This thesis deals with Computational Fluid Dynamics (CFD)simulations of the flow around delta wings at high angles ofattack. These triangular wings, mainly used in militaryaircraft designs, experience the formation of two vortices ontheir lee-side at large angles of attack. The simulation ofthis vortical flow by solving the Navier-Stokes equations isthe subject of this thesis. The purpose of the work is toimprove the understanding of this flow and contribute to thedesign of such a wing by developing methods that enable moreaccurate and efficient CFD simulations. Simulations of the formation, burst and disappearance of thevortices while the angle of attack is changing are presented.The structured flow solver NSMB has been used to get thetime-dependent solutions of the flow. Both viscous and inviscidresults of a 70°-swept delta wing pitching in anoscillatory motion are reported. The creation of the dynamiclift and the hysteresis observed in the history of theaerodynamic forces are well reproduced. The second part of the thesis is focusing on automatic meshrefinement and its influence on simulations of the delta wingleading-edge vortices. All the simulations to assess the gridquality are inviscid computations performed with theunstructured flow solver EDGE. A first study reports on theeffects of refining thewake of the delta wing. A70°-swept delta wing at a Mach number of 0.2 and an angleof attack of 27° where vortex breakdown is present abovethe wing, is used as testcase. The results show a strongdependence on the refinement, particularly the vortex breakdownposition, which leads to the conclusion that the wake should berefined at least partly. Using this information, a grid for thewing in the wind tunnel is created in order to assess theinfluence of the tunnel walls. Three sensors for automatic meshrefinement of vortical flows are presented. Two are based onflow variables (production of entropy and ratio of totalpressures) while the third one requires an eigenvalue analysisof the tensor of the velocity gradients in order to capture theposition of the vortices in the flow. These three vortexsensors are successfully used for the simulation of the same70° delta wing at an angle of attack of 20°. Acomparison of the sensors reveals the more local property ofthe third one based on the eigenvalue analysis. This lattertechnique is applied to the simulation of the wake of a deltawing at an angle of attack of 20°. The simulations on ahighly refined mesh show that the vortex sheet shed from thetrailing-edge rolls up into a vortex that interacts with theleading-edge vortex. Finally the vortex-detection technique isused to refine the grid around a Saab Aerosystems UnmannedCombat Air Vehicle (UCAV) configuration and its flight dynamicscharacteristics are investigated. Key words:delta wing, high angle of attack, vortex,pitching, mesh refinement, UCAV, vortex sensor, tensor ofvelocity gradients.

Adaptive methods in finite element analysis are essential tools in the efficient computation and error control of problems that may exhibit singularities. In this paper, we consider solving a boundary value problem which exhibits a singularity at the origin due to both the structure of the domain and the regularity of the exact solution. We introduce a hybrid mixed finite element method using Lagrange Multipliers to initially solve the partial differential equation for the both the flux and displacement. An a posteriori error estimate is then applied both locally and globally to approximate the error in the computed flux with that of the exact flux. Local estimation is the key tool in identifying where the mesh should be refined so that the error in the computed flux is controlled while maintaining efficiency in computation. Finally, we introduce a simple refinement process in order to improve the accuracy in the computed solutions. Numerical experiments are conducted to support the advantages of mesh refinement over a fixed uniform mesh.

