The last setionof this study isdevotedto theanalysisof theaerodynami

meha-nisms behindopen rotor 1P loads. Singularity methods like the lifting-linetheory

are partiularly well adapted to this type of study. Theyallow a deomposition of

theindued veloities indierent termsand, byremoving one termat eah

simula-tion, its impatan be quantied.

Desription ofthe method. Fromapurelygeometrial approah,1Ploadsan

beattributed to a dierene inthe relative veloities of the downward and the

up-wardmoving blades,whihgeneratesa diereneintheir loadsandthusanetfore

inthepropellerplane. However, the impatof induedveloities,

### ~v _{ind}

^{,}

^{on}

^{1P}

^{loads}

annotbe negletedasit hasbeen shown insetion6.2. To analyze their

ontribu-tion,

### ~v _{ind}

^{are}

^{going}

^{to}

^{be}

^{deomposed}

^{in}

^{several}

^{omponents.}

^{A}quantiation of theimpatofonesingle

### ~v _{ind}

^{omponent}

^{on}

^{thrust}

^{and}

^{1P}

^{loads}

^{should}

^{be}

^{estimated}

by removing that omponent from a HOST simulation and omparing the results

with respetto theoriginal simulation.

The indued veloities

### ~v _{ind}

^{represent}

^{the}

^{inuene}

^{of}

^{the}

^{potential}

^{wakes,}

^{the}

installation eets and the airfoil motion relative to the airow. These dierent

ontributions are added in the lifting-line methods to alulate the total

### ~v _{ind}

^{on}

eahbladesetionandthereforetoobtainthe aerodynamibehaviorofthedierent

bladesetions of the propeller.

Hene adeomposition of the

### ~v _{ind}

^{an}

^{be}

^{done}

^{as}

^{follows:}

^{(a)}

^{unsteady}

^{orretions}

due to the airfoil motion; (b) installation eets due to the hub; () auto-indued

veloities,

### ~v _{ind}

^{by}

^{the}

^{wake}

^{of}

^{a}

^{propeller}

^{on}

^{the}

^{propeller}

^{itself;}

^{and}

^{(d)}

^{}

mutually-indued veloities,

### ~v _{ind}

^{by}

^{the}

^{wake}

^{of}

^{the}

^{other}

^{propeller.}

^{Both,}

^{auto-}

^{and}

^{m}

^{}

utually-induedveloitiesaredeomposedinmeanvalue(mode0),1/revorinidenemode

(mode 1), and the Blade Passing Frequeny mode (mode BPF). Figure 6.7 shows,

foragivenradialposition(

### r = 0.75R

^{),}

^{the}deompositionoftheaxialand

**irumfer-Azimuth (deg)**

(a)Frontrotoraxial

### ~v

^{ind}

**Azimuth (deg)**

(b)Rearrotoraxial

### ~v

^{ind}

**Azimuth (deg)**

() Frontrotorirumferential

### ~v

^{ind}

**Azimuth (deg)**

(d)Rearrotorirumferential

### ~v

^{ind}

Figure 6.7: Deomposition of the indued veloities

### ~v

indfor thefront and therear

rotors.

ential omponents of the

### ~v

^{ind}

^{into}

^{its}

^{dierent}

^{terms.}

^{Note}

^{that}

^{negative}

^{axial}

### ~v

^{ind}

tend to diminish the aerodynami inidene of theblade setion, whereas negative

irumferential

### ~v

indtendto inrease it.

