»*«<< - ■J.5-:P:':-: #Pbi-=- I II H| I |l II llll||—j ||iii J||_ iiiiiiiiiiiiiinpi IN »I I i i tm ■ . .... ■' ' .^11

PTD-ID(RS)1-0099-77

FOREIGN TECHNOLOGY DIVISION

^ v.. REGULARIZATION OP THE SINGULAR INTEGRAL EQUATION FOR

A WING IN AN UNSTEADY SUBSONIC GAS FLOW

by

Yu. A. Abramov

D D C 'fn\

-■•'{ -J

Approved for public release; distribution unlimited.

-w-^SJKSWt .,,.,,

..

·•·

THIS DOCUMENT IS BEST QUALITY AVAILABLE. THE COPY

FURNISHED TO DTIC CONTAINED

A SIGNIFICANT NUMBER OF

PAGES WHICH DO NOT

REPRODUCE LEGIBLYo

'""".V':';:V^^m3KJfS™i;:'.:i-;: .V--■••; ■ -- i: ..-.,,.....,.. - .

fyp,ID(RS)I-0099~77

EDITED TRANSLATION FTD-ID(RS)T-nnqq-77 1 February 1977

REGÜLARIZATION OF THE SINGULAR INTEGRAL EQUATION FOR A WING IN AN UNSTEADY SUBSONIC GAS FLOW

By: Yu. A. Abramov

English pages: 16

Source: Trudy Irkutskogo Politechnicheskogo Instituta, Nr 52, 1969, pp 218-224, Bibliography included

Country of origin: USSR Translated by: Carol S. Nack Requester. FTD/PDXS Approved for public release; distribution unlimited.

»CCESEIÜ» tir

038

H

»HI!» Jcril«

IC\ kciion

THIS TRANSLATION IS A RENDITION OP THE ORIGI- NAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEORIES ADVOCATEDOR IMPLIED ARE THOSE OF THE SOURCE AND DO NOT NECESSARILY REFLECT THE POSITION OR OPINION OF THE FOREIGN TECHNOLOGY DI- VISION.

PREPARED BY:

TRANSLATION DIVISION FOREIGN TECHNOLOGY DIVISION WP.AFB, OHIO.

FTP ID(I:S)I-0099-77 Date 7 Feb 19 77

-y:./v;^:.;T:;iy^-r>ir^:,£^j:y^---'^^v

"■■"■:'. '■■'/ : ijiSiy.»;,;....

PPPÜT \-\:.;.i,;i,_.-;^--%:-MiVx-;::^ ::v ' '-'"■'■": msm

U.

Block

A a

E 6

B B

T r

A A

E s

W w

3 3

n H

S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Italic Transliteration Block Italic Transliteration

A

B

B

r n B

M

3

H

ft

K

n M

H

0

n

a

6

$

t

b

t

xc

i

u

a K

A

M

H

0

I)

A, a

B, b

V, v

«i 6

D, d

Ye, ye; E, e*

Zh, zh rf I-.

■f J X 3 _L

Y, y

K, k

L, 1

M, m

U, n

n n

p p

C c

T T

y y

X x

14 u

H M

UJ in

h h

hi bl

b b

3 3

a to

P

c

m

P

C

T

y y

* *

x » u V

m

h

hi

b

to

H

v

i«

%

y

I

I

»

Ji

R, r

S, s

Ts t

Ü, u

F, f

Kh, kh

Ts, ts

Ch, ch

Sh, sh

Shch, shch

Y, y

E, e

Yu, yu

Ya, ya

*y_e initially, after vowels, and after t, «; e elsewhere. When written as e in Russian, transliterate as ye or e. The use of diacritical marks is preferred, but such marks may be omitted when expediency dictates.

GREEK ALPH/ \BET

Alpha A a a Nu N V

Beta B ß Xi - 5

• amma - Y Omicron 0 0

Delta A 6 Pi n IT

Epsilon M e 1 Rho p P »

"eta 5 Sigma i a %

Eta H n Tau T T

Theta G e » Upsilon T u

Iota T i Phi * <P 4>

Kappa K H K X Chi X X

Lambda A X Psi ¥ *

Mu M u Omega ft a)

FTD-ID(RS)I-0099-77

^"*"",—""—»^— ■""•'"■"«» »*"»■>'■»«'. Ui .1 Illinium II II. II III

RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian English

sin sin

cos cos

tg tan

ctg cot

sec sec

cosec esc

sh sinh

ch cosh

th tanh

cth coth

sch sech

csch csch

arc sin , -1 sin

arc cos -1 cos

arc tg tan"

arc ctg cot"1

arc sec -1 sec

arc cosec esc"

arc sh sinh"1

arc ch cosh"

arc th tanh"

arc cth coth

arc sch . -1 sech

arc csch csch"

rot curl

lg log

GRAPHICS DISCLAIMER

All figures, graphics, tables, equations, etc, merged into this translation were extracted from the best quality copy available.

