The mimetic spectral element method is a relatively young method in numerical solutions of partial differential equations and actions that describe physical systems. Its advantage is that it takes the geometrical structure of the problem into account which guarantees consistency of the numerical scheme and the conservation of relevant quantities. This prohibits the solution to diverge into non-physical states since the discretization mimics the physical behaviour of the problem. With relation to physics the mimetic method makes use of the stationary action principle which contains all information about the system and reduces the continuous variables to discrete ones that become ready for computation. The stationary action principle requires a formulation of the Lagrangian such that it implies Newton’s laws of motion or equivalent physical laws. However these Lagrangians are only easily formulated for monogenic systems. This work investigates the class of (discrete) non-monogenic physical systems. The goal is to present a general method to numerically mimic non-conservative systems such that energy (loss) is exact. This is done by modifying a newly introduced systematic method that transforms the non-conservative problem to a system with a doubled degrees of freedom that makes it a conservative or monogenic one. In this modification process new degrees of freedom are introduced which are allowed to have arbitrary variations at the initial and final states of the new action. The system can than be mimicked with the spectral element method where algebraic dual polynomials are used as both approximating polynomials and as test functions that serve as arbitrary variations. Furthermore a gauge transformation is introduced by the modified action such that it contains the initial and final states of the system to make the solution unique.
The introduced method is generally applicable to Lagrangian mechanics of discrete systems with holonomic constraints. The method is applied to the damped harmonic oscillator (DHO) as a test case. The computational results are clearly accurate and converging. However they deviate with one order from the h-refinement spectral element theory of Gauss-Lobatto-Legendre interpolation basis polynomials. Theoretical analysis of the equations of the mimicked (single and doubled) DHO show that energy (loss) is exact. Nevertheless, the computations show an oscillation in energy that remains closely around the conserved value. This oscillation in energy in time was observed in earlier works that applied the mimetic spectral element method to conservative systems. The reason behind this behavior of energy remains a topic for further research. Furthermore, in a follow up research the method can be extended to the Lagrangian field theory and tested with simplified non-conservative fluid problems to verify it for fields.

Structure-preserving or mimetic discretisations are a class of advanced discretisation techniques derived by employing concepts from differential geometry. Such techniques can attain specific conservation properties at the discrete level such as conservation of mass, kinetic energy, etc when applied to conservation laws. However, like traditional techniques, they are not entirely robust in specific multiscale cases such as under-resolved simulations which are often encountered in industrial applications. This ties into the concept of Large Eddy Simulations (LES) of which an enticing approach is the Variational Multiscale (VMS) method. Both Mimetic methods and VMS approaches have been extensively studied independently, however, their combination offers the potential of achieving a favorable robust solver capable of handling complicated industrial problems. Moreover, the extension of the Hybridised variant of the Mimetic methods towards advection-dominated problems is an interesting avenue yet to be explored. Therefore, this thesis focuses on the extension of the Hybridised Mimetic method and its combination with the VMS theory. The said combination is tested on simple linear equations, namely the advection-diffusion equation in both steady and unsteady cases. Additionally, the Hybridised Mimetic method is studied in more complex test cases such as the Burgers' equation and the incompressible Euler/Navier-Stokes equations.

With (e-)grocery retailers striving to increase their efficiency and subsequently reduce their costs, the opportunities that lie in optimising the order picking process of the supply chain is of key importance. The complex nature of assigning articles to an optimal storage location, having efficient routing of order pickers, and optimised grouping of orders calls for an integral approach to these problems. The objective of the research is to reduce the travelled distance of order pickers and subsequently the costs of the order picking process. Next to the integral approach, a new method of Multiple Storage Locations (MSL) is introduced. Historic order data of an e-grocery retailer is used, together with information on stock keeping units (SKUs), to implement SKU allocation with MSL possibility, routing, and batching into models. This ensures that the cost saving effects of these measures can be quantified. A benchmark Mixed Integer Linear Programming model (MILP) is developed and compared to a meta-heuristic Adaptive Large Neighbourhood Search model (ALNS) to determine how much travelled distance would be saved. The ALNS has multiple destroy- and repair heuristics, some of which are novel, that are specific to the problem at hand. The ALNS is able to handle bigger instances than the MILP, whilst ensuring quality of the solution. The MILP model outperforms the ALNS for small instances, however for large instances (instances of 400 orders or more) the ALNS performs, on average, 12.5% better whilst reducing computational time by 14.8%. Finally, areas of improvement are suggested in the ALNS model as well as other effects that should be studied when introducing MSL.

