pe

U
o

__:--

iZ!|I!

_z_

.... _mr--

..... --m

--I --

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tr| -- i

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4

........._ ,_y-r-_ _-_- _. --__-_
NASA Contractor Report 4307

Unsteady Potential Flow Past

a Propeller Blade Section

M. A. Takallu

Lockheed Engineering & Sciences Company

Hampton, Virginia

Prepared for

Langley Research Center

under Contract NASI-19000

N/ A

National Aeronautics and

Space Administration

Office of Management

Scientific and Technica,

Information Division

1990

Summary

An analytical study has been conducted to predict the effect of an oscillating

stream on the time dependent sectional pressure and lift coefficients of a model

propeller blade. The assumption is that as the blade sections encounter a wake, the

actual angles of attack vary in a sinusoidal manner through the wake, thus each

blade is exposed to an unsteady stream oscillating about a mean value at a certain

reduced frequency. On the other hand, an isolated propeller at some angle of attack

can experience periodic changes in the value of the flow angle causing unsteady

loads on the blades. Such a flow condition requires the inclusion of new expressions

in the formulation of the unsteady potential flow around the blade sections. These

expressions account for time variation of angle of attack and total shed vortices in

the wake of each airfoil section. It was found that the final expressions for the

unsteady pressure distribution on each blade section are periodic and that the

unsteady circulation and lift coefficients exhibit a hysteresis loop.

Introduction

Propeller performancel and radiated noise 2 can be substantially altered by

the exposure to a non-uniform freestream, such as the flow produced by the wake of

an upstream wing or pylon. It has also been shown that an isolated propeller at some

angle of attack is also exposed to the unsteady loads experienced by the propeller

blade sections 3. While many researchers, motivated by unsteady flow encountered

on helicopter rotor blades or inside turbomachinery, have studied the physics of

airfoils exposed to nonuniform flow4-5, the present analysis addresses the problem of
propeller aerodynamicsfrom a technology base point of view. This analysis is

centered on the application of ANOPP6 (Aircraft Noise Prediction Program),

developed at NASA Langley ResearchCenter, to the unsteady flow environment.

While ANOPP is capable of coupling the aerodynamiccalculations to the analytical

noise prediction segment, only the aerodynamiccomputations leading to the

unsteady blade section pressure distribution and their unsteady lift coefficients are

considered here. The derevation of the equations for unsteady terms are descibed in

the analysis section of the report and in the final portion of the analysis, closed form

solutions have been presentedfor the special case of a thin airfoil. The thin airfoil

approximation has been used in a hybrid propeller performance code developed by

the present author. The above hybrid code has shown to be capable of computing the

performance characteristics of advanced propellers in a very efficient manner with

good accuracy7. It is intendedto extend the capabilitiesof the hybrid code to the

unsteady environment by the use of the present two dimensional analysis in a future

work.

A

B

Symbols

terms due to complex velocity potential in Z-plane

radius of the mapping circle, m/s (ft/s)

terms due to complex velocity potential in the _-plane

Joukowski transformation parameter, re a

chord length, re a

2
G section lift coefficient

Cp

F

section pressure coefficient

complex velocity potential in z-plane, re aVoo

M

T

Mach number

transformation,[d_ ' _-'']

nondimensional time, ta

Uoo

V_ free stream velocity as seen by blade section, m/s (ft/s)

W complex velocity, m/s

vortex location in Z-plane, re a

z complex plane of the airfoil, re a

section angle of attack, deg.

F unsteady circulation, re aVoo

strength of the vortex per unit length of the wake

increment of angle of attack

C 0 - _, rad.

angular measure in near circle, _' plane

perfect circle, _ = b e_to +iO
near circle, _' = b eXl/+2/ti_

angular measure in _-plane

location of vortex in the_-plane

nondimensional potential, re aVoo

Co reduced frequency, re--_-

f_ propeller angular velocity, rad./s

Subscripts:

le leading edge

te trailing edge

o steady term

unsteady componentsmultiplying A(_ cos_t

2 unsteady components multiplying Atx sincot

An_iysis

The derivation of expressions for time dependent section pressure and lift

coefficients are described in the following using the potential flow analysis. First,

4
expressionsare described for a generic airfoil; next, special cases,such as a family of

Theodorson's airfoils and in particular flat plate approximations, are discussed.

