The Effect of 3D Geometry on Unsteady Gust Response, Using a Vortex Lattice Model

Abstract

Unsteady flow response is an important consideration for a range of engineering applications, from unmanned air vehicles, where it has implications for control, to tidal turbines, where the accurate calculation of fatigue load is vital. Designers often use 2D strip-theory predictions for both steady and unsteady performance, applying Theodorsens unsteady transfer function for uniform gusts at each blade section to estimate the unsteady bending moments on the turbine blades. The purpose of this investigation is to explore the limits of the applicability of this 2D classical unsteady aerofoil theory to aerofoils with significant 3D geometry features. Using a harmonic vortex lattice model, this study shows that there are significant 3D features in the unsteady flow response, which increase with decreasing reduced frequency and with decreasing aspect ratio. The response near the blade tips is strongly 3D, and does not reach the 2D characteristic, even at high frequencies. The phase response also varies strongly along the span, leading to different blade sections responding out of phase with each other even with no spanwise gust variation. This has significant implications for bending moment calculations, with require integration of the load along the span. The observed 3D effects are shown to be caused by changes to the spanwise component of the unsteady wake, and by the presence and behaviour of a streamwise unsteady wake. The results for a model tidal turbine geometry show that the Loewy function does not capture returning wake effects adequately, but that it does model the mid-span response characteristic well at reduced frequencies over 0.8. The study concludes that using transfer functions from 2D classical aerofoil theory provides a conservative estimate of the blade loads affecting a tidal turbine, but only if no steady tip-loss corrections have been applied to the unsteady response. If tip-loss corrections are applied to the quasi-steady lift response before unsteady transfer functions are used, the resulting load amplitude will be significantly under-predicted.

Usefulness of Inviscid Linear Unsteady Lifting-Line Theory for Viscous Large-Amplitude Problems

“…unsteady vortex lattice methods [38][39][40][41] or vortex particle methods [42] can be used. However, these methods lack robust and simple LEV shedding criteria, and have numerical difficulties in modeling unsteady wakes.…”

Unsteady lifting-line theory and the influence of wake vorticity on aerodynamic loads

“…Neglecting all three-dimensional effects (for large aspect ratios) allows two-dimensional methods to be applied to the wing in strips. In instances where this is a poor approximation, low-order 3D methods such as the unsteady boundary-element method [42], unsteady vortex lattice method [30,36,44,58] or vortex particle methods [69] can be used, but at far greater computational cost due to the large numbers of interacting vortex elements required to form wake vortex structures. Eldredge and Darakananda [22] suggest forming these vortex structures with a very small number of time-varying elements instead, but this introduces difficulties.…”

Unsteady lifting-line theory and the influence of wake vorticity on aerodynamic loads

“…34 Neglecting all three dimensional effects (for large aspect ratios) allows two dimensional methods to be applied to the wing in strips. In instances where this is a poor approximation, low-order 3D methods such as the unsteady boundary-element method, 39 unsteady vortex lattice method 27,33,41,54 or vortex particle methods 65 can be used, but at far greater computational cost.…”