Turbulence modelling remains a key challenge in the field of computational fluid dynamics (CFD) for accurately resolving fluid flows. A new class of data-driven models have recently gained popularity that aim to quantitatively incorporate data from higher fidelity simulations to improve turbulent closure modelling in Reynolds Averaged Navier Stokes (RANS) models. These models have been particularly well researched in modelling the Reynold stress anisotropy tensor (RST).

This thesis re-formulates the data-driven framework in the context of resolving passive scalar fluxes. Corrections to the scalar flux vector are regressed for two different frameworks: framework 1 that uses the simple gradient diffusion hypothesis (SGDH) model and framework 2: that additionally introduces transport equations for the scalar variance and dissipation. These models are compared to each other for the Jets in-crossflow (JICF) case for resolving the scalar field where baseline RANS models struggle. Both data-driven frameworks show an improvement in modelling the scalar field and turbulent scalar flux vector than the baseline models. The study also indicates that inclusion of additional transport equations doesn’t necessarily improve regression of the corrective fields. ... Read more

Machine-learning surrogates are increasingly used to accelerate computational fluid dynamics, yet progress is limited by the lack of benchmarks capturing realistic, time-dependent turbulent flows. This thesis introduces a 13 TB dataset of high-fidelity implicit large-eddy simulations of three-dimensional turbulent wakes behind super-elliptical cylinders. Unlike existing datasets, it captures three-dimensional turbulence with an active energy cascade driven by vortex stretching, combining (i) 380 long-horizon trajectories of 400 time steps each with 3–9 million points per frame on irregular meshes, (ii) systematic variation across geometry, Reynolds number, and angle of attack, and (iii) a temporal resolution that preserves the full inertial subrange of the turbulent energy spectrum. Building on this dataset, state-of-the-art neural operators are evaluated across three prediction tasks of increasing complexity: mean-field prediction from governing parameters, the inverse pressure problem, and long-horizon autoregressive spatio-temporal forecasting. Neural operators accurately recover mean flow fields, yet fail progressively as the target fields gain high-frequency content: fine-scale spatial structure is systematically suppressed in instantaneous flow fields, and all evaluated architectures collapse for temporal predictions. These failure modes are attributed to several concurrent mechanisms: memory constraints forcing sparse point-cloud subsampling, latent-space compression discarding high-frequency spatial content, and a spectral bias of the mean-squared-error objective that de-prioritises the high-frequency residuals dominant in turbulent flows. Together, these results provide a new standard benchmark for turbulent CFD surrogates and show that progress requires methodological advances beyond architecture scaling, including physically motivated loss functions and latent representations capable of retaining fine-grained spatial content. ... Read more

Submerged ship propellers, superventilating torpedoes, and rudders on displacement and planing vessels. This research focuses on the natural ventilation of a hydrofoil strut at a yaw angle at a moderate Froude number. In such cases, air from the free surface flows towards the suction side of the hydrofoil.

A methodology was developed to extract physically meaningful features from high-speed imagery obtained during towing tank experiments. The framework combines unsupervised clustering techniques with image processing algorithms for geometric reconstruction.

It is demonstrated that the points on the hydrofoil free surface can be reconstructed algorithmically using an implementation of the Hough Gradient Method. These points have been successfully used to fit an asymmetric sigmoid function to the free surface contour. A modified implementation of the Hough Gradient Method, with adjustments to improve image contrast, enables reconstruction of the cavity closure angle. While unsupervised clustering methods show partial correspondence with known flow regimes, the results indicate that more advanced techniques are required for reliable, generalizable classification.

An additional contribution is the identification of an observable difference in free surface contour for a specific ventilation mechanism. Specifically, cases associated with a Laminar Separation Bubble exhibit a more forward free surface drawdown and a steeper depth gradient compared to other ventilation mechanisms. These findings demonstrate the potential of automated visual analysis not only for post-processing but also for advancing the physical understanding of hydrofoil ventilation. ... Read more

Vold-e is a fully electric autonomous eVTOL designed for urban and regional air mobility. Its key innovation is a folding Prandtl wing, which reduces ground footprint while maintaining aerodynamic efficiency. The aircraft carries four passengers, has a range of 200 km, and reaches speeds of 200 km/h.

