Pitching Derivatives for a Gothic Wing Oscillating about a Mean Incidence

Transcription

1 MINISTRY OF AVIATION AERONAUTICAL RESEARCH COUNCIL CURRENT PAPERS Pitching Derivatives for a Gothic Wing Oscillating about a Mean Incidence BY H.C. Garner, M.A., AF.RAe.S. and Doris E. lehrian, BSc. LONDON: HER MAJESTY S STATIONERY OFFICE 1963 Price 3s 6d net

2 c.p.no.695 Pitching Derivatives for a Gothic Wing Oscillating about a Mean Incidence - by - H. C. Garner, M.A., A.l?.R.Ae.S. and Doris E. Lehrian, B.Sc, I--- Pebrunry 1961 This note briefly introduces a non-lineor theory of wings in slow pitching oscillation with leading-edge separation in incompressible flow. The oscillatory lift and pitching moment become linear functions of mean incldence. Comparisons with measured pitching derivatives are made for a gothic wing of aspect ratio (2.75. Replaces NPL Aero Note No.lOlO-A.R.C.2&537. Published with the permission of the Director, National Physical Laboratory.

3 1. Introduction -2- It is known that leading-edge separation occurs on wings of low aspect ratio at incidence, particularly when the leading edge is sharp; in consequence, the steady aerodynamic forces are non-linear with incidence. There is a corresponding effect of mean incidence on oscillatory derivatives, as experimental data192 have already shown. For pitching oscillations of small amplitude and frequency about zero mean incidence, measured values of the damping derivative at low speeds are usually in satisfactory agreement with values as calculated by linear liftingsurface theory3. This remains true for yvings of low aspect ratio, as shown by Firs. 11 and 12 of Ref. 4 for delta (A = I) and gothic TA = 0.75) planforms. Furthermore, the comparisons of the measured1 and theoretical values for the gothic viing, reproduced in Fig. 1, show that slenderwing theory is inadequate. In steady incompressible flow, the overall forces at high incidence can be estimated theoretically by using Gersten's5 mathematical model of the flow (Fig. 2) as the basis of a non-linear theory. In Ref. 6 this vortex model is used in conjunction with Multhopp's7 linear theory for steady flow, to give a numerical method for wings of arbitrary planform. Although the vortex model fails to provide the load distri ution on swept wings with separated flow, the comparisonsk betlieen calculated and measured lift and pitching moment indicate that in this respect the non-linear theory is a decisive improvement on linear theory for wings in steady flow, The non-linear effects grow as aspect ratio demreases and become important for the delta (A = 1) and gothic (A = 0.75) wings at fairly low incidences. Clearly me need a theoretical method of calculating pitching derivatives, which allows for mean incldenoe. An extension of the lifting-surface method of Ref. 6 to lowfrequency pitching oscillations is now in preparationg. This can also be regarded as an extension of the linear oscillatory theory of Ref. 3, compatible with Gersten's5 vortex model. The basic equation is set out in Section 2, -vhile Section 3 sunnnorizes the steps of the calculation in matrix notation. In the final Section 4, comparisons of the oscillatory lift and pitching moment are made for the gothic wing tested by Bristol Aircraft Limited in Ref. 2. These experiments support the use of a theory in which effects of frequency and amplitude of oscillation are ignored, and they go some way towards justifying the assertion, that pitching derivatives should show a linear dependence on mean incidence,

