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Transcription

1 I Pp AD AOb5 913 FOREIGN TECHNOLOGY DIV WRIGHT PATTERSON AFB OHIO FIG 20/14 BOUNDARY PROBLEM OF SUPPORTING SURFACE IN NONSTATIONARY LIMITED ETC (U) FEB 77 8 N BELOUSOV. V I ORISHCHENKO UNCLASSIFIED FTD ID(RS)I NL ii I. L :2 a

2 . pn : FTD ID(RS)IO /7) FOREIGN TECHNO LOGY DIVISION V. BOUNDARY PROBLEM OF SUPPORTING SURFACE IN NONSTATIONARY LIMITED FLOW by B. N. Belousov, V. I. Orishchenko E..1I7! App roved for public release; distribution unlimited.. c J

3 FTD ID RS)I EDITED TRANSLATION FTD ID(RS)I February id 17 ooo /9 L/ BOUNDARY PROBLEM OF SUPPORTING SURFACE IN NON STATIONARY LIMITED FLOW By: B. N. Belousov, V. I. Orishchenko English pages: 7 Source: Materialy Konferentsii 0 Krayevym Za dacham, Izd vo Kazanskogo Universiteta, 1970, PP Country of or igin: USSR Translated by: Mar ilyn Olaechea Requester: PTD/PDXS Approved for public release; distribution unlimited. THIS TRANSLATION IS A RENDITION OF THE ORIGI. NM. FOREIGN TEXT WITHOUT ANY ANA LYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY: ADVOCATED OR IMPLIEDARE THOSE OF THE SOURCE AND DO NOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISION OR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISION VISION. WP.AFB. OHIO. FTD ID(Rs)I..p12 77 Date 22 Feb 19 77

4 U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Bloc k Italic Transliterat ion Block Italic Transliteration Aa A a A, a P p p p R, r 5 ö 5 6 B, b C c C c 5, s B s V, v Tr T m T, t rr r a G,g Y y Y y U, u fl 8 D, d 0 F, f E e E a Ye, ye; E, e* X x A x Kh, kh Zh, z h LI, 14 LI 1( Ts, ts 3 a 3 s Z, z H i I V Ch, ch Hu 1, 1 Ww I L I w Sh, sh R a Y, y UI, 111 Xx K, k q Shch, shc h fl A L, 1 bi bi Y, 37 M w M, m bb b e HH H N, n Sa.91 E, e 2 c 0 o 0, o o 10 Yu, yu n f7 n F, p A I Ya, ya initially, af ter vowe ls, and aftr b; e elsewhere. When written as ë in Russian, trans ce as ye or ë. The use of diacritical marks is prer. d, but such marks may be omitted when expediency dictates. GREEK ALPHABET Alpha A a. Nu N v Beta B B Xi E G amma F y Omicron 0 0 Delta 6 Fl IT Tr Epsilon E c a Rho P p Zeta Z Sigma c E t a H r Tau T T Theta 0 0 $ Upsilon T u Io ta I t Ph i Kappa K )t K z Chi X x Lambda A A Ps i 1) Mu M Omega w FTD ID(RS)I j

5 R USSIAN AND ENGLISH TRIGON OMETRIC FUNCTIONS Russ ian En glish sin c os tg ctg sec cosec sin cos tan cot sec csc sh ch th cth sc h osch sinh cosh tanh coth sec h csch arc sin sin 1 arc cos c o s arc tg tan arc ctg cot arc sec sec arc c os ec cs c 1 arc sh slnh arc ch cosh 1 arc th l tanh arc cth l coth arc sch sech arc csc h csc h rot ig curl log GRAPHICS DISCLAI M ER Al l f ig ures, graphics, tables, equa ti ons, etc. merged into this translation were extracted from the best quality copy available. FTD ID(RS)I jj L,

6 BOUNDARY PROBLEM OF SUPPORTING SURFACE IN NONSTAT IONARY LIMITED FLOW B. N. Belousov, V. I. Orishehenko (Irkutsk) In this article we examine a method of solving the boundary problem for a wing of arbitrary aspect ratio which makes periodic arbitrary oscillations of infinitely small amplitude in a limited fluid flow having a velocity at infinity of V 0. Given is a brief derivation of the main integro differentlal equations of the problems obtained in [1] and a method for solving these equations and ob taining the numerical hydroaerodynanilc characteristics of the problem Is developed. The boundary problem for the potential of accelerations 0 is formulated as [2] O F(g) ge ges, gel P(O) =O ges0. Mere Is the space occupied by the fluid; S projection of supporting surface in direction of undisturbed flow ; L trailing FTD_ID(RS)I_ 012Z_

7 edge of supporting surface ; P operator whose form Is determined by the type of limiting surface S. O The solution to the problem gives integral operator Ay, assigned in space L 1 (S ) with values in space C 2 (c ). The necessary properties of operator Ày are given in [2], and [3]. In the studied problem operator Ay takes the form of: Ai =.L i. (P) [! + sign F!_] ds, S I, where.signf ri=y(x )2 + (y + (z + + 4h) ; I solid screen +1 free surface of heavy fluid (F,.oo). The integral equation of the problem b y representing the solution In the form of 3 componen ts [2], [3] is reduced to the equation system: here : N 01 C ] dt; F 1 + C g E s (1) N 01 A,7 ThF 1 ges, (2) NA,7 F, ges. (3) 1 CC (S ) regular solut ion related to presence of break in tangential velocities during transition of surface S ; v CC (S 2 ) regular solution which describes Inertial motions ; FTD ID(RS)IO I.

