Reduced-order models (ROMs) based on Proper Orthogonal Decomposition (POD) are widely employed to reduce the computational cost of high-fidelity fluid simulations. However, their direct application to multiphase flow fields poses challenges due to the presence of discontinuities at phase interfaces. Abrupt
variations in the input data are poorly represented by truncated linear modal bases, leading to non physical oscillations and a loss of physical fidelity in reconstructed fields. This limitation is particularly critical when reconstructing flow fields in multiphase flow simulations, where accurate predictions of
interface dynamics are vital.

This work presents a physics-informed reduced-order model for multiphase flows across geometries parameterised by a scalar quantity d. High-fidelity simulations corresponding to distinct geometrical configurations are mapped onto a common reference domain, enabling the extraction of a shared modal basis.
These modes capture flow features that are common across the dataset and form the foundation for reconstructing flow fields corresponding to previously unseen parameter values. The role of training data distribution is also investigated, revealing that the ROM operates predominantly as a local interpolator
in the parameter space. In particular, predictive accuracy is found to depend strongly on the proximity of training configurations to the target case, suggesting a trade-off between local accuracy and global robustness.

To address the limitations of conventional POD, the solution fields are decomposed into smooth and discontinuous components. For the volume fraction, the interface is explicitly identified using the α = 0.5 contour and regressed across geometries using a Gaussian Process Regression (GPR) model. The full field is subsequently reconstructed from the predicted interface geometry. For the pressure field, the discontinuous capillary jump is analytically modelled using the Young-Laplace relation, ∆p = σκ, where κ represents the curvature of the interface profile. POD is then applied exclusively to the smooth pressure field, after which the discontinuity is reintroduced using the predicted interface curvature.

The performance of this adapted approach is evaluated against a baseline approach involving the direct application of POD to the raw, discontinuous data. Performance is assessed in terms of standard L²-norm based measures such as the root mean squared error, as well as newly defined discontinuity aware error metrics, including interface sharpness, volume fraction phase purity, and localised gradient and laplacian metrics for the pressure. While the direct POD yields lower global RMSE values for the volume fraction, it produces oscillatory and smeared interface representations. In contrast, the adapted approach delivers physically consistent reconstructions, yielding more accurate predictions of meniscus characteristics. For the pressure field, the adapted approach demonstrates superior performance across all metrics within the practical operating range of reduced-order models.

A comparative analysis further shows that removing discontinuities prior to applying POD leads to a more compact and efficient modal basis, with improved energy capture. The results demonstrate that incorporating physical insight into the reduced-order modelling of discontinuous flows significantly enhances both accuracy and computational efficiency. ... Read more

In nonlinear convection-dominated problems, finite-dimensional approximations cannot represent all modes generated by the solution, even though these unresolved modes can significantly affect the represented solution. This can lead to a pile-up of kinetic energy close to the approximation's effective cutoff wavenumber.
In this thesis, a closure method for the approximate Green's-function-based Variational Multiscale Method (VMS) is proposed. It is applied to the inviscid one-dimensional Burgers' equation for stationary and moving shock cases. A four-scale approach is used in which the effect of the fourth scale is modelled through a viscous term acting on the third scale only. The viscosity coefficient in this finest resolved scale is then reformulated in terms of a more physically motivated strain-rate energy target based on the second scale's strain-rate energy. This target could, in principle, be derived from spectral arguments while taking into account the scales' effective bandwidths.
This results in a clear reduction in the pile-up of kinetic energy close to the discretisation's effective wavenumber cutoff, while the model turns off when the solution lies in the coarse space. Besides this, hp-convergence and fine-scale refinement sensitivity are analysed. While overall solution hp-convergence was shown, projection errors, especially for stationary shock cases, did not show clear convergence.
Further research is required to determine the dependence of this strain-rate energy target on the discretisation and the modelling error. ... Read more

In recent years, data production in the field of fluid dynamics has increased significantly with the development of computational power and experimental measurement techniques. To improve the understanding of complex large-scale dynamical problems at feasible computational costs, the need for Reduced Order Models (ROMs) able to provide low-dimensional representations of the full-order dynamics, has never been higher. This has steered research in turbulence, both for interpretation and control, towards data-driven methods. A recent example is the Goal-Oriented Reduced Order Modeling (GOROM) framework. This modal composition approach bridges the gap between purely data-driven methods (i.e. POD) and purely operator-based modeling.

