Mutual inductance instability of the tip vortices behind a wind turbine

Abstract

Two modal decomposition techniques are employed to analyse the stability of wind turbine wakes. A numerical study on a single wind turbine wake is carried out focusing on the instability onset of the trailing tip vortices shed from the turbine blades. The numerical model is based on large-eddy simulations (LES) of the Navier-Stokes equations using the actuator line (ACL) method to simulate the wake behind the Tjaereborg wind turbine. The wake is perturbed by low-amplitude excitation sources located in the neighbourhood of the tip spirals. The amplification of the waves travelling along the spiral triggers instabilities, leading to breakdown of the wake. Based on the grid configurations and the type of excitations, two basic flow cases, symmetric and asymmetric, are identified. In the symmetric setup, we impose a 120• symmetry condition in the dynamics of the flow and in the asymmetric setup we calculate the full 360• wake. Different cases are subsequently analysed using dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD). The results reveal that the main instability mechanism is dispersive and that the modal growth in the symmetric setup arises only for some specific frequencies and spatial structures, e.g. two dominant groups of modes with positive growth (spatial structures) are identified, while breaking the symmetry reveals that almost all the modes have positive growth rate. In both setups, the most unstable modes have a non-dimensional spatial growth rate close to π/2 and they are characterized by an out-of-phase displacement of successive helix turns leading to local vortex pairing. The present results indicate that the asymmetric case is crucial to study, as the stability characteristics of the flow change significantly compared to the symmetric configurations. Based on the constant non-dimensional growth rate of disturbances, we derive a new analytical relationship between the length of the wake up to the turbulent breakdown and the operating conditions of a wind turbine.

Blind test comparison of the performance and wake flow between two in-line wind turbines exposed to different turbulent inflow conditions

“…The shear profile is implemented to match the profile given in the invitational document by Saetran and Bartl (2015). A more detailed description of the method can be found in Sarmast et al (2014).…”

“…The largest rotor investigated for wake comparison studies was the MEXICO rotor, with a diameter of 4.5 m (Schepers et al, 2010), in which the rotor performance as well as the wake flow were examined in detail. A second campaign investigating even more effects, including span-wise pressure distributions, yaw misalignment and unsteady effects, was realized at a large German-Dutch Wind Tunnel (DNW).…”

Simulation of wind turbine wakes using the actuator line technique

“…The initial derivations of the presented analysis follow the work of Ivanell et al [23] and Sarmast et al [28]. In these studies, the simulations are performed using the Tjaereborg wind turbine operating at the optimum power condition (Cp = 0.49) for a wind speed of U 0 = 10 m s −1 and tip speed ratio of λ = 7.…”

“…From the figure, it can be clearly seen that the amplitudes develop with an exponential growth rate. More information regarding the computations can be obtained from Ivanell et al [23] and Sarmast et al [28]. The curves are normalized by the amplitude at x/R = 0.5 to compare the growth from computations with different perturbations.…”

Stochastic Wake Modelling Based on POD Analysis

“…Another dynamic approach, also followed in this work, is to analyse and model wind turbine wakes using modal decompositions [21,[35][36][37][38][39][40][41][42][43][44][45][46][47] which describe the velocity field as a linear superposition of spatial modes with time-dependent weighting coefficients. It has been shown that a few spatial modes can already capture important features of the wake flow [35][36][37][38][39]41,43,47].…”