AD A FOREIGN TECI*JOLOGY DIV WR IGHT PATTERSON AFB OHIO FIG 20/; LOW ASflCT WING IN DC LIMITED FLOW OF A NONVISCOUS FLUID.

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1 AD A FOREIGN TECI*JOLOGY DIV WR IGHT PATTERSON AFB OHIO FIG 20/; LOW ASflCT WING IN DC LIMITED FLOW OF A NONVISCOUS FLUID (U) JL 76 V I KHOtYAVKO, Y F USIK LRCLASSIFIED FTD ID(RS) T O It 41Jfl_Ut IF /

2 FTD_ID(RS)T_ 0993 _7d FOREIGN TECHNOLOGY DIVISION ç LOWASPECT WING IN THE LIMITED FLOW OF A NONVISCOUS FLUID by I V I Kholyavko and Yu F DDC _{iuu uiu _u_u_1 _uuuu u u 1_ 1J Approved for public release ; distribution unlimited IIIIp V : r

3 FTD D( )T o FTD ID (F(S)T MICROFICHE NR: CSC EDITED TRANSLATION a si 7 ij t C ( LOW ASPECT WING IN THE LIMITED FLOW OF A NOhVISCOUS FLUID 7 July 1978 ODC CA J Iucl 0 P TllI fl /AY ita& u C9[S D t A / L aeid, c SI LCIAji By: V I Kholyavko and Yu F Usik English pages: 13 Source: Samoletostroyenlye I Tekhnika Vozdushnogo Flota, lthar kov, Issue 20, 1970, pp 3 10 Country of origin: U SSR Translated by: Charles T Ostertag, Jr Requester: FTD/TQTA /1Lt Michael B Overbey Approved for public release; distribution unlimited THIS TRANSLAT ION IS A QE ( IOr OF HE ORIGI NAt FOREIGN TEXT W A ALYTICAL OR EDITORIAL COMMENT STAI EMENTS OR THEORIES PREPARED BY: ADVOCA TEDORIMPLIEDARETHO SEOFTII E SO irce AND DO NOT NECESSARILY REFLECT THE POSITION TRA NSLATION DIVISION OR OPINION OF THE FOREIGN TECHNOLOGY DI FOREIGN TECH NOLOGY DIVISION VISION WP AFB OHIO FTD ID Rs T o Date 7 ju] yj9 78 k J

4 r:r U S BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Block Italic Transliteration Block Italic Transliteration A a A a A, a P p p p R, r B 5 o B, b C c C c S, S B B B v, v I i T m T, t rr r G,g Yy y U, u D, d F, f E e E a Ye, ye; E, e* X x x x Kh, kh h Zh, zh L LI Ts, ts 3a 3, Z, z l i Ch, ch k lh K u I, i Ww 11/ ui Sh, sh A Q Y, y L u IL! sq Shch, shch HH Xx K, k L i JI i /1 L, 1 bi Y, MM M, m bb b e H H H N, n 3 3 9, E, e 00 0 o O, o ki lo Yu,yu fln /7 i P, p AR 21 Ya,ya * initially, after vowels, and after i, ; e elsewhere When written as ë in Russian, transliterate as y or ë RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS Russian English Russian English Russian English sin sin sh sinh arc sh sinh cos cos ch cosh arc ch cosh 1 tg tan th tanh arc th tann_1 ctg cot cth coth arc cth coth_1 sec see seh sech arc sch sech cosec csc csch csch arc csch esch Russian ro t ig English cur l log TI: _

5 c T : LOW_ASPECT WING IN THE LIMITED FLOW OF A NONVISCOUS FLUID V I Kholyavko, Yu F Usik Aerodynamic characteristics are determined for a thin plane wing, moving near a solid or a free surface (hydrofoil) In this case the theory of a thin body is used According to it the spatial flow around a thin body which is extended in the direction of movement is replaced roughly by a two dimensional flow in transverse planes In the case un der cons ideration the task is re duce d to the study of the movement of a plane plat e near an interface (F igure 1) Assume a plane lowaspect wing, the maximum span of which coincides with the trailing edge, is mov ing at a sma ll angle of attack ci w ith a cons tant velocity V During the time dt a sector of the wing dx 1 =Vco dt passes through the fixed transverse plane ZOY In the equivalent two d imensional problem this movement of the wing corresponds to the vertical shifting of a plane plate, moving with a constant velocity V 0 =V c*, by a magnitude V 0 dt and a change in the width of the plate from 1(x 1 ) to di, 1(x,)+ _j 11 V dl where 1(x 1 ) local wing semispan Flow in the plane ZOY, caused by the vertical shift ing of the 1 1,

