aerodynamic optimization of coaxial rotor in hover and axial flight

AERODYNAMIC OPTIMIZATION OF COAXIAL ROTOR INHOVER AND AXIAL FLIGHT

Omri Rand , Vladimir KhromovFaculty of Aerospace Engineering

Technion, Israel Institute of TechnologyHaifa, 32000, Israel

Keywords: coaxial rotor, blade-element momentum theory, optimization, calculus of variation

Abstract

The paper presents an aerodynamic optimizationof a coaxial rotor system in hover and axial flight.For that purpose, a suitable theorem of calcu-lus of variations that provides a new generic re-sult is employed. The analysis is founded onthe Blade Element-Momentum Theory, which isbased on nonlinear aerodynamics and mutual (in-teractional) influences that are introduced by theinduced downwash field of each rotor over theother. The influence of non-uniformity of thedownwash distribution on the optimal design wasalso studied. The analysis supplies an importantinsight into the optimal design and efficiency ofcoaxial rotor systems. The formulation is con-sistently generic and enables the exploration of awide range of coaxial configurations.

Nomenclature

A Disc square = πR2.cd Drag coefficient.cl Lift coefficient.CP Power coefficient (= P/

(ρAΩ3R3

). All

parameters belong to the rotor under discussion.CT Thrust coefficient (= T/

(ρAΩ2R2

). All

parameters belong to the rotor under discussion.h Rotor’s clearance of the coaxial system.k Influence coefficient.Nb Number of blade.R Disc radius.r Spanwise location along the blade of the rotor

under discussion (0 < r < R).rw Upper rotor wake radius.T Rotor thrust, positive upwards.Vt Tip speed (= ΩR).

α Angle of attack.γ Numerical coefficient in the expressions

for k.ζR Coaxial system rotors radius ratio.ζΩ Coaxial system angular velocities ratio.η Lagrange multiplier.θpitch Blade pitch angle.θtw Blade twist angle.κ Induced power correction coefficient.λC Nondimensional climb velocity of the

coaxial system (normalized by ΩU RU ).λ()C Nondimensional “equivalent” climb ve-

locity (normalized by ΩR of the rotor un-der discussion).

λi Nondimensional induced velocity (nor-malized by ΩR of the rotor under discus-sion).

ρ Air density.Ω Rotor angular velocity.

()U Quantity related to the upper rotor.()L Quantity related to the lower rotor.()LI Quantity related to the inner part of the

lower rotor.()LO Quantity related to the outer part of the

lower rotor.( ) Nondimensional length quantity, normal-

ized by R.( ) Averaged quantity.()∗ Time averaged quantity.

1

OMRI RAND , VLADIMIR KHROMOV

INTRODUCTION

Different approaches to the aerodynamic analysisand optimization of coaxial helicopter rotors havebeen developed over the years, see e.g. Refs. [1],[2], [3], [4], [5]. A classical and well known re-sult shows that in hover and axial flight, an opti-mal isolated rotor is a one with chord and twistthat decrease as 1/r where r is the spanwise lo-cation along the blade. Such a design providesa uniform inflow distribution and minimizes boththe induced and the parasite power. Yet, simi-lar analysis for a coaxial rotor system is muchmore involved and the literature contains no ex-plicit closed-form formulation for the problem ofaerodynamically optimal coaxial system. In mostexisting coaxial rotor analyses, the mutual influ-ence between the two rotors is accounted for byprescribed downwash models.

OPTIMIZATION METHODOLOGY

Modeling and Fundamental Assumptions

The present paper deals with the optimal designof a coaxial rotor system in hover and axial climb.Fig. 1 shows a coaxial system of two concentric

Uiλ C

Uλ

CLIλ

LIiλ h

UR

LR

UΩ

LΩ

z

C LO LOiλ λ

Uiλ C

Uλ

CLIλ

LIiλ h

UR

LR

UΩ

LΩ

z

C LO LOiλ λ

Fig. 1 Coaxial rotor system in hover and climb.

rotors with a clearance, h, that rotate in oppositedirections. The rotors are not necessarily identi-cal in all parameters including their radius, rota-tional speed, number of blades, chord and airfoil

distribution, etc. (i.e. RU 6= RL and ΩU 6= ΩL,NU

b 6= NLb , cU(r) 6= cL(r) in the general case). We

therefore define:

ζR =RL

RU ; ζΩ =ΩU

ΩL ,

where for a constant tip speed we write ζR =ζΩ = ζ. We also define the spanwise location ofthe upper rotor’s wake radius at the lower rotorplane (i.e. the wake contraction ratio or in otherwords the ratio of the upper rotor tip vortices ra-dius at the lower rotor level over the upper rotorradius) as

rUw =

Rw

RU ; rLw =

Rw

RL =rU

wζR

.

