LES) concept where the mixing-length approach is applied to estimate the eddy-viscosity rather than employing the standard Smagorinsky model. The difficulty in reproducing steady uniform free surface flow is tackled by introducing novel inflow/outflow techniques for the situations in which the flow quantities are unknown at the inflow and outflow boundaries. The performance of these models is tested by simulating different engineering problems with an insight developed into turbulence modelling and bed/interface boundary treatment. The accuracy of the models is tested by comparing the predicted quantities such as flow velocity, water surface elevation, and turbulent shear stress with existing experimental data. The limitations of the models are mainly attributed to the macroscopic representation of the roughness layer and porous bed, difficulty in the determination of the values of the empirical coefficients in the closure terms, and limitations with the use of fine computational resolution. On the other hand, the main strength of the model is describing the complicated processes occuring at the bed using simple and practical computational treatments so that the momentum transfer is estimated accurately. It is shown that if the closure terms in the momentum equation which represent the effect of bed drag and flow turbulence are determined carefully based on the physical conditions of bed and flow, the model is capable of being employed for different civil engineering applications

Regularisation and Long-Time Behaviour of Random Systems

Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and uniqueness of a strong solution to the TMHD equations, prove that smooth data lead to smooth solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the solutions to the tamed equations converge to a weak solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a unique strong solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations

Turbulent Flows over Rough Permeable Beds in Mountain Rivers: Experimental Insights and Modeling

Steep mountain streams exhibit shallow waters with roughness elements such as stones and pebbles that are comparable in size to flow depth. Owing to the difficulty in measuring fluid velocities at the interface, i.e., from the rough permeable bed to the free surface, experimental results are rare although they are essential to improve models. Using a novel experimental procedure, this thesis attempts to improve predictions of the vertical structure of turbulent flows over rough permeable beds. To explore flows at the bed interface, I devised an experimental set-up where a fluid flowed over glass spheres (8 mm < dp < 14 mm) in a narrow flume (W = 6 cm) with slopes varying from 0.5 % to 8 %. The Refractive Index Matching (RIM) technique has been employed. This involves matching the refractive index of the fluid with that of the glass spheres, thereby allowing the interior of the medium to be examined and velocities to be measured by Particle Image Velocimetry (PIV). Vertical profiles are retrieved by employing the spatiotemporal double averaging method. In the course of this manuscript, flow processes are studied at the mesoscopic scale, i.e., by averaging quantities over distances ranging from 5 to 10 grain diameters. For open-channel flows over rough permeable beds, the spatial averaging procedure yields a continuous porosity profile. When applied to the Navier-Stokes equations, it produces a momentum equation with several terms including drag forces and three stresses: the turbulent, dispersive, and viscous stresses. The momentum equation was employed to devise a one dimensional (1D) model describing the vertical structure of unidirectional turbulent flow. A turbulent boundary layer over the rough bed was observed while experiments were performed at intermediate Reynolds numbers, i.e., Re = O (1000). In such conditions, viscosity plays a critical role through the van Driest damping effect. To model vertical profiles, the Darcy-Ergün equation is well suited to the prediction of friction forces in the permeable bed, i.e., in roughness and subsurface layers. Based on the \textit{Prandtl mixing length theory}, turbulent stress is predicted from a mixing length distribution that considers dispersive effects and assumes a continuous porosity profile. This alternative contrasts with most existing boundary layer models which postulate a discontinuous porosity profile for permeable or impermeable walls. Finally, hydraulic conditions collected by an Unmanned Aerial Vehicle (UAV) and classical flow resistance equations (Chézy, Keulegan, ...) were compared with profile simulations and demonstrate a good agreement between predictions and observations. It reveals the crucial role of fluid depth definition in equations in small submergence conditions. Furthermore, incipient sediment motion conditions have been estimated and compared to empirical results showing the importance of turbulence and lift force for grain entrainment. With regard to fluid dynamics, mountain streams are a case study of the larger scientific family of turbulent flows interacting with porous structures. Insights and developments acquired in the course of this thesis are likely to be transferable to other domains working with these phenomena such as flows over buildings, vegetal canopies or rough wings

Lattice Boltzmann Methods for Partial Differential Equations

Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions

3D coupled pore-network modelling of flow and solute transport through porous media, from laminar to turbulent flow