6 ; : 1 plate and a change of its width, leads to a change in the associated mass of the plate by a magnitude, corresponding to the increase in the lift of the wing on the length dx1, ie, 4Y d (, ) 71) V 4 ) Integrat ing this expression for the length of the chord O x 1 b, we obtain the lift of the wing (1) where m( b ) associated mass of the plate, determined in a sect ion of the wing based on maximum span The magnitude of the transverse aerodynamic moment relative to the ax is OZ 1, passing through the vertex of the wing, Is calculated by the formula M, = x 1 dx, = v!çx 1 dm(; 1) x [m (b)b m(x i) dzi1 (2) The resisting force of pressure (force of inductive resistance) of the wing taking into account the drawing in effect on the leading edge s is determined from the correlation (3) We will introduce into the consideration, in place of forces and moment s ( 1) (3), the corresponding coefficients: Al Cs 1 1; m, 2

7 ,, ; S Figure 1 Key: (1) Interface E r *1, where S wing area In a plane; dynamic head; S = 2 I(x,) dx, ; q 2 V Then the aerodynamic characteristics of a low aspect wing can be presented in such a form : in1 (i4) If the coefficient s of lift and lateral moment are known, then the position of the center of pressure of the wing relative to its vertex In shares of root chord can be calculated using the formula

8 In (5) Formulas ( 14) an d (5) are obtained from the general dependences of the theory of a thin body and are suitable in all cases when the assumptions of this theory are valid Let us note two special features which follow from formulas (24 ) and (5) 1 According to (14 ) the coefficient s of lift and moment are a linear function of the angle of attack The results of experimental investigations confirm the linear nature of these dependences for low aspect wings up to c 6 2 The coefficient of lift does not depend on the shape of the wing in the pj anc z 1 1 (x 1 ) and is determined only by the cross section of the wing on a section of the rear edge The experimental data an d calculations us ing prec ise theor ies conf irm t his spec ial feature also for wings with an aspect ratio A 1 5 Thus, in spite of the limitedness of the case X O, the theory of a thin body leads to qualitatively correct results for wings with a finite aspect ratio It can be assumed that under certain conditions this theory will also give satisfactory quantitative results It ensue s from (3) and (14) that the problem of determining the aerodynamic characteristics of a low aspect wing in a limited flow is reduced to finding the associated mass of the plate near the Interface For calculat ion of the associated mass we will use the method of singularities and we will distribute over the length of the plate within the limits of l z l an upper layer with a continuous intensity r (z) It is evident that in this case y(z) y ( z) We will take into account the Influence of the interface using the method of mirror reflection (Figure 1) The distribution of y(z) should satisfy the following condition on the plate: the particle of liquid, adjacent to the plate in any point S, should possess a constant vertical velocity V,, (Figure 1) Us ing th is boundary condit ion, we obtain an integral equation for determinat ion of the unknown function y(z): 2it V, (6) 14

9 1 s Here the plus sign refers to the movement of a wing under a free surface (hy drofo il), and the minus sign to the movement of a wing near a solid surface (ground) Values 1 and h are determined in the lateral plane of flow ZOY depending on t he coor dinate x 1, which enters into equation (6) as a parameter For the calcu lat ions o f t he coef ficients of lift and induct ive resistance of a wing the values of 1 and h should be determined in t he sect ion of max imum span We wil l introduc e a new var iable 1 from t he correlat ion z=l cos 9 (when l z l we have ir< i9 <0) and make the substitution y(z) y (l cosi9)=y (i9 ), having designated?i = h/b Then equation (6) is wr itten as: ± C T ($)qc s coss,)sine 2W J cosê cos $, (7) Let us note t he part ial solut ion o f equat ions (6) and (7), which corresponds to movement of a wing in a limitless fluid (h co) From (6) and (7) with h= we obtain from wh ich (T(z) dz z i, $ cosb cci, 0 d8 = 2xV = 2V, = 2V ctg$ (8) For the solution of equat ion (7) in a general form with h we present the unknown function y( ) as a ser ies (li) = 2V 1 [A, ctg b + Ak SIfl 2i* b ( ) in which the first term when A 0 1 is determined by the solut ion of (8), an d t he second term an d A 0 #3 characterize the additional load, developing on the plat e from the influence of the flow boundaries 5 : TTI I