The design combines six electric rotors, a lightweight composite structure, and advanced autonomous control systems with redundant safety features. Aerodynamic and structural analyses confirm that the aircraft is feasible, with strong efficiency and stability performance. Vold-e also offers zero in-flight CO₂ emissions, reduced land use, and competitive future operating costs. The main challenges are technical complexity, certification, and market adoption. ... Read more

Machine-learned surrogate time integrators promise large speed-ups over classical solvers, yet their performance is usually reported as a single aggregate error, leaving open the question of when they remain stable. This thesis determines the empirical stability limits of graph neural network (GNN) surrogates for the two-dimensional advection–diffusion equation on unstructured meshes with periodic boundary conditions, expressed directly in the dimensionless CFL and Fourier numbers.

Surrogates are trained to minimise the one-step error on fixed velocity and diffusion fields and evaluated autoregressively on unseen fields over horizons eight times the training window. Two families are compared at a fixed message passing budget: single-scale models, and multiscale models organised as V-cycles over predetermined coarsened graphs. For each model, a piecewise-linear fit of the final rollout error against the CFL and Fourier numbers yields empirical stability limits, defined by a blow-up threshold.

Within these limits the surrogates reproduce the finite element reference accurately on both seen and unseen fields and show no abrupt change beyond the training horizon, although the diffusion-dominated regime is consistently harder than the advection-dominated one. The single-scale CFL limit tracks the number of message-passing blocks and lies slightly above it. Adding coarse levels at a fixed total message passing layer budget broadens the advective stability range, decisively at the largest stride, but a two-level hierarchy trades diffusive stability and in-region accuracy for this gain. Only a three-level V-cycle removes the penalty, attaining zero blow-ups on both axes, and deeper models show no oversmoothing.

The diffusion-side limits carry large variance, traced partly to the dissipative backward-Euler reference, and should be read as indicative. The work delivers a concrete operational range for GNN surrogates and identifies how multiscale models can extend it. ... Read more

As the share of wind energy in the global electricity mix increases, it becomes more important to accurately predict wind turbine wakes in order to optimise wind farm layouts and to reduce fatigue loads on downstream turbines. Wake models based on the Reynolds-averaged Navier–Stokes equations offer a good balance between accuracy and computational cost, but their performance depends strongly on the turbulence closure used. The standard k–ε model is widely used in wind energy, but it is known to overpredict the eddy viscosity in the near wake. This leads to an overestimation of turbulent mixing and wake recovery. To address this, researchers have proposed extended closures, which have only been validated in fully elliptic three-dimensional RANS solvers. The Forced Ainslie Wake Model has improved the physical consistency of parabolic wake modelling by reintroducing part of the actuator disk force. Its turbulence closure is currently limited to a constant eddy viscosity. A transport-based turbulence model that is compatible with the parabolic marching scheme of the FAWM is therefore needed.

This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.

During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.

With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.

In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation. ... Read more

A Lattice Boltzmann (LBM) solver is presented and differentiated using the discrete adjoint method. A shape optimization pipeline is implemented based on the obtained gradients. The following features are considered, aiming for applicability in industrially relevant simulation and optimization, and possible integration with a quantum LBM code in the future:

1. Recursive regularized collision operator.
2. Volumetric grid refinement with arbitrary grid geometries.
3. A mass-conserving, differentiable boundary condition for no-slip boundaries, and a methodology to deduce physical shape gradients from the bounce-back boundary condition (for integration with a quantum code).
4. Momentum inlet and pressure outlet boundary conditions.
5. Pressure-relaxation absorption region (sponge) to dampen acoustic fluctuations.
6. Coefficients of lift and drag as quantities of interest, computed with the momentum exchange method.
7. Shape parametrization with free-form deformation.