Bayesian inference is commonly used to quantify and reduce modeling uncertainties in coastal ocean models by computing the posterior probability distribution function (pdf) of some uncertain quantities to be estimated conditioned on available observations. The posterior can be computed either directly, using a Markov Chain Monte Carlo (MCMC) approach, or by sequentially processing the data following a data assimilation (DA) approach. The advantage of data assimilation schemes over MCMC-type methods arises from the ability to algorithmically accommodate a large number of uncertain quantities without a significant increase in the computational requirements. However, only approximate estimates are generally obtained by this approach often due to restricted Gaussian prior and noise assumptions. This thesis aims to develop, implement and test novel efficient Bayesian inference techniques to quantify and reduce modeling and parameter uncertainties of coastal ocean models. Both state and parameter estimations will be addressed within the framework of a state of-the-art coastal ocean model, the Advanced Circulation (ADCIRC) model. The first part of the thesis proposes efficient Bayesian inference techniques for uncertainty quantification (UQ) and state-parameters estimation. Based on a realistic framework of observation system simulation experiments (OSSEs), an ensemble Kalman filter (EnKF) is first evaluated against a Polynomial Chaos (PC)-surrogate MCMC method under identical scenarios. After demonstrating the relevance of the EnKF for parameters estimation, an iterative EnKF is introduced and validated for the estimation of a spatially varying Manning’s n coefficients field. Karhunen-Lo`eve (KL) expansion is also tested for dimensionality reduction and conditioning of the parameter search space. To further enhance the performance of PC-MCMC for estimating spatially varying parameters, a coordinate transformation of a Gaussian process with parameterized prior covariance function is next incorporated into the Bayesian inference framework to account for the uncertainty in covariance model hyperparameters. The second part of the thesis focuses on the use of UQ and DA on adaptive mesh models. We developed new approaches combining EnKF and multiresolution analysis, and demonstrated significant reduction in the cost of data assimilation compared to the traditional EnKF implemented on a non-adaptive mesh.

We present a novel dimensionally split Cartesian cut cell method to compute inviscid, viscous and turbulent flows around rigid geometries. On a cut cell mesh, the existence of arbitrarily small boundary cells severely restricts the stable time step for an explicit numerical scheme. We solve this `small cell problem' when computing solutions for hyperbolic conservation laws by combining wave speed and geometric information to develop a novel stabilised cut cell flux. The convergence and stability of the developed technique are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). Subsequently, we develop the method further to be able to compute solutions for the compressible Navier-Stokes equations. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a full description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a wide range of test problems ranging from the nearly incompressible to the highly compressible flow regimes. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). It is the first presentation of a dimensionally split cut cell method for the compressible Navier-Stokes equations in the literature. Finally, we also present an extension of the cut cell method to solve high Reynolds number turbulent automotive flows using a wall-modelled Large Eddy Simulation (WMLES) approach. A full description is provided of the coupling between the (implicit) LES solution and an equilibrium wall function on the cut cell mesh. The combined methodology is used to compute results for the turbulent flow over a square cylinder, and for flow over the SAE Notchback and DrivAer reference automotive geometries. We intend to publish the promising results as part of a future publication, which would be the first assessment of a WMLES Cartesian cut cell approach for computing automotive flows to be presented in the literature.

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein's equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.

A major drawback in the operation of mechanical heart valve prostheses is thrombus formation in the near valve region potentially due to the high shear stresses present in the leakage jet flows through small gaps between leaflets and the valve housing. Detailed flow analysis in this region during the valve closure phase is of interest in understanding the relationship between shear stress and platelet activation. An efficient Cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three-dimensional immersed solid bodies as well as fluid-fluid interfaces. The embedded boundaries are represented using Levelsets and treated in a sharp manner without the use of source terms to represent boundary effects. The resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second-order accuracy. When dealing with problems of disparate length scales encountered in many applications, it is necessary to resolve the physically important length scales adequately to ensure accuracy of the solution. Fixed grid methods often have the disadvantage of heavy mesh requirement for well resolved calculations. A quadtree based adaptive local mesh refinement scheme is developed to complement the sharp interface Cartesian grid method scheme for efficient and optimized calculations. Detailed timing and accuracy data is presented for a variety of benchmark problems involving moving boundaries. The above method is then applied to modeling heart valve closure and predicting thrombus formation. Leaflet motion is calculated dynamically based on the fluid forces acting on it employing a fluid-structure interaction algorithm. Platelets are modeled and tracked as point particles by a Lagrangian particle tracking method which incorporates the hemodynamic forces on the particles. Leaflet closure dynamics including rebound is analyzed and validated against previous studies. Vortex shedding and formation of recirculation regions are observed downstream of the valve, particularly in the gap between the valve and the housing. Particle exposure to high shear and entrapment in recirculation regions with high residence time in the vicinity of the valve are observed corresponding to regions prone to thrombus formation.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