10 1TI It is evident that when h the coeff ic ient A 0 l, an d A 2 +O (n=l, 2) In the writing of series (9) the condit ion y ( )= _ y(7r 15 ) was used If the solut ion of (9) is known, then t he assoc iated mass is calculated in the following manner With an accuracy to constant, which subsequently is not essential, t he values of t he function of the velocity potentials are determined on the surface of the plate when f fi( ) á do, where the upper sign refers to the upper surface and the B lower to the lower (f The general formula for determination 1 ) of the assoc iated mass has the form [1] m = P, ds Since in t his case on the lower surface of t he plate and on the u pper =_j, and furthermore ds=dz, then we obta in fl 4p (z) dz T =2p /$ 1 n d ST (b)sinbde Substitut ing into this formula the expression y (Q ) from (9), Thus for determination of the associated mass of the plate it 4 is necessary to know the f irst two coef fic ien ts A 0 and A 2 of ex pans ion (9) The rema in ing coeff ic ients A 2 (n=2, 3, 14) characterize the distribution of load on the span of the plate and do no t have a direct influence on the total aerodynamic characteristics of the wing (10) 6 j

11 :: When h= from the solution of (8) it follows that A =l and 0 A =O 2n therefore accor din g to ( 10 ) m_ = p I (11) The aerodynamic characteristics of a wing with a calculation of (11) are determined using formulas (14) and (5) C I _ I C; b *=1 j,) / (X l)dxl (12) Here s aspect ratio of the wing z 1 l(x 1 ) equation for the form of t he wing in a plane Formulas (12) are the known correlations of a low aspect wing in a limitless fluid [2] We will switch to the calculation of coefficient s of expan sion A 2n (n= 0, 1, 2) Equation (7) with a calculation of (9) is transformed as: A 0 A3,, cos 2n 1 ± IQi, ) 1, (13) where t he fo llowing des ignat ions are intro duce d ICh J A o( J o _cos I 1+ _J,) cos ê, AM (J, _1 + 2 s) + A2o (J 2 2 I co,mftdb (cog ê cos )2 + 4h After the calculation of integrals and furt her conversions m of formula (13) we finally o bta in the e quat ions for the coe ff ic ients of expansion A 2n (n= 0, 1, 2,) 1 For movement of a wing near a solid surface A, c, (ia e) + i = A, [r, (k, b ) COB 2is ] (1)4 ) 7 1 4

12 2 For movement of a wing under a free surface (hydrofoil) A, [2 + r,, (iịe,)j I E A2, [CM (ii, ) Co 2ii b,j (15) The following designations tire introduced in (114 ) and (15): (, (A bs) t cos b g 2 j r; u fu s 2 C2 (h, o,) (, _ ) r,, ; = J v (sin, + 4 ji) u= VV 4+ sin o + 47t ; V = j/(sin% + 47 ) + I6i cos b,; = j _ ; 7 2, (1) = cos (2az arc cos t), where T 2 (t) Chebyshev polynomial of I family If in correlat ions (1)1 ) or ( 15) a series of values of fixed, then we will obtain a system of algebraic equat ions for the determinat ion of the coefricients of expansion A 2n (n=o, l, In this case the number of fixed point s 9 5 (i=1, 2 ) determines the number of equations in systems (114) and (15) and, consequently, t he number of unknown coe ffic ient s A 2 (n=0, J, 2 i l) The remaining coefficients (n=i, 1+1,) should be accepted as equal to zero Using (14) and (15) calculations wer e made for the coefficient s in t he case of fix in g of one po int (I a pprox imat ion), three (II approximation) and five (III approximation) Part of t he calculations for movemen t o f a wir g near a solid surface are presented in Figure 2 Based on the known coefficients A 0 an d A 2 and formula (10) it is possible to find the associated mass of a plate, and with formulas (24) and (5) characteristics of the wing is the aerodynamic 8 i 1