The need of computational power for engineering applications has been ever increasing and with classical computers approaching their physical limits, new ways of improvement have to be investigated. One of the promising solutions is quantum computing. Most engineering problems require solving a system of linear equations of higher dimensions and the Variational Quantum Linear Solver (VQLS) algorithm seems like a promising near term solution. This algorithm evaluates a cost function on a quantum machine and uses a classical optimizer to minimize this cost function. The minimum of this cost function corresponds to the solution of the linear system. This work aims at finding what the practical limitations are when solving the Poisson equation by means of VQLS. Results showed that problems small in size can be solved, however for larger problems obtaining the solution becomes effectively impossible due to barren plateaus, which are flat spots in the cost function landscape.

Advection is at the heart of fluid dynamics and is responsible for many interesting phenomena. Unfortunately, it is also the source of the non-linearity of fluid dynamics. As such, its numerical treatment is challenging and often suboptimal. One way to more effectively deal with advection is by using a Lagrangian formulation instead of the conventional Eulerian view.

This work aims to show that the Lagrangian formulation, and its underlying geometric and physical character, are fundamental in overcoming the challenges posed by (non-)linear advection. The Mimetic Spectral Element Method ensures that this geometric and physical character is kept when the equations are discretised. The advection term can then be dealt with exactly. The method is put to the test by solving the inviscid Burgers and isentropic Euler equations in one spatial dimension. The results confirm the exactness of the advection term and good overall accuracy.

The advent of global warming has brought an increased interest in non-conventional sources of energy, one of which is nuclear energy. Threatening the almost year-round functioning of nuclear power plants are Flow-Induced Vibrations (FIV). One such mechanism, Vortex-Induced Vibration (VIV), holds importance in areas of cross flow in nuclear power plants. To make fail-safe
designs, computational analysis in the domain of Fluid Structure Interactions (FSI) has been increasing over the past two decades. The thesis work aims to add to the body of knowledge by making predictions for an in-line two-cylinder configuration, set up as part of a benchmark proposed by the Organization for Economic Co-operation and Development (OECD), using the commercial code Simcenter STARCCM+ (v2020.3.1).

The main objective of this study is to test the efficacy of the URANS framework in predicting VIV which is strongly correlated with the objective of the OECD to propose recommendations for the Best Practice Guidelines. To do so, it is desired to shortlist the most appropriate turbulence model for the final application and point out gaps in the prediction of the same. The thesis work is thus carried out in three phases: code validation, turbulence model selection and final application. Key results from this study reveal the ‘Standard K-Epsilon
Low Re: Cubic’ model to be the most apt for the final application. Furthermore, gaps are also identified in the application of URANS to predict VIV resonance conditions, the primary of which is the overprediction of the vortex shedding frequency.

The aerospace engineering industry is continuously striving for faster methods to solve and optimize engineering and research problems with a higher degree of accuracy. It is therefore relevant to investigate possibilities that are expected to accelerate computational speed, such as quantum computing. For this reason, the objective of this thesis is to investigate the feasibility of optimizing 2D determinate truss structures with a quantum algorithm run on a Gate-Based Quantum Computer (GBQC). The Quantum Approximate Optimization Algorithm (QAOA) is currently expected to be the most suitable quantum algorithm candidate for optimization problems, because of its simplicity and robustness. The most important take-away from this thesis is that it is possible to map a truss structure to QAOA format to optimize it on a GBQC. The program proves to be working on quantum virtual machines. However, it is currently not possible to obtain correct results when running on real quantum hardware due to noise.