<p>This thesis deals with Computational Fluid Dynamics (CFD)simulations of the flow around delta wings at high angles ofattack. These triangular wings, mainly used in militaryaircraft designs, experience the formation of two vortices ontheir lee-side at large angles of attack. The simulation ofthis vortical flow by solving the Navier-Stokes equations isthe subject of this thesis. The purpose of the work is toimprove the understanding of this flow and contribute to thedesign of such a wing by developing methods that enable moreaccurate and efficient CFD simulations.</p><p>Simulations of the formation, burst and disappearance of thevortices while the angle of attack is changing are presented.The structured flow solver NSMB has been used to get thetime-dependent solutions of the flow. Both viscous and inviscidresults of a 70°-swept delta wing pitching in anoscillatory motion are reported. The creation of the dynamiclift and the hysteresis observed in the history of theaerodynamic forces are well reproduced.</p><p>The second part of the thesis is focusing on automatic meshrefinement and its influence on simulations of the delta wingleading-edge vortices. All the simulations to assess the gridquality are inviscid computations performed with theunstructured flow solver EDGE. A first study reports on theeffects of refining thewake of the delta wing. A70°-swept delta wing at a Mach number of 0.2 and an angleof attack of 27° where vortex breakdown is present abovethe wing, is used as testcase. The results show a strongdependence on the refinement, particularly the vortex breakdownposition, which leads to the conclusion that the wake should berefined at least partly. Using this information, a grid for thewing in the wind tunnel is created in order to assess theinfluence of the tunnel walls. Three sensors for automatic meshrefinement of vortical flows are presented. Two are based onflow variables (production of entropy and ratio of totalpressures) while the third one requires an eigenvalue analysisof the tensor of the velocity gradients in order to capture theposition of the vortices in the flow. These three vortexsensors are successfully used for the simulation of the same70° delta wing at an angle of attack of 20°. Acomparison of the sensors reveals the more local property ofthe third one based on the eigenvalue analysis. This lattertechnique is applied to the simulation of the wake of a deltawing at an angle of attack of 20°. The simulations on ahighly refined mesh show that the vortex sheet shed from thetrailing-edge rolls up into a vortex that interacts with theleading-edge vortex. Finally the vortex-detection technique isused to refine the grid around a Saab Aerosystems UnmannedCombat Air Vehicle (UCAV) configuration and its flight dynamicscharacteristics are investigated.</p><p><b>Key words:</b>delta wing, high angle of attack, vortex,pitching, mesh refinement, UCAV, vortex sensor, tensor ofvelocity gradients.</p>

This study presents an improved ghost-cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work ghost cells are mirrored through the boundary described using a level-set method to farther image points, incorporating a higher-order extra/interpolation scheme for the ghost cell values. In addition, a shock sensor is in- troduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement (AMR) is used to improve the representation of the geometry efficiently. The improved ghost-cell method is validated against five test cases: (a) double Mach reflections on a ramp, (b) supersonic flows in a wind tunnel with a forward- facing step, (c) supersonic flows over a circular cylinder, (d) smooth Prandtl-Meyer expansion flows, and (e) steady shock-induced combustion over a wedge. It is demonstrated that the improved ghost-cell method can reach the accuracy of second order in L1 norm and higher than first order in L∞ norm. Direct comparisons against the cut-cell method demonstrate that the improved ghost-cell method is almost equally accurate with better efficiency for boundary representation in high-fidelity compressible flow simulations. Implementation of the improved ghost-cell method in reacting Euler flows further validates its general applicability for compressible flow simulations.