From a methodology perspective, the GOROM framework’s key feature is the optimization of the modes via an unconstrained Lagrangian minimization problem. Modes are optimal for a user defined quantity of interest, in a Galerkin projection setting, whilst introducing a weakly enforced partial differential equation (PDE) constraint to ensure dynamic consistency in low-order representations. A segregated approach solves State, Adjoint, and Gradient equations iteratively within a Trust-Region Inexact Newton-Conjugate Gradient algorithm. The resulting modes are thus optimal in Galerkin-based ROM setting for a user specified quantity while they remain dynamically consistent.

This thesis investigates the applicability of the novel Goal-Oriented Reduced Order Modeling (GOROM) framework for analyzing and modeling wall-bounded flows, further developing it and identifying areas of improvement in the process.

The framework is tested on two primary systems:
1. A 1D forced Burgers equation surrogate for fully Turbulent Channel Flow (TCF): This provides an analogy to turbulent flows in an LES-like truncated setting, whilst retaining a low enough dimensionality to simplify interpretation of the results. Furthermore, this simplified case is perfect for the introduction of a novel simultaneous SGS closure model optimization approach, aimed at understanding the coupling between the mode-optimization framework and the closure model in a highly truncated setting where stabilization is required.

2. An optimally perturbed, 3D Incompressible Navier-Stokes-governed transitional Poiseuille flow case undergoing transient growth: a complex 3D problem governed by non-normal dynamics, where classic modal decomposition techniques tend to fail, allows for the testing of the GOROM framework in a setting where it can outperform POD-based ROMs, and where it may provide significant phenomenological insights. Moreover, it provides a real-life (complex) setting to further analyze the intricacies of the framework.

A major finding is the importance of the PDE constraint for the dynamical coherence of the mode shapes. This allows GOROM modes to successfully preserve the phase accuracy required to represent the complex dynamics of turbulent motions in the 1D surrogate model case and of non-normal interactions and transient growth mechanisms such as the Lift-up and Orr mechanisms in 3D perturbed Poiseuille flow. ... Read more

Stratospheric aerosol injection (SAI) has been proposed as a temporary measure to mitigate global warming by increasing Earth’s albedo through the injection of aerosol particles into the stratosphere. However, the effectiveness and side effects of SAI depend strongly on particle size distribution and residence time, both of which are heavily influenced by coagulation processes in the near-field plume. These processes are complex and remain highly uncertain, particularly under the intense turbulence conditions present immediately after injection.

Traditional aerosol models typically represent coagulation using additive collision kernels based mainly on Brownian motion. While suitable for low-turbulence conditions, these models often neglect turbulence-induced relative motion and particle inertia, which can significantly enhance collision rates in highly turbulent environments. As a result, conventional approaches may underestimate coagulation in the near-field region of SAI plumes.

This thesis addresses these limitations by implementing a unified collision kernel that consistently couples Brownian motion with turbulence-driven inertial effects. The kernel is integrated into a three-dimensional sectional aerosol model, allowing coagulation to be represented seamlessly across free-molecular, transition, and continuum regimes without requiring interpolation. The study evaluates the impact of this new kernel through grid refinement tests, turbulence-intensity sensitivity analyses, and comparisons of flow-field fidelity using Reynolds-Averaged Navier–Stokes (RANS) and Large Eddy Simulation (LES) approaches.

Results show that the unified kernel predicts earlier and more intense coagulation within the plume core compared to traditional models. Turbulence substantially increases collision rates, leading to faster particle growth. The quality of the underlying flow field is also critical: time-averaged LES captures turbulent structures more realistically than RANS, resulting in a more accurate downstream decay of coagulation rates. Consequently, high coagulation rates persist closer to the injection region when using LES, producing a broader particle size distribution than predicted with RANS-based simulations.

The study further examines the role of aerosol material properties. Comparisons between liquid sulfate aerosols and solid calcite particles reveal that calcite exhibits higher coagulation efficiency due to greater sensitivity to turbulent fluctuations. This leads to faster particle growth and shorter residence times for calcite compared to sulfate.