_) Unsteady Potential Flow

It has been shownl, that when the blade sections of a propeller encounter a

wake, the blade section angle of attack can be approximated to vary in a sinusoidal

manner throughout the passage. Thus, each blade is exposed to an unsteady

oncoming flow. This flow can be taken as a uniform stream oscillating about a mean

value at a certain reduced frequency.

c_ = ot0 + Aot sincot (1)

Such a flow condition requires inclusion of new expressions in the

formulation of the unsteady potential flow around the blade sections. These

expressions are needed to properly account for time variation of angle of attack and

total shed vortices in the wake of each airfoil section. Therefore, the effect of the

vortical wake behind each section is included in terms of the bound circulation

which gives rise to the magnitude of the total lift.

It is, therefore, possible to extend the steady analysis to the problem of an

oscillating flow about an airfoil. The unsteady potential flow is formulated for the

perfect circle in the k-plane. Theodorsen's transformation is utilized to obtain the

unsteady complex velocity in the Z-plane (see Fig. 1). The wake is modeled by

distributed vortices composed of pairs of an isolated vortex and its image vortex

outside and inside of the unit circle, respectively. Thus the complex velocity

potential in the k-plane is written:
_)_v_(_o-i% )_e,o a2.+_In(_ie- °)

Xlo

(2)

where the above dimensional quantities can be normalized with respect to the mean

free stream velocity, Vo., and radius of the circle, a:

F = 1-'/aV,_, x = x/a, 7 = _/V_, and _ = ae i 0

Therefore, the complex velocity in the _-plane becomes

w (_) - _(_) - ie-i0

V= 2sin(0- c0 + 2-_-+ I Q*7 dx

Xt ¢

(3)

with

___

2(1 - pcos0)

2 (4)

(1 + P - 2pcose)

Assuming that the distributed vortices all lie on the x-axis extending to infinity, the

unsteady pressure coefficient can be written as

co= /v 7t.--v:- " ]

(5)
with

Iw(z) l 2 = w(z) • w(z), (6)

(7)

and

w(z) = w(_) ° T (8)

Note that ¢, is the real part of the complex potential, F(_). In the above expression _'

is the plane of the near circle obtained by the Joukowski transformation:

b 2

z = _' +- (9)

Since the instantaneous value of angle-of-attack varies in a sinusoidal manner, it is

logical to assume that the circulation can be expressed in the following form:

1-'(0 = F 0 + AotF1 coso3t + AotF 2 sincot (10)

This expression allows for any phase lead or lag in the final expression for

circulation and, hence, lift value. Therefore, this is an adequate expression to

describe the vorticity since it can be understood as the first term in a series

representation for I-'. Since for the range of velocity defects encountered, the values

of Ao_ remain very small, terms of order of (Acz) 2 or higher can be neglected. With

7
this in mind, it can be shown that for trailing vortices to be left behind, as the airfoil

moves forward, the strength of the distributed vortices can be related to the bound

circulation so that:

and

Y=Y1Aotcoscot+ Y2Act sincot (11)

_1---_o (F1sin_oCx-xt el+F2cos_0Cx-xte_}

_2--o,tFlcos,olx- xto}-r2sin_o{x- xto})

(12)

(13)

Obviously, an expression can be found for w(_)representing a steady term and the

unsteady contributions by using Eq. (11) in Eq. (3) to obtain

w(_)

V_ - B°+ B 1Ao_ cos(ot + B2Ao_ sincot (14)

where:

= ie°I sin,0 - (15)

BI= ie-i°[F1-[2n c°(_l* Su +F2* Cu)] (16)

i

L

8
o2:ioioI [2_ 2co o+ lF .cu - F2" su1 (17)

and terms Su and Cu are given as:

su--f sin_olx-xt o)Q(p,o)dx

X_

(18)

co--_cos,oCx- x_Q(o,o__x

N.