The forward solver is validated on steady and unsteady flow conditions, as are the adjoint-based gradients, which are obtained using a general methodology by Tekitek et al. [1]. The adjoint-based gradients present relative errors to finite differences of O(10⁻⁵). A steady L/D optimization case is run, presenting improvements on the base shape on the order of 20%. It is demonstrated that the adjoint methodology supports parametrization of both the collision operator and streaming matrix in a simple, extensible manner.
... Read more

As nations broaden their renewable energy portfolios, wind power is playing an increasingly important role in the energy landscape. Although onshore wind continues to dominate the wind energy mix in the EU, rising energy demand and ambitious climate targets have driven the development of offshore wind farms and spurred research into floating offshore wind turbines.

This thesis began with the goal of developing a data-driven modeling approach that maps site conditions to the load statistics on floating offshore wind turbines. The motivation for this research emerged from a need to accelerate the site-selection process, include more site variables, and achieve accurate load estimates while keeping the computational expense low.

Before installation, all modern wind turbines must be carefully assessed for structural loads to ensure safety and performance throughout their lifetimes. Floating turbines, in particular, introduce more complexity—more environmental variables to consider, and higher uncertainty in the dynamic response. This added complexity makes analyzing structural loads expensive and time-consuming, often more so than for fixed-bottom counterparts, and it makes understanding the behavior of floating wind turbines even more important. The process of fatigue damage calculation, in particular, is computationally intensive, often requiring thousands of costly simulations. Because it is not feasible to run simulations for every possible sea state, engineers typically reduce the problem by selecting a set of variables and binning or lumping sea states to limit the number of required simulations. However, for floating wind turbines, the choice of which variables to include and how to perform this binning remains an open question. This motivates the use of reliable data-driven surrogate models that can make quick estimates of loads on wind turbines while maintaining high accuracy.

Although most existing work relies on deterministic surrogate models, offshore wind environments exhibit strong stochasticity, making turbine loads inherently uncertain. This motivates the need for uncertainty quantification to characterize not only expected loads but also estimate their conditional variability. This dissertation, therefore, develops a probabilistic data-driven methodology that propagates environmental uncertainty to the 10-minute damage equivalent loads of onshore, offshore, and floating offshore wind turbines.

Several deterministic and probabilistic data-driven models are benchmarked on onshore and fixed-bottom offshore wind turbines. The evaluation consists not only of judging the accuracy of a model, but also of practical aspects such as the robustness of the model, sensitivity to hyperparameters, and ease of implementation. Compared to deterministic approaches, which require multiple seed repetitions prior to training, it is demonstrated that with probabilistic models, this step may not be necessary to achieve high accuracy predictions (𝑅2 > 0.95), thereby saving precious computational resources needed to generate the training database. Widely used Gaussian process regression is shown to accurately estimate the conditional mean of the response with a relatively small training dataset of Q102 - 103) samples. Wasserstein-conditional generative adversarial network is used as one of the probabilistic regression models. Despite learning the functional mapping with errors comparable to the best-performing models, it is found to be very complex to implement and requires extensive hyperparameter tuning. Simpler 4th-degree polynomials are shown to make good predictions in the onshore case, but are prone to overfitting and susceptible to the additional noise introduced by the hydrodynamic features. Overall, mixture density networks are shown to provide the best combination of consistency, accuracy, and practical ease of use.

