The original work of Glauert consisted of a closed-form solution describing the optimum performance of a rotating actuator disk , where the simplified actuator disk concept for infinitely bladed propellers was extended to wake rotation of energy-extracting rotors; see also for a review on the early history of momentum theory. In order for Glauert to close the basic equations of rotor disk theory, additional assumptions had to be applied that utilized a simple relationship between the induced velocity components and pressure jump at the rotor disk . At the time of Glauert, it was assumed that the kinetic energy required for wake rotation was extracted from the freestream , and this assumed wake energy and a constant pressure jump across the rotor disk remain the fundamental assumptions in Glauert's theory. In this regard, Glauert's model differed from that of , which was built on the assumption of constant circulation across the rotor disk . Also, questioned the validity of ignoring the static pressure drop caused by wake rotation. Indeed, an additional pressure drop is required to balance the centrifugal force of rotating air parcels as a function of radial location on the disk; see also for a discussion of a general momentum theory. A complete and comprehensive summary of the validity of all rotor disk theory models can be found in . In the end, various rotor disk models all approach the accepted Betz limit for maximum rotor power coefficient at high tip speed ratio. To date, however, no exact integrals have been presented for thrust and bending moment coefficients based on Glauert's optimum solution for axial and angular induction factors.

The objective of the present work is to determine the exact integrals for thrust and bending moment coefficients, serving as an addendum to Glauert's original work deriving optimum power coefficients. In doing so, an elegant solution by means of calculus of variation reveals itself, recovering Glauert's optimum distributions for axial and angular induction factors. Exact integrals for thrust and bending moment coefficients are then derived, including proper limit behavior of performance coefficients for low and high tip speed ratio. This work is organized as follows: Sect. 2 introduces classical rotor disk theory and relevant dimensionless parameters. Section 3 discusses Glauert's optimum solution for the maximum power coefficient, as well as an alternate optimization approach by means of calculus of variation, and a formal asymptotic behavior of induction factors. Sections 4 and 5 detail the mathematics uncovering the exact integrals for thrust and bending moment coefficients, respectively. Section 6 summarizes the analytical solutions for all three performance coefficients and discusses the limitations with respect to validity at very low tip speed ratio. Section 7 contains some concluding remarks highlighting results obtained within the context of the typical range of tip speed ratio for modern wind turbines.

The classical rotor disk formulation according to has been documented in various texts . In the following, only the primary relations relevant to this work are summarized.

Consider an axi-symmetric streamtube model that encompasses a wind turbine. The cross section, where the rotor is located, can be represented by a thin rotor disk of area, A=πR2, where R is the disk radius. The axial induction factor, a, determines the reduction in wind speed, V0, at the disk. This coefficient is defined as 1a=1-uV0, where u is the axial speed at the rotor disk. As a consequence of rotor thrust, there exists a discontinuity in pressure, Δp, at the disk. Therefore, Bernoulli's equation is applied both upstream and downstream of the disk. Adding these two equations together allows solving for Δp, i.e. the pressure jump arising from disk theory, 2Δp=2ρV02a(1-a), in terms of the fluid density, ρ, V0, and a as shown. Next, disk rotation is added at an angular speed, Ω. The wake angular velocity component, ω, just downstream of the disk, however, is affected by a rotor disk torque in the circumferential direction. The angular induction factor, a′, relates the wake angular velocity to the angular speed of the rotor disk: 3a′=ω2Ω.

Bernoulli's equation is applied once more upstream and downstream of the rotor disk. Note that in Glauert's theory, the assumption is that the same pressure jump, Δp, that generates thrust also generates rotor torque and power. The resulting Δp equation arising from added wake rotation becomes 4Δp=2ρV02a′(1+a′)λr2.

A practical relation between a and a′ is obtained by equating Eqs. () and () such that 5λr2=a(1-a)a′(1+a′), where λr=rRλ is the local tip speed ratio, with rR being the non-dimensional blade radius and λ=ΩRV0 being the tip speed ratio. In rotor disk theory, the rotor power coefficient, CP=P/(12ρV03A), with P being rotor power, is obtained by the following integral: 6CP=8λ2∫0λa′(1-a)λr3dλr.