(19)

Similarly, the following integrals are defined as:

CNu=f cosolx-x,ol_('O,0)a_

Xtc

(20)

where V is the argument

SNu--f s_,o(x-x_o)_(o,O)dx

X t©

of the expression:

(21)

iv WL--IWLle

(22)

9
These are used in the expression for time rate of change of potential:

--= _'l Ao_ coscot + t_'2Aa sincot

_)t

(23)

(24)

(25)

With eq. (15), (16), (17) and (7) let:

A0 = B0T (26)

A 1 = B1T

A2= B2T

Then the final expression for the unsteady pressure distribution is obtained as a

steady term plus unsteady components in the usual manner

Cp = Cp0 + AaCpl coscot + AaCp2 sincot

(27)

(28)

(29)

-%

10
with

Cpo=1- AoAo (30)

Cp_ -2_'1- A0/_1-/(oA1 (31)

Cp2= -2(_' 2 - A0 A2 - A0 A2 (32)

where A stands for the conjugate of A.

In the above expressions, values of the unsteady circulation components I-"0,

F1, F2 need to be specified before the pressure coefficients can be computed. The

correct amount of the circulation is, however, related to the vorticity production and

dissipation which itself is inherently a viscous phenomenon. For steady flow past

thin airfoils at small angles of attack, the Kutta condition is merely a specification of

the points of separation at the trailing edge. For unsteady flows, the rate of change

of bound circulation must be related to the vorticity flux into the wake at the points

of flow separation (on both upper and lower surfaces). This relation is known as the

MRS (Moore, Rott, Sears) Criterion. This procedure would require computation on the

viscous boundary layers on both surfaces up to the points of separation 8. As was

indicated above for airfoils at moderate angle of attack, the Kutta condition can be

understood as trailing edge separation where the removal of a singularity due to

transformation specifies the circulation value uniquely. Therefore, an unsteady

extension of the Kutta condition can be understood as trailing edge separation for all

times. This requirement specifies the circulation components uniquely, when w(_)is

forced to vanish at a point corresponding to the trailing edge.

F0= 4_ sin (_o- 0at ) (33)

1I
2o,Cutcos(o Er, =(1/2 -0.1Sut)2+(coCut 

(34)

2(1/2_: - co Sut) cos(OTE- ot3

F 2 = (35)

(1/2_ - co Sut)2+ (to Cut) 2

In this manner then, the components of the unsteady circulation can be used to

obtain a final expression for the unsteady lift coefficient as an unsteady extension of

Blasius relation:

Cl = Cto + AotCll cos cot + A(3tC/2 sin cot (36)

indeed, the abo,¢e expression will account for any phase lead or lag in the final

expression for total lift on each blade section, to order of Aot, as the sections

experience the oscillating streams which are caused by blade wake encounter.

b) Family of Th¢0rdorsen Airfoils

In the previous section, the expressions for lift and pressure coefficients were

obtained in terms of the mapped flow past the unit circle and the metrics of the

mapping function. In numerical analysis these constitute the Jacobian of the

transformation. In analytical methods, such as for a family of Theodorsen's airfoils,

the airfoil is mapped into a near circle _' and, subsequently, the near circle is

7= : 7:7

mapped into the unit circle. Thus, for a known airfoil section, the coordinates and

r: :25 :i -_ i: =:: :: : _; :'5 : ili:-

leading edge radius=0f the airfoil are given and the task is to obtain parameters of the

near circle and the radius of the circle in terms of the airfoil data.