Based on this analysis, the study extends the use of mixture density networks to a more sophisticated application of spar-type floating offshore wind turbine. It is shown to successfully capture the conditional response in terms of the normalized 2-Wasserstein distance despite the added complexity. The surrogate is further used to make probabilistic estimates of the lifetime damage equivalent loads on four potential floating wind turbine sites. Since the surrogate model is fast (order of milliseconds once trained), load predictions can be made over all sea states within seconds, without the need to lump or bin the sea states beforehand. The uncertainty in the aggregated lifetime fatigue loads due to stochastic inputs is extremely narrow, with variability on the order of only 0.1–0.5% of their mean values. This results from summing the 10-minute damage equivalent loads over a million occurrences, effectively nullifying the impact of the outliers. The use of a probabilistic surrogate that correctly captures the conditional distribution is still useful, as it minimizes the aggregation of error in the final response.

Through these analyses, it is demonstrated that surrogate models can be powerful tools for fatigue estimation in the site analysis process, especially for floating wind turbines, where the choice of variables and binning methods is still an open question. Additionally, using probabilistic surrogates like mixture density networks helps reduce bias in calculating the aggregate mean fatigue, as the conditional distributions are heteroscedastic and not always normally distributed.
... Read more

Flow control plays a vital role in improving efficiency in aerospace, maritime, and energy systems by delaying transition to turbulence and suppressing instabilities such as Tollmien–Schlichting waves. This work uses the Kuramoto–Sivashinsky (KS) equation as a model to study convective instabilities in boundary layers. A finite-difference discretization yields a state-space formulation, enabling systematic application of control strategies. Linear Quadratic Regulator (LQR) and adjoint-based optimization methods are developed to minimize flow perturbations. Results show LQR effectively suppresses disturbances, while adjoint-based control scales efficiently to nonlinear cases, highlighting promising avenues for future turbulence management. ... Read more

Computational Fluid Dynamics has been widely used to model flows for engineering applications. The challenge has been to model or resolve turbulent flows accurately and there exist different approaches such as Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds Averaged Navier Stokes (RANS). With limitations on the available computational power and time, RANS is the favorable choice to model such flows despite significant modeling errors.
With recent advancements in machine learning (ML) algorithms and availability of high-fidelity (accurate) data, the turbulence community has been actively working to improve RANS modeling by informing it from the reference data and developing corrections. This is also known as data driven turbulence modeling. There exist several algorithms that have improved RANS modeling but their computational cost and lack of physical interpretability remains an issue to gain an understanding of flow physics and the relation between different flow quantities.

The present study implements two data driven methods to provide corrections ad improve RANS modeling, focusing on Turbulent Kinetic Energy and the Reynolds Stress Tensor. However, the goal is
to also obtain symbolic models and physically interpretable corrections, while ensuring computational cost remains minimum.

A priori analysis, where only the reference data is used to develop corrections, shows significant re- ductions in the computational cost (function calls to the CFD solver) while converging to acceptable
correlations as well. The superior of the two methods is selected for a posteriori study where it is
coupled with a CFD solver. The correction is now formulated by differentiating the CFD code. This
information is given to the ML algorithm which adjusts its parameters accordingly to optimize the objective function and improve turbulence modeling. Results show that, with regularization, a generalized symbolic formula can be obtained which when tested on different geometries improves the prediction of flow quantities and turbulence modeling, as a whole. ... Read more

The transition to renewable energy sources is essential to mitigate climate change, and floating wind turbines (FOWTs) present a promising solution to harness offshore wind resources. Light Detection and Ranging (LiDAR) systems mounted on nacelles provide a cost-effective and efficient means to measure wind fields, critical for turbine performance, control and load simulation. However, FOWT's motion introduces complexities in LiDAR measurements due to velocity and positional changes. This thesis focusses on developing a correction method for LiDAR measurements on FOWTs, addressing the influence of motion on wind velocity, position, and direction. The accuracy and uncertainty of these corrected measurements are quantified.