Subsurface hydrology including flow and solute transport modelling is essential for designing many engineering processes such as seepage, remediation of contaminated groundwater, improved oil recovery, etc. The processes involved in such activities are observed across a wide range of length and timescales; from nanometres to kilometres and from nanoseconds to years. The recent growth in imaging technologies has shown that the size of a single pore in a porous medium may range from 0.1 nm to a few centimetres (Marry & Rotenberg, 2015). Therefore, to perform reliable field-scale simulations, a deep understanding of the processes happening at the pore-scale level and their consequences at larger scales is needed (Mehmani, 2014). Most of the previous work that modelled flow and solute transport at the pore-scale assumed laminar flow and applied Darcy’s law. However, in some cases, such as the flow of gases through porous media, flow near wellbores, and flow through the hyporheic zone, non-Darcy flow can be observed. It is not clear how solute transport processes are affected by the flow behaviour in the non-Darcy (Forchheimer) and turbulent flow regimes. In this work, a pore-network model (PNM) capable of simulating flow and solute transport within the Darcy, Forchheimer and turbulent flow regimes was developed. One of the aims of this work is to determine the onset of non-Darcy flow and the onset of turbulence, after which Darcy’s law loses its validity. Using PNM, any porous medium can be simplified into large pores connected to each other’s by narrow pores, then analytical or semi-analytical equations can be implemented to model the flow and transport processes through the medium. The proposed model was verified against experimental data of a packed spheres sample and other data in the literature. X-ray Computed Tomography scans of the packed spheres, sandstone and carbonate samples were used to extract the equivalent pore-network. It was found that the onset of non-Darcy flow is highly dependent on the medium degree of heterogeneity, and in heterogeneous media, the onset velocity could be up to three orders of magnitude smaller compared to the homogenous media. In porous media with coarse particles, the assumption of fully developed flow in each pore is not valid and using the Hagen-Poiseuille equation does not predict the flow behaviour properly. After the onset of non-Darcy flow, if Darcy’s law is applied, this causes overestimation (up to ~10 times) of the Péclet number and the longitudinal dispersion coefficient (DL). In the turbulent flow regime, DL increased, due to the effect of turbulent diffusion, by a factor up to 1.6 compared to the DL value obtained under the Forchheimer flow conditions

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Improved modeling for fluid flow through porous media

Petroleum production is one of the most important technological challenges in the current world. Modeling and simulation of porous media flow is crucial to overcome this challenge. Recent years have seen interest in investigation of the effects of history of rock, fluid, and flow properties on flow through porous media. This study concentrates on the development of numerical models using a ‘memory’ based diffusivity equation to investigate the effects of history on porous media flow. In addition, this study focusses on developing a generalized model for fluid flow in packed beds and porous media. The first part of the thesis solves a memory-based fractional diffusion equation numerically using the Caputo, Riemann-Liouville (RL), and Grünwald-Letnikov (GL) definitions for fractional-order derivatives on uniform meshes in both space and time. To validate the numerical models, the equation is solved analytically using the Caputo, and Riemann-Liouville definitions, for Dirichlet boundary conditions and a given initial condition. Numerical and analytical solutions are compared, and it is found that the discretization method used in the numerical model is consistent, but less than first order accurate in time. The effect of the fractional order on the resulting error is significant. Numerical solutions found using the Caputo, Riemann-Liouville, and Grünwald-Letnikov definitions are compared in the second part. It is found that the largest pressure values are found from Caputo definition and the lowest from Riemann-Liouville definition. It is also found that differences among the solutions increase with increasing fractional order

Structural, Magnetic, Dielectric, Electrical, Optical and Thermal Properties of Nanocrystalline Materials: Synthesis, Characterization and Application

This book is a collection of the research articles and review article, published in special issue "Structural, Magnetic, Dielectric, Electrical, Optical and Thermal Properties of Nanocrystalline Materials: Synthesis, Characterization and Application"

Etude du colmatage des systèmes carburant de turboréacteurs par des suspensions denses de particules de glace

Water, which exists naturally in jet-engine fuel, may freeze within theaircraft fuel pipes under certain temperatures and flow rates. The ice particles released by these deposits are entrained by the flow, and clog the hydraulics downstream. The understanding of this phenomenon, highlighted by the crash of a Boeing 777 in 2008, is an important issue for the aviation industry. Therefore a device has been designed to reproduce this threat in a controlled and quantified way. Water is atomized in low temperature jet-engine fuel and the droplets crystallize. The resulting slurry clogs different kinds of perforated targets. Temperatures, flow rates and pressure drops are monitored, and the phenomenon is filmed by a high frequency camera. A model was constructed based on these observations and data from literature and feedbacks. For the fluid phase, the incompressible Navier-Stokes equations are solved within a finite volume framework. The pressure-velocity coupling is achieved using the SIMPLE algorithm and high order of accuracy thanks to the MLS method. The solid phase is simulated using discrete elements. The fluid-particle interaction is based on a porous medium approach. A CFD-DEM parallel code has been developed to run the model. The first simulations of flow through granular media are in good agreement with experimental results.Dans certaines conditions de température et de débit, l’eau naturellement présente dans le kérosène va givrer l’intérieur des conduites du système carburant avion. Ces dépôts peuvent libérer des particules de glace qui sont entrainées par l’écoulement, et provoquent le colmatage des équipements hydrauliques situés en aval. Ce phénomène fut mis en évidence suite à l’accident d’un Boeing 777 en 2008, aussi sa compréhension est un enjeu important pour les acteurs de l’industrie aéronautique. Un dispositif a été spécialement conçu pour reproduire cette menace de façon quantifiée. De l’eau est atomisée dans un écoulement à basse température, puis cristallise pour former une suspension qui vient colmater différentes cibles perforées. Les températures, débits et pertes de charge sont mesurées, et le phénomène est filmé par une caméra haute fréquence. Un modèle a été réalisé à partir de cesobservations, complétées par des données issues de la littérature et de retoursd’expérience. Pour la phase fluide, les équations de Navier-Stokes incompressibles sont résolues par une approche volumes finis. Le couplage pression-vitesse est obtenu par l’algorithme SIMPLE et l’ordre élevé au moyen de la méthode MLS. La phase solide est simulée par éléments discrets. L’interaction fluide-particules repose sur une approche de type milieu poreux. Un code CFD-DEM parallèle a été développé, et les premières simulations d’écoulement en milieu granulaire sont en bon agrément avec des résultats expérimentaux