Numerical study of wind turbine wakes using modal-decomposition techniques and stability analysis

This thesis is focused on numerical analysis of wind turbines wakes. The flow over wind turbines is simulated performing Large Eddy Simulations (LES), where the rotor blades are modeled using the Actuator Line Method, whereas the Immersed Boundary Method is employed for tower and nacelle. The effect of tower and nacelle on wake dynamics is investigated by means of Proper Orthogonal Decomposition (POD) of numerical velocity data produced by two LES of a model wind turbine: one accounts only for the blades effect; the other includes also tower and nacelle. The turbine operates at Reynolds number Re = 6.3 × 10^5 and tip-speed ratio λ = 3. The two simulations are analysed and compared in terms of mean flow fields and POD modes that mainly characterize the wake dynamics. In the rotor-only case, the most energetic modes in the near wake are composed of high-frequency tip and root vortices, whereas in the far wake, low-frequency modes, mostly located in the wake shear layer region, are found. When tower and nacelle are included, low-frequency POD modes are present already in the near wake, linked to the von Karman vortices shed by the tower. These modes interact non linearly with the tip vortices in the far wake, generating new low-frequency POD modes, some of them lying in the frequency range of wake meandering. An analysis of the mean kinetic energy entrainment of each POD mode shows that tip vortices sustain the wake mean shear, whereas low-frequency modes contribute to wake recovery. This explains why tower and nacelle induce a faster wake recovery. The proper orthogonal decomposition, despite being able to isolate energetic flow structures in the wake, does not provide any physical information on their origin. In an attempt to determine the physical mechanisms responsible for the emergence of these flow structures, the numerical data obtained without tower and nacelle are further analyzed performing two-dimensional modal and non-modal stability analysis of the turbulent mean flow developing downstream of a wind turbine rotor. Linear stability and optimal forcing analyses have been carried out in different cross-sections sufficiently far from the rotor, where nonparallel effects are rather weak. The frequency content and spatial structure of the most amplified perturbations are compared with that of the most energetic coherent structures recovered by POD analysis. Results show that most unstable modes computed relatively close to the rotor resemble large-scale oscillations isolated by the POD. Moving downstream, this matching is no longer verified; however, restricting the stability analysis to waves having streamwise wavenumber consistent with that of the POD analysis, we find three slightly stable eigenmodes bearing a strong resemblance with the most energetic POD modes. The analyses described above are based on a model wind turbine; however utility-scale wind turbines operate at far larger Reynolds number, of the order of 10^8 and higher tip-speed ratio. These differences can lead to a different wake dynamics. For this reason a reference utility-scale wind turbine (i.e. the NREL 5-MW) is simulated and analyzed using Proper Orthogonal Decomposition and Dynamic Mode Decomposition (DMD) in its sparsity promoting variant, which selects a limited subset of dynamically relevant modes. In contrast to the model wind turbine, the wake meanflow is, in this case, essentially aligned with the rotor axis and axisymmetric, suggesting a weaker impact of the tower. The coherent structures isolated by the two modal decomposition techniques are similar to those observed for the model turbine, but a weaker interaction of tower’s wake and tip vortices is confirmed and a faster breakdown of the latter is reported. Furthermore, POD and DMD of the flow field provide rather different results. Large-scale, low-frequency oscillations are not present among the most energetic POD modes. On the contrary, sparsity-promoting dynamic mode decomposition suggests that large-scale structures, developing far from the rotor, are relevant to the flow dynamics, despite their energetic content is not sufficiently high to overcome that of the tip vortices and their harmonics, which are among the first POD modes. This result demonstrates that while Proper Orthogonal Decomposition is efficient at identifying coherent structures, it may not be suitable for building a low-dimensional model of a wind turbine wake, while sparsity-promoting DMD can be a better choice.