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.