Combined+Free+Wake+%2FCFD+Methodology+for+Predicting+Transonic+Rotor+Flow+in+Hover

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details of the near and middle flow field of the ro-

tor wake may be obtained by CFD simulation,

t here are still cer tain difficult ies in p redict ing t he

far flow f ield of th e rotor w ake. T her efore, a free

wake / CFD methodolo gy , w hich combines CFD

simulation f or t he near and m iddle flow field o f t he

rotor with realist ic rotor far field wake effects, is

requir ed. In this paper, a combined free w ake /

CFD method for predicting transonic rot or f low is

co nceived. T hro ug h sample calculations g ood re-

sults are obtained and it is p roved that the com-

bined free wake / CFD method developed here can

include realistic wake effects in CFD solutions for

the rotor flow .

1 Free W ake A nalytical T echnique

1. 1 Generalized wake model

T he t ip vortex flow v isualization data can be

w el l sim ulated by a series o f linear and exponent ial

functions. For a hover ing rotor , its p rescribed and

generalized w ake can be expressed by th e follo w ing

relations [ 2] .

The tip vortex axial coordinates and radial co-

ordinates are respectively

Z =

k1w 0 w 2/ Nb

k1 2/ Nb + k 2( w - 2/ Nb )w 2/ Nb

( 1)

r = A + ( 1 - A ) e-

w ( 2)

The vortex sheet axial coordinate is

Z= Zr = 0 + r( Zr = 1 - Zr = 0 ) ( 3)

T he v ort ex sheet rad ial coordinat e is

r = r CrB/ rD ( 4)

1. 2 Vortex core effects

Us ing a f ree w ake analytical idea, the tip vor -

tex evolves freely in the air envir onm ent . T he tip

vortex element's induced velocity is contributed by

the blade bound vortex , inner vortex sheet and tip

vortex itself. Based on Biot-Savart law , th e nondi-

mensional induced velocity caused by a vortex ele-

ment at an y calculated point can be calculated. It s

radial, circum ferent ial and ax ial induced velocit y

components in cylindrical polar coordinates are re-

spectively expressed as follows ( see Fig. 1)

u-

=C!F

C!{x2

tansin( #- #) +

[ xt ansin( #- #) + xcos(#- #) ]

( Z- Z) }d#p

3 ( 5)

v- = C!F

C!{x2t an[ x - xcos(#- #) ] -

[ xt ancos( #- #) - xsin( #- #) ]

( Z- Z) }d#p

3 ( 6)

w-

=C!FC!{x

2- x xcos( #- #) -

x xt ansin( #- #) } d#p

3 ( 7)

where

p2

= x2

+ x2

- 2x xcos( #- #) +( z - z)

2+ %2

Fig. 1 Schematic of vortex core element

T he variabl es w ith prime repr esent t he co-

ordinates at any point of the vortex element centric

line and respectiv e pitch angl e. T he variables w ith-out prime represent the coordinates at the cal-

culated points. %is the core diameter of the vortexelement.

In this paper, the inner sheet is modeled as a

zero-thickness vortex sheet , and the tip vortex is

represented by a space helix with a constant vortex

core diameter. T he axial induced v elocity due to

the vortex core effect at its middle point

[ 6, 7]

is

w-

c =2cos&

pL

ln4pL% - 1/ 4 - k

2sin

2

cos2

1 - k2cos

2

12

-

ln1 + cos

2

sin2

+ cos2

( 8)

66 CA O Y i-hu a, YU Zhi-qiang, SU Yuan, K A N G K ai CJA


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The contribution of the vortex core effect to

the circum ferential induced velocity can be ex-

pressed as

v-

c = - w-

ct an& ( 9) According to Eq. ( 2) , the co ntribution of t he

vortex core effect to the radial induced velocity

may be approximately expressed by the following

formula

u-

c = u- C

Ae

- #w ( 10)