In 1935, aerodynamicist Hermann Glauert approached the mathematical optimization problem of maximizing CP as a function of λ. Glauert's formal definition of the objective function f is given as 7f(a,a′)=a′(1-a).

For the sake of brevity, Glauert's detailed derivation is not presented here but is included in modified form in Appendix A. It concludes in a third-order polynomial for a(λr): 816a3-24a2+(9-3λr2)a+(λr2-1)=0, which can be solved iteratively using, for example, a Newton–Raphson algorithm. Glauert also found a simple expression for a′(a) that reads 9a′=1-3a4a-1.

Next, Glauert substituted the optimum solutions for a(λr) and a′(a) back into Eq. () and solved for the exact integral for the maximum power coefficient, CPmax, as a function of tip speed ratio, λ: 10CPmax=1λ2⋅(29)3[645x5+72x4+124x3+38x2-63x-12ln⁡x-4x]x2=1-3a2x1=14. where a2 is the corresponding solution of a(λr) when evaluating Eq. () at λ. The exact CPmax integral from Eq. () converges to the theoretical Betz limit at 1627≈0.5926 for high λ. The reader is referred to Appendix A for details.

In the present paper, Glauert's work is amended first by finding a simple alternative solution based on calculus of variations and formally showing the behavior for low and high λ and then by deriving corresponding exact integrals for thrust and bending moment coefficients, CT and CBe.

The pressure jump, Δp, generates a rotor thrust, T=ΔpA, across the rotor disk in the axial direction. In differential form, this becomes dT=ΔpdA=4πρV02a(1-a)rdr, using the definition of Δp from Eq. () and dA=2πrdr. A dimensionless rotor thrust coefficient is then defined as 29CT=T12ρV02A.

The incremental thrust coefficient, dCT, can be computed by dividing the incremental thrust, dT, by 12ρV02A, resulting in the following expression for dCT: 30dCT=8a(1-a)rRd(rR).

Recall that the definition for local tip speed ratio is λr=rRλ. Substituting the expression for non-dimensional blade radius, rR, into Eq. () leads to the final expression for dCT: 31dCT=8λ2a(1-a)λrdλr.

To write CT exclusively in terms of a, differentiating Eq. () yields a relation for λrdλr with 32λrdλr=3(4a-1)(1-2a)2(1-3a)2da,

such that an integral for CT can be written as 33CT=8λ2∫0λa(1-a)λrdλr=-24λ2∫0λa(1-a)(1-4a)(1-2a)2(1-3a)2da.

The same substitution as used by Glauert is carried through, where x=1-3a, resulting in 34CT=1λ2⋅8243∫x2x1[(1-x)(2+x)(1-4x)(1+2x)2x2]dx=-1λ2⋅8243∫x1x2(16x3+28x2-20x-25-1x+2x2)dx,

which can be easily integrated to yield the following: 35CT=1λ2⋅8243[4x4+283x3-10x2-25x-ln⁡x-2x]x2=1-3a2x1=14.

To better understand the behavior of the CT integral, Eq. () is rewritten again in terms of a. The integration bounds x1 and x2 are substituted in as 14 and (1-3a2), respectively, where λ2=λr2|a2 such that 36CT=(1-3a2)(1-a2)(1-4a2)2⋅8243[-13.3272-(4(1-3a2)4+283(1-3a2)3-10(1-3a2)2-25(1-3a2)-ln⁡(1-3a2)-2(1-3a2))].

As λr→0, or a2→14, there exists a singularity and CT is not defined (note: a similar behavior was found earlier for CPmax). Indeed the limit for CT as λr→0, or a2→14, becomes 37lim⁡a2→14CT=00.

Since evaluating this limit results in the indeterminate form of 00, L'Hôpital's rule can be applied to determine the true limit. The theorem equates the following limits, where functions f and g are differentiable: 38lim⁡a2→cf(a2)g(a2)=lim⁡a2→cf′(a2)g′(a2).

For ease of reference, the functions f and g extracted from Eq. () are explicitly stated below: 39f(a2)=8243[-13.3272-(4(1-3a2)4+283(1-3a2)3-10(1-3a2)2-25(1-3a2)-ln⁡(1-3a2)-2(1-3a2))],40g(a2)=(1-a2)(1-4a2)21-3a2.

