On the velocity at wind turbine and propeller actuator discs

Abstract

<jats:p>Abstract. The first version of the actuator disc momentum theory is more than 100 years old. The extension towards very low rotational speeds with high torque for discs with a constant circulation became available only recently. This theory gives the performance data like the power coefficient and average velocity at the disc. Potential flow calculations have added flow properties like the distribution of this velocity. The present paper addresses the comparison of actuator discs representing propellers and wind turbines, with emphasis on the velocity at the disc. At a low rotational speed, propeller discs have an expanding wake while still energy is put into the wake. The high angular momentum of the wake, due to the high torque, creates a pressure deficit which is supplemented by the pressure added by the disc thrust. This results in a positive energy balance while the wake axial velocity has lowered. In the propeller and wind turbine flow regime the velocity at the disc is 0 for a certain minimum but non-zero rotational speed. At the disc, the distribution of the axial velocity component is non-uniform in all actuator disc flows. However, the distribution of the velocity in the plane containing the axis, the meridian plane, is practically uniform (deviation &lt;0.2 %) for wind turbine disc flows with tip speed ratio λ&gt;5, almost uniform (deviation ≈2 %) for wind turbine disc flows with λ=1 and propeller flows with advance ratio J=π, and non-uniform (deviation 5 %) for the propeller disc flow with wake expansion at J=2π. These differences in uniformity are caused by the different strengths of the singularity in the wake boundary vorticity strength at its leading edge. </jats:p>

An impulse-based derivation of the Kutta–Joukowsky equation for wind turbine thrust

“…for the circumferentially uniform case. van Kuik (2020) directly corroborates Eq. (24), showing it to hold for his model of rotor flow.…”

“…When the wake is circumferentially uniform, the magnitudes of the radial velocity and the axial induction factor must be equal somewhere on the plane containing the rotor, and we cite previous studies that show this to occur near the rotor tip across a wide range of thrust coefficients. Equation ( 24) was verified in the simulations of van Kuik (2020).…”

“…The circumferential average of v or the value at the blades can, therefore, be small while the average of v 2 can be significant. The flow in the tip region of a rotor with a finite number of blades can include motion of the tip vortex towards, rather than away from, the axis of rotation (van Kuik et al, 2014).…”

“…Although the KJ equations are generally introduced as assumptions in BEM theory, they can be derived using the unsteady Bernoulli equation. This was shown as early as Glauert (1935) and can also be found in Sørensen (2016) and van Kuik (2018). Assuming the trailing wake to be infinitely thin to permit the use of the Bernoulli equation, and representing the rotor's power extraction by a sudden drop in pressure across the rotor plane, sectional thrust can be expressed as follows:…”

Dynamic inflow model for a Floating Horizontal Axis Wind Turbine in surge motion

“…The results of the semi-free wake vortex-ring model show a larger oscillation of induction closer to the actuator edge. This is not a finite-blade tip effect, nor the radial variation of induction previously found in a steady actuator disc with uniform loading (van Kuik, 2018). It is actually an effect of blade (actuator) vortex interaction due to the motion of the actuator and unsteady loading.…”