Figure6.8plotstheontributionofeah

### ~v _{ind}

^{omponent}

^{on}

^{thrust,}

^{1P}

^{load}

^{norm,}

and1Ploadphase. Notiethatthe perentagevalueaountsfortheimportaneof

theimpatofeahomponent,whereasthesignaountsforitspositiveornegative

ontribution. Moreover, perentages in redshould be onsidered arefully, asthey

are a onsequene of non-linear eets, as it will be explained hereafter. These

resultshavebeenobtainedbystoringthe

### ~v _{ind}

^{eld}

^{for}

^{a}

^{omplete}

^{CROR}

^{simulation}

and then removing one

### ~v _{ind}

^{omponent}

^{at}

^{eah}simulation, inorder to quantify its

**T** **h** **ru** **s** **t ** **F** **R**

Figure6.8: Contribution ofthe

### ~v _{ind}

^{terms}

^{on}

^{the}

^{thrust,}

^{the}

^{1P}

^{load}

^{norm,}

^{and}

^{the}

1P loadphaselag offront and rearrotors.

Aerodynami mehanisms of Thrust. The mean value of

### ~v _{ind}

^{is}

^{omposed}

mainly of an axial and a irumferential omponent, i.e. the swirl. In the present

methodboth omponentsareonsidered together. Notiethemeanvalueof

### ~v _{ind}

^{is}

themain mehanism impating thepropeller thrust. However, the impatof

auto-indued and mutually-indued veloities is not the same. On one side,in thease

ofauto-induedveloities,bothaxialandswirlomponentsdereasetheloalangle

of attak and henethethrust. Onthe otherside, inthease of mutually-indued

veloities,theswirl inreasestheangleofattakwhereasaxialomponentdereases

it, but the overall eet tendsto inreasethe angleof attak.

The front rotor thrust predited by HOST is almost not aeted by the rear

rotor wake. Thus we an onsider that its response will be similar to the ase

of a single propeller in inidene. On the ontrary, the rear rotor thrust is more

impatedbythefrontrotormean

### ~v _{ind}

^{(bar}

^{5)}

^{than}

^{by}

^{its}auto-induedveloity(bar 2). Furthermore, the ontribution of the swirl of thefront rotor wake inreases the

bladeinidene and thus therear rotorthrust.

Finally,asthehub tendstoaelerate theairowintheaxialandupward

dire-tions,ithasanegativeimpaton thethrust ofboth rotors(bar1),though itisless

important than the wake omponents (

### −9%

^{).}

Aerodynami mehanisms of 1P load norm. The main

### ~v _{ind}

^{mehanism}

^{}

im-pating 1P load norm is the rst mode of the wake (bars 3 and 6), due to the

propellerinidene. Indeed,the vortiityshedinthewakeislinked totheevolution

aninreaseintheangleofattakgeneratesawakewithmorevortiityandmore

### ~v _{ind}

whih tend to redue the angle of attak. As the 1P load norm is diretly related

with the osillation of the angle of attak of the blades, both auto- and

mutually-indued veloities tend to redue this 1P load norm. As for the thrust, the most

important ontribution on the 1P load norm of the rear rotor is the rst mode of

the front rotor wake. This helps to explain what was notied in setion 6.2: that

front rotor wake redues the rear rotorinidene andthus the1P loadnorm.

Notie the important ontribution of the mean

### ~v _{ind}

^{(bar}

^{2).}

^{As}

^{plotted}

^{in}

^{the}

shemeof Fig.6.9,thisis dueto non-linear airfoil data,whih isnot desiredinthe

present linearanalysis ofthe 1P loads. Indeed,when removing the mean

### ~v _{ind}

^{from}

theinputperturbationles,theloalinideneinreasesuptovaluesaftertheairfoil

stall angle. As itis shown inthegure, for a given inideneosillation and when

removing the mean

### ~v _{ind}

ontribution, the lift oeient does not almost osillate.
Consequently, the method predits a very important but unphysial ontribution

of themean

### ~v _{ind}

^{on}

^{the}

^{1P}

^{load}

^{norm}

^{and}

^{phase}

^{lag.}

^{This}

^{would}

^{not}

^{be}

^{the}

^{ase}

if the angle of attak remained around the working point and inside the range of

inidenes where theairfoildataislinear. Therefore,theperentagesof theimpat

of mean auto-indued and mutually-indued veloities on the 1P load norm and

phase lag(bars 2and 5) shouldbeonsidered arefully.