Applying L'Hôpital's rule once results in the following as a2 approaches 14: 41lim⁡a2→14f′(a2)g′(a2)=lim⁡a2→14-24a2(16a24-36a23+28a22-9a2+1)(3a2-1)26(4a2-1)(4a22-4a2+1)(3a2-1)2=00.

It is valid to apply L'Hôpital's rule a second time, as functions f and g are differentiable. This yields 42lim⁡a2→14f′′(a2)g′′(a2)=lim⁡a2→14-80a24+144a23-84a22+18a2-112a22-10a2+2=34, where it is interesting to note that the thrust coefficient, CT, converges to 0.75 as λ→0. Though seemingly a surprising result at first, it will be reconciled with actuator disk theory in a later section.

Note again that the high tip speed ratio limit of Eq. () for λr→∞ (or a2→13) is more readily shown with 43lim⁡a2→13CT=lim⁡a2→132(1-a2)(1-4a2)2⋅8243=89, which is also fully consistent with actuator disk theory, as will be shown further below.

The bending moment, Be, is an important structural parameter when assessing the loading of wind turbine blades. A dimensionless bending moment coefficient is defined as 44CBe=Be12ρV02AR.

In differential form, dBe is essentially the product of thrust dCT and lever arm rR of the local annulus such that 45dCBe=dCT⋅rR=8λ2a(1-a)λrdλr⋅rR=8λ3a(1-a)λr2dλr, where Eq. () was used. The exact integral for the total bending moment coefficient, CBe, is hence defined as 46CBe=∫0λdCBe=8λ3∫0λa(1-a)λr2dλr.

Substituting a combination of Eqs. () and () yields 47CBe=-24λ3∫0λa(1-a)3/2(1-4a)2(1-2a)2(1-3a)5/2da.

Once more, the integration substitution is performed, where x=1-3a, so that CBe can be solved analytically to 48CBe=-1λ3⋅8243⋅271/2∫x1x2(1-x)(2+x)3/2(1-4x)2(1+2x)2x5/2dx=1λ3⋅8243⋅271/2[-24ln⁡((x+2)1/2+x1/2)-(x+2)1/2(192x6+408x5-532x4-890x3+585x2-260x+20)15x3/2]x2=1-3a2x1=14.

To better understand the behavior of the CBe function, Eq. () is rewritten solely in terms of a. This means substituting λ3=λr3|a2 in the denominator by raising Eq. (), which is evaluated at the integration bound a2, to the power 32 such that 49λ3=(1-a2)32(1-4a2)3(1-3a2)32.

The values for x1 and x2 are substituted in as well, being 14 and (1-3a2), respectively, where a2 is simply a(λr) such that 50CBe=-(1-3a2)3/2(1-a2)3/2(1-4a2)3⋅8243⋅271/2[2.5457-(-24ln⁡[(3-3a2)1/2+(1-3a2)1/2]-(3-3a2)1/2⋅(192(1-3a2)6+408(1-3a2)5-532(1-3a2)4-890(1-3a2)3)15⋅(1-3a2)3/2-(3-3a2)1/2⋅(585(1-3a2)2-260(1-3a2)+20)15⋅(1-3a2)3/2)].

As λr→0 (or a2→14), the bending moment coefficient, CBe, yields the following indeterminate form: 51lim⁡a→14CBe=f(a2)g(a2)=00.

For ease of reference, the functions f and g extracted from Eq. () are explicitly stated below: 52f(a)=8243⋅271/2[2.5457-((-24ln⁡[(3-3a2)1/2+(1-3a2)1/2]-(3-3a2)1/2⋅(192(1-3a2)6+408(1-3a2)5-532(1-3a2)4-890(1-3a2)3)15⋅(1-3a2)3/2-(3-3a2)1/2⋅(585(1-3a2)2-260(1-3a2)+20)15⋅(1-3a2)3/2)],53g(a)=-(1-a2)3/2(1-4a2)3(1-3a2)3/2.

Using L'Hôpital's rule once leads to the indeterminate form of 00 with f′(a2)=3359232a27-11757312a26+16166304a25-11127456a24+4041576a23-734832a22+52488a231321-3a252(3-3a2)1/2,g′(a2)=9(1-a2)1/24a2-124a22-4a2+11-3a252, and 54CBe=lim⁡a2→14f′(a2)g′(a2)=00.

