A. Palha

Master thesis (4)

4 records found

Finite element method for wind turbine wake modelling

Master thesis (2024) - J. Paddeu (author) , Delphine De Tavernier (mentor) , A.R. Meyer Forsting (mentor) , N.G.W. Warncke (mentor) , Simon J. Watson (graduation committee member) , A. Schroll (graduation committee member) , A. Palha da Silva Clerigo (graduation committee member)

As the demand for renewable energy rises, the impact of wake effects on wind farm performance and efficiency has become a primary focus for both industry professionals and academic researchers. However, to effectively mitigate these wake effects and improve wind farm efficiency, there is a critical need for an accurate wake representation. This requires well-established medium-fidelity wake models, which are currently lacking. To achieve this goal, it is essential to develop a wake model capable of accurately generating the axial wind profile within the wake. This model is crucial for precisely assessing the loads on downstream wind turbines and evaluating their energy production potential. Moreover, it provides valuable insights into optimising methods to mitigate wake effects and improve overall wind farm efficiency. For the model to be truly beneficial to the research effort and industry applications, it must demonstrate an increased accuracy with a limited increase in computational cost.

This thesis delves into medium-fidelity wake models to enhance the precision of calculations and predictions concerning wind turbine fatigue and aeroelastic loading. It specifically examines the use of a finite element method with Hermite interpolation basis functions as a numerical approach to solve the Ainslie wake model and achieve these objectives. Additionally, the study employs a downstream marching scheme to solve the partial differential equations and introduces the Newton-Raphson method to address non-linearities within the model. The potential benefit of using a finite element method lies in its potential for improved stability compared to the finite volume method, which shows satisfactory performance but limited stability, and superior performance compared to the spectral method, which has shown to exhibit poor conservative properties and instabilities for higher mode numbers. Furthermore, this thesis aims to bridge the gap between modelling and reality by implementing a pressure Poisson equation. Specifically, the focus has been on implementing the pressure component resulting from the forcing in the pressure Poisson equation.

Implementing Hermite interpolation basis functions posed several challenges. Increasing their order caused significant ill-conditioning and greater sensitivity to mesh quality, reducing their effectiveness for this application. Additionally, higher-order Hermite functions proved ineffective for solving the continuity equation due to the presence of odd basis functions. In contrast, piecewise linear basis functions have shown to be compatible with the continuity equation and were effective in solving the diffusion term. Despite this, stability issues emerged, likely due to the Ladyzhenskaya-Babuka-Brezzi condition not being met.

The implementation of forcing in the finite volume Ainslie wake model demonstrated mass and momentum conservation, and the axial velocity has been shown to align with literature. However, the dependency of the vortex strength on the domain size complicates the determination of a correct converged value for $\Gamma$. Furthermore, comparisons of the velocity field with the IEC 61400 standard implementation without the inclusion of a forcing term revealed that the forcing term implementation more accurately reflects the real flow dynamics, capturing both the blockage effect and the edge force effect.

Overall, this thesis demonstrates that the investigated method does not provide a stable solver for the Ainslie wake model. However, this does not rule out the potential of the finite element method for solving the Ainslie wake model. With adjustments to the basis functions and the application of Taylor-Hood elements, this method is still anticipated to deliver satisfactory results compared to existing methods. Furthermore, this thesis has laid the foundation for the inclusion of a Poisson solver. Specifically, it outlines the integration of pressure gradient resulting from actuator disk forcing and presents various formulations for the wind turbine forcing. However, further research is needed to address the calculation of the pressure induced by the velocity field.

Comparing MSEM and conventional CFD for incompressible 2D Navier-Stokes

Master thesis (2024) - J.M. Plutecki (author) , M. Gerritsma (mentor) , SJ Hulshoff (graduation committee member) , A. Palha da Silva Clerigo (graduation committee member) , Y. Zhang (graduation committee member)

Computational Fluid Dynamics (CFD) has become important in designing aerospace and transport products. It allows for predicting the key flow properties. The CFD techniques were developed in the late 20th century. Mimetic schemes are relatively new in the realm of CFD, although they have been applied to a wide range of Partial Differential equations (PDE) problems. Some attempts have been made to draw comparisons between Finite Volume Methods and Finite Element Methods along with other conventional CFD techniques but not much attention has been paid towards the comparison of the Mimetic Spectral Element Method (MSEM) with these most conventional techniques. Further analysis is required in order to make judgments regarding the effectiveness, stability and accuracy of MSEM against conventional methods.