Simulated six degrees of freedom (6DOF) motion and a power law wind field are inserted in a numerical LiDAR model, in which corrected and uncorrected measurement position, direction and line of sight velocity are constructed. The corrected outputs are validated through reconstructed wind fields and the uncertainty of the correction is quantified. In this study, significant motion-induced bias is identified in the reconstructed wind fields. The dominant motion affecting measurement accuracy was identified as pitch motion, especially when it exhibits a non-zero mean. The relative error of the reconstructed power law wind field parameters is reduced by 3 orders of magnitude. Despite an increase in uncertainties associated with the correction method applied, the correction remains effective in reducing the error in LiDAR measurements induced by FOWT motions. The findings highlight the necessity and feasibility of motion correction for LiDAR measurements, offering substantial improvements in the accuracy and reliability of reconstructed wind fields for floating wind turbine applications. ... Read more

Solving the incompressible Navier-Stokes equations is computationally heavy, with the pressure Poisson equation being the most time-consuming step. Iterative linear solvers are typically utilized to solve this equation. Since most solvers are iterative and rely on an initial guess, an opportunity emerges to use machine learning to improve this initial guess, such that fewer iterations are needed, consequently saving time.
A novel graph neural network is designed that employs a custom convolution algorithm, message-passing scheme, and pooling algorithm to maximize its performance. First, a convolution algorithm is proposed that uses interpolation to make a discrete (3x3) CNN kernel continuous. Then, instead of directly computing the kernel weight from the function, an integral over specified bounds is applied to account for the geometrical inhomogeneous distribution of the source nodes. The integral is embedded as the weighted sum of a vector containing learnable parameters, computed through a dot product with the edge attribute vectors. Next to the convolution operation, a message-passing scheme is designed that is compatible with the data format of the finite volume method whilst performing well in terms of the distance information can travel over the mesh. To conclude the design of the model, a custom pooling algorithm is designed that is equivalent to average pooling in CNNs.
A normalization procedure is established that ensures consistency in the model's magnitude. Notably, the ground truth output pressure is normalized using its standard deviation, which is unknown. To estimate this normalization factor, a correction model is established that uses the same convolution algorithm but employs an architecture inspired by classification CNNs.
The model's performance is evaluated based on the reduction in number of iterations required to reach convergence. This is done for both the Preconditioned Conjugate Gradient (PCG) solver and the multigrid Geometric Agglomerated Algebraic Multigrid (GAMG) solver
Across the various tests conducted, the number of iterations needed to reach convergence is reduced by approximately 40%, with the PCG solver performing slightly better than the GAMG solver. However, the PCG solver yields less consistent results, performing very well at samples that closely align with the training data, leading to a reduction of up to 60%. However, its performance drops significantly when tested on data that does not closely resemble the training data, sometimes even increasing the number of iterations. The GAMG solver demonstrates consistent performance, with almost no difference between the training and evaluation data.
In terms of generalization, the model demonstrates promising results, achieving similar performance across datasets with varying levels of complexity. This is interesting as the root mean square error differs significantly across the datasets and individual samples. This suggests that there is no direct relationship between the reduction in the number of iterations and the accuracy of the prediction. Furthermore, the model performs well on unseen meshes, showing that it can handle the unstructured nature of the meshes used throughout this research. This demonstrates the model's ability to be trained on a diverse dataset, after which it can be applied to unseen cases.
... Read more

Turbulence modelling remains a major challenge in computational fluid dynamics (CFD), as Reynolds-Averaged Navier-Stokes (RANS) closures rely on empirical relations introducing significant model-form uncertainty. Symbolic regression offers a path toward interpretable data-driven closures, but conventional Deep Symbolic Regression (DSR) struggles to efficiently explore the vast space of physically consistent expressions.

We propose a Multi-Agent Deep Symbolic Regression (MADSR) framework that reformulates turbulence model discovery as a cooperative multi-agent reinforcement learning (MARL) problem. Each agent discovers one scalar coefficient function in the tensor-basis expansion of the Reynolds-stress anisotropy tensor, sharing a common reward derived from frozen RANS evaluations. This cooperative setup promotes coordinated learning among model components.

