Using the concept of impulse in control volume analysis, we derive general expressions for wind turbine thrust in a constant, spatially uniform wind. The absence of pressure in the impulse equations allows for their application in the near wake, where the flow is assumed to be steady in the frame of reference rotating with the blades. The assumption of circumferential uniformity in the near wake – as applies when the number of blades or the tip speed ratio tends to infinity – is needed to reduce these general expressions to the Kutta–Joukowsky (KJ) equation for blade-element thrust. The present derivation improves upon the classical derivation based on the Bernoulli equation by allowing the flow to be rotational in the near wake. The present derivation also yields intermediate expressions for thrust that are valid for a finite number of blades and trailing vortex sheets of finite thickness. For the circumferentially uniform case, our analysis suggests that 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. The derivation reveals one further complication; when deriving the KJ equations using annular control volumes, the existence of vorticity on the lateral control surfaces may cause the local blade loading to differ from the KJ equation, but the magnitude of these deviations is not explored. This complication is not visible to the classical derivation due to its neglect of vorticity.

Blade-element theory (BET) for wind turbines uses the fundamental assumption that the forces acting on the elements comprising the rotor blades are given by the Kutta–Joukowsky (KJ) theorem. The thrust and torque are balanced by the change in the axial and angular momentum, respectively, of the flow through a control volume (CV) enclosing the rotor; the combination of BET and momentum theory gives rise to blade-element momentum (BEM) theory. BEM is developed in many texts on wind turbine aerodynamics, such as

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

In keeping with typical notation in BEM analysis, we define

When the flow is circumferentially uniform, Eq. (

The remainder of the paper is organized as follows. Since the wind energy research community is likely to be unfamiliar with the concept of “vortical impulse” or “hydrodynamic impulse” – usually referred to simply as “impulse” – we offer a brief introduction in Sect.

Impulse theory expresses fluid-dynamic forces in terms of the first moment of vorticity (e.g.

The introduction of the concept of impulse removes the pressure and introduces vorticity to the equations of linear momentum conservation. This approach can be traced at least as far back as

Derivation of an impulse-based force formulation begins with the conventional application of the Reynolds transport theorem to the momentum equation and proceeds by manipulating the equations using vector calculus identities. For a full presentation, we recommend the

For an inertial CV in an incompressible fluid, the equation for force,

The momentum integral – the first integral on the right-hand side of Eq. (

We consider a wind turbine rotating steadily at a tip speed ratio,

Control volumes (CVs) for the present analysis. In both variants, the upstream face is well upstream of the rotor, where the velocity is equal to the freestream, and

We now consider the contributions of each term in Eq. (

The first two terms in the surface integral on the right-hand side of Eq. (

So, we have seen that the impulse term's thrust contribution vanishes, the two velocity terms in Eq. (

Velocity (in the rotating frame of reference) and vorticity vectors in the near wake. If diffusion is neglected and the wake is assumed to be rigid, velocity and vorticity triangles on each orthogonal plane are geometrically similar.

Equation (

Since a future goal of this research program is unsteady turbine modelling, an alternative derivation of Eq. (

If the downwind face of the CV is moved to be just ahead of the rotor – replacing integrals on

Equation (

For a turbine with a finite number of blades, the situation is more complex. The early near-wake measurements of

Assuming that

To determine the blade-element version of Eq. (

Annular CVs used for the blade-element force analysis. The downstream face of width

As noted above, the circumferential velocity ahead of the rotor vanishes for a circumferentially uniform flow, in which case only radial and axial velocities contribute to the integral in Eq. (

Recall that the classical derivation of Eq. (

When the flow on

The sum of blade circulations at radial station

Circumferential uniformity is approached as

In blade-element analysis, the Kutta–Joukowsky equation is usually introduced as an assumption, but this can only be valid as

Recall that we neglected the contribution of the vortex terms on the lateral boundaries of the annular CV to arrive at Eq. (

This paper describes the application of vortical impulse theory to determine the thrust on a steadily rotating wind turbine in a steady, spatially uniform wind. The principal attraction of the impulse approach is the absence of pressure in the force equations, which allows them to be applied anywhere in the flow, including immediately behind the rotor. We assume that the vorticity field is steady when viewed by an observer rotating with the blades, so the vortex lines and streamlines in the near wake are aligned in this frame. Subsequently, we rederive the Kutta–Joukowsky (KJ) equation for blade-element thrust under the special condition of circumferential uniformity in the wake (i.e. as the number of blades or the tip speed ratio tends to infinity). Our derivation improves upon the classical derivation – which is based on the Bernoulli equation – by allowing the wake to be rotational and three-dimensional and is exact for any amount of flow expansion through the rotor. In addition, intermediate steps en route to the KJ equation provide thrust expressions which allow for circumferential non-uniformity and trailing vortex sheets of finite thickness.

The derivation also yields insight into the radial velocity based on Eq. (

An important avenue for future work is an investigation of the effect of vorticity on the lateral boundaries of the annular control volumes in the immediate proximity of the blades. The blade-element KJ equation was recovered only when these terms were neglected, but the validity of this neglect was not established. This complication is not visible to the classical derivation of the KJ equation due to its neglect of vorticity.

Assume that, at all times, the fluid elements along a streamline passing through the rotor remain irrotational. The trailing vorticity in the wake is assumed to be infinitely thin. The unsteady Bernoulli equation along this streamline is

When the polar co-ordinates are attached to the blades rotating at

In Noca's derivation of Eq. (

For a steadily rotating CV, Eq. (

Thus the reduced form of Eq. (

The vortex terms are better expressed in the inertial frame,

Curiously, three integrals with the integrand

No data sets were used in this article.

The work was conceived, executed, and written up equally by both authors.

The authors declare that they have no conflict of interest.

David H. Wood's contribution to this work is part of a research project on wind turbine aerodynamics funded by the NSERC Discovery Program. Eric J. Limacher acknowledges receipt of an NSERC Post-Doctoral Scholarship. The authors thank the anonymous referees and Gijs van Kuik for their valuable comments.

This research has been supported by the NSERC (Discovery Program and PDS).

This paper was edited by Jens Nørkær Sørensen and reviewed by three anonymous referees.