1. 3 Free wake analysis

In cylindr ical polar coordinates, new w ake g e-

ometry ( see Fig. 2) coordinates can be defined as

new( i+ 1) = ol d(i) + v ( i)(t/ r ( i)

rnew( i+ 1) = r

old( i) + u ( i)(t

znew( i+ 1) = z

old(i) + w (i) (t

( 11)

Fig. 2 Rot or wake geomet ry in hover

The wake geometry convergence criterion is

replaced by a circulation converg ence criterion in

this paper. T he circulation conv erg ence criterion

can be described as

error = N

e

i= 1

!n ew(i) - !old( i) 2 N

e

i= 1

!new( i) 2 &0

( 12)

where for &0 , its selection depends on the compro-mise bet w een the calculating amount and accuracy

requirement.

2 CF D F ormulation

2. 1 Governing equationsIn a Cartesian coordinate sy stem ro tating w ith

angular velocity )o f t he rotor, the co mpr essibleEuler equations are formulated as follows ( see

Fig . 3)

W

t+

Fx

x+

Fy

y+

Fz

z+ u)

W

x+ w )

W

z= G

( 13)

where

W=

uvwe

, Fx =

uu2 + puvuwuh

, Fy =

vvu

v 2 + pvwvh

,

Fz =

ww uw v

w 2 + pw h

, G =

0

- )w0

)u0

L et

F = Fxex + Fyey + Fzez ,

U)= u)ex + v )ey + w )ez

where ( ex , ey, ez ) are the unit vector s in the ( x ,

y , z ) coordinate system . Then Eq. ( 13) can be

written as

W

t+ F + U)W= G ( 14)

F ig. 3 T he rotor system

( a ) the coordinates of the rotor blade;

( b) The grid of rotor blade

2. 2 Numerical procedure

2. 2. 1 Finite volume method

T o apply th e f inite v olume m eth od, Eq. ( 14)

is written in the integral form

67Ma y 2002Combined Free Wake / CFD Methodology for

Predicting Transonic Roto r Fl ow in Hover


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tV

Wdv + V

nFds + V

U)nWds = V

Gdv

( 15)

for a domain V with a bounding surface V, w h e r e

n denotes the unit outward normal vector to the

surface element ds.

By applying Eq . ( 15) separ at ely to each cell,

one can obtain a system of ordinary differential e-

quations

ddt

( Ji, j, kWi, j, k) + P i, j, k + Qi, j, k = Ji, j, kGi, j , k

( 16)

T o prevent th e appearance o f oscillations in

regions cont aining severe pr essure gradient s n ear

shock waves and stagnation points , the finite vol-

ume scheme is augmented by the addition of artifi-

cial dissipative terms D ij k. Eq. ( 16) is replaced by

ddt

( Ji, j, kWi, j , k) + Ci, j, k - D i, j, k = 0 ( 17)

where C i, j, k= P i, j , k+ Q i, j, k- Ji, j, kGi, j , k; Jij k is t he cell

volume; Pij k repr esent s th e net absolut e flux out of

the cell; and Q ij k is the rotational flux .

2. 2. 2 T ime-stepping scheme

T h e class ical f ou rt h-o r der Ru ng e-K u t t a

Scheme is us ed t o integrate Eq . ( 17) . Suppressing

the subscripts ( i, j, k) gives

dW

dt+ R( W) = 0 ( 18)

where R ( W) =1

Ji, j , k( Ci, j ,k - D i, j ,k ) .

Then, Eq. ( 18) is solved by the f ollow ing

stages

W( 0)

= Wn

W( 1)

= W( 0)

- +1(tR ( W( 0) )

W( 2)

= W( 0)

- +2(tR ( W( 1) )

W( 3)

= W( 0)

- +3(tR ( W( 2) )

W

( 4)

= W( 0)

- +4(tR ( W( 3)

)

Wn+ 1

= W( 4)

( 19)

2. 2. 3 Boundary conditions

T he f low tangency is applied at th e blade sur-

face. In t he plane containing the blade root, the

solid bo undary condition is al so u sed, an d w/ z

are implemented to cut off the flux in t he spanw ise

direct ion. T he far-field boun dary is t reat ed accor d-

ing to the nonreflecting boundary conditions.