Computational fluid dynamics is nowadays one of the pillars of modern aircraft design, just as impor­tant as experimental wind tunnel testing. Very ambitious goals in regards to performance, efficiency and sustainability are being asked of the aviation industry, the kind that warrant a virtual exploration of the edges of the flight envelope. High­ order has come to be regarded as a necessary ingredient to achieve breakthrough advances in this direction. So much so, in fact, that the major aircraft manufac­turers, governmental aerospace research agencies and top universities worldwide have been acting in coordination to increase the technology readiness level of these approaches, for the last ten years. In this work, I study in significant detail the one aspect that makes high-­order methods ideal candi­dates for enabling the use of more advanced turbulence models in industry: the cost­-efficiency of their discretization—they offer minimal amounts of dispersion and dissipation errors, for a given number of degrees of freedom. I consider three research objects: discontinuous Galerkin spectral method (DGSEM), flux reconstruction (FR) and a novel B-­spline-based discontinuous Galerkin formulation stabilized via algebraic flux correc­tion (DGIGA­-AFC), and characterize their order of accuracy, linear and nonlinear stability characteristics, as well as dis­persion and dissipation errors as a function of wavenumber. Afterwards, I experimentally investigate their relative suitability towards scale­-resolving simulation of compressible and turbulent flows, by solv­ing a number of simple test cases of increasing difficulty (linear advection, inviscid Burgers and Euler equations; all in 1D) using a purpose­-made MATLAB implementation. The proposed isogeometric method (DGIGA) has been found to be at least as viable as the other two, a priori, for the resolution of high­-speed turbulent flows. Moreover, I have found that low dispersion and dissipation need not always be associated to high order, but to a high number of degrees of freedom per patch instead; these two coincide in the more conventional schemes, yet not necessarily in DGIGA. As a nonlinear stabilization mechanism, however, the proposed combination of DGIGA with AFC has turned out to be inferior to existing limiters.

Physical systems in the continuous domain are often solved using computer-aided software because of their complexity. Preserving the physical quantities from the continuous domain in the discrete domain is therefore of utmost importance. There is however a broad range of techniques that can accomplish the translation between the continuous and discrete domain, e.g. finite difference, volume and element techniques, fourth order Runge-Kutta or Störmer-Verlet to name a few. Accompanied with the aforementioned come strengths and weaknesses but have the common thread to try maintaining the physical behaviour of the continuous system closely. The mimetic spectral element technique is used to develop an energy-conserving spectral element scheme through a Lagrangian formulation. This new formulation of the mimetic spectral element technique allows for solving time-dependent problems and the simple harmonic oscillator serves as the sample problem in this thesis. The solution has been derived from Lagrangian mechanics in a variety of ways. Discrete Lagrangian formulations have been investigated at rst and their respective equations of motions have been tested against the exact solution of the simple harmonic oscillator. This method achieved marching in time and no damping of the solution, yet energy was only bounded and not exactly conserved. The mimetic spectral element formulation of the Lagrangian formulation showed diculties when using variational analysis, i.e. boundary treatment in the future. Arbitrary domain mapping was among the possible solutions, but this formulation was found to be unreliable and unsuccessful. It was found that a more robust formulation, i.e. the spectral marching method, was most suitable. Throughout this thesis the focus was put on the conservation of energy using a Lagrangian formulation. Except the spectral scheme using arbitrary domain mapping, all schemes kept the energy of the system bounded, but no energy was conserved up to machine precision. Using arbitrary domain mapping, the energy seemed to grow over time.

The demand for accurate and efficient simulations in order to test the geomechanical effects is a reality for the entire geoscience community. The motivation that arises from that need is the development and the evolution of modelling methods to study these effects. Deep understanding of any problem in fine scale is crucial, especially when it extends to much coarser scales.
In this work the finite volume method (FVM) is used for mechanical modelling of deformation in elastic media. The momentum balance equation is solved as the governing equation for mechanics, assuming linear elasticity for the stress tensor. Here, displacement is mapped onto a vertex-centred grid in three dimensions (3D). A set of eight trilinear basis functions are used to locally interpolate the value of displacement within each grid cube. In the finite volume method, the discretized form of the equations are obtained by integrating the governing equation over control volume surfaces, since in 3D the control volume is a cube. Hence, discretized forms are obtained by considering 24 surfaces, which form between a displacement node and its neighbouring displacement cells. This required extensive derivation. The implementation of the numerical model was carried out by writing a code in MATLAB. Several numerical test cases are presented to demonstrate the capability of this model. In the first place, the consistency of the model is checked through comparison with synthetic analytical solutions, which are compared to the numerical solutions. Furthermore, the simple test case of uniaxial compression, has been carried out with this model, but also compared to the results with a 2D FVM model and a 3D finite element (3D FEM) one. In another test case, ground plain strain subsidence is studied in a real hydrocarbon field with a heterogeneous map for elasticity parameters. It is shown that 3D FVM is in close agreement with 3D FEM in predicting the subsidence due to field depletion. Last but not least, displacement and stresses, in a faulted reservoir in which fluids are injected, are modelled and the results are shown to coincide with a robust analytical solutions for the system. Finally, the aim of this work is to shed some more light on the finite volume method for mechanics and bring it closer to the audience of science.

