On stability of vortices and vorticity generated by actuator lines

Abstract: Vortices are present in nature and in many flows of industrial importance. The stability of configurations of vortices can have real-world consequences, because vortices play a crucial role in accelerated mixing. In particular, vortices are present in the wake of wind turbines and other rotors. Their blades create a system of multiple helical tip and hub vortices in the wake. The stability of the tip vortices greatly influences the wake recovery behind a turbine and, consequently, can affect the power production and fatigue of a downstream wind turbine in clustered wind farms. Also, concentrated vortices can cause vortex-structure interaction which increases vibration and noise. In this work, the stability of vortices is studied by analytical models and Navier-Stokes simulations. The vorticity generated in these simulations was studied in order to develop improvements to the numerical methods used to simulate blades and wings.Numerical simulations of a moving rotor, representing a floating offshore wind turbine, showed that the wake is dominated by the stability modes predicted by the linear stability theory. Also, the observation that the stability of helical vortices has properties that can be related to the stability of a two-dimensional row of vortices, also noted previously in other works, motivated the development of a new formulation to study the stability of two-dimensional potential flows, based on the bicomplex algebra. Models based on vortex filaments and the Biot-Savart law were developed to study the stability of the system of multiple helical vortices created by turbine blades. The results indicate that the linear stability of the tip vortices is independent of the linear stability of the hub vortices (and vice-versa). For more complex configurations, such as two in-line turbines or blades that create multiple vortices near the tip, the numerical simulations and analytical studies indicate a more complex scenario, with multiple vortices interacting.The Navier-Stokes simulations employ the actuator line method (ALM), which is a method used to model blades that allows coarser grids, reducing computational costs. In this method, the blades are represented by body forces that are calculated from the local flow velocity and airfoil data. However, until recently, the actuator line method misrepresented the forces near the tip of the blades. The recently developed vortex-based smearing correction resolved some of these limitations. In this work, the understanding of the vorticity generated by actuator lines is used to develop more accurate corrections for the velocity induced by a smeared vortex segment and for the magnitude of the vorticity generated in the simulations. Also, a non-iterative procedure for the smearing correction is proposed based on the lifting line method. These modifications improve the agreement of the ALM with a non-linear lifting line method. For the first time, configurations typical of airplane aerodynamics are simulated with the ALM, such as a wing with winglets and a combination of horizontal and vertical tails. The accuracy of these results may motivate other communities to adopt the ALM for a diverse set of applications, beyond rotor aerodynamics.

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