3 Results and Analyses

3. 1 Rotor blade airfoil calculation

( NACA0012 Airfoil)

A t w o dimensional airfoil O-typ e mesh ( 99

35) is generated by solving Possion equations and

Laplace equations for the airfoil.

Figs. 4 ( a) an d ( b) show the pressure distri-

bution of NA CA0012 airfoil. T he computational

results for the conditions ( Ma= 0. 63, += 2) an d( M a= 0. 72, += 0) agr ee w ith the ones in Ref.

F ig. 4 NACA0012 airfoil calculation

( a ) M a= 0. 63, += 2;

( b) M a= 0. 72, += 0;( c) M a= 0. 80, += 0

[ 8 ] . When t he angle of attack is incr eased, t he

difference of the pressures between the upper and

low er surfaces is incr eased, and th e lift is r aised.

U nder tr ans onic condition, the capture of shock-

w ave place, the abrupt chang e of pressure and t he

distribution of pressure, w hich are shown in Fig . 4

( c) , agr ee w ith th ose in R ef. [ 9] .

3. 2 Caradonna and Tung Rotor result



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T he O-H m esh s ize of the rot or blade is 99

3551. T he initial data for Caradonna and T ung

rot or co mput ation is got f rom t he t w o-bladed mod-

el rotor repo rt ed in R ef . [ 10] with non-tor sion and

non -tapered NACA 0012 blade, ,= 6. 0, radius ofblade R = 1. 143 m, and airfoil cho rd c= 0. 1905m.

Fig s. 5( a) , ( b) and ( c) show the s ho ck posi-

Fig. 5 Comparison of the calculation r esults

and ex periment dat a

( a) The result report ed in Ref. [ 10] ;

( b) Calculation r esult ( r = 0. 84R , M a= 0. 8,+= 2) ;( c) Calculation result ( r = 0. 8R , M ati p= 0. 877, += 2)

tion, the abrupt chang e of pressure and t he distri-

bution of pressure. It can be found that the calcu-

lated results are in agreement w ith th e experiment

data in Ref. [ 10] .

3. 3 Different blade tip shapes

In order to investig ate the ef fects of different

Blade T ip shapes on the rotor f low , a model r otor

for the Dolphin 2 Helico pter ro tor is adopted. T he

blades with different blade tip shapes are shown in

Fig. 6.

Fig. 6 Schem atic of r ot or blade shapes

( a) Rectangular ta pered blade;

( b) Rectangular tapered blade with sweptback tip;

( c) Rectang ular tapered blade wit h anhedr al t ip;

( d) Schematic of rotor blade

3. 3. 1 Rectangular tapered blade

As shown in Fig. 7 and Fig. 8, it is found that

w hen the angle of attack is r aised fr om 3t o 6

( the M ach N umber of the blade t ip Ma tip= 0. 877) ,

the maximum pressure difference at r = 0. 951R in-

creases fr om 0. 2 t o 1. 0; furthermore, when the

M ach Nu mber of the blade t ip is raised from 0. 877

to 0. 9 ( angle of attack += 6) , the m ax imum pres-sure difference increases f rom 1. 0 to 1. 2.

3. 3. 2 Rectangular tapered blade with sweptback

effect ( with a constant swept angle )When the blade has a constant swept angle




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F ig. 7 Rectangular t apered blade result

( a) M ati p= 0. 877, += 3; ( b ) M a tip= 0. 877, += 6

F ig. 8 Rectangular t apered blade resu lt(M ati p= 0. 9, += 6)

( a) r = 0. 788R ; ( b ) r = 0. 951R ; ( c ) r = 0. 976R ; ( d ) r = 0. 984R

30, the shockw ave beco mes w eaker, w hich is

shown in the contrast betw een Fig . 8 and Fig . 9.