In MADSR, the effectiveness of 2 MARL techniques, proximal policy optimization (PPO) and centralised training decentralised execution (CTDE), is investigated on symbolic turbulence modelling. Several MADSR variants are developed and tested, including a vanilla multi-agent DSR, a proximal policy optimization (PPO) based MADSR, an actor-critic MADSR, and a MAPPO-DSR inspired by multi-agent proximal policy optimization (PPODSR).

Applied to the Explicit Algebraic Reynolds-Stress Model (EARSM) and k-corrective RANS formulations, MADSR outperforms single-agent DSR in frozen RANS evaluations of the periodic-hill benchmark. The multi-agent structure enhances exploration efficiency and enables discovery of more consistent and interpretable turbulence closures. MADSR thus represents a promising step toward fully end-to-end, reinforcement-learning-based symbolic turbulence modelling. ... Read more

This thesis explores the use of Bayesian Deep Learning to improve uncertainty quantification in Reynolds-Averaged Navier-Stokes (RANS) turbulence models. While RANS models are commonly used in computational fluid dynamics due to their efficiency, they are often criticized for inaccuracies in certain flow conditions, primarily due to the challenges in modeling the Reynolds stress term. The thesis acknowledges the limitations of traditional turbulence models, which rely heavily on empirical parameters and often fail to generalize across different flow scenarios, leading to significant uncertainties.

To address these issues, the research introduces a data-driven approach, leveraging Bayesian Neural Networks (BNNs). BNNs are particularly suitable for this task because they not only improve prediction accuracy but also provide a mechanism to quantify uncertainties arising from both the model and the data. This dual uncertainty quantification is critical, as it helps to address the inherent ”black box” nature of machine learning models, which can introduce additional uncertainties into the predictions.

The methodology involves correcting traditional turbulence models and integrating them with BNNs to capture both aleatoric (data-driven) and epistemic (model-driven) uncertainties. The thesis demonstrates the effectiveness of this approach through various flow case studies, comparing the results against more accurate but computationally expensive methods like Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS).

The research concludes that the integration of Bayesian Neural Networks into RANS turbulence models not only enhances predictive accuracy but also provides a more comprehensive uncertainty quantification, making it a promising direction for future work in turbulence modeling.
... Read more

Reynolds averaged Navier Stokes (RANS) models are the industry standard when it comes to computational fluid dynamics (CFD). RANS models have notable shortcomings as the
Reynolds stress is modelled locally and does not include temporal behaviour. Therefore,
current RANS models fail to make accurate predictions in areas of out-of-plane straining,
high streamline curvature and significant anisotropic Reynolds stresses.
Data augmented RANS models have shown promise to improve accuracy, with sparse regression and machine learning algorithms being able to train models between high fidelity
data and RANS. The current data driven models have problems, as they lack generalisability between data-sets, are difficult to interpret and only provide a local, non-temporal
correction or prediction for the Reynolds stresses.
The current work will explore sparse regression routines, in particular PDE FIND, for
the purpose of creating a non-local, temporal transport equation from data. Rather than
using high-fidelity turbulence data, a self-developed model problem, dubbed the ψ − ϕ
system, is created to replicate the uncertainties related to turbulence modelling. Multiple
candidate libraries for PDE FIND are tested, with the greatest emphasis on libraries
lacking a part of the true functional form, known as incomplete candidate libraries.
Incomplete candidate libraries are found to benefit from using highly correlated replacement functions for the missing function with an exception being a candidate library lacking
the correct production term. Even if models were trained poorly, PDE FIND was able to
identify the most important functions with an accurate coefficient.
PDE FIND could potentially create an universal transport equation to model the residual
between high fidelity data and RANS, but only if there are enough highly correlated
candidate functions and the production terms are well known or well represented in the
candidate library.
... Read more

In recent years, computational fluid dynamics (CFD) has become an essential design tool across various industries, allowing engineers to tackle complex fluid dynamics problems that would otherwise require costly and time-consuming real-life experiments. For Formula 1 teams, who must experiment within strict time limits in the wind tunnel and on track, the ability to simulate airflow around their race cars under various conditions is crucial for maintaining competitiveness in the fast-paced world of Formula 1 racing. Reynolds-Averaged Navier-Stokes (RANS) simulations remain the industry standard for simulating turbulent flows, as they allow engineers to conduct simulations efficiently. However, this efficiency comes at the expense of accuracy, as RANS cannot resolve all turbulence scales, leading to uncertainties.