Overall, the findings demonstrate that neglecting turbulence-driven coagulation can lead to substantial inaccuracies in predicting the early evolution of injected aerosols and, therefore, in assessing the climatic effectiveness of SAI. Accurate near-field modelling must account for coupled Brownian and turbulent effects to improve reliability in climate intervention assessments.
... Read more

This thesis investigates how surface deformations can delay the transition from laminar to turbulent flow in aircraft boundary layers, thereby reducing drag and improving fuel efficiency. This is important because aviation contributes significantly to greenhouse gas emissions, and improving the lift-to-drag ratio is one of the most effective ways to make aircraft more sustainable.

The focus is on the early stages of boundary layer transition, particularly the growth of stationary crossflow instabilities in swept-wing boundary layers. These instabilities play a major role in triggering turbulence on swept wings. While experiments and Direct Numerical Simulations (DNS) can study this process accurately, they are computationally expensive. Therefore, this thesis uses flow stability analysis methods, which are much faster.

Three stability-analysis approaches are discussed. The classical Orr–Sommerfeld method is computationally efficient but assumes a locally parallel flow and only works for small perturbations. The Parabolized Stability Equations (PSE) improve on this by including streamwise development and nonlinear interactions, but they are only valid for slowly varying flows. The Harmonic Navier–Stokes (HNS) equations retain all streamwise derivatives and can therefore handle strongly non-parallel flows, although at a higher computational cost than PSE.

To exploit these advantages, a new computational framework, the Delft Harmonic Navier–Stokes Solver (DeHNSSo), is developed. DeHNSSo can analyse the effect of both smooth and sharp surface deformations, such as humps and steps, on boundary layer instabilities. The solver uses a Fourier-based representation of perturbations, spectral discretisation in the wall-normal direction, and finite differences in the streamwise direction. Nonlinear interactions between perturbation modes are included iteratively.

The framework is validated using several standard instability cases, including Tollmien–Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept-wing boundary layer. In all cases, DeHNSSo agrees closely with DNS and with other stability methods such as PSE and Adaptive Harmonic Linearised Navier–Stokes.

The main application of the solver is to investigate the effect of a shallow smooth surface hump on crossflow instabilities. The hump creates a local region of reversed crossflow without causing flow separation. Away from the hump, the boundary layer quickly returns to its original state.

For small perturbation amplitudes, the hump reduces the growth of crossflow instabilities over a large downstream region. Although there is some local destabilisation near the hump, the overall effect is stabilising because the perturbation shape is altered in a way that weakens the lift-up mechanism responsible for instability growth.

At larger perturbation amplitudes, however, the hump becomes less effective. A second unstable mode appears near the wall and transfers energy to the main instability. This can lead to locally increased disturbance amplitudes and earlier quasi-saturation, potentially accelerating transition.

The results suggest that smooth surface humps are most effective when the incoming disturbances are still small and approximately linear. Under those conditions, they can delay transition and reduce drag. The study therefore provides a promising basis for using optimised surface humps on aircraft wings. Future work could further optimise hump shape and arrangement and improve computational efficiency by coupling the HNS approach with faster methods such as nonlinear PSE.
... Read more

Modern aircraft design requires efficient prediction of complex, time-dependent flow phenomena. While high-fidelity datasets offer accuracy, they come with high computational costs. To address this, NATO’s research group AVT-351 is investigating reduced-order models (ROMs). Proper Orthogonal Decomposition (POD) is widely used in ROMs, but truncated POD bases struggle with sharp gradients and shocks. The enriched POD-LSTM neural network ROM
(ePOD-LSTM-ROM) has demonstrated improved accuracy for airfoil pressure distributions
and was extended to full wing surfaces. This approach was effective for sharp discontinuities
moving significantly in time. For realistic cases, such as the ONERA M6 wing with multiple shocks, domain decomposition allowed for multiple enrichments in separate subdomains,
but at the cost of rapidly increasing degrees of freedom. To improve LSTM-NN accuracy,
enrichments were optimized using mutual information, filtering, and error trade-offs, reducing the total ePOD-LSTM-ROM error by 23.8%. Finally, a goal-oriented reduced-order model
(GOROM) was developed to alter POD modes optimized for a specific goal function, improving
shock accuracy by 5.0% with minor losses elsewhere. ... 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

