Actuator disc theory is the basis for most rotor design
methods, albeit with many extensions and engineering rules added to make it a
well-established method. However, the off-design condition of a very low
rotational speed

Although the concept of the actuator disc is more than 100 years old, it is
still the basis for rotor design codes using the blade element momentum
theory developed over these 100 years (see

For a Joukowsky actuator disc, the swirl of the wake is induced by a discrete
vortex at the wake centre line, leading to an infinite azimuthal velocity and
pressure for the radius

A failed attempt to reproduce the results of

The flow is governed by the Euler equation:

Pressure distributions acting in the momentum balance. The arrows
give the direction of the pressure fields acting on the flow. The meaning of

Only the pressure and the azimuthal velocity will be discontinuous across the
infinitely thin disc, so integration of the axial and azimuthal component of
Eq. (

The thrust

With the conservation of circulation,

The momentum theory results are very sensitive to the
choice of

Both analyses used the vortex core boundary as a lower limit in the integration
of momentum and energy on the control volume used in momentum theory. This
implies that the vortex core is excluded, motivated by its vanishing
dimension in the limit

With

This result will be used in Sect.

The momentum equation drawn on the stream tube as control volume (see Fig.

Streamlines with

Streamlines with

Figure

constant pressure jump across the disc giving the jump in Bernoulli
parameter

pressure distribution due to the jump in

the same pressure distribution in the far wake due to the

constant pressure to achieve

the contribution by the vortex core cross sections (Eq.

When all contributions are expressed in

By mixing Eqs. (

The velocity components at

For large values of

For the limit

Strength
of the vortex sheet as a function of the distance

The Joukowsky momentum theory results compared with potential flow calculations.

The
Joukowsky actuator disc

The computer code described in

As shown in Figs.

The radial component of Eq. (

Figure

Now that the pressure at the upstream side of the disc is known to be constant,
the radial derivative of Eq. (

In

The balance for the entire stream tube is defined by Eq. (

Figure

An actuator disc momentum theory including wake swirl has been developed
resulting in the physically plausible result that

The novelty in the momentum theory is the removal from the momentum balance of the singular behaviour of the pressure near the wake centreline vortex, giving rise to non-physical results in several previously published methods. This removal is done by including the vortex core in the momentum balance.

The momentum theory results are very accurately confirmed by potential flow field calculations.

At the actuator disc the velocity in the meridian plane is constant.

The Joukowsky momentum theory results are higher than the equivalent results for rotors with an infinite number of blades optimized for Betz–Goldstein solutions.

The dataset “Background data for Joukowsky actuator disc
momentum theory wes-2016-55”,

The author declares that he has no conflict of interest.

Thanks go to Jens Norkær Sørensen, DTU, for the discussions about the
modified momentum theory and to Valery Okulov, DTU, and David Wood,
University of Calgary, for a discussion on the Betz–Goldstein solution and
for providing the data shown in Fig.