In this model, the upper rotor model takesinto account the lower rotor induced velocity (inaddition to the entire configuration climb veloc-ity) as “equivalent climb speed” and similarly,the lower rotor model takes into account theupper rotor induced velocity as an “equivalentclimb speed” as well. The present model isdeveloped and presented in two parallel courses.The first course is founded on a simplifiedversion of the above described interaction andtherefore will be founded on mean (constant)influence coefficients. The second course will befocused on deriving a detailed description of theabove described mutual influences.

The derivation of the above models wasfounded on the following sets of assumptions:

The simplified model:

a) It was assumed that the lower rotor down-wash over the upper rotor disc area is uniform.Note that in some existing models it is evenassumed to be small enough to be neglected, seee.g. Ref.[4].

b) The velocity induced by the upper rotorover the lower one is confined to a uniformdownwash over its inner part (which is inside the

2

Aerodynamic Optimization of coaxial Rotor in Hover and Axial Flight

upper rotor’s wake). The upper rotor’s inducedvelocity over the outer part of the lower rotor isneglected.

The detailed model:

a) Experimental data and numerical freewake studies show that the downwash of thelower rotor over the upper rotor disc is notuniform, it has a relatively large value at thehub and it tends to diminish towards the discperimeter. The effect of such non-uniformitywill be demonstrated in what follows.

b) The upper rotor’s induced velocity overthe outer part of the lower rotor is usually ofsmall magnitude.

c) Generic distribution of the upper rotordownwash over the lower inner rotor disc areawill be studied. In some models it is assumedto be nonuniform and distributed along theradius as it is distributed over the upper rotorradius. Consequently, in hover, it is assumedthat λLI

C (rL)= kULλUi(rL/rU

w)

where 0< rL < rUw .

Assumptions that will be applied to both models:

a) The effect of the interaction of the upperrotor’s wake with the lower rotor’s wake isneglected.

b) Sharp boundary between the region in thelower rotor that is inside the upper rotor wakeand that which is outside it is assumed. Despitethe sharp changes in the induced velocity andthe effective angles of attack, two-dimensionalanalysis is assumed to remain valid in thistransition area as well.

The Mutual Interaction Between the Rotors

As indicated above, the upper rotor is submergedin the downwash that is induced by the lower ro-tor. This downwash is written as kLU λL

iζRζΩ

where

kLU is an influence coefficient and λLi is the av-

eraged nondimensional induced velocity over thelower rotor. In general, kLU is a function of rU ,and therefore, a constant (mean) value of kLU

stands for uniform influence of the lower rotoron the upper one. Similarly, the inner part ofthe lower rotor (which is inside the upper ro-tor’s wake) is submerged in the downwash thatis induced by the upper rotor and is written askULλU

iζΩ

ζRwhere kUL is an influence coefficient

and λUi is the averaged nondimensional induced

velocity over the upper rotor. Clearly, kUL is afunction of rL. Hence in the general case, theequivalent climb velocities over the upper and thelower rotors are:

λUC (r) = kLU(r)λL

iζR

ζΩ

+λC, (1)

λLC(r) =

[kUL(r)λU

i +λC] ζΩ

ζR. (2)

λC is the climb velocity of the system (normal-ized by the tip speed of the upper rotor). Notethat λL

i and λUi are the averaged induced velocity

over the disc areas and are therefore expressedas:

λUi = 2

∫ 1

0rλ

Ui (r)dr; λ

Li = 2

∫ 1

0rλ

Li (r)dr.

The simplified model: In the simplifiedmodel discussed above, we assume that kLU isconstant, and kUL is constant for rL < rL

w andvanishes for rL

w < rL < 1. As will be provedlater on, in such a case, λU

i (r) of the optimaldesign turns to be also constant. Similarly,λL

i (r) becomes constant for rL < rLw (and will

be denoted λLIi there), while it takes a value of

different constant for rLw < rL < 1 (and will be

denoted λLOi there). Hence, in such a case, the

equivalent climb velocities are given by:

λUC = (3)

kLU(

rLw)2

λLIi +

[1−(rL

w)2]

λLOi

ζR

ζΩ

+λC,

and

λLIC =

[kUL

λUi +λC

] ζΩ

ζR, (4)

3


λLOC = λC

ζΩ

ζR. (5)

Estimation of the influence coefficientsand wake contraction ratio in hover

The estimation of the influence coefficientskUL(rL) and kLU(rU) and the wake contractionratio rL

w will be discussed in what follows bytwo stages. For the simplified model, we shallfirst deal with the averaged values of kUL(rL) andkLU(rU) and rL

w as functions of the rotors clear-ance h = h/R. Later on, a more detailed distri-butions of the above coefficients and a modifiedwake contraction function will be discussed forthe detailed model.