Recent advances in data-driven RANS turbulence modeling have enabled partial correction of these uncertainties. However, obtaining a correction that is generalizable under different geometries and flow conditions remains a challenge. Turbulence models are calibrated to fit specific flow regimes, so correcting these models across the entire domain can disturb these calibrations, worsening performance. A solution is to divide the domain into regions based on identifiable physical phenomena and apply local corrections without disturbing calibrated regions. In Formula 1 race car design, the most critical region is the shear layer, where RANS shows the largest discrepancies.

In this thesis, a classifier was developed to distinguish the shear layer from the rest of the domain based on the ratio between turbulent kinetic energy production and destruction, as well as turbulence intensity. Within this classifier region, corrections to the k-ω SST turbulence model are made by extracting model form errors from high-fidelity data using k-corrective-frozen RANS. These corrections include a residual term added to both the k and equations and a term for the anisotropy of the Reynolds stress tensor. The Spars Regression of Turbulent Stress Anisotropy (SpaRTA) framework, based on elastic-net regularization, was used to regress symbolic expressions for the corrections, enabling their application to simulations of unseen test cases.

The models discovered with the SpaRTA framework for the shear layer show promising results, improving the prediction of separation and reattachment positions. These models were tested on various geometries and simulations at different Reynolds numbers, demonstrating a certain level of generalizability. While there is room for further improvement, this thesis shows that integrating targeted model corrections into RANS simulations, informed by isolated shear layer data, can enhance the understanding and prediction of shear layer dynamics in 2D-separated flows. ... Read more

Reducing anthropogenic climate change is a significant challenge requiring a global response to prevent tipping points in the climate system, such as the disintegration of ice sheets, and thawing of permafrost, among others. The rapidly growing air transport sector, which carried 4.5 billion passengers in 2019, is projected to emit nearly 2 Gt CO2 by 2050—about 2.6 times the emissions in 2021. Decarbonising aviation is challenging due to its reliance on fossil fuels, and while technological, operational, and regulatory measures have reduced fuel consumption, they are insufficient to mitigate aviation’s overall climate impact. The non-CO2 effects are significant, accounting for about two-thirds of aviation’s warming impact in terms of Effective Radiative Forcing (ERF). These effects include contrails, contrail-induced cirrus clouds, nitrogen oxides (NOx ), and water vapour emissions, collectively contributing to approximately 4% of anthropogenic forcing since the pre-industrial era. Given their spatio-temporal variability, climate-optimised flight planning canmitigate these impacts by avoiding sensitive regions, but this faces several challenges. These include the inherent chaos of weather, low scientific understanding of non-CO2 effects, and the large computational expense of calculating sensitive regions using climate change functions (CCFs).

To address these issues, this thesis first analyses algorithmic climate change functions (aCCFs), a simple surrogate model obtained by regressing the CCFs against local atmospheric variables. The aCCFs are computationally inexpensive to run since they only use few meteorological inputs to estimate climate impact, enabling real-time flight trajectory optimisation on arbitrary days. However, aCCFs are applicable only in parts of the Northern Hemisphere and require thorough verification before implementation. The focus is narrowed down on local aviation NOx effects on climate change, which largely causes warming via short-term increase in tropospheric ozone (O3) and is characterised by large variability. This necessitates a detailed investigation of NOx-O3 effects in isolation and its mitigation, which is a previously unexplored area. After verifying the O3 aCCFs through complex climate-chemistry model simulations, it is concluded that while it enables a reasonable first estimate, there are a few discrepancies.

