Factor analysis of coaxial rotors aerodynamics in hover

The 18th European Rotor-cr-aft. For-um

B-18

Paper No 99

FACTOR ANALYSIS OF COAXIAL ROTORS AERODYNAMICS IN HOVER

VADIM

N.

KVOKOV

Kamov Helicopter- Scientific

&

Technology Company

Avignon,Fr-ance September- 15-18,1992

FACTOR ANALYSIS OF COAXIAL ROTORS

AERODYNr"it1 I CS IN

HOVER

Vadim N.Kvokov f<amov Helicopter· Scientific 2, Technology Company

An aerodynamic design method for coaxial r· otor· s oper· at i ng at axial flight regimes was developed in Kamov Helicopter Scier1tific

& Technology Company on the basis of a disc vor .. tex thc;or·y. The method consists in the following.

A rotating , .. otor· is preser1ted as a disk wiLI1 uni fonnl y distributed radial votices with radius-variable circ~latirn~.The ait·flow passes UwoLtgh the disc and the blade vor·tices <we aligned v1ith stnea111lines. The str·eantlines fonn a spir·al surfdce fm· a gener·E<l CC<se of a var·iable pitch. ThE: gener··ating 1 ine of this

Vor· t ices spatial positioll is consider·ed to be fi:<ed <Fig. 1). If necessar·y coor·dina·tes oT vortices can be r·efined

kno~~~r; r·elalionships.

(r··,on-1 inear· theory) using

y

Fig. 1.

Thus the vcw teJ< str·uc LLwes of up pet·· and

patterned. The induced velocities 1.11 ·tt·l·e· r·ntor·· _{~ p} 1 _{ane and along}

uptional section of the air-flow are lcikten as the sum of the induced velocity proper and additional v _ya of a r1eigt1bouring

It is assumed

vys

that f r·ee b 1 a de vor·t ices move with cl speed which is mean along the disc and vc.~riable along ll1e airflow axis. Blade section airflow is assumed to be t\t~o-di mensi onal (hypothesis of flat sections). In deter·mining induced velocities the ai r··f 1 ow is regar·ded as ideal <Re= oo ) , non-compr·essi Ul e

and the function Cy(O<)-as linear·.

When a blade element lifl/drag are viscosity and

compr·essbili·ty of the air-flow are taken

into

ac c uunt:, as well cis non-1 i necw i l y Cy( ()() takt::?n fr·om thE::.~ pr··of i 1 e ~'IIi nd-tunnel

results Cy, Cx=f'CO<,

t-1,

Re). \<Ji thin U1e scope of t he adopted model, the fonoulae to detenuine ,,,ean induced velocity yener·ated by tile r·otcw V(Jr·te>: str~uctur·e in an c::H"b.i·tr·ary poirit

take the following forms:

f<<p,r-l'-"

I

I

-q/(

L" -q 0 q"'1 y

*

Y:;:

-

Y'q -2

p~

+ ' 1 1 I p+l

uf

F

~-­ P _dp 1

I

F

p (p) Up,' ) dp Yq ) 2

Po

"'

G

L~ q q + L2 q q<p+l q~p+1 1 in "IJace Cx y z l ';;::> • "

*'

*

p ( v ) c 'q

fF

p\p)

f:(p,;, )dp

r pu 99. 2

v

mec.n

- v

00 abW

=-..,--~. () ymear. ly=Ol

- v

00 V -_q V<y~ol• G ly l -_q constlv ,y _{. q· q-1} l R _{c .} ly) :'> 1 ;

y is the distance of the wake c,.·oss section f,.·om the ,.._otor r·ote~tional plane r·elated to the r·otot· r·adius.

The

function te~kes into e~ccoLmt the change in a jet

r··e~ctius along the height of a vorlei< culumn and lh<2 function GCy)

- lh<2 chanqe in the vortices velocity. The,;e fum:t.i.ons define the

rcJtur .. vor·te;< str-L(ctur-e and can I.Je t-efinEtd dLtr-ing the calculations. Viith account for· inductivt-1 velocity d. blade' sectiun angle of

attctck ~ is found 1Fig.3l, which is necessar··y to defint? l i f t and dr··a~l cCJeff:icients. Then:.: by way of nunH:::r· :i.c.::::•,l

the total coefficients of C and tor·que

T Coa>: i al

lies within t!1e linear· model ar·e d~ter·mined by wey of sol-vir·ty a systen1 of lir1ear· irllegr·o--differ·erltial equatior1s • .() - <() + 0 yn ysn yetn V ( ) _v.~~ ~ 1 _-v