12
It can be shown (ref. 9) that if

for the near circle:

and the perfect circle is described by:

b2

z = _' + _, (37)

_' = b e_tei_ (38)

= b e_o ei0 = a ei0 (39)

One can write the deviation of the near circle from the perfect circle as:

= (40)

In order to compute the deviation, E, and radius of the perfect circle, a, in terms of z =

x + iy, the relation between _ and _' is obtained by division of equation (38) by (39):

_'= _ e(_-Vo) ei(_-0) (41)

This can be expanded in a Fourier Expansion as:

_' = exp (An + Bn) / an e i n e (42)

1

Thus, coefficient of the expansion can be written as:

2/_

An= lJ an _- _cos nO dO

o (43)

2_

B__.o_n = 1 [ _ sin nO dO

an J

0 (44)

2_

o (45)

13
In an iterative manner, then, real and imaginary parts of (42) are computed

for the following relations:

oo

_-_o = _ (Ancosn0+B--n--sin n0)

an an

n=I

(46)

BI. [

n=l

cos nO - Aa--_nsin nO )

(47)

In the limiting case for 0 x, the leading edge radius is obtained as:

_le- 2 b sinh 2 _ (48)

cosh Xl/

and the Joukowski transformation parameter per chord length becomes:

_" = l. rj__e (49)

4 8

and the expression for the position of the isolated vortex, per radius of unit circle,

becomes:

(x + _x 2- 4b 2) (50)

P= 2

Note that in the above expression:

b = "b. c (51)

14
c) Flat Plate Approximation

In many cases, the airfoil sections for advanced turboprops are very thin

airfoils and the calculations above can be approximated for a flat plate airfoil section.

In this case, the limiting values for influence functions becomes:

or

Let 1"1= x-2 then:

Q= _ (52)

-4

_-(x-2 + x_VV_-4) (53)

" cos (o,'rl)

Cuo= -41 drl (54)

o T1 + 5/r12 + 4rl

Suo= -41 sin (carl) drl

° rl + _/_12 + 4rl (55)

Thus, the expression for the unsteady lift coefficient becomes:

Cl (t) = F (t)/2 (56)

and expressions for Ct o' Cl 1' and C/2 are written as:

Cl 0 = 2_x sin cto (57)

15
. co Cuo cos O_o

(2/1 = (1/2 x-coSuo) 2 +(coCuo) 2 (58)

_ (1/2 x- co Suo) cos cto

(2/2 = (1/2 x- co Suo_ + (oK2uo) 2 (59)

Results and Discussions

The analysis described above has been employed to predict the steady state and

unsteady lift coefficient components for blade sections of a straight bladed scale

model (SR-2) propeller. The physical dimensions and the operating conditions are

given in reference 1, and they are not repeated here. However, the practical

application of the above analysis is shown by the variation of the unsteady lift

coefficient for a value of mean angle-of-attack as described by reference 1.

Accordingly, the section angle of attack varies in a sinusoidal form due to the passage

of the propeller blade through the wake of the pylon. The pylon, in this case, is

positioned upstream of the propeller in a so-called pusher configuration. The wake

momentum deficit can cause an immediate change in angle of attack as it is seen by

the propeller blade section. This change is periodic and can be approximated by

sinusoidal behavior. Thus, the amplitude and reduced frequency of the motion are

obtained from the kinematics of blade/wake interaction. A typical variation of angle

of attack is illustrated in Fig. 2. It can be seen that the amplitude of oscillation is of

order of 1°. Thus, the present analysis, which is restricted to small Aoc, is valid for

this type of application.

16
Next, considerthe unsteadylift behavior shown in Fig. 3 for a blade section at

37% radial distance from the propeller pitch axis. The lift coefficient was obtained

by conformal mapping of Theodorsen'smethod. A modification of the aerodynamic

module of ANOPP was required to obtain the unsteadycomponentsof lift, drag and

pressure coefficients under an unsteady environment. Note that the hysteresis

behavior of the unsteady lift coefficient is recoveredby this analysis without

inclusion of unsteady boundary layer analysis. This can be explained, as discussed

earlier, by the fact that for airfoils at small anglesof attack, the unsteadyKutta

condition is understood as trailing edge separationand uniquely defines the

magnitude of circulation and consequentlythe lift curve behavior. This has

paramount implications in understanding structural and acoustic problems.

