Mathematical Architecture for Models of Fluid Flow Phenomena

This thesis is a study of several high accuracy numerical methods for fluid flow problems and turbulence modeling.First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step. We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought. Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost. We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. Magnetically conducting fluids arise in important applications including plasma physics, geophysics and astronomy. In many of these, turbulent MHD (magnetohydrodynamic) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations. Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests

An immersed boundary method for particles and bubbles in magnetohydrodynamic flows

This thesis presents a numerical method for the phase-resolving simulation of rigid particles and deformable bubbles in viscous, magnetohydrodynamic flows. The presented approach features solid robustness and high numerical efficiency. The implementation is three-dimensional and fully parallel suiting the needs of modern high-performance computing. In addition to the steps towards magnetohydrodynamics, the thesis covers method development with respect to the immersed boundary method which can be summarized in simple words by From rigid spherical particles to deformable bubbles. The development comprises the extension of an existing immersed boundary method to non-spherical particles and very low particle-to-fluid density ratios. A detailed study is dedicated to the complex interaction of particle shape, wake and particle dynamics. Furthermore, the representation of deformable bubble shapes, i.e. the coupling of the bubble shape to the fluid loads, is accounted for. The topic of bubble interaction is surveyed including bubble collision and coalescence and a new coalescence model is introduced. The thesis contains applications of the method to simulations of the rise of a single bubble and a bubble chain in liquid metal with and without magnetic field highlighting the major effects of the field on the bubble dynamics and the flow field. The effect of bubble coalescence is quantified for two closely adjacent bubble chains. A framework for large-scale simulations with many bubbles is provided to study complex multiphase phenomena like bubble-turbulence interaction in an efficient manner

Direct Numerical Simulation of Transition and Turbulence in Magnetohydrodynamic Flows in Rectangular Ducts

In this work, a flow of an electrically conducting fluid is driven through a rectangular duct by a constant pressure gradient in the presence of a transverse, externally applied magnetic field: the flow is studied using the method of Direct Numerical Simulation (DNS). This particular Magnetohydrodynamic (MHD) flow investigation is important in the development of liquid metal blankets design, which is the proposed cooling system within nuclear fusion reactors. The duct walls parallel to the magnetic field are ideally electrically insulating, while the walls perpendicular to the magnetic field are ideally electrically conducting. This flow is referred to as a Hunt’s flow. In this work the emergence of time dependent flow and its transition to a fully developed turbulent regime is explored. By fixing the strength of the magnetic field and increasing the fluid velocity, a number of time-dependent flow regimes have been observed in the side layers, which includes Ting-Walker vortices, elongated vortical structures, fully turbulent side-wall jets, as well as singular and multiple side-wall jet detachments. It has been found that at low velocities, the time-dependant flow takes the form of TingWalker vortices, which develop in the side layers of the duct. For all but the lowest magnetic fields studied, the Ting-Walkers vortices completely disappear after a short initial transient time, being replaced by new, higher energy, complex, anisotropic vortical structures. Additionally, a number of new flow regimes involving jet detachment have been identified. This study also demonstrates that Hunt’s flow exhibits hysteresis behaviour, where different unsteady states are possible for the same flow parameters

Adaptive time-integration for goal-oriented and coupled problems

We consider efficient methods for the partitioned time-integration of multiphysics problems, which commonly exhibit a multiscale behavior, requiring independent time-grids. Examples are fluid structure interaction in e.g., the simulation of blood-flow or cooling of rocket engines, or ocean-atmosphere-vegetation interaction. The ideal method for solving these problems allows independent and adaptive time-grids, higher order time-discretizations, is fast and robust, and allows the coupling of existing subsolvers, executed in parallel. We consider Waveform relaxation (WR) methods, which can have all of these properties. WR methods iterate on continuous-in-time interface functions, obtained via suitable interpolation. The difficulty is to find suitable convergence acceleration, which is required for the iteration converge quickly. We develop a fast and highly robust, second order in time, adaptive WR method for unsteady thermal fluid structure interaction (FSI), modelled by heterogeneous coupled linear heat equations. We use a Dirichlet-Neumann coupling at the interface and an analytical optimal relaxation parameter derived for the fully-discrete scheme. While this method is sequential, it is notably faster and more robust than similar parallel methods.We further develop a novel, parallel WR method, using asynchronous communication techniques during time-integration to accelerate convergence. Instead of exchanging interpolated time-dependent functions at the end of each time-window or iteration, we exchange time-point data immediately after each timestep. The analytical description and convergence results of this method generalize existing WR theory.Since WR methods allow coupling of problems in a relative black-box manner, we developed adapters to PDE-subsolvers implemented using DUNE and FEniCS. We demonstrate this coupling in a thermal FSI test case.Lastly, we consider adaptive time-integration for goal-oriented problems, where one is interested in a quantity of interest (QoI), which is a functional of the solution. The state-of-the-art method is the dual-weighted residual (DWR) method, which is extremely costly in both computation and implementation. We develop a goal oriented adaptive method based on local error estimates, which is considerably cheaper in computation. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical results verify these results and show our method to be more efficient than the DWR method