The estimation of the averaged values is de-scribed in details in Ref.[3] with comparison toexperimental data. Subsequently, the averagedinduced velocity are given by

kUL = 1+(

d√1+d2

)γUL

; kLU = 1−(

d√1+d2

)γLU

where d =∣∣∣h/RU

∣∣∣ for the upper rotor and d =∣∣∣h/RL∣∣∣ for the lower rotor, γUL = 0.6 and γLU =

0.3÷ 0.5 - see Fig. 2. Hence, momentum argu-ments show that the wake contraction is given by

rUw =

√1

kUL .

Fig. 3 presents a free wake geometry of a sin-gle rotor. The wake contraction is clearly ob-served. The induced velocity of such a wake isnot uniform in the general case. The resulting in-fluence coefficients in such a case are expressedas:

kUL(rL) =λ∗L(rL)

λUi

ζΩ

ζR

; kLU(rU) =λ∗U(rU)

λLi

ζRζΩ

where λ∗U(rU) and λ∗L(rL) are the time averagedvalues as functions of the radial stations.

0r =

0.3γ =

0.6γ = 0.3γ =

0.5γ = 1.0γ =

0

i

i d

λλ

=

d

Test data “Region of interest” averaged data 0.75R data

(Biot-Savar t law based analysis, )0r =

Above rotor Below rotor

0.3γ =

0.6γ = 0.3γ =

0.5γ = 1.0γ =

0

i

i d

λλ

=

d

Fig. 2 Averaged induced velocity distribution be-low and above a single rotor.

The Proposed Optimal Coaxial System Analy-sis

The proposed analysis is based on a calculusof variations theorem that exploits the BladeElement-Momentum Theory with nonlinear aero-dynamics including mutual rotor influences andrigorously solves the problem of aerodynami-cally optimal coaxial rotor.

For aerodynamically optimal rotor, two basicconditions should be fulfilled: (a) the induced ve-locity distribution should be the one that mini-mizes the induced power; (b) each cross-sectionshould work in its optimal angle of attack to max-imize its cl

/cd

. The optimization of the inducedpower results in the induced velocity distribu-tion which thereafter, together with the parasitepower minimization, leads to the optimal chordand twist distributions.

Induced Power: It is well known that con-stant induced velocity minimizes the inducedpower for a single rotor that is subjected to a uni-form (along the blade span) external downwash.Assuming that kLU is constant, or in other words,the “equivalent climb” of the upper rotor (dueto the lower rotor) is uniform, the above condi-tion of constant induced velocity may be applied.However, for the lower rotor, the external down-wash induced by the upper rotor is not uniform

4


wrwr

Fig. 3 Free Wake Geometry in Hover.

and exhibit a large change between the regionwhich is inside the upper rotor’s wake and thepart which is outside the upper rotor’s wake. Inother words, kUL(rL) is not constant (and as al-ready discussed, the most simplified acceptableassumption may be kUL = c1 for rL < rL

w andkUL = c2 for rL > rL

w, where c1 and c2 are con-stants).

To rigorously obtain the optimal conditionwhen kUL is not constant, a new calculus of vari-ations based optimal condition which is based onthe Blade Element-Momentum Theory has beenderived. To clarify and support the following dis-cussion, it should be mentioned that the BladeElement-Momentum Theory determines the in-duced velocity by equating the thrust distribu-tion per unit length as separately obtained by the“momentum” and the “blade element” theories,namely:

4πρr (VC + vi)vi =12

NbρΩ2r2ccl(α),

where the effective angle of attack is given by

α = θpitch +θtw(r)− tan−1(

λC +λi

r

).

In a nondimensional form we write the above as

8π(λC +λi)λi = Nbrc cl(α), (6)

where in the general case, the nondimensionalchord, c, and the nondimensional induced veloc-ity, λi, are functions of r. In the coaxial analy-sis case, and as will be shown in what follows,λC should be considered as function of r as well.Note that cl is also a function of r due to thenonuniform distribution of airfoils, the variationin the effective angle of attack, α, and the de-pendency of the aerodynamic characteristics onMach and Reynolds numbers.