PTD-ID(RS) 1-0099-77 ii

...™».

-■■:: :3-X/J/ ^ ,:^^;^^'. '^p|^ .ipnwu^UMtW^1« «^

DOC = 0099 PAGE 1

CO 99

REGDLARIZATION OF THE SINGULAR INTEGRAI EQUATION FOR A WING IN AN

ÖNSTEAEY SUBSONIC GAS FLOW

Yu. A. Abramov

This study attempts to regularize the singular integral equation

for a wing in an unsteady subsonic plane-parallel gas flow in

acceleration potential space, obtain the Fredhola equation for

deteraining the distribution function TC§I$:<« and find the asymptotic

solution.

It is «ell-known that the singular integral equation for the

probleu of the oscillations of a profile in an unsteady subsonic gas

flow can be obtained for acceleration potential space in the for»

FTD-ID(RS)I-0099-77

■■ ■ ■'■■■■. .; ::■

DOC ■ 0099 PAGE

' ' "S^Mf *"

$$0mm ■:-:liwv;i:::'v;:"1-":K ;;; .'- :: .-*-'::

4*W»

■--;-■ -- -cf MI» ,i .-.',-...!:...;..■.-■..-

»here the kernel H^|Ä!^^|i-ii; depends on the Mach number « and

the Strouhal number P, while function f fr) is determined by the

for» cf the profile oscillations. We know that in the acceleration

potential space kernel K(/*\;*-0 ' »*s the form:

^,(M*fK m

where

«CfcHti].

Then, using expression (2), we can give equation (1) the form;

x f)d|-F(X* H>

UBiÄi mm

Transferring the second ten on the left side of equation (l) to th*

right side, we can obtain

FTD-ID^RS)1-0099-77

=^:^m~-y^^i^:-^--^P:i ;--/vj-/^ ^^F^YY^:^ ;:siTi^--^~.~r-^A/:^: ,::;..r, -■::.,-. Y-:;:-.,- -.!■.■■; ,:v--^y^^y^;&=^>--KiiAS:^:

,v-,•,'-■'■;"-'..-:'; >-:--;-,-■ 7 7T;-T-;'.'•:■:?*•--«;

DOC ■ 0099 PAGE 3

i^fM^f «H (!>

The integral of the differential equation which relates the

acceleration potential 6 and the velocity potential f , which

satisfy the condition of the absence of perturbations at an infinite

distance ahead for a coapressirle flow, is as fellows (5):

tt* 0e* M «if. (6)

Then we can represent the right side of equation (4) as follows;

(6)

where :V"-;-

HP1

t^lie,

FTD-ID(RS)I-0099-77

apijiijjji^ie^j^y". «!^WHKiMP::3^E

DOC » 0099 PAGE 4

Thus, we have arrived at the problem of solving singular integral

equation (4) with the right side of (6). Equations of this typ? are

solved in (2) .

Finding the solution according to (2], »e can write the genpral

expression for distribution function flf/ *

j=„ /itirf 2 '«I *4 » r

I ?:■'£

I /?:3P"

j

^=^^:---^:;:^--^-;^-^^:^:r:-^;;^>^ :r>.v--. ..^^■.;H;.:^^-^'-^^^::i;;^ ■■:Lv^^:'-^:::\: ^-^-- ;^ ^:>vv:M^M;i^ --^M^

f|j|P?pB8jp^||p

DOC = 0099 PAGE 5

«here

Us)-, the fheodorsen function,

Q * 2 € C "jt I [ t * *i* I"" C j ; E^ *'" i §■

A

7i I it *%

Further, by eliminating that part of ^<u» which is related to

distribution function T[\) » "« can write expression (7) in the

foro cf the Fredholn integral equation for determining pCf*

COC = 0099 PAGE 6

(8)

«her« !r is determined fcy formula (7) for the function of the form

cf the profile oscillations and Kernel ^(W j ?) - is the following:

One of the main advantages of equation (8) for determining the

distribution functionAs that the zero approximation for solving

equation (8) is the distribution function <f# for unsteady flow,

while for all the other equations, this 2ero approximation serves as

the distribution function tf for steady flew.

The form of the kernel K^tfl^Ä-jy is obtained with

--V-- v>:-:^-^~->-7'^ '"::'.•'- . ' "'.r.. •-.■::-. '•» ■"»

DOC « 0099 PAGE 7

consideration of the linearized gas-dynamic movement equation for the

acceleration potential, which can assume the following well-icnown

form:

V# tX #«6, C#)

where J% £ *H « * f > ■ K u4 ih

In the notations we are using, we have the flow condition

i *■!