Fo r example, in the pl ace r= 0. 951R, the pressure

co efficient w ithin the shockw ave zone for the rect-

angular blade is steeply decreased from 1. 3 to 0. 4,

while the one for the blade with a sweptback 30

ang le is d ecreased slowly from 1. 25 t o 0. 7. Addi-

tio nally , th e m ax imum pressure difference betw een

the upper and lo w er blade su rfaces for the abo ve

tw o cases is close ( approximately 1. 1 in r =

0. 976R ) .

3. 3. 3 Rectangular tapered blade w ith an an-

hedral effect ( with an anhedral angle &=20)

T he effect of an anhedral blade tip is shown in

the contrast betw een F ig. 8 and Fig . 10. At r =

0. 984R, t he pressur e co ef ficient w ithin the shock-

w ave zone f or th e rectangular blade is decreased

from 1. 25 t o 0. 5, while the one of t he blade w ith

an anhedral angle ( &= 20) is decreased slow lyfrom 1. 25 to 0. 55. Furthermore, the maximum

pressure differ ence between t he upper and low er

blade surfaces is close. Ho wever, t he pressure dif-



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ference of the w ho le upper and low er blade sur -

faces for the blade with an anhedral effect is a little

bigger than that of the rectangular blade.

Fig. 9 Sw eptback b lade result

(M ati p= 0. 9, += 6, = 30, &= 0)( a) r = 0. 951R ; ( b) r = 0. 976R

Fig. 10 Th e r esult of r ectangular t apered

blade w ith an anhedral effect

(M ati p= 0. 9, += 6, = 0, &= 20)

References

[ 1] Gray R B. Vort ex modeling fo r rot or aerodyn amics[ J ] . Jour-

nal of the American Helicopter Society, 1992, 37( 1) : 8- 10.

[ 2] Landgrebe A J. The wake geometry of a hovering helicopter

rotor and its influence on rotor performance [ J ] . Journal of

the A merican He licopte r So ciety, 1972, 17( 4) : 3- 15.

[ 3] Felker F F, Quackenbush T R, Bliss D B, et al . Compar-

isons of predicted and measured rotor performance in hover

using a new free wake an alysis[ J] . Vertica, 1990, 14( 3) :

361- 383.

[ 4] Caradonna F X, Isom M P. Subsonic and transo nic potential

flow over helicopter blades [ J] . AIAA J, 1972, 10 ( 12) :

1606- 1612.

[ 5] Srinirasan G R, Raghavan V , Duque E P N, et al. Flowfield

analysis of modern helicopter rotors in hover by Navier-

Stok es m ethod[ J] . Journal o f the A merican Helicopter S oci-

ety , 1993, 38( 3): 3- 13.[ 6] Lamb H. Hydrodynamics [ M ] . N e w Y o r k: Dover Publica-

tions, 1945. 210- 241.

[ 7] Samant S S, G r a y R B. A semi -empirical correction for the

vortex core effect on h overing roto r w ake geomet ries [ A ] .

In: Proceedings of 33rd Annual Forum of the A mer ican Heli-

copter Society [ C ] . W ashington D C: American Helicop ter

S ociety , 1977.

[ 8] [ M] .:, 1983.

( Handbook of aerodynamics[ M] . Beijing: Nat ion al Defence

Industr y Pre ss, 1983. ( in Chinese) )

[ 9] Deese J E. Num erical experiments w ith the split flux vector

fo rm of t he Euler equ ations [ R] . AIAA 83-0122, 1983.

[10] Caradonna F X, T ung C. Experimental and analytical studies

of a model helicopter r otor in hover [ R] . NA SA T M-81232,

1981.

Biography:CAO Yi-hua Born in 1962, professor ,

doctoral super viso r , he received his Ph D

degree in Aircraft Engineering from Nan-

jing U niversit y of A er on au tics and A st ro -

nautics in 1990. He st udied at the Insti-

tute of Flight Mechanics at TU Braun-

schw eig, Germany during 1996-1997.

His resear ch inter ests include fl uid m echanics, aerodynam-

ics, flight d ynamics and flight cont rol. He is a member of

t he A merican Helicopt er So ciety.



Combined+Free+Wake+%2FCFD+Methodology+for+Predicting+Transonic+Rotor+Flow+in+Hover

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