Consequently, one may express the thrust andinduced power coefficients as

CT = 41∫

0

r (λC +λi)λidr, (7)

CPi = 41∫

0

r(λC +λi)2λidr. (8)

For a given climb velocity distribution, λC(r), theinduced power optimization task is focused onthe selection of λi(r), that for a given thrust coef-ficient, CT , as given by Eq.(7), will minimize CPias given by Eq.(8). For that purpose we employthe calculus of variations technique and adopt theminimization process of an integral of the form

J =

1∫0

F(r,λi,dλi

dr)dr,

while for the present case of optimization withconstraint (minimum induced power for a giventhrust) we define:

F =CPi +ηCT ,

where η is a Lagrange multiplier. Hence, J takesthe form:

J =

1∫0

4r(λC +λi)

2λi +η [4r (λC +λi)λi]

dr.

The above integrand shows that λi(r) shouldfulfill the following Euler equation

∂F∂λi

=∂

r(λC +λi

)2λi +η [r (λC +λi)λi]

∂λi

= 0

5


or

r(λ

2C +ηλC +4λCλi +2ηλi +3λ

2i)= 0. (9)

Parasite Power: As far as the parasite poweroptimization is concerned, each blade cross-section should operate in its optimal effectiveangle of attack, αopt , where the ratio cl/cd ismaximal, and, hence, we define copt

l = cl(αopt),copt

d = cd(coptl ).

Blade Design, Thrust and Power

Chord Distribution: In the general case, Eq.(6)shows that

cX(r) =8π[λX

C(r)+λXi (r)

]λX

i (r)NX

b

1rcopt

l (r), (10)

where 0 < rX < 1 and X = L,U .The simplified case: For the upper rotor

where λUC is constant, the above Euler equation

(Eq.(9)) shows that λUi should be constant as

well. Thus, Eq.(6) shows that the chord distri-bution should take the form:

cU(r) =8π(λU

C +λUi)

λUi

NUb

1rcopt

l (r)(11)

for 0 < r < 1, while for the lower rotor, the aboveEuler equation shows that different constant val-ues should be assigned to the inner and outerparts of the lower rotor, and therefore:

cLI(r) =8π(λLI

C +λLIi)

λLIi

NLb

1rcopt

l (r),

for (0 < r < rLw) and

cLO(r) =8π(λLO

C +λLOi)

λLOi

NLb

1rcopt

l (r)

for (rLw < r < 1). As will be shown in the exam-

ples presented in what follows, an introduction ofrelatively small cut-out removes the singularity atr = 0 but practically does not damage the optimalefficiency.

Twist Distribution: To ensure that αXopt(r)

matches the required distribution over the entireupper and lower blades (as dictated by the air-foil distribution and Mach and Reynolds numberdistributions) and by requiring

(θX

tw)

r= 34= 0, the

twist distribution and the pitch angle are givenby:

θXtw(r)=α

Xopt(r)−θ

Xpitch+tan−1

(λX

C(r)+λXi (r)

r

),

θXpitch =

(α

Xopt)

r= 34+tan−1

((λX

C)

r= 34+(λX

i)

r= 34

3/4

)where X = L,U .

The simplified case: for the upper

θUtw(r) = α

Uopt(r)−θ

Upitch + tan−1

(λU

C +λUi

r

),

θUpitch =

(α

Uopt)

r= 34+ tan−1

(λU

C +λUi

3/4

).

Similarly, to ensure that αLopt(r) matches the

required distribution over the entire lower bladeand by requiring

(θL

tw)

r= 34= 0, the twist distribu-

tion over the lower rotor and its pitch angle aregiven by:

θLItw(r) = α

Lopt(r)−θ

Lpitch + tan−1

(λLI

C +λLIi

r

),

θLOtw (r) = α

Lopt(r)−θ

Lpitch+ tan−1

(λLO

C +λLOi

r

),

θLpitch =

(α

Lopt)

r= 34+ tan−1

(λLY

C +λLYi

3/4

),

where Y ⇒ I for rL

w > 3/4

Y ⇒ O for rLw < 3/

4.

As already indicated regarding the chord dis-tribution, in the actual design the twist does notreach infinite value at r = 0.

6


Thrust: In the general case Eq.(7) should beemployed.

The simplified case: Since λUi and λU

C areconstants, the thrust of the upper rotor is givenby, see Eq.(7),

CUT = 2

(λ

UC +λ

Ui)

λUi . (12)

Similarly since λLIi , λLO

i , λLIC and λLO

C are con-stants, for the lower rotor we define

CLT =CLI

T +CLOT ,

where

CLIT = 2

(λ

LIC +λ

LIi)

λLIi ·(rL

w)2,

CLOT = 2

(λ

LOC +λ

LOi

)λ

LOi

[1−(rL

w)2].

Induced Power: In the general case Eq.(8)should be employed.

The simplified case: Using Eq.(8), the powercoefficients for the upper rotor is defined as

CUPi = 2

(λ

UC +λ

Ui)2

λUi ,

and therefore by employing Eq.(12)

CUPi =

(λ

UC +λ

Ui)

CUT . (13)

Similarly, for the lower rotor:

CLpi =

(λ

LIC +λ

LIi)

CLIT +

(λ

LOC +λ

LOi

)CLO

T .