_'Y

, ) _-.v ( _y;~v , _f;HIE:'C\rl 11 • ro ysn yan n - - - 1 , . v < v l ~v < v~o l •G < y l ~ -

.Yc

ysn · ysr1 · · n 2 T 1 2rr

V

+v 00 ya

v

ye; • G ( y) n

;:;

(y)~[-....::.1

__

_

yan . 2rr(1-p

f f ;:;

(~=0,~,8Jd8d~ y c.:t,n ' · · D p 0 0

v

00 <op - - - l +

E

n 1

v

_{meat! rn} 99.3

] *G <yl

...

n

-

1 !' 2 ; m

-

_{1' 2} fln (

,-

)

-

abW ₂ ) _n f2m (r) ~flrn(r··) 2rr p+l-·'"' _{p~ -p} - -_r·cose f< (y ,p, ; ) nm on· · =

J

E

qrn Ll"' 2 L t 1 I iO p+1 qu1 qff1 G 0 q~t q<p+1 q"--p+ 1

-p

~cose qm Her··e y

0n- :is the Jist,=:tnc~ i:Jf the n-Lh

fr-om the nelghbour·ir\g roto(·: rt=l

---

up pet'· n;:..:.2

-

1 t:.:Jwer· 2-817 r

y

I'" - ·"'

>

-·e

q111 (\!Y! .L q-1 u r·olati onai r·otor·

-

y _on

<

r·otor

-

_Yon> ; r· _·-'

The solution of this syslem of r:L..Juations ~~imoun·ts to SC.tlving 2,

set of 1 ine~r· algel:Jr··e .. \ic. equations. The mathen•atical pr-IJyt·aJnme dev~loped

fa.i. f'l y

U.i~tant:.e. I t ensut-t::!S the specifieJ tL.fft-~t c=:H..:.~ J..fl Lur .. yut2 al::.f;ut:ltC:-:. :..rF

ttv.::~ U}-)J.3~r- a.nd 1 Uv..J~::r- r·utLJ('S MK:n- MKl .:..;;co!-IS t ..

As a result we gel

tt1e

Lot6l tt1r1~s·t CT• CTl murnent CL.H:?ffic:ients mK=mK.l + mKn:o 1tJh2r·e

+

c

i"·1 _{K. (_II} l - ) rn , C ₁

=

---:::--..:....-,---K.:. 11~ .E._ ~'u)J=.·', . .::.p:::· 2 ' '' " <.:l.iHJ

ctr·e coc•f-ficienl·::::. of the 1o~·H-=r and uppet r·cd..:or·~~ r t~·::.pec..t.:iv:=ly. Tu a.i".l-t:~,lize the l'·esul ts of L!"·ltt col"nj.Jul.dlioris d.ild tf.J c.:o:-ntJCir·t,t

w:i.l1·1 the e:<pet·i111e!·1t.al c.idi..:a lht2 r·c;lur fiyLwt? of wt:'r i L ;.';".) LiSc.·d~

fr·om

99.4

computations are the first approximations because there

exists

a

number of factors which are not accounted for in the

mathe'llatical

model.

To take into account this condition, the coefficients

M,

Mn

are introduced into

Crm~

and

n

0 :

Then

C =C _{T T}

*

M (C M)3/2 T 2m k

n

_o ~n _o

*

M

_n

Here

hm~p

-is an

additional

value

of

a

profile

component

of tm·que moment coefficient, arising when the aerodynamic pr··ofile

operates on a r·otating blade. To determine

hm

special

r·ig tests

~p

were

condLtcted

using

rotor

models.

Flat

blades

of

specific

pr·ofiles wer·e installed at an angle such

that

rotor

thrust

was

zero and simultaneously the torque moment

1

Then

m~po=Ta expo

was subtracted

from

it,

was measur-ed.

where

c

xpo

profile drag coefficient received for ti1e given

profile

in

t

unn<= •

1

Th

e

d1

'ff

erence

'-'

*m

·

=

m~-1

m

~p ~ ~po was

added

is

wind

to

the calculated value:

m~=m~+hm~p •

The e:<periment

has

revealed

that

D.m~p

is practically constant within the

oper·ating

tip Mach nUinber·s.

of

Ti;e

,.

value

is usually

r·elated

to the so-called thr·ust tip

losses. We can asswne aftt:t" Pr-,£\fldtl:

2

.j2C;

M=B

=

1- • _ _::.__

!(

Then tht: influence of the rest neglected factors

(due

·to

1

oad

non-uniformity along the blade length and three-dimetlsiortal rtaturt:

of the flow over the blade) is concentrated in ,.