8 singular solution, determined from equation (3) transferred to space of velocity potential by operator N. Equation (1) Is not used, since it contains an in determinant constant C for the case of a supporting surface of arbitrary aspect ratio. Thus, the inertia component of the solution y 2 is determined from equation (2), which In form coincides with the equations of the stationary theory, while the vortex components y 1 and y 3 are determined from equation (3). The kernel equations (2) and (3) K (x,, y i) = V(x _ )2 + ( i) ( y 1) (x )(y I)) is represented by the Leydlo Sobolev approximation for a wing of arbitrary aspect ratio in the form of two singular kernels with singularities w ith respect to (x ) and (y 11), respective ly, which from the theoretical standpoint Is incorrect, although it does lead to satisfactory f inal results. In th is study the approxima tion for this kernel contains a regular term In addition to the singular components, and this makes It possible to preserve the main properties of the kernel K(x,, y, ii) [1]. K(x,, y, i) A ( ) sign (y ) e 2 sin k (x E) dk]+ + si n(x_e)[.l_ji e 2 k sin k (y _.) dk], (4) where P (A) O, A.0; p (T) o 1 x., P (A) oo, ).oo; P (+) i oo, X O. FTD ID(RS)I i. 0 0.

9 00 0 The kernels of equations ( 2 ) an d (3), which consider the limited nature of the flow, are approximated by functions which have the form of the regular terms in the kernel approximation K(x,, y, 11 ). The system of equations (2) and (3) based on approximation (4) is reduced to the system of one dimensional integro dlfferential equations of the Prandtl type r (y) c (k) t; (A) 1C [a yy...! (k) S 0(y_,i) d _ (6) Here IkP,(k) 0 0(Y_ii) =,. ç t3i:;,, (). _2P (J_) k fr A or () e J h s1n k (y i )dk, 0.2P (.1. ) k 2 O [(..L)+211J (y i ) = e A + e 4 k e sin k(y v 1) dk. From equation (6) when k equation. 0 we get the stationary theory The equation for the inertial component of the solution is represented in the form of y) 0(x E) de +.L ) _ e 2 21 }slnk(x ) dk. (7) 0 0_

10 As an example of solving equations of this type let us analyze the method of solving the equation of the stationary theory J 1(,y) O (x )d.! S L 0(y )d L S ) G1(y ) di1=f x, y). (8) Here : P (x, ) = (, i ) d ; r ( ) = y (t, ) dl The solution y(e, r ) is sought in the form of a series = v A 1 i P. n O p U Using the method of moments with weight factor 31I y V 1 x, equation (8) is reduced to a system of algebraic equations 0 A op B nvqs =0qa, (9) n 0 n o where coefficients B pq5 are represented in the form of series in powers of functional parameters pq) = j/ p 2 (A) p( )] 2 1 2[h +.$L]. In order to solve equat ion system (9) a program was compiled on the BESN 4 computer. Calculation results in the form of the distribution of circulation r over the wing span are shown in Fig. 1. FTD ID(RS) I L i

11 r4 ) (,.cz sí.f O i Fig. 1. The solution to an analogous equat ion by the Mu ltopp method, which did not consider the distribution of circulation over the chord, was given in the work by I. P. Tkachenko [4]. of the calculations agree. The results Equation (7) for inertia component y 2 is solved by the method used to solve equation (8). The solution of y (E, n) is sought in the form of (Z, ) ). 1 i i 2 E E n 0 n O The weight factors are used in the same form. are shown in Fig. 2. Calculation results Lr y,i ! CS C 03 0/ 90 b Pig. 2. FTD ID (RS)I O

12 BIBLI OG RAPHY 1. Be o y co B. II., fl a ii e K 0 A. II. Tcopiiø tiecyuteii IIOCTIt B uecta ij oii a flom not oke, ocil onalillan Ila oj ulome )li ux IinTerpo.Jui I ). l lepe nuhajib.iwx ) pabiieiiiirx, C6O IIItK.ACIIMHTOI lj ieckti e M 1oI I4 B TCO IIII HecTaw,o IIapuiwx Ii pot ec Os, ll3.t n0. IIay Ka (ii ne aiii). 2. Ii a II i e u K o u A. I I. I lecyala, IIouepx uioctl. n IleCialuloliaplI om II,. Toice OrpaHu..eliHofl )KIiSKOcTli (cia I), coolnl IlK. FIIll po)1li IIaNIIICa GOA bill II cko pocreil, 9b111. 4, H32iBO.Hayico na syiiic a, K a eli K 0 u A. H. I Iecyuja ii nobepxhoc il. B liectawr o,iapu oic LI.,. TOice orpalin lenhioi I *IISKOCTU (cta?l.r 11), C6O{flIIIK. rimpwuul am.ika 6OJIbWLI ck OpoCTeii. DM H no. }Iayic ooa 1IyMKa, K Tx a q e Ii K o H. II. BJI IIRIIIIe 4,oIiMw n uj ial ue na. a paxr epuc u i, xpmjia B6aII3II 3u( palia, )LolcJta)u.I K XVIII I3CeCOlo3IIoii IIay liio.texuuul ueck..it x oii4epehitxra no eop i,ii icofao (Kpiiii oui c ici,e.iienit a),.rieiiuiii rpaj. 193S FTD ID(R 5)I 0l

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