Parasite Power: The parasite power

Pd = NbΩ

∫ R

0

12

ρΩ2r2cr copt

d dr,

may be written in its nondimensional form as:

CXPd =

NXb

2π

∫ 1

0r3cX(r)copt

d (r)dr,

where X =U,L.

The simplified case: In the simplified case,using Eqs.(7,11,12)

CUPd = 2CU

T

∫ 1

0

coptd (r)

coptl (r)

r2dr, (14)

which for a uniform blade becomes

CUPd =

23

CUT

coptd

coptl

. (15)

Similarly for the lower rotor as:

CLPd =

2CLIT

(rLw)

2

∫ rLw

0

coptd (r)

coptl (r)

r2dr+

2CLOT

1− (rLw)

2

∫ 1

rLw

coptd (r)

coptl (r)

r2dr.

The downwash distribution effect

Figure 4 shows various downwash distributionsalong the blade while the average downwash ve-locity is the same for all cases. The table in this

(0)Cλ (1)

Cλ (2)Cλ

(3)Cλ

(4)Cλ

(5)Cλ

Cλ

rR

(0) (1) (2) (3) (4) (5)

(0)

(0)

(0)

1 0.997 0.974 0.997 0.994 0.9971 0.948 0.880 1.051 1.075 1.0401 0.988 0.957 1.007 1.008 1.004

C C C C C C

Pi Pi

Pd Pd

P P

C CC CC C

λ λ λ λ λ λ

(0)Cλ (1)

Cλ (2)Cλ

(3)Cλ

(4)Cλ

(5)Cλ

0.10

0.02

0.08

0.06

0.04

0.2 0.4 0.6 0.8 1.00.0

Cλ

rR

(0) (1) (2) (3) (4) (5)

(0)

(0)

(0)

1 0.997 0.974 0.997 0.994 0.9971 0.948 0.880 1.051 1.075 1.0401 0.988 0.957 1.007 1.008 1.004

C C C C C C

Pi Pi

Pd Pd

P P

C CC CC C

λ λ λ λ λ λ

Fig. 4 Various downwash distributions along theblade with a comparison of the required power.

figure shows the induced and parasite power re-quired for the same thrust value. Note that foreach downwash distribution - a different optimalrotor design is required.

Figure 5 shows the effective angle attack forvarious downwash distributions along the bladewhen the blades were design for uniform (aver-age) downwash velocity distribution. The result-ing power values in this case are clearly higherthan those shown in Fig. 4.

7


rR

, (0) (1) (2) (3) (4) (5)

(0)

(0)

(0)

1 1.021 1.145 1.021 1.047 1.027

1 1.012 1.058 1.013 1.020 1.007

1 1.019 1.129 1.020 1.042 1.024

C C C C C C

Pi Pi

Pd Pd

P P

C C

C C

C C

(0)C

(1)C

(2)C

(3)C

(4)C

(5)C

Optimal lookup table values

20

10

0.2 0.4 0.6 0.8 1.00rR

, (0) (1) (2) (3) (4) (5)

(0)

(0)

(0)

1 1.021 1.145 1.021 1.047 1.027

1 1.012 1.058 1.013 1.020 1.007

1 1.019 1.129 1.020 1.042 1.024

C C C C C C

Pi Pi

Pd Pd

P P

C C

C C

C C

(0)C

(1)C

(2)C

(3)C

(4)C

(5)C

Fig. 5 Effective angle of attack for various down-wash distributions.

THE SOLUTION SCHEME

In the general case, the above described optimiza-tion of a coaxial rotor system is expressed as anonlinear system of 4 equations an unknowns,namely:

fi(x1 . . .x4) for i = 1 . . .4

where the unknowns are x1 = λUi and x2 = λL

iand x3 = ηU x4 = ηL. For a given set of thesefour values, the following calculations should beexecuted:

a) Calculate λUC (r) and λL

C(r) by Eqs.(1,2).b) Calculate λU

i (r) and λLi (r) from Eq.(9) as:

λXi (r) =

13

(−Aλ +

√Bλ

)where

Aλ = 2λXC(r)+η

X

Bλ =(λ

XC(r)

)2+η

Xλ

XC(r)+

(η

X)2

and X =U,L.

(c) Calculate the residual functions:

f1 = CT −CTOTALT

f2 = CUPi +CU

Pd−[CL

Pi +CLPd] ζ5

R

ζ2Ω

f3 = λUi −2

∫ 1

0rλ

Ui (r)dr

f4 = λLi −2

∫ 1

0rλ

Li (r)dr

Thus, by employing a nonlinear solver, thevalues of xi that yield fi = 0 are found.