Finally, similar to what has been obtained for the single propeller, the hub in

inidenehasapositiveontributiononthe1Ploadnormasitaeleratestheairow

upwards,inreasingthediereneinrelativeveloityseenbythedownwardmoving

bladeandtheupward movingblade,andthusinreasingtherotorvertialforeand

onsequently inreases the 1Pload norm(bar1).

Aerodynami mehanisms of 1P load phase. One of the most important

ontributions omes from the installation eets (bar 1). Indeed, as it has been

explained inthe previousparagraph, thehubinreases thevertial foregenerated

by the propeller without modifying its side fore. This is why installation eets

tendto redue 1P loadphaselag.

Notiealso that theunsteady airfoilmodelhasan important positive

ontribu-tion on the phase lag (bar 8). Inaddition to the

### ~v _{ind}

^{from}

^{the}

^{airfoil}

^{motion,}

^{this}

modelinludes theeet ofthenearwake. Therstone tendstoaelerate thelift

evolution when the inidene is modied, whereas the seond one tends to lag it.

The overall ontribution isan inrease inthelag between inideneand lift, whih

generates an inreasein1P loadphaselag.

As explained before, the mean

### ~v _{ind}

^{has}

^{a}

^{non-linear}

^{impat}

^{on}

^{1P}

^{load}

^{norm}

^{and}

phase lag,and should be onsideredarefully (bars 2 and5).

Therstmodeof

### ~v _{ind}

^{(bars}

^{3}

^{and}

^{6)}

^{tend}

^{to}

^{derease}

^{the}

^{amplitude}

^{and}

^{lag}

^{the}

blade loading osillation due tothe propeller inidene. Thisexplains why1P load

phaselagoftherearrotorisaroundtwotimestheoneofthefrontrotor. Figure6.10

tries to illustrate howthe front rotor indued veloities ontribute to lag theangle

**Incidence, [ ]** **L** **if** **t ** **C** **o** **e** **ff** **ic** **ie** **n** **t,** ** C** **L**

### -4 0 4 8 12 16

### -0.4 0 0.4 0.8 1.2

** ** **w/o mean v** _{ind} **C** _{L}

_{ind}

_{L}

**C** _{L} ** ~ 0**

_{L}

Figure 6.9: Example of two-dimensional airfoil dataand theosillation ofangle of

attak dueto theinidene alongablade yle. Comparisonbetween theasewith

and the asewithout meanindued veloities.

angleofattakofasinglepropellerinduedbyitswake;itsminimumvalueisfound

in the downward moving blade zone, beause there the wake has more vortiity.

Seond, the dash-dotted urve represents the angle of attak indued by the front

rotor wake; its minimum value is plaed around

### 270 ^{◦}

^{,}

^{where}

^{the}

^{front}

^{rotor}

^{wake}

hasmorevortiity. Finally,thesolidurveshowstheadditionofbothontributions,

asin the ase of a CROR. It presents a redutionof the amplitude of the

### ∆α

^{and}

an inreaseinits phasewithrespetto thesinglepropellerase.

Finally, the ontribution of the BPF modes in HOST simulations is negligible

(bars 4and 7). However, ananalysisof thesensitivity of1P phaselag totimestep

presented by François et al. [François2013b℄, puts forward that the BPF modes

have animpat ofaround

### 3.5 ^{◦}

^{for}

^{the}

^{front}

^{rotor}

^{(}

### −17%

^{)}

^{and}

### 3 ^{◦}

^{for}

^{the}

^{rear}

^{rotor}

(

### −7%

^{).}

^{These}

^{osets}

^{an}

^{be}

^{a}

^{onsequene}

^{of}

^{the}

^{three}

^{reasons}

^{exposed}

^{in}

^{setion}

6.2: usingdierent timesteps,negletingvisouseetsinHOST,andreduingthe

bladeto its quarter-hord line inHOST.