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Wind Energ. Sci., 5, 807–818, 2020

https://doi.org/10.5194/wes-5-807-2020

© Author(s) 2020. This work is distributed under

the Creative Commons Attribution 4.0 License.

Top-level rotor optimisations based

on actuator disc theory

Peter Jamieson

Centre for Doctoral Training in Wind and Marine Energy, University of Strathclyde, Glasgow, G1 1XW, UK

Correspondence: Peter Jamieson (peter.jamieson@strath.ac.uk)

Received: 11 September 2019 – Discussion started: 14 October 2019

Revised: 20 April 2020 – Accepted: 7 May 2020 – Published: 23 June 2020

Abstract. Ahead of the elaborate rotor optimisation modelling that would support detailed design, it is shown

that signiﬁcant insight and new design directions can be indicated with simple, high-level analyses based on

actuator disc theory. The basic equations derived from actuator disc theory for rotor power, axial thrust and

out-of-plane bending moment in any given wind condition involve essentially only the rotor radius, R, and the

axial induction factor, a. Radius, bending moment or thrust may be constrained or ﬁxed, with quite different

rotor optimisations resulting in each case. The case of ﬁxed radius or rotor diameter leads to conventional rotor

design and the long-established result that power is maximised with an axial induction factor, a=1/3. When the

out-of-plane bending moment is constrained to a ﬁxed value with axial induction variable in value (but constant

radially) and when rotor radius is also variable, an optimum axial induction of 1/5 is determined. This leads to

a rotor that is expanded in diameter 11.6 %, gaining 7.6 % in power and with thrust reduced by 10 %. This is

the low-induction rotor which has been investigated by Chaviaropoulos and Voutsinas (2013). However, with an

optimum radially varying distribution of axial induction, the same 7.6% power gain can be obtained with only

6.7 % expansion in rotor diameter. When without constraint on bending moment, the thrust is constrained to a

ﬁxed value, and the power is maximised as a→0, which for ﬁnite power extraction would require R→ ∞.

This result is relevant when secondary rotors are used for power extraction from a primary rotor. To avoid too

much loss of the source power available from the primary rotor, the secondary rotors must operate at very low

induction factors whilst avoiding too high a tip speed or an excessive rotor diameter. Some general design issues

of secondary rotors are explored. It is suggested that they may have the most practical potential for large vertical

axis turbines avoiding the severe penalties on drivetrain cost and weight implicit in the usual method of power

extraction from a central shaft.

1 Introduction

Two quite different innovative rotor concepts have been con-

sidered previously. These are the low-induction rotor and the

secondary rotor.

A low-induction rotor in optimal operation is designed

to operate with lower values of axial induction than 1/3,

the ideal value according to the according to basic actu-

ator disc (AD) theory to maximise power at a ﬁxed cho-

sen diameter. The primary motivation for the low-induction

concept is to lower the cost of energy in scenarios where

sacriﬁcing some power in reducing design induction val-

ues leads to relatively more signiﬁcant load reductions that

are of overall economic beneﬁt to the design. Discussion

of the low-induction concept appears in Johnson (2019),

where Christopher L. Kelly of Sandia National Laboratories,

in an unpublished presentation at the Wind Energy Science

Conference of 2017, had noted that the ﬁrst low-induction

design with constrained blade rotor bending moment was

due to Ludwig Prandtl and is reproduced in Tollmien et

al. (1961). Snel (2003) observed that when the power coefﬁ-

cient, Cp, is stationary at its maximum value associated with

an axial induction of 1/3, the thrust coefﬁcient, Ct, is still

strongly increasing. Simple actuator disc theory determines

that dCt

da=4

3when dCp

da=0. There is therefore, for a very

Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.

808 P. Jamieson: Top-level rotor optimisations based on actuator disc theory

small power penalty, a relatively large reduction in thrust,

and there are associated bending moments to be gained from

reducing induction levels a little below the theoretical opti-

mum for maximum power, and independent blade manufac-

turers have long been aware of this. The potential beneﬁts of

yet more radical reductions in induction to around 0.2 were

highlighted by Chaviaropoulos and Voutsinas (2013). Work

on low-induction rotors continued in the Innwind.EU project

(Chaviaropoulos et al., 2013), and structural design issues

of a low-induction rotor were reviewed by Chaviaropoulos

and Sieros (2014). Development of lower lift aerofoils that

may suit a very large rotor (Chaviaropoulos et al., 2015) was

of value in addressing the problems of stability of aerofoil

characteristics at a very high Reynolds number, a topic rel-

evant for all very large rotors, but aerofoil design require-

ments for low-induction rotors can be better deﬁned follow-

ing the optimisation of the spanwise distribution of induc-

tion. Low-induction designs with expanded rotor diameter

continued to be explored by Chaviaropoulos et al. (2013),

Bottasso et al. (2014) and Quinn et al. (2016), but again

all of this work including associated cost of energy analy-

ses was predicated on non-optimum largely constant span-

wise distributions of induction. The Technical University of

Denmark (DTU) 10 MW reference turbine was further de-

veloped as an International Energy Agency (IEA) reference

turbine by Borlotti et al. (2019). This work involved sophis-

ticated multi-variable numerical optimisations with complex

constraints, entirely appropriate for a detailed reference tur-

bine design, but with the result that the role of reduced in-

duction could not be clearly seen in isolation. However the

rotor diameter was regarded as a free variable, and this will

be shown to be of the essence of the low-induction concept.

The secondary-rotor concept involves extracting power us-

ing a rotor generator system mounted on the blades of an oth-

erwise conventional primary turbine. The secondary rotors

operate at high speed in much-elevated relative air speeds

leading to much smaller and lighter power conversion equip-

ment than with a conventional centre-shaft-based drivetrain.