Water accumulating in the fuel tanks of any aircraft presents an unacceptable safety hazard. This accumulation, which is due to a combination of physical phenomena, is difficult to assess and therefore not well known. The purpose of this project is the development of a lightweight computational prediction method for water accumulation in helicopter fuel tanks. This model delivers a better understanding of the physical mechanisms involved and allows prediction of the accumulated water quantity under given conditions.

Cavitating tip vortices of marine propellers are the main contributors to underwater radiated ship noise due to their dynamic behavior and broadband sound spectrum. Experimentally, it is challenging to get a detailed insight into the vortex structure which makes it difficult to understand the noise generating mechanisms. Although Computational Fluid Dynamics (CFD) could offer a solution to this, knowledge regarding the modelling of the inception and dynamics of tip vortex cavitation is insufficient to obtain reliable numerical results in relation to noise predictions. To this end, an extensive verification and validation study in the context of transition and turbulence modelling of an elliptical planform with NACA66₂−415 cross-section (`Arndt’ wing) is carried out. The finite length of the lifting surface causes the generation of a tip vortex without the additional complexity of secondary motion (propeller rotation). CFD simulations are carried out using the community based open-usage code ReFRESCO.

Mimetic discretisation techniques are a growing field in computational physics research. Among these techniques, the recently developed mimetic spectral element method allows for exact discretisation of metric independent relations. This has been proven numerically in various mixed formulations, for instance the mixed velocity-vorticity-pressure formulation for Stokes flow, where mass conservation was point-wise strongly satisfied by the solution in the computational domain. Another example is the mixed stress-displacement formulation for the linear elasticity equations, where the balance law of linear momentum was point-wise strongly satisfied as well. A recent extension to a hybrid method leads to additional attractive features, such as the ability to decompose a large part of the computation of the solution into smaller problems. The aim of the research is to find a formulation for linear elasticity that is hybridisable while strongly satisfying conservation of linear and angular momentum as well, where the combination of linear momentum conservation and symmetry of the stress tensor is equivalent to angular momentum conservation. The proposed formulation has a mixed basis of both primal and algebraic dual nodal and edge basis functions. It fulfils the requirements as it is shown to be hybridisable, to satisfy point-wise linear momentum, and the discrete representation of the stress tensor is point-wise symmetric, hence angular momentum conservation is point-wise satisfied as well. The thesis furthermore functions as an overview of the method applied to elliptic problems, showing the results for previous formulations, and as a starting point for the next steps towards applying the method to fluids. A first step is proposed on extending the new formulation to a Stokes flow formulation with the stress as primary unknown, aimed at satisfying both linear and angular momentum conservation as well as mass conservation.

Being able to solve numerically partial differential equations is fundamental for engineers to evaluate, optimize and improve industrial equipments. The framework of mimetic finite element methods allows engineers to find solutions characterized by strong conservation properties: this may result in a pointwise divergence free-flow field. However, sometimes, the computation of the solution of partial differential equations is time consuming, as a results, to reduce the computational time, engineers and mathematicians have developed hybrid methods.
The objective of this thesis is the development of a hybrid mixed finite element formulation of the vector Laplace equation without spurious modes. Discontinuous elements permit an higher degree of parallelism, and, at the end, a lower computational time. Lagrange multipliers are used to impose continuity between discontinuous elements. These turn out to be not only mathematical features but they are connected to the physical variables of the problem. Furthermore, it has been found that the usage of a new Lagrange multiplier, on the intersection of 4 or more elements, removes the spurious modes. Therefore, the associated system of equation is non-singular. The usage of the hybrid finite element methods reduces the computational time while maintaining the pointwise divergence constraint and the optimal convergence rate of all variables.
At the end, the mixed hybrid formulation is modified to solve the Stokes equations. Lagrange multipliers are used as boundary conditions. Solution of the lid-driven Stokes flow is shown.