Among error estimation methods, adjoint-based approaches are considered the most accurate but have the highest computational cost. For unsteady non-linear problems such as the Navier-Stokes equations, substantial storage requirements arise, as the full primal solution must be stored to solve the adjoint problem. As a result, careful management of storage resources is essential. The purpose of this research was twofold: to develop a surrogate model for the primal solution and to compute an accurate adjoint-based error estimate with the developed surrogate primal.
This study compared three methodologies to create a surrogate model of the primal solution while reducing the storage requirements of unsteady adjoint-based error estimation: a Convolutional AutoEncoder (CAE), an Echo State Network (ESN) and a combination of the first two, referred to as CAE-ESN. A benchmark Proper Orthogonal Decomposition (POD) served as a baseline for comparison with the deep learning techniques. Three numerical test cases were analyzed, where the finite element method was used for spatial discretization and implemented with the \texttt{FEniCS} computational framework. The first test case involved a manufactured solution to verify the implemented solver and methodologies. The remaining test cases used a turbulent channel flow dataset to force the unsteady viscous Burgers' equations in 1D (wall-normal component) and 2D (spanwise and wall-normal components).
For the manufactured solution, the ESN and POD outperformed the remaining approaches for the lowest and highest spatial resolutions, respectively. The success of the ESN was linked to its training being a linear regression problem. As established in previous studies, the smooth nature of the solution rendered the POD optimal. For the 1D case, the CAE was optimal, particularly for lower spatial resolutions. This method offered equivalent compression ratios to the POD while being more efficient in terms of computational cost and accuracy. In contrast, the ESN-based methods failed to accurately capture the error estimate, as they were not able to accurately compute the primal residual. However, these methods offered a higher compression than other approaches, along with a decrease in accuracy. Moreover, the error indicators produced by the ESN-based methods continued to effectively pinpoint the elements required for mesh adaptation. In the 2D case, only the POD and CAE were investigated. The ESN was excluded due to the high-dimensional nature of the test case. The CAE-ESN was not applied because of the limited time interval, which provided insufficient data for training. The CAE again proved optimal due to its efficiency and higher compression capabilities than the POD. While both methods provided accurate error estimates and indicator fields, the CAE outperformed the POD due to its superior compression. The CAE was also able to compute the adjoint solution and primal residual more accurately than the POD for most spatial resolutions. This research highlighted the potential for the CAE to outperform more conventional methods, such as POD. ... Read more

Adjoint-based error estimation is among the most accurate methodologies for quantifying numerical errors, forming the foundation of Adaptive Mesh Refinement (AMR) strategies in turbulent flow simulations, particularly within Large Eddy Simulations (LES). However, in unsteady problems, the need to access fully time-resolved primal fields during the backward-in-time adjoint computation, along with the corresponding injected residuals, leads to prohibitive storage requirements. This thesis addresses this bottleneck by developing and assessing data-driven compression frameworks for both injected-residual and primal fields, aiming to substantially reduce storage demands while preserving the accuracy and consistency of adjoint-based error estimation.

Two machine learning–based surrogate modeling techniques, the Convolutional AutoEncoder (CAE) and the Echo State Network (ESN), were investigated to assess their ability to capture the spatial and temporal dynamics of the compressed fields, respectively. Their performance was compared against Proper Orthogonal Decomposition (POD), which served as the benchmark method. The computational framework was implemented in OpenFOAM for both primal and adjoint solvers, while the compression and reconstruction models were trained and evaluated in Python using PyTorch and ReservoirPy packages. Two test cases were considered: a smooth Manufactured Solution (MMS) of the 1D viscous Burgers’ equation, used to verify the framework and ensure consistency under controlled conditions, and a DNS-forced 1D viscous Burgers’ problem derived from Turbulent Channel Flow (TCF) data, used to evaluate the method’s robustness in representing complex, unsteady turbulent flow dynamics.

The MMS results confirmed that the framework can accurately reconstruct both the primal and injected-residual fields without altering the adjoint-based output error estimation. When the compressed residuals were used in the adjoint computation, the recovered error indicators were in close agreement with the fully resolved reference, verifying that the compression and reconstruction stages preserve consistency. Minor differences were observed only when surrogate primals were employed for adjoint solution computation, yet these deviations remained negligible, confirming the robustness of the overall formulation before applying it to more complex unsteady cases.