4 2. Basic Theory -3- A non-linear lifting-surface theory for wings in slow pitching oscillation can be developed with the aid of Gersten's simplified model of steady incolflpressible separated flop (Pig. 2). The trailing vorticity is supposed to be convected from the wing in planar sheets inclined at half the instantaneous angle of incidence, i.e. &[u + e(t)], to the wing surface. It is possible to derive an expression for the upwnsh at the wing, in lvhich cts, ~128, uv, e* and higher order terms are neglected, but cr.2 and ae are included as T/e11 as the linear terms in a and 8. In fact the nondimensional vring loading is written as ahere T 1 4 = 4,a +.C,,a* + Re (z,'cia + z,,as,,)e [ a is the mean incidence, are complex quantities. iwt 1,... (1) 0 = Re[O,eiwt], z, and Let the Yving describe pitching oscillations of smplitude e2 and angular frequency w about an axis x = h:, where c is the geometric mean chord. Then the boundary conditions at the wing surface require that the upwash angle - Re i( 1... (2) over the planform S. upwash angle 1s related In Ref. 8 it fill be shown that to the wing loading in equation the (1) by w - = & U (8,~ + C,,aa)K(x - x', y - y')dx'dy' S - a - --_ a2 8 ay2 4,a(x - x')dx' > (2, + T,,a)rc(x - x', y - y')dx'a.y' S (2, +??,,a) - xl, y - Y')dX, dx'ay' - ~okxl~ U a a2 (i X )I ,(x - x')dx' eoeiwt,. -m.. (3)

5 -4- where K(x-x', y-y') = &r5, Li~xl..._~ -T, [(x-x')2 + (y-y')21" 3 It can be seen thnt equation (3) involves four distinct operators Al+?] = - & e(x',y')k(x-x' ) y-y')dx'dy S Bid] = - & 4(X', y') K(xo-x', y-y')&... (4) Di&] = - -& $55 C(x', y) (X-X')%X The first of these,.4[d], arises in steady linear liftingsurface theor ;r and is fully discussed in Ref. 7. The operator Bl.6 together with A[8{, forms the basib of Ref. 3. The t&d, C{e] is derived from Qersten's model of steady separated flow k-d, together with A 81, forms the basis of Ref. 6. The remaining operator, DIE I is peculiar to the unsteady non-linear theory and will be discussed. in Ref Solution in Matrix Notation In terms of the operators of equation (4), equations (2) and (3) combine to give

6 -5- = a + Re (1 + iv& - ivh)f&eiwt c 1..., (5) where 4; = x/o and the frequency parameter v = G/U. The steady and oscillatory parts of equation (5) become respectively and At?,] t A[8,,]a - Cje,]a = 1... (6) - iv(b[z,] t?3[?,,]a - Di?,]a) = 1 t iv(t;-h).... (7) In a collocation method, equations (6) and (7) are replaced by sets of equations and the operators (4) are treated as matrices. Then the right hand side of equation (6) becomes a column representing unit values at specified collocat,iop~~ixt~: Similarly the right hand side of equation (7) becomes (1 - ivh)ja,] t iv[a,], where Ia,] denotes a column of values of c = x/s at the specified collocation points. Equation (6) leads to the steady solution of Ref. 6, where 141 = l%la + l~,,la2,... (8) I&,] = A- ia,] - le,,l 3 k CIP,,j i * Similarly the real parts of I? ] by set ling v = 0 in equation 17)) and 214,,]. Hence we may write and IZ,,], determined are respectively 14, ]

7 (9) where the real quantities Ie:i and IG,l remain to be chosen so that equation (7) is satisfied identically to f lrst order in v. Thus = B[4,] + 2B&,{or - D[~,]cx - hfa,l + [cx,~. The terms independent of c1 in equation (10) lead to the linear solution of Ref (IO) It then folloms that [b;,] = A- C)&J] + 2A- Bie,,] - A-ID!&,],... (12) where!e,l, h,l and Ie:l and (11). are given in equations (8) The unsteady part of the non-dimensional wing loading in equation (1) is expressed as... (13) where 6 = de/at. The pitching derivatives are then derived from the lift and moment integrals l.t (14)