The simplified case:In the simplified case, the above optimization

problem may be expressed as the following non-linear algebraic system of 9 equations and un-knowns:(

λUC)2

+ηU

λUC +4λ

UC λ

Ui +2η

Uλ

Ui +3

(λ

Ui)2

= 0,(λ

LIC)2

+ηLλ

LIC +4λ

LIC λ

LIi +2η

Lλ

LIi +3

(λ

LIi)2

= 0,(λ

LOC

)2+η

Lλ

LOC +4λ

LOC λ

LOi +2η

Lλ

LOi +3

(λ

LOi

)2= 0,

λUC = kLU

(rL

w)2

λLIi +

[1−(rL

w)2]

λLOi

ζR

ζΩ

+λC,

λLIC =

[kUL

λUi +λC

] ζΩ

ζR,

CUT = 2

(λ

UC +λ

Ui)

λUi ,

CLT =CLI

T +CLOT ,

CTOTALT =CU

T +CLT

ζ4R

ζ2Ω

,

CUPi +CU

Pd =[CL

Pi +CLPd] ζ5

R

ζ2Ω

, (16)

where the unknowns are

CUT ,C

LT ,λ

UC ,λ

LIC ,λLO

C ,λUi ,λ

LIi ,λLO

i ,ηU ,ηL,

and

λLOC = λC

ζΩ

ζR.

Note that CTOTALT represents the total required

trust and that the last equation of Eqs.(16) standsfor the torque balance condition which is requiredin a coaxial system.

Illustrative Limiting Cases

At this stage it is worth mentioning few limitingcases:

a) For very closed rotors (zero clearance,h/R = 0) two identical rotor are first discussed.

8


Each rotor produces the same thrust (CT/2) andinduced velocity (λi) and is submerge in the otherrotor’s downwash which should be considered as“equivalent climb”. Hence, in this case, kUL =kLU = 1 and rU

w = 1 and the induced velocity fora rotor in climb is

λi =−λC

2+

√λ2

C4

+CT/2

2

Substituting λC = λi yields:

λi =12

√CT

2

Hence the total inflow, 2λi =√

CT2 , is identical

to the one produced by a single rotor of CT . Theinduced and parasite power components become(see Eqs.(13,15)),

CPi = 2× (λi +λC)CT

2=

√C3

T2

CPd = 2× 23

CT

2copt

d

coptl

. (17)

which is exactly the value created by a singlerotor of CT .

b) For tandem configuration, where two iden-tical rotors, each produce thrust of CT/2 andkUL = kLU = 0

λi =

√CT/2

2,

and hence,

CPi = 2×λiCT

2=

1√2

√C3

T2

and CPd is identical to the one shown in Eq.(17).

Rotor Efficiency

Two different quantities may be used for theexamination of rotor efficiency. The first one

is the rotor’s Figure of Merit (FM) that isdefined as the ratio of the ideal induced power(Pi =

√T 3/(2ρA)) over the total power. In all

cases discussed, T stands for the total system’sthrust and A stands for the total disc areas in thesystem. Yet, as will be shown later on, a muchmore important parameter is the (dimensional)power loading defined by the ratio of the thrustover the power.

Single non-optimal rotor: For a (non-optimal) rotor of constant chord one maywrite

T =Nb

6ρΩ

2cclR3; PD =Nb

8ρΩ

3ccdR4,

where cl and cd are the averaged drag and lift co-efficients given by

cl =3

R3

∫ R

0r2cldr; cd =

4R4

∫ R

0r3cddr.

Hence, PD = T 34Vt

cdcl

where Vt = ΩR, and

FM =

√TA

κ

√TA + 3

4√

2ρVtcdcl

.

κ is a correction that applies to the actual inducepower in a non-optimal rotor. It mainly accountsfor the inflow nonuniformity:

κ =1√2

∫ 10 rλ3

i (r)dr[∫ 10 rλ2

i (r)dr]3/2

.

Substituting a linear variation of λi in the aboveyields a value of κ' 1.13, which is close to typ-ical measured values.

The ratio of the thrust over the power is thengiven by

TP=

FM√T

2ρA

=

√2ρ

κ

√TA + 3

4√

2ρVtcdcl

.

Single optimal rotor: Based on Eqs.(13,14),one may write for a single optimal rotor

FM =λi

λi +2∫ 1

0copt

d (r)copt

l (r)r2dr

,

9


where λi =√

CT2 . By noting that T

A =CT ρV 2t , we

write

FM =

√TA√

TA +2Vt

√2ρ

∫ 10

coptd (r)

coptl (r)

r2dr,

which yields (note that κ = 1 in this case)

TP=

FM√T

2ρA

=

√2ρ√

TA +2Vt

√2ρ

∫ 10

coptd (r)

coptl (r)

r2dr.