This idea emerged in designs such as the space frame turbine

of Watson (1988) and the airborne system of Jack (1992),

where the driver was to have an ultra-lightweight wind tur-

bine. Thus the motivation for secondary-rotor systems has

always been to reduce drivetrain mass and potentially also

cost. The secondary-rotor concept was considered further

by St-Germain (1992) and Madsen and Rasmussen (2008).

Jamieson (2011) highlighted it as a possible solution to the

design challenge faced by large vertical-axis wind turbines

(VAWTs), where a very low optimum speed leads to high

drivetrain torque, weight and cost if the power is extracted

in the most usual way from the central shaft. Leithead et

al. (2019) employed secondary rotors for power take-off in

an innovative X-rotor VAWT design.

This paper shows that the low-induction and secondary-

rotor concepts have a common origin in basic optimisa-

tions derived from actuator disc theory. As already discussed,

these design concepts are not themselves new, but their fun-

damental connection to elementary actuator disc theory has

not previously been highlighted. More signiﬁcantly, once this

connection is made, it much facilitates high-level analyses

that can usefully guide preliminary design. A key assump-

tion in blade element momentum theory (BEM) is that the

rotor plane may be analysed as a set of annular rings that are

regarded as mutually independent. This enables AD theory

to be generalised to deal with a spanwise variation of induc-

tion. AD theory and BEM are very long established, and the

form of equations used here often follows Jamieson (2011).

The underlying actuator disc optimisations are now pre-

sented, followed by their application to more detailed anal-

yses guiding top-level design of the low-induction rotor and

secondary rotor respectively.

2 Basic optimisations from elementary actuator

disc theory

2.1 Actuator disc equations

Actuator disc equations for power, thrust and out-of-plane

bending moment as related to ambient wind speed, U0; air

density, ρ; and rotor radius, R, are presented in Table 1. The

coefﬁcients of power and thrust, Cpand Ct, depend only on

the axial induction factor, a, and are in widespread use. A

companion out-of-plane bending moment coefﬁcient, Cm, is

also deﬁned as in Jamieson (2011). The standard assump-

tion of blade element momentum theories is that each annular

ring of the actuator disc can be treated as independent. Thus,

when the axial induction varies radially, rotor area-averaged

values of the coefﬁcients may be deﬁned as in the right col-

umn of Table 1.

Three distinct optimisations are now considered with the

objective in each case of maximising power:

a. the rotor radius is ﬁxed and axial induction is to be de-

termined;

b. the out-of-plane bending moment is ﬁxed but rotor ra-

dius and axial induction are variable;

c. the rotor thrust is ﬁxed but rotor radius and axial induc-

tion are variable.

2.2 Optimisations with radially constant induction

The optimisations are ﬁrst considered in the context of an

axial induction that does not vary spanwise. Case (a) is

the familiar one where, with radius Rﬁxed and power

P∝a(1 −a)2, which is consequently maximised with a=

1/3. This represents conventional design and is the basis

of a reference design used in subsequent comparisons. In

the reference design, R=R0,P=P0,T=T0and M=

M0, where the reference values, P0,T0and M0are all

based on R=R0and a=a0=1/3. In case (b), the out-

of-plane blade bending moment is ﬁxed and M=M0=

Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020

P. Jamieson: Top-level rotor optimisations based on actuator disc theory 809

Table 1. Basic actuator disc equations for power, thrust and out-of-plane bending moment.

Variable Actuator disc Rotor power coefﬁcient Rotor power coefﬁcient

equation (radially constant axial (radially variable axial

induction) induction)

Power P P =0.5ρU 3

0πR2CpCp=4a(1 −a)2Cp=81

R

0

a(1 −a)2xdx

Thrust T T =0.5ρU2

0πR2CtCt=4a(1 −a)Ct=81

R

0

a(1 −a)xdx

Moment M M =0.5ρU 2

0πR3CmCm=8

3a(1 −a)Cm=81

R

0

a(1 −a)x2dx

0.5ρU 2

0πR3Cm=4

3ρU 2

0πR3a(1 −a)=8

27 ρU 2

0πR3

0, which

on solving for Ryields

R=(3M0

4ρU 2

0πa(1 −a))1/3

.(1)

Substituting for Rfrom Eq. (1), the power equation P=

0.5ρU 3

0πR2Cp=2ρU 3

0πR2a(1 −a)2becomes

P=2ρU 3

0π(3M0

4ρU 2

0πa(1 −a))2/3

a(1 −a)2.(2)

From Eq. (2), the power, P, now varies only with aand

P∝a1/3(1 −a)4/3.(3)

Differentiating Pin Eq. (3) to ﬁnd a maximum leads to

(1 −a)(1 −5a)=0, and hence Pis maximised at a=1/5.

Comparing with a standard rotor design, when a=1/5 and

Pis maximum,

R

R0=a0(1−a0)

a(1 −a)1/3

=1.116,(4)

P

P0=a(1 −a)2

a0(1−a0)2R

Rs2

=1.076,(5)

T

T0=a(1 −a)

a0(1−a0)R

Rs2

=0.896.(6)

As in Jamieson (2018), general trends of R,P,Mand Trela-

tive to unit values of the standard rotor are presented in Fig. 1.

The analysis indicates that a rotor designed for an axial

induction factor of 0.2 that is 11.6 % larger in diameter can

operate with 7.6 % increased power and 10 % less thrust yet

at the same level of blade rotor out-of-plane bending moment

as the baseline design. In case (c), the thrust is maintained

at a constant value, T0. Since power ∝R2a(1 −a)2and T∝

R2a(1−a) is constant, it is evident that the power P∝(1−a)

and is maximised as a→0. However, for the power to be

ﬁnite and positive when the axial induction and hence the

power coefﬁcient are zero requires R→ ∞.

