Stability of helical tip vortices in a rotor far wake

Abstract

As a means of analysing the stability of the wake behind a multi-bladed rotor the stability of a multiplicity of helical vortices embedded in an assigned flow field is addressed. In the model the tip vortices in the far wake are approximated by infinitely long helical vortices with constant pitch and radius. The work is a further development of a model developed in Okulov (J. Fluid Mech., vol. 521, p. 319) in which the linear stability of N equally azimuthally spaced helical vortices was considered. In the present work the analysis is extended to include an assigned vorticity field due to root vortices and the hub of the rotor. Thus the tip vortices are assumed to be embedded in an axisymmetric helical vortex field formed from the circulation of the inner part of the rotor blades and the hub. As examples of inner vortex fields we consider three generic axial columnar helical vortices, corresponding to Rankine, Gaussian and Scully vortices, at radial extents ranging from the core radius of a tip vortex to several rotor radii.The analysis shows that the stability of tip vortices largely depends on the radial extent of the hub vorticity as well as on the type of vorticity distribution. As part of the analysis it is shown that a model in which the vortex system is replaced by N tip vortices of strength Γ and a root vortex of strength − N/Γ is unconditionally unstable.

Natural snowfall reveals large-scale flow structures in the wake of a 2.5-MW wind turbine

“…NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5216 ARTICLE manifesting a drastic reduction of vortex strength, is a result of the breakdown of the trailing vortex sheets, which are needed to sustain the tip vortices 22 . The unsteadiness of the trailing sheets can quickly trigger tip-vortex instability and cause vortex core collapse and, subsequently, void disappearance.…”

Multiple helical modes of vortex breakdown

“…To compare the results to the stability analysis described by Okulov (2004) and Okulov & Sørensen (2007) we analyse the PIV-processed flow field using (2.2)-(2.4). To illustrate this procedure, consider two steady flow fields at two different positions in the parameter range, which are both close to a critical Reynolds number.…”

“…The instability makes it a very difficult task to detect and study multiple helix modes during vortex breakdown. We base our search on the results from the stability analysis by Okulov & Sørensen (2007) showing that the ambient swirl flow can significantly enhance the stability of vortex multiplets.…”

“…Each vortex is determined by an identical core with dimensionless radius σ = ε/R, dimensionless pitch τ = 2πl/R, and circulation Γ . The stability analysis (Okulov 2004;Okulov & Sørensen 2007) leads to the following stability condition:…”

“…We base this search on the results from a stability analysis (Okulov 2004;Okulov & Sørensen 2007) showing that an assigned vortex flow can significantly enhance the stability of vortex multiplets. First, a full theoretical analysis of the effect of an intensively swirling assigned flow is performed to determine the stability properties of multiplets.…”