The Kutta condition is a boundary condition applied at a wing’s trailing edge (TE), which enables the calculation of the fluid-dynamic lift produced by a wing in potential flow. This condition is actually a stable equilibrium resulting from a transient flow. The main question of this research is: how is the Kutta condition flow-field created when a wing is brought in motion from rest? This is investigated using an experimental apparatus, in which a wing model is accelerated from rest and the transient TE flow is analysed using high-speed Particle Image Velocimetry techniques. Initially, fluid moves from the airfoil’s pressure side, along the TE, towards the suction side, and a boundary layer is created. Then, boundary layer stagnation occurs, a vortex separates from the TE, and a transformation in the presence and locations of stagnation/separation occurs. The observed process confirms classical theories which describe the creation of the Kutta condition flow-field.

In Pressurized Thermal Shock (PTS) applications, there is a co-existence of different flow regimes. The regimes can be divided in two main groups namely, dispersed two-phase flows and the separated two-phase flows having relatively larger interface. Simulating a two-phase flow itself is already a complex task. It gets even more complex when the flow is turbulent. Each phase can be laminar or turbulent depending which one is taken as point of reference. Primary phase may be turbulent compared to secondary phase but may be laminar compared to the vessel.
The objective of this master’s thesis project is Computational Fluid Dynamics (CFD) modeling of two-phase stratified flow settings relevant to Nuclear Reactor Safety (NRS) scenarios more specifically, the focus is placed on the modeling of Taylor bubble co-current flow (airwater) in a vertical pipe in Two-Dimensional (2D) and Three-Dimensional (3D). This is done through the application and validation of i) OpenFOAM’s standard VOF solver in combination with a conventional RANS model to mimic the turbulence behavior of the flow, and ii) NRG’s Runge-Kutta (RK) VOF solver in combination with a modified RANS model to capture the turbulent two-phase flow with large interfaces. For the purpose of this project two data sets are selected against which the morphology of the Taylor bubble is compared, namely the analytical model of Tomiyama et al. [1] and the high resolution Taylor bubble LES data of Frederix et al. [2].
It is shown that in case of i), where standard k−ε is applied to model the turbulence behavior, the model fails to predict the interface of entire Taylor bubble adequately. This is because standard RANS models, which are initially developed for single-phase flows, overpredict the turbulence intensity near the interface. This is due to excessive turbulence production. This problem is addressed a.o. by Egorov et al. [3] who developed a model with a turbulence damping term in the turbulence model. This term is activated locally at the interface. Frederix et al. [4] have extended the Egorov approach in order to make it even more stable and robust. Frederix’s model is applied in ii) where it is shown that for a damping term, length scale, δ/D = 3.9×10−3 the turbulence model is damped at the interface while in the rest of the flow the turbulence behavior is modeled properly by k−ω SST model.

In order to solve linear system of equations obtained from numerical discretisation fast and accurate, preconditioning on the coefficient matrix is needed. In this thesis research, a preliminary study on preconditioning techniques suitable for Mimetic Spectral Element Method (MSEM) will be presented. The spectral limit change of some important discrete operators that appear in MSEM are studied, with respect to the discretisation order change. Some of the limits are proven by theorems and some are estimated by reasoning. Preconditioners and preconditioning techniques are developed and performed on a Poisson problem using both p-version and hp-version MSEM. Lastly in conclusion, the cause of ill-conditioning of MSEM-related systems are discussed; a common framework of preconditioning study for MSEM is proposed, which will be helpful for future studies and application on different problem types.