Figure 1. Design parameters related to axial induction.

As opposed to the conventional solution of power take-off

from a central shaft, additional (secondary) rotors are set on

the blades or other support arms at a radial distance from the

central axis of the primary rotor, thereby experiencing a high

relative wind speed. The ideal optimisation at zero induction

and hence inﬁnite radius cannot be realised, but it will be

shown that very low induction values are feasible without un-

acceptably large secondary rotors. The secondary rotor may

be therefore be considered as an ultra-low-induction rotor. In

the system of Fig. 2, the torque reaction to the primary rotor

is provided by thrust on the secondary rotors, and a speciﬁc

value of thrust on each secondary rotor is therefore required

to optimise power extraction from the primary rotor. The sec-

ondary rotors are small, high-speed rotors, and the sum of

design torques of all secondary rotors can be much less than

the design torque associated with power take-off in the con-

ventional way from a central shaft. This property can offer

a solution to a key problem of large VAWT design where an

inherently lower shaft speed than any equivalent horizontal-

axis wind turbine (HAWT) puts a large premium on drive-

train torque, mass and cost.

https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020

810 P. Jamieson: Top-level rotor optimisations based on actuator disc theory

Figure 2. Rotor with secondary rotors.

3 Low-induction rotor design

For a radially constant axial induction distribution and ﬁxed

out-of-plane bending moment, M=M0, it was established

in Sect. 2 that a=0.2 maximises power, giving a 7.6 %

power gain for 11.6 % radius expansion compared to con-

ventional design. The question then arises of whether an op-

timised radially varying distribution of axial induction can

realise greater power gains or, for example, the same 7.6 %

power gain at reduced rotor expansion. Related to this is

the question of what may be a suitable, efﬁcient generalised

model of the radially variable axial induction. A representa-

tion in the form a(x)=a(1 −xn)pis found to be versatile

and highly effective. With arbitrary values of only two free

variables, nand p, a wide range of distributions can be gen-

erated (Fig. 3). This even includes approximations to con-

stant values of axial induction less than 0.333, for example,

a=0.2. The curve (yellow trace) of Fig. 3 illustrates this

although a much more accurate approximation than shown

can be obtained. More general optimisation methods could

be employed to determine optimum distributions of axial in-

duction subject to varied constraints, but the simple approach

adopted here is highly effective.

Now there can never be beneﬁt in a > 1/3 as the bend-

ing moment would be increased and power decreased. Also

as x→0, the bending moment M→0, and so in the limit

x→0, which is approaching the shaft centre, it is logical

that a→1/3 in any design that seeks to constrain only bend-

ing moment. In the following analyses, a,nand pare all

treated as free variables although, as expected, the value de-

termined for ais usually very close to 1/3. This tends to

conﬁrm that the optimisation, although in effect having only

two free variables, nand p, is quite accurate. Polynomial

representations by comparison are far inferior. A quadratic,

for example, a2x2+a1x+a0, with a0=1/3, would have two

free variables, a2and a1, but could only represent linear or

parabolic shapes. In order to have results that are likely to be

realistic for typical rotors with small ﬁnite blade numbers, a

Figure 3. Distributions of axial induction for arbitrary choices on n

and p.

tip loss effect is introduced using the Prandtl tip loss factor,

F(x)=2

πacosne−(1−x)Bλ

2(1−a)o. The question of an overall max-

imum in power regardless of required diameter expansion is

now addressed. Using the generalised forms of Cpand Cm

from Table 1, the power is expressed as

P(a, n, p)=

4ρπ U 3

0M

2

3

0

1

R

0

a(1−xn)p1−a(1−xn)p2xF (x)dx

(4ρπ U 2

0

1

R

0

a(1−xn)p1−a(1−xn)px2F(x)dx)2/3.(7)

Using a maximisation routine such as available in PTC Math-

cad 15, an overall maximum in power P(a,n,p) is obtained,

with values a=0.331, n=1.504 and p=1.125 giving an

axial induction distribution as in Fig. 4. The gain in power

(see Fig. 4, Pmax) is found to be 11.9 %, which is much

greater than the 7.6 % for a radially constant axial induc-

tion but requiring a radial expansion of 34%. This is too

large a radial expansion to be of practical beneﬁt considering

the implications in increased tip speed or drivetrain torque.

In the next analysis the radial expansion is constrained (see

Fig. 4, Pcon) to a value such that the power gain is 7.6 % as

for optimum constant induction. The associated axial induc-

tion distribution has parameters a=0.333, n=0.417 and

p=0.136 as illustrated in Fig. 4. Note that all the distri-

butions of Fig. 4 maintain the same constant value of out-

of-plane bending moment at the shaft centre line. The strik-

ing result however is that this same power gain of 7.6 %

is realised with a radius expansion of only 6.7 % (diamond

marked point of Fig. 5) as opposed to the 11.6 % (triangular

marked point of Fig. 5) required with a constant axial induc-

tion of 0.2.

Also shown in Fig. 5 is the ratio of power gain to expan-

sion which maximises around 3 % expansion. Above this low

level, the required rotor expansion rises more rapidly than the

gain in power although the most economic beneﬁt will prob-

Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020

P. Jamieson: Top-level rotor optimisations based on actuator disc theory 811

Figure 4. Axial induction distributions giving rise to the same out-

of-plane bending moment, M0, at rotor centre.

