David H. Wood

Maximum wind turbine performance at low tip speed ratio

Wind turbines can approach the Betz-Joukowsky limit on maximum power only at sufficiently high tip speed ratio: in practice, for ratios in excess of about seven. This paper analyses the performance of a turbine with an infinite number of blades as the tip speed ratio decreases to zero, beginning with the two traditional ways of determining the maximum power. The first is the “Glauert” optimization of the power extracted at every radius and the second is the “Betz-Goldstein” optimization of the whole rotor with the wake represented as a rigid helicoidal sheet of constant pitch. At high tip speed ratio, the two methods give very similar power and both asymptote to the Betz-Joukowsky limit. As the ratio approaches zero, the differences become significant. It is shown that Glauerts analysis does not account for effect of the varying pitch on the axial velocity. In addition, there is a large and unphysical energy extraction by a stationary rotor, and an unphysical constant circumferential velocity. Glauerts analysis, however, gives positive torque on a stationary rotor, which is necessary to start a wind turbine, but the torque at very low tip speed ratio seems too high. In contrast, the Betz-Goldstein analysis implies no torque on a stationary rotor but the circumferential velocity is zero on the axis of rotation. A modification is proposed to the Betz-Goldstein analysis to yield positive torque on a stationary rotor. The modified Betz-Goldstein torque coefficient never exceeds 1/2 and the power is less than the Glauert optimum. Finally, the analysis for varying pitch is corrected and the power maximized numerically. The modification applied to the Betz-Goldstein rotor is required to produce torque on the stationary rotor. The maximum torque coefficient was 0.55, and the maximum power was again less than the Glauert maximum. As tip speed ratio increases, the power from all methods asymptotes to the Betz-Joukowsky limit.

Wake Dynamics Behind a Normal Thin Flat Plate at Moderate Reynolds Numbers

The wake behind an infinite span (2D) thin flat plate normal to uniform flow was examined using Direct Numerical Simulation (DNS) at \(Re = U_0 h/\nu = 1200\) and 2400. Three distinct flow regimes were identified in the wake due to the interruption of regular anti-symmetric Karman shedding. Disruption of the regular shedding (Regime M) was followed by a period of delayed roll-up (Regime L) and a longer duration of high intensity shedding (Regime H) before regular shedding resumes again. Size of the wake mean recirculation region changed with progression of the wake patters from a long-term mean length of 2.90–3.60h (L) and then 2.16h (H). Moreover, the wake turbulence characteristics were effected with the wake evolution. These effects were quantified by variations in the turbulence kinetic energy (TKE) magnitude, production, dissipation and diffusion. Spanwise instabilities were responsible for appearance of the three flow regimes. A projection of these instabilities was observed in the pressure field behind the plate.

Ideal wind turbine performance at very low tip speed ratio

At very low tip speed ratios, wind turbine rotors behave similarly to stationary wings for which the well-known lifting line analysis gives the optimal loading. Lifting line analysis is applied to a stationary rotor of N blades for N = 1, 2, and 4. Analytic (for N = 1 and 2) or semi-analytic solutions (for N = 4) agree with the classical results obtained by conformal mapping. The present solutions compare well with numerical solutions for the Goldstein function for optimally loaded propeller or wind turbine rotors at low tip speed ratio. In all cases, the induced velocity is linear with radius. Assuming this result applies for all N, lifting line analysis is recast as a singular integral equation whose solution agrees with Goldsteins obtained using the same conformal mappings as in his general analysis for any tip speed ratio. The implication is that the assumed induced velocity distribution is correct, and is, therefore, fundamentally different from that at high tip speed ratios when the induced velocit...

Evolution of Vortex Formation in the Wake of Thin Flat Plates with Different Aspect-Ratios

The effect of aspect-ratio (AR) on the formation and interaction of vortical structures in the wake of normal thin flat plates is examined at Re \(=\) 1200 using an infinite span (2D) plate and rectangular plates (\({ AR}=1.0\), 1.6 and 3.2). The vortex shedding frequency significantly increased for \({ AR}=3.2\) compared to the 2D plate, while it dropped for \({ AR}=1.6\) and 1.0. The lowest frequency of vortex shedding was observed for \({ AR}=1.0\). Shear layers rolled to form vortices closer to the plate leeward surface at higher AR. The mean recirculation length was longer for the 2D plate compared to rectangular and square plates. The interaction of shear layers originating at the edges of the rectangular plates led to the formation of vortex loops in the wake. The wake appeared less organized for the lower AR plate (1.6) with a higher turbulence energy compared to \({ AR}=3.2\). The magnitude of turbulence kinetic energy was lowest at \({ AR}=3.2\) and it increased with decreasing AR. The mean drag coefficient was \(\approx \)2 for the 2D plate and \(\approx \)1 for \({ AR}=1\), which suggested major differences in the wake structures.

The Aerodynamic Characterization of Generic Tail Fin Shapes

This paper describes the experimental characterization of the yaw response of a range of generic tail fin shapes: delta wings, right triangles, rectangles, and ellipses. The fins were tested in a wind tunnel without the complication of blades and nacelle. They were released from an initial angle, usually 45°, and the subsequent response recorded using an accurate potentiometer. Unsteady tail fin performance can be analysed using unsteady slender body theory and the quasi-steady analysis originally developed for wind vanes. Both predict a linear, second order response so the measured responses are quantified in terms of the natural frequency and damping ratio. It is shown that unsteady slender body theory is slightly more accurate, but both fail to describe the full complexity of the response.