For the DNS-forced 1D viscous Burgers problem, representing the wall-normal velocity fluctuations of a TCF, the framework was evaluated under more realistic, nonlinear, and temporally evolving conditions. In this case, the ESN demonstrated the highest capability in reconstructing temporally varying residuals and maintaining consistent adjoint sensitivities, benefiting from its recurrent dynamics. Across all refinement levels, the time-averaged adjoint-based error indicators remained closely aligned with the reference, even at compression ratios exceeding two orders of magnitude, confirming the framework’s ability to drastically reduce storage without compromising estimation accuracy. The CAE effectively captured spatial features but showed mild instability and localized artifacts when surrogate primal reconstructions were employed for adjoint computation. While the CAE provided better reconstruction of mean values, the ESN more accurately reproduced the second-moment statistics, leading to improved representation of unsteady dynamics. The POD, used as a benchmark, yielded stable reconstructions but was limited in representing nonlinear temporal behavior and performed notably worse than the ESN when compared at equivalent compression ratios.

Overall, the results demonstrate that the proposed compression framework can reliably reduce storage demands in unsteady adjoint-based error estimation while preserving high fidelity in the output error indicators. The findings emphasize that while POD remains the most stable baseline, CAE provides strong nonlinear spatial encoding and ESN achieves the most accurate temporal reconstruction. The framework establishes a practical and scalable foundation for applying adjoint-based AMR in LES and other turbulent flow analyses where unsteady data storage remains a critical limitation. ... Read more

Adaptive meshing is important to many practical problems, though it presents many challenges. This work focuses primarily on mesh refinement within the context of Mimetic Spectral Element Method (MSEM). This is a method that approaches formulating the discrete system of equations in the style of Finite Element Method, but through application of differential geometry and exterior calculus. This results in the method satisfying the inf-sup condition by construction, allowing it to correctly solve any saddle point problem, such as the incompressible Navier-Stokes equations. This work also examines the applicability of using results given by Variational Multi-Scale (VMS) theory as an error estimator. The solver that was written based on the theory in this work and then later used to produce the results can be found at the remote repository under the GNU Public License 3.

MSEM in this work is formulated in a hybridized way, where each element is considered to have separate degrees of freedom, with continuity being enforced through Lagrange multipliers. This allows for neighboring elements to have different polynomial orders and even levels of refinement. Local refinement is achieved using raising polynomial order of elements (p-refinement) and hierarchically dividing elements (h-refinement). To this end, the theory and procedures to implement combined hp-refinement within the existing MSEM framework are formulated and verified.

From there, a refinement criterion for deciding between h-refinement or p-refinement for each element involved in a refinement sweep is introduced and validated. This refinement criterion is based on estimating the change in the L² error norm of the solution and is computed by using the Legendre coefficients of the error estimate for each element. This is in contrast to many other similar criteria present in literature, which consider only the solution and the current mesh state as the basis of the decision to apply either h-refinement or p-refinement.

To obtain an error estimate to use for the refinement criterion, several different estimators are proposed, notably including VMS. The main appeal of using VMS is that a good error estimate is offered by the unresolved scales, which are obtained by the method normally. These error estimates were tested on steady, two-dimensional test problems of increasing complexity, from mixed formulation Poisson equation, to linear advection-diffusion, and lastly incompressible Navier-Stokes equations. Based on these tests, VMS appears second only to knowing the exact error, though the computational cost associated with each refinement criterion was not compared. ... Read more

Goal-oriented reduced-order models (GOROM) have the potential to represent complex dynamics in an interpretable way. In the GOROM procedure, modes are found which when used in a Galerkin projection of the governing equations, return the desired goal function with minimum error. Determining such models leads to a high-dimensional optimisation problem, which may have multiple extrema, making purely local optimisation inapplicable.
In this thesis, a global optimisation algorithm was derived for GOROM. The optimisation response surface was approximated with a surrogate model, built using the Kriging technique. In each iteration, the surrogate model was improved by including the objective function of the three points with the highest expected improvements. This iterative process was stopped when the expected improvement fell below a threshold.
First, the performance of the algorithm was confirmed for a simple non-GOROM optimisation. Then the algorithm was applied to GOROM optimisation for the forced Burgers equation, initially using a complex forcing function giving multiple minima and later to a forcing function derived from a DNS of a turbulent channel flow. The initial forcing required on the order of tens of degrees of freedom to be optimised. The corresponding response surface proved similar to that of the simple problems, showing these optimisations to be tractable.
For the DNS forcing, hundreds of coefficients needed to be optimised. Using the developed algorithm a better optimal was found than with the local algorithm, although substantially more computational resources would be required to confirm global optimality. ... Read more