Coaxial system: assuming constant tipspeed, Vt (where ζR = ζΩ = ζ), we write the“T/A” and “T/P” quantities as

TU +T L

AU +AL =1

1+ζ2

(CU

T +CLT ζ

2)ρV 2

t ,

TU +T L

PU +PL =CU

T +CLT ζ2

CUP +CL

Pζ21Vt.

which yields

FM =

√1

2ρ

TP

√TA

=

√1

2(1+ζ2)

(CU

T +CLT ζ2) 3

2

CUP +CL

Pζ2 .

For identical rotors (ζ = 1), this value maybe interpreted as the ratio between the ideal in-duced power required by a single (isolated) rotorof area 2A (or, since tip speed is assumed con-stant, two (isolated) rotors each of area A) to theactual required power in a coaxial system.

TYPICAL RESULTS

Figure 6 presents power loading vs. disc loadingfor a non-optimal rotor, optimal rotor and a coax-ial optimal system. In this figure, ρ = 1.2Kg/m3,Vt = 180m/s, copt

d /coptl = cd/cl ' 1/40 (a typical

value of NACA0012) and h/R = 0.2.It is first shown that FM is increasing with

T/A for all configuration while T/P is decreas-ing. Hence FM is certainly not a comprehensiveenough criterion for efficiency. Three working

2Disc Loading, N m

Power Loading, N W

( )*T A( )* 2T A

Optimal-Non optimal

or

( ) 0.2Coaxial optimal h R =

1.0FM =0.8FM =

h

50 100 500 1000

0.20

0.10

0.04

0.06

0.08

2Disc Loading, N m

Power Loading, N W

( )*T A( )* 2T A

Optimal-Non optimal

T/2 T/2

AA

A

T

T

A

2A

Tor

( ) 0.2Coaxial optimal h R =

1.0FM =0.8FM =

h

Fig. 6 Power loading vs. Disc loading for non-optimal rotor, optimal rotor and a coaxial system.

point are shown in Fig. 6: a single rotor with atypical disc loading, (T/A)∗, a tandem configu-ration with an extra identical rotor (and thereforewith a half of the disc loading) and a coaxial opti-mal system with the same disc loading (T/2A)∗.Clearly, moving from a single to a tandem con-figuration yields a much higher T/P ratio due tothe decreasing disc loading. The contribution ofthis paper at this particular context is in the rig-orous evaluation of the interference effects thatreduce this T/P ratio to a lower level as shownfor a coaxial system in Fig. 6.

Fig. 7 presents the variation of the optimal an-gle of attack in a typical NACA0012 airfoil asfunction of the radial station. This variation ismainly due to the Mach number variation alongthe blade. As shown, the exact data that is ob-tained from the (discrete) tables is piecewise-constant and is not continuous due to the em-ployed lookup table technique. It is thereforesmoothed and feeded into the analysis.

Figs. 8 presents the variation of cl vs. angleof attack and Mach number for the NACA0012airfoil. The red cross-curves show the optimalvalues that maximize its cl

/cd

.Fig. 9 presents the torque coefficient of a sin-

gle optimal rotor compared with a variety of typ-ical rotors of constant chord (in the range of 5 -8% of the radius) and linear twist (in the range

10


,α °

r R

Lookup table values

Smoothed values

,α °

7.5

5.0

0.5 1.00.0 r R

Fig. 7 Optimal angle of attack variation as func-tion of the radial station.

of 5 - 30). In all optimal rotor calculations,the (optimal) twist and chord distributions werenot changed while cut-out was introduced. Asshown, optimal conditions may not be achievedwith constant chord and linear twist. Note that inthis example, for a cut-out that is larger than 0.3,the optimal values lose their applicability.

Similar comparison for a coaxial configura-tions is presented in Fig. 10. As expected, op-timal conditions may not be achieved with con-stant chord and linear twist in this case as well.

Fig. 11 presents a comparison of the inducedvelocity distribution in optimal and non optimalcoaxial configuration. As shown, in the optimalcase (with and without cut-out), the induced ve-locity is uniform for the upper rotor and piece-wise uniformly distributed for the lower rotor. Inthe case of constant chord and linear twist, the in-duced velocity is a function of the radial station.

Typical chord and twist distributions for opti-mal coaxial rotor are present in Fig. 12,13.

Finally, Fig. 14 presents two typical singleand coaxial optimal rotor designs.

