Subsonic flow past a thin airfoil in a wind tunnel

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/91 $3.00 + .00

Vol. 18, (2/3),129-134, 1991 Printed in the USA Copyright (c) 1 9 9 1 Pergamon Press plc

SUBSONIC FLOW PAST A THIN A I R F O I L

L.Drago~ Faculty of M a t h e m a t i c s , Bucharest, Romania

IN A bliND TUNNEL

University

of

Bucharest

"Received 20 June 1990; accepted ~ r p f i n t 6 Augustt99~ Introduction

In this paper the integral equation flow in wind tunnel is o b t a i n e d and grate this e q u a t i o n is given.

of thin airfoil in subsonic a numerical method to inte-

Formulation

In [I] we

showed

system

the plane

bles

of

that

the

fundamental

subsonic

solution

aerodynamics

of

the

linearized

in d i m e n s i o n l e s s

varia-

is 2

p(x,y):

Xofl+~ Yof

J

2~

2 2 2 Xo+B YO

B Xof-

, v(x,v):

2~

2 _2Y°f12 '

(1)

Xo+~ Yo

where

Xo=X-~, This

solution

bounded

gives

stream pressure

ted at

the point

p~ and

(1)

The

produced

is the

p~,

global

unit

by a force motion

in a uniform vector (fl,f)

is defined

of

un-

the Ox-

concentraby the pres-

field

pl=p In

U'~ (?

density

(~,Q).

velocity

(I' )

the p e r t u r b a t i o n

of v e l o c i t y

axis),

sure and

yo=y-q.

v represents

+P U2p,

the

V1:U(i'

component 129

on

the

(2)

+ ~). Oy-axis

of

the

veloci-

30

L. DRAGOS

ty and

~ = ( I - M 2 ) I/2,

M being

the

Mach

number

of

the

unoerturbed

motion. The

thin

airfoil

tained

by

on

chord

the

theory

imposing of

distribution

on

the

such

in

the

the

airfoil

that

above-defined

wing and

the

of

determining

following

v(x,~O)

uniform

a distribution

the

boundary

flow

forces

is ob-

defined

intensity

condition

of

this

holds:

= h'(x) th{ (x),

(3)

where

y=h(x)± h 1 (x) are as

the

boundary

the

of

of

the

conditions

definition

half of

equations

the

of

(3)

The

are

chord).

aperture

on

variables

When

a

functions

defined

dimensionless

airfoil

dimensionless

wing.

(3')

the

winq

(Fig.l)

we

h(x),

hi(x)

the

segment

the

reference

is s i t u a t e d

have

also

to

as

well

[-l,l]

(in

lenth

is

in a tunnel imnose

the

conditions

v(x,+a/2) for

all

x

= 0

(4)

in [ - l , l ] .

U~

5" i'l -t

Fig.1.The Integral

On

the

coordinate

system

equation

basis

of

formulae

duced

by a c o n t i n u o u s

in an

bounded

flow,

(1)

we

deduce

distribution is d e t e r m i n e d

of by

that forces

formulae

the

perturbation

defined

on

~ro

[l,+l]

THIN AIRFOIL IN SUBSONIC FLOW

f_+l Xofl (E)+B2Yf(E)

p(x,y)=

1 21~B

v(x,y)=

B +1 X o f ( E ) - Y f l (E) T~I_I 2 lE32y~ dE

In order

I --

x

to satisfy

Considering

x

2 o

132 2

+

d~F,

y

also conditions

and

(5)

+

o

symmetric

the planes ~a/2

distributions

symmetric

(4) we use the on the

distributions

image in the plane ~3 a/2 etc., we obtain representation of the perturbation: _

p(x,y)-

] +'/+I 2~B S +~ - I

~

Xof

1

images

image method. of the wing

on the

images

the following

in

of the

general

(~)+(-l)nB2 (y-na)f(E) 2+_2 dE x ° ~ ( y - n a ) 2"

(6)

n

-~ v(x,y)=

131

(-l) Xof(E)-(y-na)fl (~)

~. f~l

x2+02(y_na)2 o

d~

I t is easy to see t h a t v g i v e n here s a t i s f i e s ( 4 ) . In o r d e r to impose a l s o c o n d i t i o n (3) we t a k e the l i m i t f o r y ~ O , - l < x < + l . The s i n g l e

integrals

d i n g to n=O w h i c h formulae [ 2 , 3 ] : 1 n=1 A 2 - - ~2n

which

become s i n g u l a r

are c a l c u l a t e d

~ cosh~A = ~-~ "s~nh~A

as in [ I ] .

are

those correspon-

Taking

into

l ~ (-I) n ~ 1 2A 2' n=1 A2+n 2 : ~ i m h ~ A

] 2A 2

account

(7)

we deduce

p(x,z0)=

. I I , +I f l ~ ) m2f+l[~coth( ~ ~L ~f(x)+ 2~ -I - --dS+ ~-6~Bj_I m Xo)]fl (E)d~ o o

v ( x , 4 * 0 ) = t_1 ~ f 1 ( x ) + T~~ ' f +-1l

f (xE°) d ~

+ ~

lf(E)K(Xo)dE '

(8)

where

m=al3,

K(Xo)= 11; s i n h - 1 (~Xo)_ m

1 ~--"

(9)