Figure 5. Power gain related to rotor radius expansion ratio.

ably arise with power gains and rotor expansions in a 5% to

10 % range.

Comparing (see Fig. 4) the optimum axial induction distri-

bution (for 7.6 % power gain) with the constant value of 0.2,

it is evident that more power is being obtained over most

of the span except near the blade tip. Consistent with these

higher power levels, there is only a 3.5% reduction in ax-

ial thrust for the radially variable axial distribution (Pcon in

Fig. 4) as opposed to approximately 10 % reduction for con-

stant induction at a=0.2. If cost of energy modelling sug-

gests that there is beneﬁt say from reduced wake impacts in

a thrust reduction greater than 3.5 %, say at the same power

gain of 7.6 %, with appropriate constraints on the power max-

imisation procedure, the necessary rotor expansion can then

be related to thrust reduction as in Fig. 6.

Tip loss has no effect in comparing distributions where the

axial induction is constant radially because it cancels in the

power, moment and thrust ratios, provided the low-induction

rotor is compared with a reference rotor having the same

number of blades. It has a small effect (Fig. 7) for designs

with rotor expansions below about 15% and a more notice-

able effect at large expansion ratios which however may be

of little practical interest.

The distributions in Fig. 7 are very similar and that is what

matters most. On account of the sensitivity of the power law

relationships, the associated values of nand pwill often dif-

fer considerably. For no tip loss n=0.416 and p=0.136,

Figure 6. Rotor expansion related to thrust reduction for a ﬁxed

power gain (7.6 %).

Figure 7. Effect of tip loss on optimum axial induction distributions

for a power gain of 7.6 %.

with tip loss n=0.295 and p=0.112. Another main issue of

practical relevance is that blades are never aerodynamically

active near the shaft centre line. They may become cylindri-

cal near the root contributing only drag and connect to a hub

having a conical cover or spinner. To approximate the loss of

aerodynamic performance in the hub area, some of the anal-

yses were repeated, with lower limits on integrals such as in

Eq. (7) changed from 0 to 0.15. As with tip loss, effects were

only very noticeable at large (impractical) expansion ratios.

Figure 8 compares the results for maximum possible power

gain with and without exclusion of the ﬁrst 15 % of span.

Table 2 presents data relating to axial induction distribu-

tions of Fig. 8. Although the power gains differ only ∼1 %,

there is a noticeable difference in the axial induction distribu-

tions of Fig. 7 and a large difference in the rotor expansions at

34 % for the complete span being aerodynamically active and

25 % when the innermost 15 % of span is excluded. When

designs in a more realistic range of parameters are consid-

ered, for example, as in Fig. 7 with power gain restricted to

7.6 %, there is no signiﬁcant difference between cases with

and without exclusion of the inner 15 % of the rotor.

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812 P. Jamieson: Top-level rotor optimisations based on actuator disc theory

Figure 8. Axial induction distributions for maximum power gain.

Table 2. Parameters of the distributions for maximum power gain.

Fraction of span inactive aerodynamically 0 0.15

a0.331 0.333

n1.504 1.130

p1.125 0.674

Radius expansion factor 1.343 1.246

Power gain 1.119 1.109

4 Secondary-rotor design

4.1 Power extraction using secondary rotors

Secondary rotors near the tip of HAWT blades will experi-

ence a higher relative ﬂow velocity and may thus be smaller

in diameter than those of a VAWT of similar rated power.

However the tip region of a large HAWT is subject to large

deﬂections and a torsional stiffness that is relatively reduc-

ing with upscaling. Thus reacting the total edgewise load of

a blade near the tip may pose problems for aerodynamic sta-

bility and structural stiffness. Even more problematic may

be preserving alignment of secondary rotors on a pitching

blade. The classic issues with VAWTs which had led to

them being uncompetitive historically are (a) an intrinsically

lower optimum speed leading to factors of 2 or 3 on driv-

etrain torque, weight and cost and (b) reduced power per-

formance associated with intrinsically lower average lift-to-

drag ratios per cycle of rotation leading to maximum power

coefﬁcients ∼0.4 when large HAWTs have power coefﬁ-

cients ∼0.5. Power take-off using secondary rotors may

avoid the torque penalty intrinsic in a conventional VAWT

design, providing a more effective drivetrain solution that

may breathe new life into VAWT technology.

For these reasons the focus in the following analyses is

on secondary rotors for a primary rotor of VAWT design al-

though much of the analyses are directly relevant or easily

adapted to HAWT design. The secondary rotors are always

assumed to be HAWTs. In the following analyses upper case

symbols refer to a primary rotor and lower case to a sec-

ondary rotor. Where there are multiple secondary rotors, the

parameters of 1 of nrotors will have the subscript n. The

aerodynamic torque of the primary rotor is reacted by the to-

tal thrust of the secondary rotors acting (under the present

simpliﬁed assumptions) with a moment arm at the maximum

radius R0of the primary rotor. The relative wind speed inci-

dent on the secondary rotors is equal to the tip speed of the

VAWT ∼40 m s−1, and as a further simpliﬁcation, the ambi-

ent wind speed which is small in comparison is ignored. The

power generated by the primary rotor is then

P=ntnR0.(8)

The total power, p, extracted by the secondary rotors is then

p=ntnR0(1 −a).(9)

Now with the usual assumption that each annular ring of

the actuator disc can be analysed independently, then Eq. (9)

applies to the elemental power and thrust contributions of

each annulus, and a radially varying axial induction, a(x),

will have exactly the same performance as a constant induc-

tion of a, the area-averaged value of a(x). For this reason

only radially constant values of axial induction are consid-

ered although, in a detailed design embracing all aspects of

structure and loads, there may be some beneﬁts from radi-

ally varying axial induction. This result is of course quite

different from the case of the low-induction rotor where the

bending moment is constrained and radial variation of axial

induction is very signiﬁcant. As an example, to focus dis-

cussion of secondary-rotor design issues, parameters as in

Table 3 are selected for a VAWT rated at 5MW.