CONCLUSION

An aerodynamics optimization methodology fora coaxial rotor system in hover and axial flightthat is based on clear and basic physical reason-

lOptimal c

lc

M,α º

1

0

-1

0.80.6

0.40.2

-20

20

0

lOptimal c

lc

M,α º

Fig. 8 Lift coefficient of the example airfoil data.

ing, is offered. The optimization study is evalu-ated by suitable theorem of calculus of variationsand provides a new generic result. The method-ology is based on introducing the interactionalinflow fields into the Blade Element-MomentumTheory while using real nonlinear aerodynamictables.

REFERENCES

[1] Nagashima T. and Nakanishi K. Optimum per-formance and wake geometry of a co-axial rotorin hover. Vertica, 7(3):225–239, 1983.

[2] Bourtsev B.N., Kvokov V.N., Vainstein I.M., andPetrosian E.A. Phenomenon of a coaxial heli-copter high figure of merit at hover. In 23rd Eu-ropean Rotorcraft Forum, Germany, 1997.

[3] McAlister K.W., Tung C., Rand O., KhromovV., and Wilson J.S. Experimental and numeri-cal study of a model coaxial rotor. In AdditionalProceedings Papers of the American HelicopterSociety 62nd Annual Forum, Phoenix, Arizona,2006.

[4] Leishman J.G. and Ananthan S. An optimumcoaxial rotor system for axial flight. Journal ofthe American Helicopter Society, 53(4):366–381,2008.

[5] Juhasz O., Syal M., Celi R., Khromov V., RandO., Ruzicka G.C., and Strawn R.C. A comparisonof three coaxial aerodynamic prediction methodsincluding validation with model test data. In Pro-

11


Optimal

QQ

cut outx R−

0.05 0.08 5 30

0.0065Twist

T

Range of chord R RRange of

C

θ== ° °

≅

……

Optimal

QQ Optimal chord & twist, non-zero cut-out

0.0 0.1 0.2 0.3 0.4

1.0

1.1

1.2

Optimal single rotor

Chord = const, linear twist, non-zero cut-out

cut outx R−

0.05 0.08 5 30

0.0065Twist

T


C

θ== ° °

≅

……

Fig. 9 Torque coefficient of a single optimal andnon-optimal rotors.

ceedings Papers of the 2010 AHS Aeromechan-ics Specialists’ Conference, San Francisco, CA,2010.

Copyright Statement

The authors confirm that they, and/or their company ororganization, hold copyright on all of the original ma-terial included in this paper. The authors also confirmthat they have obtained permission, from the copy-right holder of any third party material included in thispaper, to publish it as part of their paper. The authorsconfirm that they give permission, or have obtainedpermission from the copyright holder of this paper, forthe publication and distribution of this paper as part ofthe ICAS2010 proceedings or as individual off-printsfrom the proceedings.

cut outx R−

0.05 0.08 5 30

0.013Twist

T


C

θ== ° °

≅

……

Optimal

QQ

0.0 0.1 0.2 0.3 0.4

1.0

1.1

1.2

Optimal coaxial rotor

Optimal chord & twist, non-zero cut-outChord = const, linear twist, non-zero cut-out

cut outx R−

0.05 0.08 5 30

0.013Twist

T


C

θ== ° °

≅

……

Optimal

QQ

Fig. 10 Torque coefficient of a coaxial optimaland non-optimal rotors.

iλ

r R

( )( )( )

0.0, ,

0.2 , ,

0.2 , 12.5 , 0.06

CO

CO

CO

tw

tw

tw

x opt c r opt

x R opt c r optx R c r R

θ

θ

θ

= = =

= = =

= = ° =

COxLower inner

Upper

Lower outer

iλ

0.08

0.06

0.04

0.02

0.0 0.5 1.0

r R

( )( )( )

0.0, ,

0.2 , ,

0.2 , 12.5 , 0.06

CO

CO

CO

tw

tw

tw

x opt c r opt

x R opt c r optx R c r R

θ

θ

θ

= = =

= = =

= = ° =

COx

Fig. 11 Comparison of the induced velocity dis-tribution in optimal and non optimal configura-tion.

12


r R

c R

Upper

Lower

r R

c R0.2

0.1

0.5 1.00.0

Fig. 12 Coaxial optimal rotor chord distribution.

r R

,Twist °

Upper

Lower

r R

15

10

0.5

1.00

5

,Twist °

Fig. 13 Coaxial optimal rotor twist distribution.

0.0065TC = 0.013TC =

Lower

Upper

0.0065TC = 0.013TC =

Fig. 14 Two typical single and coaxial optimalrotor chord distributions (top view, no twist).

13

aerodynamic optimization of coaxial rotor in hover and axial flight

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