Applying L'Hôpital's rule a second time still results in the indeterminate form of 00 as f′′(a2)=120932352a28-519001344a27+925888320a26-893765664a25+509553504a24-175414896a23+35429400a22-3779136a2+15746431321-3a272(3-3a2)3/2,g′′(a2)=3456a25-7776a24+6624a23-2736a22+558a2-45(1-3a2)7/2(1-a2)1/2, and 55CBe=lim⁡a2→14f′′(a2)g′′(a2)=00.

Differentiating functions f and g a third time results in a definite value for the limiting case of CBe with 56f′′′(a2)=3265173504a29-16597965312a28+36147435840a27-44231007744a26+33530384160a25-16363658880a24+5158048248a23-1017532368a22+114791256a2-566870431321-3a2923-3a252,57g′′′(a2)=10368a26-31104a25+36288a24-21312a23+6642a22-1026a2+631-3a2921-a232, and 58CBe=lim⁡a2→14f′′′(a2)g′′′(a2)=12.

Note again that the high tip speed ratio limit of Eq. () for λr→∞ is more readily shown with 59lim⁡a2→13CBe=lim⁡a2→13-(3-3a2)12⋅43(1-a2)32(1-4a2)3⋅8243⋅2712=1627.

For the high λr limit, the value of CBe approaches 1627≈0.5926. Here it is interesting to note that both CPmax and CBe tend to the Betz limit as λr→∞, which is reconciled in the next section.

Figure  shows the three coefficients of interest, CPmax, CT, and CBe, and their tabulated values over a range of design tip speed ratios between 0 and 10. Note that Glauert's original work showed exclusively CPmax based on optimum flow conditions. Exact analytical solutions for CT and CBe, on the other hand, constitute an amendment to Glauert's original work.

Beginning with the high tip speed ratio limit of λ→∞, both power and bending moment coefficients, CPmax and CBe, converge to the Betz limit 1627≈0.5926. This may at first seem surprising for CBe; however, it is a direct consequence of the respective limit for the thrust coefficient, CT. In fact, CT→89 for λ→∞, which describes the limit of actuator disk theory with CT=4a(1-a)=89 for the optimum a→13. In addition, one consequence of a constant pressure jump, Δp, across the rotor disk is a linear thrust distribution dT=Δp2πrdr, whose center of pressure is known to be at 23rR. From this simple thought experiment, it is indeed understood that CBe=23CT=1627 (Betz limit) for λ→∞.

For the low tip speed ratio limit of λ→0, it is known that CPmax→0; however, it has been formally proven for the first time as part of this amendment using L'Hôpital's theorem. On the other hand, thrust and bending moment coefficients, CT and CBe, tend towards non-zero values. The thrust coefficient, CT, remains consistent with a non-rotating actuator disk such that CT=4a(1-a)=34 for a→14. The center of pressure stays at 23rR in the limit such that CBe=23CT=12, all of which is consistent with Fig.  and the limit of actuator disk theory. Note, however, that for rotor disk theory where 0<λ<∞, the ratio CBe/CT≈23, though not exactly, as a(λr)≠ constant in Glauert's solution (see Fig. ).

This work derived several amendments to Glauert's original optimum rotor disk solution. First, an alternative approach by means of calculus of variations was pursued to solve the underlying classical objective function for CPmax in wind turbine aerodynamics. Second, Glauert's optimum rotor disk solution was used to derive exact solutions for the thrust and bending moment coefficients, CT and CBe. Here, L'Hôpital's theorem was employed to determine the convergence behavior of all three coefficients for λ→0, while the high tip speed ratio limit of λ→∞ was more readily shown. Some surprising results included that both power and bending moment coefficients, CPmax and CBe, approach the Betz limit for λ→∞ and that thrust and bending moment coefficients, CT and CBe, have non-zero values for λ→0. Using a simple thought experiment, it was shown that all observations are indeed consistent with the limit of a uniformly loaded (constant pressure jump) rotor disk, whose validity holds for tip speed ratios typical of modern utility-scale wind turbines but has, due to its inherent assumptions, limited validity at very low tip speed ratio as discussed. Nevertheless, this work presents an interesting addendum to one of the pioneering works in wind turbine aerodynamics.