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

The start of rotor aerodynamics dates back more than 100 years, when
the concept of the actuator disc to represent the action of
a propeller was formulated by

The constant circulation model of Joukowsky as well as the constant
velocity model of Betz represented the ideal rotor. It was not yet
possible to compare the models and to conclude which was
best. Both models were valid only for lightly loaded rotors as wake
expansion or contraction was neglected. A solution for the wake of
Betz's rotor, still restricted to lightly loaded propellers, was
presented by

The Joukowsky and Froude discs are still subjects for research as many
modern design and performance prediction codes are based on them; see
e.g.

The present paper addresses the topic which received the least
attention: the velocity distribution at the disc. The paper is part of
a sequence of papers, starting with

Section

Figure

The coordinate system of an actuator disc acting extracting energy.

The stream tube of a propeller disc from cross sections

The pressure and azimuthal velocity are discontinuous across the disc when

The power

The power and thrust have the same sign as

The velocity in the far wake is characterized by

The momentum theory presented in

The axial velocity

Figure

Definition of actuator disc flow cases

Several particularities can be observed in Fig.

For values of

For wind turbine discs having

For propeller discs having a very high

For low rotational speed (low

For lower rotational speeds the pressure jump at the edge has become

In

Table

The flow patterns of wind turbine discs

In Fig.

With

The velocity distribution at the disc, for flow cases

Figure

As shown in Fig.

Disc load

Discs with

The non-uniformity in

In all flow cases the axial velocity is far from uniform, as was
already shown by

The radial velocity receives little attention in actuator disc and
rotor publications compared to the axial velocity. Some exceptions are

Recently

The radial position where Eq. (

Equation (

The Euler equation of motion (Eq.

Qualitative observations regarding the increase or decrease in

At the upwind side of the streamline, when moving towards the disc, the distance to

At the downwind side of the disc the streamline is to be distinguished in two parts: upstream and downstream of

For flow cases with a contracting wake the same reasoning is valid,
with an appropriate change of signs, leading to a minimum

However, these qualitative considerations miss the effect that
a vortex ring induces a non-zero

The radial velocity induced by a unit vortex ring positioned at

The distribution of the vortex sheet strength

Curved lines: the radial velocity along the streamline passing the disc at

For a vorticity tube things are slightly different, as is easily shown
by the example of a tube of constant strength with a semi-finite
length. Each elementary vortex ring

The argument of non-zero

At

According to (a) and (b), the position where

This qualitative line of arguments (a)–(d) requires a numerical
validation and quantification. The calculated wake vorticity

The absolute value of the slope of the tangents is lowest in flow case

With respect to the average velocity at the actuator disc, the following applies.

For Joukowsky discs in wind turbine and propeller mode, the average velocity has been found, from

For a very high

Propeller disc flows without wake expansion or contraction are possible for specific values of

In the propeller as well as wind turbine flow regimes the velocity at the disc becomes

the differences in uniformity are caused by the different strengths of the leading edge singularity in the wake boundary vorticity strength.

The dataset

The author declares that there is no conflict of interest.

The author thanks the reviewers David Wood, University of Calgary, Canada, and
the anonymous referee. Their comments improved the manuscript significantly. The
same holds for the discussion with David Wood and Eric Limacher, Federal
University of Pará, Belém, Brazil, about the significance of
Eqs. (

This paper was edited by Alessandro Bianchini and reviewed by David Wood and one anonymous referee.