The flow upwind of an energy-extracting horizontal-axis wind turbine expands as it approaches the rotor, and the expansion continues in the vorticity-bearing wake behind the rotor. The upwind expansion has long been known to influence the axial momentum equation through the axial component of the pressure, although the extent of the influence has not been quantified. Starting with the impulse analysis of

Conservation of axial and angular momentum are fundamental principles for wind turbine analysis. They are applied using control volumes (CVs) such as those in Fig.

Control volumes (CVs) to be used in the present analysis. In both variants, the upstream face extends in

When a turbine extracts kinetic energy from the wind, the flow must expand both upwind and downwind of the rotor. As noted on p. 185 of

the flow upwind of the rotor and outside the wake is inviscid, steady, and spatially uniform;

the total energy of the wake is reduced instantaneously at the rotor, after which it is conserved;

viscous and/or Reynolds stresses can be neglected on the CV surfaces;

the axial,

viscous drag is negligible;

the vorticity in the wake is concentrated in line vortices or vortex sheets aligned with the local streamlines in the rotating frame of reference – in other words, the wake vortices rotate rigidly with the blades and vortex lines and streamlines coincide;

to derive the local or differential form of Eq. (

Although Eq. (

Equation (

For context, we now examine the connection between the impulse- and momentum-based approaches to turbine thrust, which requires a relationship between

In the next section, we express the contribution of pressure on the expanding upstream streamtube to actuator disc thrust. The section thereafter analyzes the local form of the thrust equation. It contains our second main result about the behaviour of

Some results of the impulse analysis can be converted easily to conventional equations containing the axial velocity and the pressure on the CV surface even when the flow expands through the rotor. For example, Bernoulli's equation for

In the far wake, the pressure and circumferential velocity are related by

To recover the classical thrust equation and to provide a comparison to the analyses of

In considering the local equation for

The results for the far wake can be used to estimate the conventional thrust when

Having considered the thrust for the complete rotor, we now consider the local contribution at radius

We now consider the consequences of the exact Eq. (

The preceding analysis shows that

A more definite statement about

We now consider the far wake in more detail to determine the vortex pitch and its relation to

Outside the hub vortex core of a Joukowsky wake,

The KH equations for a doubly infinite helical vortex of constant radius and pitch lead to

We assume

Without loss of generality, let the lifting line representing one blade lie instantaneously along the

The simplest calculation of

Performing the

If

As with any Biot–Savart analysis, the behaviour of Eqs. (

The numerical evaluation of

It was found that

The Biot–Savart integrands in Eq. (

Integrand,

Integrand,

Solving Eq. (

Axial induction,

The mass flux through the rotor, using Eq. (

To find the unique

The results in Table

Results for the expanding Joukowsky wake with constant pitch.

From Table

The integrands

Ratio of axial induction at the rotor,

Figure

In the final figure, Fig.

The pressure in the expanding flow ahead of a wind turbine contributes to the axial force on the rotor and a momentum deficit in the flow outside the rotor. Researchers have been aware of these two effects for many years, but the present analysis provides the first quantitative determination of them in Eqs. (

The impulse analysis of

The first three sections of the paper used only the standard form of control volume (CV) analysis for axial momentum to determine the thrust of the rotor and the incremental thrust of the blade elements comprising the rotor. To clarify the effects of expansion, most analysis in this paper used CVs with downwind faces in the immediate vicinity of the rotor, as opposed to their common placement in the far wake. The rotor and the flow are assumed to be circumferentially uniform. We argued in the Introduction that the impulse analysis provides a simple and novel perspective on the role of the pressure. The thrust equations derived in Sect. 3 for the rotor, and in Sect. 4 for the local flow at any radius, contain the pressure acting on the downwind face of the actuator disc, which must be removed to make the equations suitable for actual blade analysis. Removal can be done accurately only for very low tip speed ratios where the expansion and its effects are small.

To the rotor thrust, the pressure along the bounding streamsurface adds a term containing the integral of

Including

The role of the radial velocity and flow expansion is probably more complicated in rotors with a limited number of blades than the actuator discs considered here.

This analysis started from the impulse-derived Kutta–Joukowsky equation for wind turbine thrust, which does not involve the axial velocity. The equation is valid for any amount of expansion in the upwind flow and the wake and any distribution of bound circulation on the rotor. We were able to

demonstrate the conventional thrust equation containing the axial velocity can be correct only when the tip speed ratio is large;

derive an exact expression for the effects of flow expansion on the conventional momentum equation – this involves the axial induction factor and the radial velocity;

apply the conventional and impulse thrust equations in the far wake to give the pitch of the tip vortices in the Joukowsky wake in terms of the tip speed ratio and the far-wake induction;

find a semi-analytic solution of the Biot–Savart law for the induced velocities at the rotor by assuming the tip vortex had constant pitch – the axial velocity near the rotor tip approached the far-wake value, but was prevented from exceeding it as an alternative to using the familiar cut-off in the Biot–Savart integrals, and the increase in the rotor value contradicts the familiar relation that the axial induction factor everywhere at the rotor is half that of the far wake; and

derive in Sect. 5 the following results from the model of constant pitch, expanding tip vortices:

the angle of the tip vortex surface to the wind direction was 53

because it is neither very small nor very large, this expansion leads to an error of around 6 % in the conventional thrust equation, which would be accurate for both extreme expansions;

the resulting wake expands less than the familiar Betz–Joukowsky wake – for two pitch values corresponding to tip speed ratios of 7 and 14, the far-wake area was 1.59 times the rotor area;

we find the reduction in the rotor power and thrust due to expansion – the maximum power coefficient and corresponding thrust coefficient were 6 % less than the values given by the Betz–Joukowsky limit; and

we quantify the influence of the expansion on the flow outside the rotor – for example, the radial velocity at three rotor radii is still 3 % of the wind speed when the rotor is producing maximum power, and the axial induction factor decays to zero more rapidly than the radial velocity as radius.

All the MATLAB codes and data used in this analysis are available from the first author.

This study was conceived by both authors, who shared the analytic development. DHW completed the numerical analysis and the first draft of the manuscript, with both authors contributing equally to its revision thereafter.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper was inspired partly by an anonymous referee of LW who doubted the value of Eqs. (

This research was funded by a Canadian Natural Sciences and Engineering Research Council Discovery Grant.

This paper was edited by Jens Nørkær Sørensen and reviewed by Gijs van Kuik and Emmanuel Branlard.