This thesis presents a data-driven closure model for the variational multiscale method. The model is trained and tested in the context of a turbulent channel flow with Reτ = 180. Focus is on predicting the closure terms of the momentum equations. The model is trained using norms which only take into consideration local flow accuracy. For this reason long term stability is not addressed explicitly and is not the focus of this work. A convolutional neural network is chosen for its ability to efficiently make use of local spatial and temporal data. Different architectures and inputs are considered for the model. The performance is analyzed in an a priori sense by plotting correlations between the predicted closure terms and the exact closure terms. Using integrated forms of the momentum equation as input produces the highest a priori correlations. Taking multiple time steps as input also proves to be important. The model is then integrated into the simulation of a turbulent channel flow. The closure terms produced by the model are more accurate than those of a modern algebraic model. The flow field produced by the model is thus closer to that of the DNS than the flow field of the algebraic model. ... Read more

An approach to data-driven closure modelling in the framework of variational multiscale method for large eddy simulation is presented. A turbulent channel flow at the friction Reynolds number of 180 is used as a case study. Challenges in the modelling linked to the continuity closure terms force the scope of the study to be narrowed to only momentum Navier-Stokes equations. By using integrated forms of the resolved flow solution to train a multi-layer perceptron closure model, high a priori correlations with true closure terms can be achieved. Validation of the data-driven closure models a posteriori shows that the models are suitable for the computation of the velocity field, but fail to obtain a physical pressure solution. Accumulation of the model error leads to a decay of the kinetic energy of the mean flow, while the resolved turbulent fluctuations become excessively energized. Novel techniques such as data augmentation, mini-batch sum loss and filtering of the closure terms are presented and evaluated via large eddy simulations. ... Read more

Maneuvering or Stability and Control (S&C) characteristics of an aircraft constitute the most challenging and expensive phase of its design process. The S&C design phase extends into the development process, sometimes leading to unexpected aerodynamic issues. Thus, identifying aerodynamic issues and their consequences, e.g. transonic buffet, in the early stages of development, is crucial. Recent advancements in Computational Fluid Dynamics (CFD) have expanded its utilization. However, despite improvements in CFD, the comprehensive representation of dynamic effects for all possible maneuvers is still unfeasible due to the high computational expense.

Reduced Order Models (ROMs) have been combined with CFD data to predict an aircraft's dynamics in all possible maneuvers. ROMs enable the efficient utilization of high-fidelity CFD data, providing valuable insights into flight dynamic effects. This thesis project took place at the Netherlands Aerospace Center (NLR). The NLR in cooperation with TUDelft, has developed a ROM method for predicting unsteady aerodynamic loads of air vehicles. The current ROM approach combines the Proper Orthogonal Decomposition (POD) of pressure distribution with a Long Short-Term Memory (LSTM) type Neural Network (NN). So far, the POD-LSTM ROM method predicts the pressure distribution well in the incompressible flow regime. However, the increased number of spatial POD modes required to accurately represent the shock discontinuities in a pressure distribution poses challenges to POD-LSTM ROM. This leads to high computational costs, rendering the application of the POD-LSTM ROM in transonic flows impractical. Therefore, this thesis aims to set the foundation for expanding the POD-LSTM ROM for predicting the pressure distribution over sections of the DLR-F22 model in transonic conditions.

This research introduced a novel approach to address the increased number of spatial POD modes needed to approximate shock discontinuities in transonic flows. The enriched Proper Orthogonal Decomposition (ePOD) method introduces an enrichment basis into the standard truncated POD basis. The enrichment basis explicitly accounts for pressure discontinuities caused by shock waves, allowing the standard basis to focus on representing the remaining pressure distribution. The results confirm that the ePOD reduces the DoF required to approximate pressure distribution in transonic flows.