4.2 Sizing of secondary rotors

The power produced by the primary (VAWT) rotor is P=

0.5ρU 3

0(2R0L)CP, and the total power extracted by nsec-

ondary rotors is p=npn=P(1−a). For each secondary ro-

tor, pn=0.5ρ(R0)3πr 2

n4a(1−a). Hence the ratio of radius

of one of nsecondary rotors to that of the primary rotor can

be expressed as

rn

R0=LCP

2nπλ3a(1 −a)R00.5

.(10)

The ratio of secondary- to primary-rotor radius deﬁned by

Eq. (10) is shown in Fig. 9 as based on the data of Table 3.

The curve is symmetrical about a=0.5 although this is not

obvious as a logarithmic scale is employed in order to show

more clearly the variation of rn/R0at very low axial in-

duction values. The vertical line of Fig. 9 marks a=0.333.

There is no interest in greater values of a, and the optimum

design value for an effective system will certainly be much

less than 0.333 as this would imply a sacriﬁce of 1/3 of

primary-rotor power. In the data of Table 3 the number of

secondary rotors is chosen as six, which may be two on each

of three blades or three on each of two. A value of aof 0.05

is chosen for further illustration of secondary-rotor design is-

sues. This implies a sacriﬁce of 5 % of primary-rotor power,

Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020

P. Jamieson: Top-level rotor optimisations based on actuator disc theory 813

Table 3. Parameters of a primary Htype VAWT rotor and of secondary HAWT rotors.

Primary Secondary Unit

Ambient wind speed U0m s−1

Rated power P5/{η(1 −a)}pn5/n MW

Design tip speed Vt40 vt160 m s−1

Rotor power coefﬁcient CP0.4 Cp4a(1 −a)2

Rotor thrust coefﬁcient CTCt4a(1 −a)

Rotor radius R065 rnm

Rotor angular speed ωnrad s−1

Design tip speed ratio 34λ ωrn/ R0=4

Blade length L100 m

Drivetrain efﬁciency η

Number of rotors 1 n6

Rotor thrust T tnN

Rotor torque Q nR0tnqnpn/ωnN m

Blade chord cnm

Figure 9. Ratio of secondary-rotor radius to that of primary rotor.

and the associated radius fraction is 0.084. Each secondary

rotor then has a radius (∼5 m) that is 8.4 % of the primary-

rotor radius (60 m).

4.3 Torque beneﬁt of secondary rotors

A major issue with large VAWTs especially is a very high

level of drivetrain torque. In a conventional drivetrain solu-

tion with power take-off from a central shaft, the torque, Q,

of the primary rotor would drive mass and cost of the drive-

train. To assess the beneﬁt in secondary-rotor power take-off

the ratio of the sum of secondary-rotor torques to Qis now

compared.

nqn

Q=pn

ωnQ=(1 −a)

nωn=(1 −a)rn

λR0

(11)

For a design with a=0.05 and parameters otherwise as

in Table 3, the torque ratio nqn

Q=(1−a)rn

λR0has a value

0.95×0.084

4=0.02 showing that the sum of secondary-rotor

torques is ∼1/50th of primary-rotor torque. As a power

take-off system, each secondary-rotor system comprises both

bearings and generator but also an aerodynamic rotor sys-

tem. The estimates of secondary-rotor diameter and torque

reduction factor are realistic, provided it is accepted that at

a=0.05 the fraction of available primary-rotor power ex-

tracted will be less than 95 % to an extent, depending on the

effect of parasitic drag losses. For conventional large HAWTs

and possibly more so for VAWTs, rotor cost is generally less

than the drivetrain cost, but even at ratios 2/50th, 3/50th or

much more, there are potentially very large savings in cost

and weight of power conversion with secondary rotors. The

further beneﬁts of multiple rotors are in rnreducing as 1/√n,

with the torque ratio of Eq. (11) similarly reducing.

4.4 Design characteristics of secondary rotors

Does the design of the secondary rotor differ much from con-

ventional HAWT designs considering the unusually high rel-

ative wind speed and unusually low design levels of axial

induction? This is initially assessed by deriving an equation

for rotor solidity. From Jamieson (2011) a non-dimensional

lift distribution, with Cldas design lift coefﬁcient (lift value

at maximum lift-to-drag ratio), is determined as

cnCld

rn=8πa(1 −a)F(x)

Bλ(1 +á)p(1 −a)2+λ2x2(1 +á)2.(12)

In Eq. (12) the tangential induction factor, á, is determined

as

á≡á(x)=4a−4a2+λ2x20.5−λx

2λx .(13)

Considering an annular ring of the rotor swept area of span-

wise width, dr, the local solidity is the sum of planform el-

emental areas of Bblades within the ring as a ratio of the

complete swept area of the ring. Thus the local solidity at

radius ris given as

https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020

814 P. Jamieson: Top-level rotor optimisations based on actuator disc theory

σn(r)=Bcndr

2πrdr=Bcn

2πr ,(14)

and the solidity of the whole rotor is then

σn=2

πr2

n

rn

Z

0

πrσn(r)dr=2

1

Z

0

xσn(x)dx

=2

1

Z

0

4a(1 −a)F(x)

xλCld(1 +á)p(1 −a)2+λ2x2(1 +á)2dx . (15)

The right-hand side of Eq. (15) is obtained using Eq. (12) to

substitute for cnin Eq. (14). A tip loss factor, F(x), appropri-

ate to a three-bladed rotor is applied; the inner rotor region

where the solidity would become inﬁnite is omitted; and a

typical aerofoil design lift coefﬁcient, Cld, of 0.8 is assumed.