Reference2 points out that lift hysteresisof aerodynamicloads, obtained under

nonuniform inflow conditions, increases loading noise and the associated radiated

noise by a propeller blade.

Considerthe results for the special caseof a fiat plate airfoil, describedabove

under part c) of the analysis.This case is directly applicable to the work of reference

7. In this work, the present author has developeda code to predict the performance

of advancedpropellers by a simple strip method, where the induced blade velocities

are obtained by an iterative procedure. Thus, sectional angle of attack and Mach

numbers are obtained through direct computation of section aerodynamic

characteristics using thin airfoil theory. This analysis has shown excellent

agreementwith wind tunnel results for advancedpropellers, such as SR-7. In this

case, the contribution of the thin airfoil to the unsteadylift coefficient can be

approximatedby equations58-60. A parametricbehavior of the unsteady lift

coefficient is simulatedin Fig. 4. The angle of attack was chosento be the sameas

shown in Fig. 2. The values of unsteady lift coefficient, C l are plotted for various

17
reduced frequencies, co. It can be seen that the lift hysteresis loop becomes smaller

with increasing value of w and changes the inclination of the loop accordingly. In

addition, Fig. 5 shows that for higher mean angle-of-attack the lift coefficient and

the size of the hysteresis loop both decrease with oc0 relative to their mean value.

Furthermore, there is no apparent change in the direction of the loop. This is in

accord with the explanations of reference 8, since the present analysis is limited by

the potential flow assumption and cannot account for flow separation, consequently,

the above results do not apply to airfoils near stall. Nevertheless, the analysis

captures the essence of excess loads encountered under time dependent flow

conditions.

Conclusions

A method of solutionhas been developed to compute the unsteady potential flow

past airfoil sections of arbitrary shape. Even though the study is limited to

perturbations of angle of attack from a mean value, the analysis accounts for any

lead or lag, of the order of small amplitudes, in the final expressions for the unsteady

aerodynamic coefficients. These coefficients then properly represent the hysteresis

behavior encountered in a periodic flow such as the flow over advanced propeller

blades in an unsteady flow environment.

References

18
I.

o

Takallu, M.A. and P. L. Spence, "Prediction of Unsteady Thrust and Torque

Coefficients for a Pusher Propeller", AIAA-87-2630-CP, 5th Applied

Aerodynamics Conference, Monterey, CA, 1987.

Takallu, M. A. and P. J. W. Block, "Prediction of Added Noise Due to the Effect of

Unsteady Flow on Pusher Propellers." AIAA-87-0255, 25th Aerospace Sciences

Meeting, Reno, NV, 1987.

.

.

Algebri, A. S., "Wind Tunnel Tests on a One-Foot Diameter SR-7LK Propfan

Model". AIAA-87-1872, 23rd Joint Propulsion Conference, San Diego, CA, 1987.

Poling, D. R. and D. P. Telionis, "The Response of Airfoils to Periodic

Disturbances -- The Unsteady Kutta Condition". AIAA Journal, Vol. 24, No. 2, pp.

193-199, 1986.

, Adamczyk, J. J., "Analytical Investigation of Compressibility and ThreeDimensionality on Unsteady Response of an Airfoil in Fluctuating Flow Field".

AIAA 73-683, July 1973.

° Nolan, S. K., "Aircraft Noise Prediction Program Propeller Analysis System

IBM-PC Version User's Manual, Version 2.0". NASA CR-181689, August 1988.

. Takallu, M. A., and D.J. Dunham,"A Hybrid Method of Prediction for Propeller

Performance", AIAA-90-0440, 28th Aerospace Sciences Meeting, Reno, Nevada,

1990.