An LSTM neural network was utilized to forecast the time-dependent coefficients and parameters of the enriched reduced-order basis in unseen flow conditions. The results also showed that the ePOD reduced the complexity of the time-variant parameters of the reduced-order basis compared to the standard POD with the same number of degrees of freedom (DoF), facilitating more efficient training of the neural network. ... Read more

Wings with leading-edge (LE) tubercles have gained increasing attention over the past decade. Despite their impressive aerodynamic performance, the underlying flow control mechanisms of tubercles remain controversial. In this thesis, both experimental and theoretical approaches are employed to investigate the flow patterns of a tubercled wing at pre-stall and post-stall angles of attack (AoAs).

In the experimental study, 2D Particle Image Velocimetry (PIV) was used to measure flow patterns at cross-flow planes along the chord. At a pre-stall AoA, high-vorticity regions generated by the tubercles appear in an alternating pattern near the LE. A quantitative comparison was conducted to examine the similarities between a tubercle and a delta wing. The results show that tubercles cannot be regarded as small delta wings in terms of vortex generation. The leading-edge vortex (LEV) sheets are convected downstream, where they interact with laminar separation bubbles (LSBs), creating complex flow patterns in the downstream regions. At a post-stall AoA, stall cells (SCs) appear along the span, with their formation dependent on both Reynolds number (Re) and tubercle amplitude. However, the spacing of SCs is relatively independent of AoA, Re, and amplitude, consistently ranging between 5 to 7 tubercle wavelengths.

In the theoretical study, the lifting line theory (LLT) approach was first used to predict the LEV strength but proved ineffective due to the absence of thickness effects. A subsequent analysis using the panel method in xflr5 showed that the Kutta condition should also be applied to the leading edge (LE) rather than only to the trailing edge (TE). Crow’s model was adapted by taking LEVs into consideration. However, a global description of the instability was not obtained due to difficulties in representing LEVs and related mathematical challenges.

This thesis contributes to a further understanding of the tubercle’s role in flow control. The LEVs generated by the tubercles are identified as key factors influencing flow evolution, yet these effects are not captured by LLT-based models or a conventional panel method. Future reduced-order models (ROMs) should account for the influence of LEVs to provide accurate representations of tubercled wing flow dynamics. ... 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

This thesis presents an energy-conservative data-driven approach in modelling the closure terms of the Navier-Stokes equations casted through the Variational Multiscale (VMS) framework. For context, the VMS framework is applied in designing stabilised finite element methods for multiscale phenomena in which stability is not guaranteed. Under standard applications of this framework with appropriate cubature rules and iteration routines, the only error source is model related in the assumptions for the fine scales, thereby influencing the closure terms of the coarse-scale equations. This issue can be rectified by predicting the perfect closure terms with the application of neural networks. To maintain energy conservation (as a generality, not as a guarantee), a constrained objective function was formulated, which was then translated into an unconstrained problem suitable for present day neural network training tools via the Augmented Lagrangian method. The features of the eight neural networks used were extracted from two spatiotemporal stencils. A priori evaluation of the fine-scale model demonstrated a tendency to overpredict closure terms for the Re_tau = 180 turbulent channel flow test case, yielding desired energy dissipative behaviour. This was true for both regimes of the flow in which the closure terms yield energy gain and energy loss via backscatter and cascade respectively. Unverified a posteriori simulation of the test case with the model displayed excessively high energy dissipation, in addition to high frequency oscillations, which were undampened by the time marching method used. ... Read more

This research investigates the formation, growth, and size distribution of H2SO4 aerosols in aircraft engine plumes. It aims to enhance calculations by incorporating spatial variation and turbulence modeling. The study develops a toolchain using AER 3-D software, sets up a CFD model of an engine wake, and couples it with microphysics and advection schemes. The toolchain compares the AER 3-D microphysics model with state-of-the-art box models, showing its effectiveness in handling high H2SO4 concentrations and producing similar size distribution results. Using ANSYS, an axisymmetric engine wake field is calculated, revealing that condensation and nucleation occur in the injection area and wake boundary layer, while coagulation primarily occurs inside the wake. Turbulence influences aerosol diffusion and growth, resulting in size variations with distance from the wake center. The research emphasizes the importance of incorporating spatial variation and turbulence modeling for accurate aerosol predictions. ... Read more