An estimate of secondary-rotor solidity with a=0.05 and

otherwise consistent with the values of Table 3 is determined

as

σn=

1

Z

0.15

8a(1 −a)F(x)

xλCld(1 +á)p(1 −a)2+λ2x2(1 +á)2dx=0.072.(16)

The dependence of rotor solidity on aerofoil design lift coef-

ﬁcient is illustrated in Fig. 10. An aerofoil such as NACA 63-

418 has been used on wind turbines and (with some variation

according to data sources) may provide a lift-to-drag ratio

of ∼125 at Cld∼1. According to Fig. 10 this may yield a

solidity ∼6 % at a design axial induction ∼0.05, which is

only a little higher than values of 4 %–5 % most common

in large HAWT designs. Thus the secondary rotor need not

differ much from conventional designs of large HAWTs in

respect of solidity. Light loading from a very low design ax-

ial induction value and very high relative ﬂow velocities have

mutually compensating impacts on rotor solidity, whereas a

secondary-rotor design for the usual design values of axial

induction, a∼1/3, would have solidity ∼30 %.

The next consideration for secondary-rotor design is the

range of Reynolds number, Re. For a solidity ∼0.07 as in

Eq. (16), the chord at around 80 % span will be

cn∼σnrn

0.8B=0.072 ×5

0.8×3=0.15,(17)

and the associated Reynolds number is

Re =0.8ρvtcn

µ=0.8×1.225 ×160 ×0.15

1.8×10−5=1306667.(18)

Considering the high tip speed of the secondary rotor, us-

ing vtas the resultant velocity in the estimate of Eq. (18), and

by implication neglecting the ambient wind speed, will give

a good approximation. Equation (18) shows that Re values

of the secondary rotor will be in a normal range for medium

to large HAWTs although the rotor diameter is small ∼10 m.

Figure 10. Rotor solidity related to design axial induction and de-

sign lift coefﬁcient.

Another important design consideration is the level of op-

erational loads on the secondary rotor. Assuming a rated

wind speed of Ur=11 m s−1, and a relative wind speed for

the secondary rotors of 160 m s−1, then, compared to a con-

ventional rotor of similar diameter, rotor thrusts and out-of-

plane bending moments are both in the same ratio:

tn

T0=mn

M0=v2

ta(1 −a)

U2

ra0(1−a0)=1602×0.05 ×0.95

112×0.333 ×0.667 =45.2.(19)

This is a huge increase in steady operational loading com-

pared to conventional design. Also the steady and turbulent

components of the ambient wind speed will introduce cyclic

and random disturbances to secondary-rotor inﬂow, which

may increase available power (Leithead et al., 2019) but will

inevitably introduce fatigue loading. Now it is vital for the

secondary rotors to minimise parasitic drag in the hub region

as torque from this will absorb power from the primary rotor

that cannot be recovered. It is of no beneﬁt to have a spin-

ner that may deﬂect the central ﬂow outwards, augmenting

ﬂow over the inboard blade sections, and, equally, it is of no

beneﬁt to have ducted secondary rotors that produce any ﬂow

augmentation. This is because any augmentation contributes

to added thrust (drag) on the spinner or the duct that will con-

sume irrecoverable primary-rotor power. This suggests that

the secondary-rotor system may beneﬁt from having blades

of more ideal proﬁle than is usual near the hub centre line not

because any very signiﬁcant gain in secondary-rotor power

can be obtained but in order to minimise drag in that area. In

this scenario the blades would twist to near 90◦out of plane,

bringing the blade roots very close each other and to the axis

of rotation. The large chord widths nearly parallel to the axis

would be exploited for structural strength of the whole rotor,

which would most probably use a lot of carbon in its con-

struction and have titanium leading-edge erosion protection.

Another idea aiming to reduce parasitic drag, perhaps too far-

fetched, would be to engineer a rotor generator system with

a hollow centre although there would still be issues of drag

on the internal surfaces.

Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020

P. Jamieson: Top-level rotor optimisations based on actuator disc theory 815

Figure 11. Twin rotor secondary-rotor system.

4.5 Secondary rotors on a common axis

Returning to actuator disc theory, the idea of twin rotors

counterrotating on a common axis enabling a doubling of

relative velocity at the generator air gap has been considered

(Shen, 2017, and Rosenberg et al., 2014). According to sim-

ple actuator disc theory, the ideal maximum Cpwith the twin

rotors in series, assuming they are sufﬁciently apart for com-

plete pressure recovery near the downstream rotor, increases

from the Betz limit of 0.593 only to 0.64 (see Newman, 1986)

or decreases to 0.32 if the swept area of both rotors is ac-

counted for. The situation is very different for very lightly

loaded secondary rotors (Fig. 11) where the downstream ro-

tor may operate almost as efﬁciently as the upstream. The

potential beneﬁt of a secondary-rotor pair in a series arrange-

ment is not only that the design torque and weight of the

power train may be reduced compared to a single equiva-

lent rotor but perhaps that a slimmer generator and hence a

slimmer centre body with less parasitic drag may be realised.

Any kind of multi-rotor, secondary-rotor system has obvi-

ous advantages in torque and weight reduction, but having a

physical arrangement of support structure and connection to

the primary rotor that minimises parasitic drag will be very

important.