TheO3 aCCFs are replaced by using a more comprehensive dataset comprising global NOx-O3 impacts, identifying additional physical variables that influence this impact, and using this information to train stochastic surrogates based on homoscedastic and heteroscedastic Gaussian processes. These models provide mean and uncertainty estimates for the climate impact of NOx on O3, for the first time. The heteroscedastic model more accurately reproduces the data distribution and its ease of use in predicting the climate impact of individual flights is demonstrated. Defined as probabilistic aCCFs (paCCFs), these models demonstrate superior accuracy over aCCFs, provide valuable insights for aviation’s non-CO2 effects, and offer broader implications for climateoptimised flight planning. The thesis concludes with limitations and recommendations to furthermitigate aviation’s environmental impact. ... Read more

Site analysis to determine the loads experienced by wind turbines based on site-specific environmental conditions is typically done using either coupled aero-servo-elastic simulations for onshore wind turbines or coupled aero-servo-hydroelastic simulations in the case of offshore wind turbines. These simulations become computationally expensive when multiple load cases are needed to be taken into account, together with the numerous possible combinations of turbulence inflow patterns that result in the same mean inflow conditions. Probabilistic surrogate models offer a cheap alternative to these expensive simulations for predicting load statistics. This thesis explores a type of neural network called Conditional Generative Adversarial Networks (CGANs) as a potential candidate for such a surrogate model. Originally developed for image generation, CGANs have seen success in other applications. However, most applications of GANs to date are high-dimensional, with relatively low research focused on low-dimensional problems such as wind turbine load statistics. Multiple experiments are conducted using various multimodal and heteroscedastic datasets to assess its ability to model such characteristics
accurately. The conditional log-likelihood andWasserstein-1 distances were used as metrics. The results show that CGANs can indeed model such low-dimensional datasets. Finally, the CGANs are trained on data from simulations of onshore and offshore wind turbines in OpenFAST and compared with predictions from Mixture Density Networks (MDNs). The results from CGANs are comparable to MDNs, showing its potential as another alternative surrogate method, although more research needs to be performed. ... Read more

Unsteady numerical simulation has been proven to be an essential tool for research. The quality of the results can be improved by using mesh adaptation. Mesh adaptation uses error indicators to refine the mesh in regions with high errors. The error indicators used are output errors with the most accurate output error estimation method being adjoint-based error estimation. However, for this method, the primal solution needs to be stored, which is storage intensive, especially for large unsteady simulations. The method proposed in this thesis uses a neural network autoencoder to compress and reconstruct the primal solution. This solution is compared to a reconstructed solution using Proper Orthogonal Decomposition (POD).

The one-dimensional unsteady Burgers equation is used as validation for the methods using a manufactured solution while the lid-driven cavity flow is investigated using the proposed method. The manufactured solution of the one-dimensional Burgers case could be exactly reconstructed using two POD modes. For the autoencoder a small latent space was used. For low resolutions, the small latent space did not prove to be a problem as the primal and residual could be captured accurately. However, for higher resolutions, the reconstruction error of the autoencoder became dominant for the residuals and resulted in erroneous adjoint-based error estimates while the primal remained qualitatively similar.

For the lid-driven cavity flow, the POD was still able to capture the solution using a low number of modes due to the smoothness of the solution. This resulted in an unfair comparison between the POD and autoencoder reconstructed solutions. The reconstructed autoencoder error estimates for lower resolutions were more accurate due to the latent space being large enough to capture the residual of the discrete primal accurately enough. When moving to higher resolutions, the autoencoder was not able to reconstruct the residual accurately enough leading to erroneous error estimates. Therefore, the latent space of autoencoders should be sufficiently large in order to gain an accurate reconstruction of the residual. If the latent space is large enough, the error estimate is accurate and the local error estimates can be used as a first iteration error indicator for mesh refinement. ... Read more