19
° Sears, W. R., "Unsteady Motion of Airfoils with Boundary Layer Separation".

AIAA Journal, Vol. 14, pp. 216-2220, Feb. 1976.

, Theodorsen, T. and I. E. Garrick, "General Potential Theory of Arbitrary Wing

Sections". NACA Rep. 452, 1933.

20
II "_ _,

.--/

N

N

_]..1

ffl

N

O

t:m

O

O

&

.p,,I

21
9

8

¢x,deg 7

6

- Span = .373

Xw=.I C

! I I I

1.571 3.142 4.712 6.283

cot, rad

Fig. 2 Variation of angle of attack during the wake passage.

22
1.010

1.005

Cl/Clo 1.000

0.995

- Span = .373

X w = ,1 C

Q = 11400 rpm

_ a o = 6.837 deg

CI

0.990 I I I

0.6 0.8 1.0 1.2 1.4

Fig. 3 Normalized lift hysteresis for a propeller blade section.

23
Cl(t)

1.1

1.0

! 1 I I

0.6 0.8 1.0 1.2

!

1.4

.05

.10

.15

.20

I

1.6

Fig. 4 Unsteady Lift Curve

(xo=3 cl o=0.33 co=0.05-0.5

24
Cl(t)

1.1

1

0.6

I I

0.8 1.0

(_(t)

.O5

.10

.15

I I !

1.2 1.4 1.6

Fig. 5 Unsteady

o_o=3 %=0.657

Lift Curve

co = 0.05 - 0.5

25
_La_J_rk_ _Cr_au_cS an(_

1, Report No,

NASA CR-4307

4. Title and Subtitle

Report Documentation Page

2, Government Acces_on No. 3. Recipient's Catalog No.

5. Report Date

Unsteady Potential Flow Past a Propeller Blade

Section

7, Author(s)

M. A. Takallu

9, Performing Organization Name and Addre_

Lockheed Engineering & Sciences Company

144 Research Drive

Hampton, Virginia 23666

12. Spon_ring A_ncy Name and Addre_

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665-5225

July 1990

6. Performing Organization Code

B. Performing Organization Report No.

10. Work Unit No,

535-03-01-02

11. Contract or Grant No.

NASI-19000

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supplementaw Notes

Langley Technical Monitor=

Final Report

Dana J. Dunham

16. Abstract

An analytical study has been conducted to predict the effect of an oscillating

stream on the time dependent sectional pressure and lift coefficients of a model

propeller blade. The assumption is that as the blade sections encounter a wake,

the actual angles of attack vary in a sinusoidal manner through the wake, thus

each blade is exposed to an unsteady stream oscillating about a mean value at a

certain reduced frequency. On the other hand, an isolated propeller at some

angle of attack can experience periodic changes in the value of the flow angle

causing unsteady loads on the blades. Such a flow condition requires the inclusion

of new expressions in the formulation of the unsteady potential flow around the

blade sections. These expressions account for time variation of angle of attack

and total shed vortices in the wake of each airfoil section. It was found that

the final expressions for the unsteady pressure distribution on each blade

section are periodic and that the unsteady circulation and lift coefficients

exhibit a hysteresis loop.

17. Key Words(Suggested by Au_or(s))

Unsteady Airfoil Theory

Unsteady Potential Flow

Propeller Performance

Propulsion Integration

i19. SecurrtyClassif.(ofthm _po_) !_.S_ur_ Ctassif,(_ _mpage) 21. No. of page=

Unclassified Unclassified 28

NASA FORM I(QIOCT

_rsa3.e _ t_ Nation_ T_hnicalI_ormationService,Sprm_eld, V_ginia22161-2171

18. D_tri_t_n Statement

Unclassified - Unlimited

Subject Category 02

22 Pr_e

A03

NASA-Lm_ley, 1990