'I)"

By

""'Y

of comparing computa·ti onal resul·ts and ex peri men·tal

data

~~e get

..

n

experiment.

as

a

discrepancy

between

tile

computation

and

For a single rotor the

M

val ctes are taken under· the following

n

conditions. Vortex dr·ift velocities dependant on

the

function

and the '"ake shape <wake t·adius) - on the continuity equation R

'

tr.

7

y • 1IJJ !Jy thi'=l method. <Fig.

4>.

0

~t

2G- 0

~

-I

I\

-I

G=iiL.

j,J_ !)" i • _R -2 -2

y

y

cc ")

T Y] ~ ~.-:----0 ZmK Fig. 4. 0 5 I 0

RD

I

--

-Rc-Rc

R

va1Lies we get fut l\\':2 Sj'-t(:::tLifiec..:i r .. ·uLur·

L:ur~-figu(·alion anJ mude of uper-at.iLJll:

By way of illu·;:::,t.r·a.tion Fi~J.S~6 show lht::o plotted Jep(i•r\det1c.i.e~~

w+

" '"fCC / <>) +or· tliffe,-ent values: ll.'f> -blade <J8Ui\\8lr:ic twist;

vJR-YI T

-tJlade tlp speed.

1'/o 0. . . 0. 7 0.

o.

x,

1 . 1.

1.

0. X, 1.0 0.0 0 1 11.. A . 0. 0 1.0 . r 0. 9 A

0. 9

0.0 It

.,

/ /

i'

v::

exp.

226

11~~

187

•

0. 1

0.2

~

_{~ 4}

~ ~ • 0. 1

0.2

Fig. 5. GJR=119

m/s

~

""'

~

...

...

....

0. 1

0.2

Fig •. 6. ·

r!

_*

..

A· A -T < A 0 0

"'

~

~

/

cole. c.JR

Mo

• ... 119 00000 187 AAAAA 226 0. 3

Gr/u

... 119 00000 153 00000 187 AAAAA 226 0. 3

Gr/u

Arp 0.347 0.55 0.665 0.347 0.45 0.55 0.665

'*

0. 3

Gr/u

It i s assumed for a coaxial r·otor that tip lo5ses value~ ~ and

~ will be tl1e san1e for coaxial r-otor·s as for a single r·otor·~ and

'I)

the wake shape is taken as an identification factot·. For· this purpose a multiplier A is introduced into the function G of the upper r·otor:

-*

-:!:

G =1- ; y = A

_*Y

/l+y*z'

whereas the boundar·y wake 1 ine of the 1 ower· r··otor· is taken equidistant to the wake boundary of the upper r-otor.

For the investigation the experimental data obtained from wind tunnel testing performed in TsAGI for a coaxial rotor model were

u:;.,d. Fig.7 two e~<peri (!\ental curves "I) • f (

c /()')

0 T for

coaxial rotor and for the equivalent single rotor las to the rotor solidity 0' _s: = a _c ) . In this case the single r·otor· mathematical model

was i.denlifi-=d for· "r) with the exper·imental dependency uf a single rotor-. Then wlth the kno~.o.~n function 2t •f'CC,; a ) " such

r) T .

A • fCC/ 0 ' ) were selected so as Lhe design values uf T

values uf uf coa>-:ial rotors became close tu the e;-{pc:!ri mente<.l ones <Fiy. 7) ..

7Jo 0.8.---~,---,---, A 1.6

0. 0

1.4 1.2 1.0

0.8

0.0

•

•

0. 1

•

•

0.1

coaxial

0. 2

•

_•

*

0.2 ~

.

•

.

R=1.26 m K=2x3 u=0.15 G<JR=60 m/s 1.40=0.177 Arp=6°

calculation

*"

* * •

A=var(Cr/ u)

_ _ A=1

This func...:l.i.un i s nul dl):=.ulult~ S.i.lH::e i t cor-r·e=.poncJs to o specific

e~<per·i!r,ento.l rutr..::w c.:onfigur-cd:ion .. Wha·L is HK;r·e impor-tiJnt i s that

tiny) l:.hat the figur·e of merit u-f a. C..Dd}~idl t olur

higher· as COf!lpd(·ed wi·LI·I d '..~~if\yle ~"OLor·.

is dr·aslically

Slmileu-ly the r·esults wf oth~r· H10del te':::l-t:.S ~·Jet"e hc.~ndled •

.and A.t.-Jhen used far· calc..ulating .aer·udyna,nic: char.acterislics of the r·otor"·s these mater·ials impr·ove the auther1Lici\:.y uf the r·esLtlts.