"3f. .1 I ~. I | 11} r- ('f H „; * v(* if * »/ 1 *^ ? -1

The solution to equation (9) can be written as

i - l* JLJL

Then the kernel of equation (1) will be expressed as

«*.«-l>"$[? C«M xrrt xVU~?f*r C£0)

^.KSSS^ü^a-^ -

: '"VJ:1 ,:iÄuii!i ■■^^=v-.;-v:/-^;--;-!T-;;;-:;-.- i..r™?77?K??l3^ffS^ ... . , U-jyiLUiLU- ^wff^^p.

DOC * 0099 PAGE 8

MM K- *} - It Is easy to sho* that the ketnel^has structure (2) and that kernel

ltf|M, *" 1) * „hich corresponds to kernel K(Mt*-*>„ »iH be as

fellows:

M •51« s . V t * :: " J* -

% I i tt )'

whereas kernel M '* is obtained in the foris:

£_

■ "--'"'

-;v-:-"': ^v- ■ ■ : -■■ -^v/. ■ -'.'-.-■-•.■.-■

........... . ,.,... •"".''

i-r

'a

t *: i

fl

COC » 0099 PAGE

l;

V?: q1

tu. K i

* l*

'* * ?<

1 % ' .'

>,J* ../*'. t?i ?

ihis integral and the €Xpressions for the kernel

differ when y * 0 and x = 0; however, it can fce replaced by an

integral which agrees by introducing the following relationship:

DCC = 0099 PAGE 10

&$it?w*~ &fem-*WK>

then the kernel J^gfcN^f,^) , after certain trar.sformations with

consideration of the value of the integral

I T»>%%dmk#.

is written as follows:

*-P TT r Hc **/ #' n —«3—" *

<* - —

DOC = 0099 PAGE 11

We will consider the approximate solution to equation (Q \ for

small values of parameter ■*•(*•<£<> *t small values of --,

will write the asyaptotic expression for the Hankel function

Vf-

ECC = 0099 PAGE 12

M# it) - fc in jr. i A - H - yl J« 2 > ^ fH

<jff* pj ^H^k" XH;8U), .. .JJ

Then kernel K.,(M,|.Pi assuues the fori

*<

1?***"

¥ f -- t«J

J 1

1 »! it

* t< /,

*■ pill*

f » i ilx

IUJiJPUJLyi!lliUJJ...y.l! wmmmmmmmmmm

DOC = 0099 PAGE 13

At saall values of «. the last integral in square brackets in

expression (11) can be represented according tc [4]

M* \ c IX

;i-tirtetf''"f»l«N.

'■'■'■■ ■■■!' :^^^~^-~v^,^^^^-^^^m '■■■■■'"

DOC = 0099 PAGE 1U

Substituting (12) in (11) and nailing the cbvious abbreviations, we

Mill obtain the following for small values of parameter it of kernel

*{

**».*»»♦& JlMfrV i*f /{# • f£| i-1 VT^

M y7 i * - 'i <>

/■;»

(Ö)

* j f { -! ~--~ /aft

Thus, equation (8) can be written as follows for saall K:

FTD-ID(RS)I-0O99-77

■■: - : ■ ' : ' ÜB

DOC ■ 0099 PAGE 15

•<l*

«*>,«* ,V '- fid ' t "'

Js

Integrating expression (11) from -1 to *1» we can immediately obtain

the formula for the lift of an oscillating wing in a subsonic

compressible flow, retaining the terms whose order of magnitude is

not higher than rr

w

where X-'t-Hi At II - 0, we will obtain the well-known result [1,

2] for an incompressible flow from formula (15):

*t

p--vlWiH^fi^MP|^rf|^rJ#

The comparison of the well-known numerical results with those

obtained by formula (15) indicates good agreement.

FTD-ID(RS)I-0099-77

- ^ .V^ÄiflW»M ■■.....«,«......... ,,,!.■;. .V .... ■ ...V,-.

DOC * 0099 PAGE 16

Eiblicgcaphy

Ban.7, 196?.

4, IU»41H H.E» 'JV'ty'/iHlk :; ■••<!■'"<*•.- '■'>'■■■ - 1 •«-' >1:I-HC AM LOOP, iil-,

i'i-IS, 6» BaH~M~BypH A j'. '■■■.<>>■:< * •• -1 m n ur i'emoi nm&mti

KVü»a. nportJiewu MRxaiWi'M. !«»■'-. . fv=«» - -■'•/- rru pugataDfell

7. BB, viüMHDro'ftj' P.il.,:.'Ui4;« X.« X.-R -H :•, & .jjoynpyrocTt,.

PTD-ID(RS)I-0099-77

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tFTD-ID(RS)I-Q099-77 «. TITLE fand Subtitle)

REGULARIZATION OF THE SINGULAR INTEGRAL EQUATION FOR A WING IN AN UNSTEADY SUBSONIC GAS FLOW

REPORT DOCUMENTATION PAGE 2. GOVT ACCESSION NO

7. AUTHORfs)

|Yu. A. Abramov

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