Based on wind tunnel tests on actuator discs represented

as porous screens, Newman (1986) concluded that his theory

for multiple actuator discs in series, in the particular case of

two actuator discs, became inaccurate only at spacings closer

than a disc radius. This gives conﬁdence that at the very low

disc loadings applicable to a pair of secondary rotors in se-

ries, spaced about a diameter apart, there should be complete

pressure recovery between the rotors. A single rotor of radius

5 m could be replaced by two rotors side by side as in a multi-

rotor arrangement of radius 5/√2=3.536 m. When the ro-

tors are twins in series on the same axis, the radius required

to have the same total thrust at an equivalent axial induction

of 0.05, thereby extracting 95 % of primary-rotor power, is

related to the velocity recovery approaching the downstream

twin. In the analysis following, pressure recovery is assumed

and the velocity approaching the downstream turbine is taken

as the far wake velocity of the upstream turbine, R0{1−δ},

where the velocity deﬁcit ratio is δand would be 2a=0.1

for a single ideal actuator disc in inviscid ﬂow. The axial in-

duction factors are selected in an optimisation constrained so

that the twin rotors provide the speciﬁc total thrust required

for primary-rotor power extraction and also extract the same

total power as a single secondary rotor with the design axial

induction value, a=0.05. This is accomplished as follows.

The thrust, t1, on a single rotor that would be replaced by the

twin system is proportional to the square of the radius, r1; the

square of the relative velocity, Vt=R0; and a thrust coef-

ﬁcient based on the axial induction, a=ae=0.05. Consider

now the equivalent twin rotor system, with axial induction au

on the upstream turbine, adon the downstream turbine, rela-

tive velocity Vton the upstream turbine and Vt{1−2(1−z)au}

on the downstream turbine. The wake velocity deﬁcit ratio is

δ=2(1 −z)au, where zis a factor measuring the extent of

velocity recovery being 0 when, as for a single actuator disc

far wake, the deﬁcit is 2aand 1 if there is complete velocity

recovery. For the twin to produce the same total thrust as the

single rotor with thrust, t1, requires

r2

1ae(1−ae)=r2

uau(1−au)+r2

dad(1−ad)

{1−2(1 −z)au}2.(20)

In addition, if the same total power is required, then, with

power being proportional to the square of the radius, to the

power coefﬁcient and to the cube of the relative velocity,

r2

1ae(1−ae)2=r2

uau(1−au)2+r2

dad(1−ad)2

{1−2(1 −z)au}3.(21)

For given values of z, Eqs. (20) and (21) are solved with the

additional assumption that the upstream and downstream ro-

tors have the same radius, ru=rd, that is to be minimised.

The results in Fig. 12 show the variation of secondary-

rotor radius; upstream rotor induction factor, au; and down-

stream rotor induction factor, ad, with velocity recovery fac-

tor, z. Conventional wake models, such as assessed in a com-

parative study of velocity deﬁcit by Luong et al. (2017), sug-

gest little velocity recovery will take place between rotors 2

to 3 radii apart. However such models may be conservative

and it is also difﬁcult to gauge their applicability. The very

high relative wind speed would imply a very low turbulence

intensity, which would not assist velocity recovery. However,

the loading on the secondary rotors is necessarily very light

to avoid too much loss of primary-rotor power, and the weak

wake may be skewed by centrifugal force. Quite close spac-

ings ∼1 radius may be beneﬁcial because of the interaction

of the rotating wake which is not accounted for in any sim-

ple actuator disc modelling. A considerable amount of re-

search into various counterrotating rotor systems has taken

place since Newman (1986). Tests on a small 6 kW contra-

rotating rotor discussed in Shen et al. (2017) indicated that,

at the relative high loadings of conventional turbines, 30 %

more power (as opposed to 8 % on the basis of an ideal Cp

of 0.593 rising to 0.64) can be obtained. Numerical mod-

elling (also Shen et al., 2017) of a counterrotating pair of

https://doi.org/10.5194/wes-5-807-2020 Wind Energ. Sci., 5, 807–818, 2020

816 P. Jamieson: Top-level rotor optimisations based on actuator disc theory

Figure 12. Secondary-rotor radius for no power loss related to ve-

locity recovery factor.

Nordtank 500 kW wind turbines using the EllipSys3D code

developed at the Technical University of Denmark (DTU)

with reference to a particular site predicted 43.5 % more en-

ergy than for a single turbine. None of the existing literature

considers the very light loadings appropriate to a pair of sec-

ondary rotors, but experiments and computational ﬂuid dy-

namics (CFD; Koehuan et al., 2017) analyses generally pro-

vide encouragement that performance in real ﬂow will ex-

ceed, sometimes greatly exceed, the performance predicted

by simple actuator disc inviscid ﬂow models. Even with little

velocity recovery where the required diameter of the twin ro-

tors approaches that of a single equivalent rotor, there may be

still be net advantage from lighter blade loading, lower gen-

erator torque and reduced generator diameter with associated

reduced centre body drag. The velocity recovery that may oc-

cur is evidently speculative and may only be better assessed

by CFD modelling of a speciﬁc design arrangement.

5 Concluding remarks

Three quite distinct design directions have emerged from op-

timisations relating to basic loads predicted by actuator disc

theory. These are (a) conventional design with rotor radius

predetermined, which has been used as a reference; (b) the

low-induction rotor arising from constraint on out-of-plane

bending moment; and (c) the secondary-rotor concept aris-

ing from constraint on rotor thrust loading.