The purpose of this report is to present the findings and outcomes of 2D incompressible Navier-Stokes flow simulation techniques. Then, the comparison between the MSEM and conventional CFD will be made. The mass, energy, enstrophy and vorticity conserving (MEEVC) was chosen as a MSEM solver. MEEVC employs mixed finite element discretization which allows for the preservation of mass, energy, enstrophy and vorticity. The phyem package for Python created by Yi Zhang et al. is used to carry out MEEVC. Because a non-linear system needs to be solved, a Newton linearization is implemented. ANSYS Fluent represents the conventional CFD due to its built-in 2D solver. A coupled solver is selected as it solves for both pressure and velocity simultaneously compared to usually used segregated methods.

Four test cases were conducted using both software, and the results were compared with the literature for validation. They were selected in order to capture different flow conditions, including internal and external, steady and unsteady flows. The study validates both methods against the literature and analytical solutions. Then, a cross-case study is performed in order to answer the research questions.

Development and Application of bundle-valued forms in hybrid mimetic spectral element method

Application to Linear elasticity

Master thesis (2023) - R.K. Sharma (author) , Marc I. Gerritsma (mentor) , Artur Palha da Silva Clérigo (graduation committee member) , Bo Yang Chen (graduation committee member) , Alexander H. van Zuijlen (graduation committee member)

One of the novel methodologies in computational physics research is to use mimetic discretisation techniques. Among these, the mimetic spectral element method holds special promise as it not only has the benefits of mimetic methods but also the additional benefit of higher-order discretisations using higher polynomial degrees. These methods are aided by the development of algebraic dual polynomials, resulting in a sparser system for better computational efficiency. This combination was used to develop a formulation that would result in topological relations for equilibrium of forces as well as the symmetry of the stress tensor for linear elasticity as well as the first steps for Stokes flows in an orthogonal domain. As a result, this study was extended to look at how a modified formulation would behave for an unsteady linear elastic solid, with the intention to extend this method to Fluid-Structure Interaction cases. However, the choice of both primal and nodal basis functions makes it impossible to undertake this challenge, demanding a rethink in strategy towards looking at linear elastic solids when the physical domain is not orthogonal. With the use of bundle-valued forms to represent physical quantities, a new hybrid formulation is developed where the equivalent of the physical problem is computed on a reference domain, which is orthogonal and thus can utilise the spectral bases defined before. The physical problem is defined on a skewed domain, where partial transformation of components results in a formulation that can conserve linear momentum point-wise, but not conservation of angular momentum, although angular momentum does converge on refinement of polynomial degree and mesh parameters. A change in bases with partial transformation aiming to make angular momentum conservation topological is not fruitful, although the value of the error decreases in the process. The final attempt is through full transformation, which results in a formulation with an inherent error in the formulation, owing to erroneous assumptions.

A mimetic dual-field discretization with exact conservation properties for the incompressible Navier-Stokes equations with periodic or Dirichlet boundary conditions

Master thesis (2023) - W.N. Nederpel (author) , Deepesh Toshniwal (mentor) , A Palha (graduation committee member)

The Navier-Stokes equations govern the flow of viscous fluids such as air or water. Since no general solution is known, computer simulations are used to obtain approximate solutions. As computers are unable to handle continuous representations of fields, finite-dimensional projection of fields is required, resulting in a loss of information. A difficulty in constructing these discrete solutions, is to obtain discrete solutions that have the same conservation properties as continuous solutions. Other challenges include the efficient handling of the nonlinear term in the equations. In this thesis we aim to construct a numerical method whose solutions satisfy the conservation of mass, kinetic energy and helicity exactly. To achieve this, a structure preserving (mimetic) spectral element method is employed. Such methods aim to represent properties of the continuous problem exactly in the discrete problem. These methods have their theoretic roots in differential geometry, where there is a clear connection between fields and the geometric objects they are associated with. From this association we note that the fields in the Navier-Stokes equations show a dual nature, where each field can be associated to two types of geometric objects. In [1] this dual nature is used to construct a mass, kinetic energy and helicity conserving method that handles the nonlinear term efficiently by using a leapfrog scheme for problems on periodic domains. In this thesis this work is extended to handle Dirichlet boundary conditions. When considering periodic boundary conditions, the constructed numerical method exhibits the same conservation properties of energy and helicity as the Navier-Stokes equations. When considering Dirichlet boundary conditions, some additional contributions to the dissipation rates of energy and helicity are noted in the cases of inflow into the domain and nonzero normal vorticity at the boundary respectively. These additional contributions cannot be meaningfully quantified within the employed approach, and are a result of strongly enforced boundary conditions on velocity. Furthermore, the numerical method constructs a pointwise divergence free velocity field at every other time step. The mimetic spectral element method allows for high order spatial approximation of fields. Optimal spatial convergence orders are observed for all fields. Temporal integration was done via a combination of the trapezoidal and midpoint rule and the expected second order temporal convergence is observed.