Prediction of emissions from combustion systems is a complex problem involving the coupling between the flow field and chemistry. CFD analysis is the most commonly employed approach. However, it has the drawback that a very high computational cost prevents the use of detailed chemistry models. This master thesis, which focuses on NOx emissions, uses a more unconventional method of emissions prediction: Chemical Reactor Network (CRN). The advantage of this method is that, as it does not use fine discretisation, closure models nor fluid dynamics equations, it allows the implementation of detailed chemistry mechanisms.

A CRN is developed first for a single-element Lean Direct Injection (LDI) combustor and then the CRN is adapted for a Multi-Point LDI (MPLDI) combustor. CFD and experimental results are used to set up the CRN. In the base case scenario, in which kerosene is the fuel choice, the NOx emissions predicted are very close to experimental measurements. This is particularly meaningful given the high uncertainties of modelling this highly complex turbulent combustion process.

The developed CRN is also used to predict NOx emissions for the same combustor when the fuel choice is varied. Namely, the alternative fuels considered are kerosene enriched with hydrogen, methane and methane enriched with hydrogen. In comparison to kerosene, higher Lower Heating Values (LHVs) of these fuels lead to lower combustor temperatures for the same power input. Consequently, the thermal NOx pathway is weakened and the NOx mass flow generated is reduced. Nevertheless, these fuels with a higher LHV come with an increased operational risk that must be overcome before their implementation in aviation becomes possible.

The Royal Netherlands Navy (RNN) operates four diesel-electric submarines, the Walrus class. The submarines sail submerged on electric engines and periodically recharge the batteries with diesel engines at periscope depth. The exhaust gases are dispelled at the back of the sail below water level. During the sea trials it was observed that the rising exhaust gases elevated the surface locally with a height of 1 to 2 m. This elevation limited visibility through the periscope, water flooded the air intake and the submarines could be easily detected. To remedy this problem model tests were performed. To enable evaluating several exhaust configurations for a replacement of the Walrus class a numerical model is studied to predict the surface elevation.

To model this situation a numerical method is used. The method applied in this work is the Volume of Fluid code ReFRESCO, which is a Reynolds Averaged Navier-Stokes (RANS) solver. Three test cases are studied, a rising bubble, a buoyant jet and an exhaust plume from a submarine sail.

The simulations for the buoyant jet show that the momentum in the jet is dominated by buoyancy rather than by the initial inflow momentum. The influence of different turbulence models on the width of the jet and the corresponding surface elevation are investigated. The choice of turbulence model influences the distribution of air in the domain, but does little to the surface elevation. It is concluded that the KSKL model yields the most physical results where the air spreads out in the domain. This model also has a satisfactory convergence behavior. It is concluded that both the density of the exhaust gas and the velocity profile at the nozzle have little influence on the final result. A parabolic velocity profile instead of a uniform outflow does improve the convergence. The dominant numerical error is the discretization error, which is in the order of 12% for the surface elevation. To this end the L∞ norm of the iterative error must be reduced to a satisfactory value, in the order of 10^-3 or smaller. Comparing the results with experiments is difficult due to a lack of proper experimental data, however it is concluded that the results are in a similar order of magnitude.

Finally the exhaust gases on a submarine sail are modeled. For the modeling three simplifications are made: only the flow surrounding the sail is modeled (the hull of the submarine is not modeled), the control planes on the sail are not modeled, and no incoming waves on the free surface are taken into account. Both in the experiments and in the simulations a pulsating behavior can be observed in the rising air. L∞ norms for the submarine modeling are generally in the order 10^-3, but occasionally less. The numerical results are validated against the experimental results. It is concluded that the use of the RANS code ReFRESCO is possible for the modeling of a submerged exhaust of a submarine. The estimated uncertainty for the result is in the order of 15%.

This study is aimed at modelling of the unsteady interacting boundary layer around wind turbine rotor blades. It is shown that nonconservative mechanisms are present in this system, affecting the definition and approximation of solutions to the problem. The objective is to overcome the inconsistency of classical (dissipative) schemes, and find a suitable approximation method. A path-consistent discontinuous Galerkin method is proposed to correctly model the dynamics of this system. In the future, the current implementation is to be joined with the existing part of the RotorFlow simulation code. This will enable the incorporation of unsteady aerodynamic phenomena in the early design stage of new and large wind turbine rotors, which is currently unfeasible due to the high computational cost.