The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Karman vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Karman vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

The system of vortices created by the hub and tip vortices of rotors and propellers is composed of two subsystems of helical vortices that have different radii and pitches. A system of external and internal vortices can also be created by some blade devices proposed to destabilize the tip vortices of helicopters and a similar system was observed in an idealized simulation of two in-line wind turbines. The steady solution of these systems of vortices was recently described. However, their stability was not studied. The stability of a system of multiple helical vortices is studied in this work using two methods: a complex-step technique combined with a time-stepping algorithm used to linearize the Biot-Savart law and the vorticity transport equations; and an approximate analytical method. It was noted that the hub and tip vortices do not interact and their linear stability can be treated separately if the velocity field induced by one system is considered in the stability analysis of the other. Strong interaction between the vortices was observed for internal vortices closer to the external vortices, with an out-of-phase mechanism appearing as one of the main phenomena.

The hydrodynamic stability of a vortex system behind two in-line wind turbines operating at low tip-speed ratios is investigated using the actuator-line method in conjunction with the spectral-element flow solver Nek5000. To this end, a simplified setup with two identical wind turbine geometries rotating at the same tip-speed ratio is simulated and compared with a single turbine wake. Using the rotating frame of reference, a steady solution is obtained, which serves as a base state to study the growth mechanisms of induced perturbations to the system. It is shown that, already in the steady state, the tip vortices of the two turbines interact with each other, exhibiting the so-called overtaking phenomenon. Hereby, the tip vortices of the upstream turbine overtake those of the downstream turbine repeatedly. By applying targeted harmonic excitations at the upstream turbine's blade tips a variety of modes are excited and grow with downstream distance. Dynamic mode decomposition of this perturbed flow field showed that the unstable out-of-phase mode is dominant, both with and without the presence of the second turbine. The perturbations of the upstream turbine's helical vortex system led to the destabilization of the tip vortices shed by the downstream turbine. Two distinct mechanisms were observed: for certain frequencies the downstream turbine's vortices oscillate in phase with the vortex system of the upstream turbine while for other frequencies a clear out-of-phase behaviour is observed. Further, short-wave instabilities were shown to grow in the numerical simulations, similar to existing experimental studies [1].

Floating offshore wind turbines (FOWTs) are subjected to platform motion induced by wind and wave loads. The oscillatory movement trigger vortex instabilities, modifying the wake structure and influencing the flow reaching downstream wind turbines. In this work, the wake of a FOWT is analyzed by means of numerical simulations and a comparison with linear stability theory. Two simplified models based on the stability of vortices are developed for all degrees of freedom of turbine motion. In our numerical simulations, the wind turbine blades are modeled as actuator lines and a spectral-element method with low dispersion and dissipation is employed to study the evolution of the perturbations. The turbine motion excites vortex instability modes predicted by the linear stability of helical vortices. The flow structures that are formed in the non-linear regime are a consequence of the growth of these modes and preserve some of the characteristics that can be explained and predicted by the linear theory. The number of vortices that interact and the growth rate of disturbances are well predicted by a simple stability model of a two-dimensional row of vortices. For all types of motion, the highest growth rate is observed when the frequency of motion is one and a half the frequency of rotation of the turbine that induces the out-of-phase vortex pairing mechanism. For lower frequencies of motion, several vortices coalesce to form large flow structures, which cause the high amplitude of oscillations in the streamwise velocities, which may increase fatigue or induce high amplitude motion on downstream turbines.

The actuator line method (ALM) is extensively used in wind turbine and rotor simulations. However, its original uncorrected formulation overestimates the forces near the tip of the blades and does not reproduce well forces on translating wings. The recently proposed smearing correction for the ALM is a correction based on physical and mathematical properties of the simulation that allows for a more accurate and general ALM. So far, to correct the forces on the blades, the smearing correction depended on an iterative process at every time step, which is usually slower, less stable and less deterministic than direct methods. In this work, a non-iterative process is proposed and validated. First, we propose a formulation of the non-linear lifting line that is equivalent to the ALM with smearing correction, showing that their results are practically identical for a translating wing. Then, by linearizing the lifting line method, the iterative process of the correction is substituted by the direct solution of a small linear system. No significant difference is observed in the results of the iterative and non-iterative corrections, both in wing and rotor simulations. Additional contributions of the present work include the use of a more accurate approximation for the velocity induced by a smeared vortex segment and the implementation of a free-vortex wake model to define the vortex sheet, that contribute to the accuracy and generality of the method. The results present here may motivate the adoption of the ALM by other communities, for example, in fixed-wing applications.

Two configurations typical of fixed-wing aircraft are simulated with the actuator line method (ALM): a wing with winglets and a T-tail. The ALM is extensively used in rotor simulations, modeling the presence of blades by body forces, calculated from airfoil data and the relative flow velocity. This method has not been used to simulate airplane aerodynamics, despite its advantage of allowing courser grids. This may be credited to the failure of the uncorrected ALM to accurately predict forces near the tip of wings, even for simple configurations. The conception of the vortex-based smearing correction showed promising results, suggesting such failures are part of the past. For the non-planar configurations studied in this work, differences between the ALM with the original smearing correction and a non-linear lifting line method (LL) are observed near the intersection of surfaces, because the circulation generated in the numerical domain differs from the calculated corrected circulation. A vorticity magnitude correction is proposed, which improves the agreement between ALM and LL. This second-order correction resolves the ambiguity in the velocity used to define the lift force. The good results indicate that the improved ALM can be used for airplane aerodynamics, with an accuracy similar to the LL.