In comparison to conventional design, the design chal-

lenges in realising a low-induction rotor are not radically

new. The present work highlights that the power gain in rela-

tion to required rotor expansion (a cost) and thrust reduction

(a beneﬁt for turbine loads and wind farm wake impacts) is

sensitive to the radial distribution of axial induction and dis-

cusses optimisation around these factors. In particular it is

shown that the same power gain of 7.6% with an optimum

radially constant axial induction of 0.2 that required a rotor

expansion of 11.6 % can be achieved with an expansion of

only 6.7 % when the axial induction varies radially and is

optimised. The modelling developed here enables the deﬁni-

tion of a space of all self-consistent combinations of power

gain, rotor expansion and thrust reduction with each associ-

ated axial induction distribution. This could enable a prelim-

inary determination of an overall optimum axial induction

distribution using a combined wind turbine and wind farm

cost of energy model.

An expanded rotor of standard design could be operated

at low induction using pitch control, thereby restricting the

steady-state blade root bending moment, but this would not

be satisfactory. It is vital to contain all loads of the expanded

rotor, steady state, dynamic and loads when idling in extreme

wind conditions by limiting the lift and drag of the rotor to

the levels of the non-expanded reference rotor. This calls for

lower lift aerofoils or reduced solidity or both. There is much

less of a design challenge in the low-induction rotor with a ra-

dially varying optimised axial induction distribution (Fig. 4)

as compared to the constant induction of 0.2. The required ro-

tor expansion is much less, and the progressive reduction of

axial induction towards the blade tip is sympathetic to blade

structural design with a natural taper in strength and solidity

from rotor to tip. The graded reduction in spanwise axial in-

duction is also much more favourable than a global reduction

to 0.2 for limiting tip deﬂection to maintain acceptable tower

clearance without having undue added cost in stiffening the

blade.

Overall the results suggest there may be great value in

treating the axial induction distribution and rotor diameter as

free variables in a basic system optimisation for the lowest

cost of energy where direct power gains, rotor loading and

reduced wake effects from thrust reduction can all be traded

in the design optimisation.

Secondary rotors have not been used on an operational

wind turbine although a design is now being developed (Leit-

head et al., 2019). The main aim in using secondary rotors is

to have a drivetrain with much reduced design torque com-

pared to the usual transmission system based on power take-

off from a central shaft. That can certainly be achieved, with

torque reduction of 1 to 2 orders of magnitude being pos-

sible depending on speciﬁc design choices. Although the

design of secondary rotors is much more demanding than

that of conventional rotors of the same diameter, the design

torque reduction is so great that it seems certain that sub-

stantial savings in drivetrain cost can be realised. The fo-

cus of the secondary-rotor design exploration is on VAWTs

as the primary rotor rather than HAWTs because it solves

a key problem with VAWTs of relatively low shaft speed

leading to high drivetrain torque and expensive drivetrains,

whereas, as applied to HAWT design, it could introduce ma-

jor problems for primary-rotor blade design. It emerges that

the radial distribution of axial induction is not critically im-

portant for secondary-rotor design as all distributions with

the same area-averaged axial induction will lead to the same

size of secondary rotor. The high relative wind speed com-

pensates for relatively small rotor diameter and very low de-

Wind Energ. Sci., 5, 807–818, 2020 https://doi.org/10.5194/wes-5-807-2020

P. Jamieson: Top-level rotor optimisations based on actuator disc theory 817

sign axial induction in a way that for primary rotors in the

multi-megawatt range maintains a Reynolds number ∼106

and suggests a solidity only a little higher than that typi-

cal of large HAWTs is required. However with secondary

rotors, very high tip speeds are desirable to limit drivetrain

torque and to limit the overall scale of the rotor and generator

system. Also steady-state operational loads are exceptionally

high in relation to rotor diameter. Having multiple secondary

rotors (more than one per blade) has the usual beneﬁts of

multi-rotors (Jamieson, 2011) in reducing net torque, weight

and cost of secondary-rotor systems, but, as was mentioned,

it is particularly important with secondary rotors to minimise

losses from parasitic drag or degradation of primary blade

performance depending on their physical mounting arrange-

ment. The idea of realising multi-rotors as a twin set in series

on a common axis looks promising considering the very low

axial induction levels required of secondary rotors to avoid

wasting primary-rotor power. Whether this is a particularly

good idea cannot be resolved without evaluating speciﬁc de-

sign arrangements and developing a greater understanding of

the ﬂow ﬁeld around the twin secondary-rotor system.

The preliminary evaluation of the X-rotor VAWT design

(Leithead et al., 2019) suggests that use of secondary rotors

will lead to more competitive VAWT designs. Another in-

novative VAWT design, the DeepWind VAWT of Paulsen et

al. (2015), has major savings through integration of the ro-

tor blade shaft and support structure into a single element.

On the other hand, substantial challenges remain for the de-

sign and maintenance of the underwater electrical generating

system. Could an adapted variation of this design with modu-

lar secondary rotors that can form a more economical power

train to be accessed and maintained above sea level be ad-

vantageous?

In summary, three quite different rotor optimisations are

shown to arise naturally from long-established actuator disc

equations and can usefully guide high-level design of the in-

novative rotor systems described as the low-induction rotor

and the secondary rotor.

Data availability. No data sets were used in this article.

Supplement. The supplement related to this article is available

online at: https://doi.org/10.5194/wes-5-807-2020-supplement.

Competing interests. The author declares that he has no conﬂict

of interest.

Special issue statement. This article is part of the special issue

“Wind Energy Science Conference 2019”. It is a result of the Wind

Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.

Financial support. The author acknowledges the support of the

Centre for Doctoral Training in Wind and Marine Energy of the

University of Strathclyde, Glasgow.

Review statement. This paper was edited by Katherine Dykes

and reviewed by two anonymous referees.

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