13 8 \ 7 4 (Iã? img,l jp (8 \ T/eoja i ox cmo( ) I 8 1 I QI,42 t?8 41 i I Figure 2 Key: (1) approximation; (2) Solid s irface Figure 3 gives the magnitudes of the derivative of the coefficient of lift based on the angle of attack C, related to the value of in limitless flow (12) Here also the dots designate the average values of C,, obtained using the theory of a vortical carrying surface for rectangular wings with x=014l6 [4] Analysis of the calculations made shows that satisfactory results right up tot 01 can be obtained already in the III appro x imation for a wing close to a solid surface, an d in II f or a hydrofoil If we are limited by the values IT)oLi (wing close to the ground) and rt>,o25 (hydrofoil), then the aerodynamic characteristics of a low aspect wing can be determined using the I approximation In this case we have simple analytical dependences, which follow from (114) and (15) when = /2 and A = 0 (n=l 2, 2, 3) 9 i

14 ruir I _ z 1 For a wing close to a solic surface _ Vi+ ii m p Since then 2 For a hydrofoil 2 1 Vi+ m== EpI (17) 2 Vii V I + 4h Calculations using formulas (16) and (17) are shown b y the solid lines Let us note that if in expression i I +(2#i ) we replace 2 by a larger number (3 for example), then formulas (16) and (17) give satisfactory coincidence with calculations using the III approximation up to F 0l0 Thus it is possible to recommend the following approximation formulas for calculation of the associated mass of a plate In a limited flow when FT,0l 0 1 Near a solid surface 2 Under a free surface u p! (18) (19) 1 10

15 t / fli?1it$leilwawe 8 k I, ; 5Of4J T W7Q 0 I 0 ChIod# aw nakj& w cii, Figure 3 Key: (1) approximation; (2) Solid surface; (3) Free surface; ( 14) according to t4] Formulas (18) and (19) are suitable for the calculations of moment characteristics and the position of the center of pressure of wings (5) and (18) we find For example, for a wing close to a sol id surface from Xô 1 b! (b)j/i+ ii Before the integral =h/l(b ), h position of the rear edge of the wing relative to the interface, 1(b) wing semispan In the integrand expression the magnitudes of h and 1 depend on the coordinate x 1 At small angles of attack h (x,)= h+(b x 1)a= In the assumptions of the theory of a thin body r y4 and the second term in the right side under certain conditions can be of the order of the first term, therefore the posit ion of the center of pressure on a plane wing close to an interface depends on the angle of attack 11

16 _ We will b e limited to the case c 0, when h(x 1 )=l (b)i =const and the position of the center of pressure is determined by the formula 1_ VI± 1/1 (x 1)Vis (x *)+9h % dx t (20) where T(x1) =f relative local wing semispan ;, Let us consider a family of wings, in which the shape in the plane is assigned by the equation (x ) xt The exponent m change s within the limit s When m=l we obtain a delta wing, when m 0 rectangular According to formula (20) we will have Xo 1_ V 9 p ç?my 2 97 $d (21) If similar transformat ions are fulfilled for a hydrofoil, then from (5) and (19) when c 0 it follows that X = 1 For a family of wings 1(x,) 4 ( (22) \ V i (23) When = (flow of a limitless fluid) from (21) and (23) (214 ) This result can also be determined from formula (12) In a general case the integrals in formulas (21) and (23) expressed by hypergeometric funct ions [3] F igure 24 gives the are calculations of X = for wings with m= 1/2, 1, 2 (soli d lines for a wing close to a solid surface, broken for a hydrofoil)

17 uui, I ( 41 4\ 4f I t Figure 24 An analysis of the results obtained shows that with the approximation of a wing to a solid surface (ground) the center of pressure is shifted to the trailing edge, and this shift is greater, the smaller the exponent m In particular, for a rectangular w ing (m O) the greatest shift should be observed The influence of the boundary o f a free surface for a hydrofoil is opposite to the effect of the ground BIBLIOGRAPHY I H Si a 6 p u xa UT A poiumauuuia $4g go Hayxa N Lt * 11 HJI Ce II A2po4uuIasaHKa ynpaajihemw* chapajwa Oøupqjws N H C r pa j w r e A H H M P H K Ta6aHu HIsTerprn u,os cy pa w I npoh3b esehha H9MaTrH3 M, C M beao uep ko Rcic i i TOHK*M N CYUWI IOSep HUCTh I *U SyIOSOM 00 toae rua Haiso chayl a, N, :::

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