0

From (7) we deduce

the meaning

of

the f u n c t i o n

f,

namely

32

L. DRAGOS

f(x)=p(x,+O)-p(x,-O) This

will

serve

With

conditions

to c a l c u l a t e (3)

from

the

(8),

lO)

lift

and m o m e n t u m

it follows

coeff

cients,

that:

fl (x)=-2hl(X) 2--~

Formula

(ll)

Equation The

I x-~

For a ~

determines

(12)

kernel

"" + ~

ll)

1 f ( ~ ) K ( x - ~ ) d ~ = h ' (x)"

fl and

is the e q u a t i o n

K has

(7) and

no

equation of

thin wing

singularities

(12)

reduce

(12)

12)

the function

theory

f.

in a tunnel.

for ~=x.

to the

known

case

of

the

unbounded

fluid.

Numerical

Integral

equation

Reference thin

solution

[4],

(12)

and

is of

the

represents

type

of

integral

the g e n e r a l i z e d

equation

eauation

of

(9) o the

airfoils.

Its s o l u t i o n s Imposing putting

the

depend

on

the

Kutta-Jukowski

h in the

form

tion of e q u a t i o n

-~(E

(12) will

behaviour

imposed

condition

at

being have

the

I-~

Gauss-type

i f i i-

2--~

i "'f-~"

i/2

quadrature

the points

trailing

dimensionless,

~ f = - s (T-+-~E)

Using

the

at

6<
edge the

if. and solu-

form

I/2 F ( E )

•

13)

formulae n

G(~)d~= ~ l

r, (l-Ec~)G(E(z) 14)

I f+1(l~+ E) I/2 ~F(~) d[= 2~ -I

] ~ 2n+l

_x_____j_F(~) E( I - ~ ) ' =I

where

~ =cos

2~ 2n+1'

xj=cos

2"-I z2n+l j-~

15)

THIN AIRFOIL IN SUBSONIC FLOW

~=~,

j=1~,

equation

(12)

reduces ko

the

133

linear algebraic

sys-

tem n

R AjccFo=H j o.=1

,

i=in,

(16)

~here l

[5

Aj = 2n+----Tm sinh with

the notation

FI,...,F n from obtained

F =F(~),

(16),

from

-I

the

E~ ]

R

~(xj-~)

xj_E

Hj=-h'(xj).

(17)

Determining

lift and moment

the unknown

coefficients

will

be

formulae

CL=_~+I ~ +I I-E I/2 F 2~ if(E)dE = ~ /_i (]--~-~F) (E)dE= ~ +1

1

CM=- ~ f - I

(18

Nl

~

f(~)d~= - 2"'~ N2 '

where n

n

NI= 2n+ll Z] (I-~cx.)FoL, N2

2n+12 }3 ~OL(I-~o)FoL.

c~=I The pressure taking

distribution

into account

p(~,L0)=

~-~

(19

~=I on

(ll) and

the wing

(13). We obtain

~ 2n+l

Ftg

is determined

+

from

formulae

~_~1,/+I h~ (x) -I ~ dx

2 m

f-

which may System

(20 +I[ 2 coth ~ ( ~ - x )

~

(8

1 m

m

~

be e x p e r i m e n t a l l y

(16) may

be solved

1

-x

]h~ (x)dx

checked

by the aid of pressure

on a high-speed

comDuter.

tubes

It is seen

that Ao do not depend on the form of the wing such that the J~ program is general (some change oecurs only in the column matrix H.). For instance for the flat plate of incidence ~ (H=-~) we J have Hj=I. In this case formulae (18) Drovide C L = C L N I , C M = C M N 2 ~ C L and C M being

the c o e f f i c i e n t s

der to e s t i m a t e

the order

performed

the following

in the unbounded

of m a g n i t u d e

numerical

fluid.

of the tunnel

determinations

In or-

effect we

34

L. DRAGOS

Table

a=~ In

of

M=0

M=0.6

1.1296 I . 0310 the

the when

aperture,

the

presented

here

lift

2.

Values

of

N2

I

1.1621 I . 0458

tunnel

it d e c r e a s e s

N

Table

I. V a l u e s

lift the

M=0.8

M=0

1.848 1 . 0740 is

larger

tunnel

is an

provides

a=2 a=5

0.9639 1 . 0000

than

aoerture

increasing numerical

M=0.6

in

the

0.9204 0. 0998 unbounded

increases. function

estimates

At

of of

M.

0.8006 0. 0993 fluid;

the The

these

M=0.8

same theory

effects.

References

[i]

L.Drago~, M e t h o d of linear aerodynamics,

[2]

T.J.Bromwich, ries, L o n d o n ,

[32

Y.S.Gradstein, Y.M.Ryzhik, T a b l e s of i n t e g r a l s , summs, series and p r o d u c t s (in R u s s i a n ) , 5 th e d i t i o n , Moscow, Nauka 1979.

[43

L.Drago~, A numerical s o l u t i o n of the e o u a t i o n of thin air foil in g r o u n d e f f e c t s , accepted for o u b l i c a t i o n in A I A A Journ.

fundamental solutions in p l a n e Acta Mechanica 4_/7(1983) 277.

An i n t r o d u c t i o n to MacMillan Co.1949.

the

theory

of

steady

infinite

se ~