8 -7-4. Comparisons with Experiment Oscillatory cxpcriment s on (* = several 0.75) sharp-cd5ed models of the same Gothic planform are described in R&s. I and 2. The earlier mensurcments of Ref. 1 were mnde on two uncambered models of diffcrcnt maximum thicknesses (0.082 cr and cr) by observing the decay of pitching oscillations about each of three axes h = , 0.724, 0.949: only the dampin derivative was obtained. The later mcasurcments of Ref. 2 on another uncambered model of the smaller maximum thickness cr were carried out by forced pitching oscillation about two axes h = and 0.949; the results are presented as complete sets of pitching and heaving derivatives for the axis h = The effects of amplitude and Prequcncy parameter on the pitchln5 dcrivatlvcs are small and within experimental error, but significant erfccts of mean incidence are found. Relative to the leading apex the planform 1s defined as follows: leadin edge x = x4 = c, [I - '/-1-(y/s)] trniliny ed5e x = xt = cr semi-span s = &c r mean chord c = 2C 3 1~ t... (15) where cr denotes the root chord. Calculations for this planform have been made with 7 spanliise and 3 chordwise collocation stations, i.c. 12 collocation points on the half wing. The results in Figs. 18 and 19 of Ref. 2 include static measurements of lift and pitchin moment about the nxio x = he = 0.724:. The coefficients... (16) from experiment and from equation (8) are compared in Fi5. 3. The two sets of experim_ntal data correspond to some previous balance measurements ('A) about the axis h = and a

9 -8- mean (x) from the v&nvt,unnel balance measurements of Ref. 2 about the two axes h = and h = The graphs of CL against a and Cm show ho< greatly the non-linear steady theory of Ref. 6 improves on the results of linear theory. As we have already seen in Fig. 1, the linear pitching damping r"rom the theory of Ref. 3 is in good agreement with the various experimental results from Refs. I and 2 for a zero mean incidence. Also shol;m in Fig. 1 is a less satisfnctory curve from slender-wing theory, which gives -2 8 =+lca -z- 8 = w(2.3 - h) -in 0 = ha(0.7 - h) -m* 0 = +xa(1.5 - h12... (17) It is important to realise how wide of the mark this attractively simple linear theory can be, even for an aspect r&i0 as low as A = Relating to the gothic planform, Figs. 12, 13, 20, 21, 40 to 42 of Ref. 1 show little consistent effect of yrequency parameter over the available range o<v = WC/u < 0.7, but there are indications of a systematic dependence of -m* on a. The values of the pitching damping for a = 20 arg plotted against axis position in Fig. 4. The formula (13) corresponding to the non-linear theory of Ref. 8 represents a significant improvement on the linear theory of Ref. 3 for the experlmental range of oxis position. For the rearmost axis h = 0.949, hol.:ever, the new theory is not entirely satisfactory. The nature of this deficiency becomes clearer l'rhen both lift and pitching-moment derivatives are considered. Figs. 10 to 13 of Ref. 2 give the four derivatives for the pitching axis h = 0.724; the full curves against mean incidence adequately represent smoothed experimental values for any frequency parameter in the range O<v<O.?. The lift derivatives -zc and -26 of equation (14) are presented in Fig. 5, the experimental points (x) corresponding to the full curves in Figs. 10 and 12 of Ref. 2. For zero mean Incidence the stiffness derivative -zc shoivs satisfactory agreemen t with equation (17) from slender-wing theory and with the non-linear lifting-surface theory which for a = 0 reduces to Ref. 3; non-linear theory and experiment sholv fairly consistent large effects of a, and

10 -Y- values of -se at ct=l5" are more than double those at a = 0. On the other hand, the cross damping derivative -z 6 is less satisfactory. Only fair agreement is obtained between linear theory and experiment at a. = 0, and slenderwing theory gives a serious overestimate; moreover, linear theory, giving -zi, = for all a, SCtXlS preferable to the non-linear theory in this case. The direct derivatives -me and -mg of equation (14) are presented similarly in pig. 6. Both dcrivativcs show fair agreement between theory and experiment at a = 0, except that slender-wing theory greatly overestimates the pitching damping. Nonlinear theory and experiment show identical cffccts of a on -m*. For both pitching-moment derivatives the non-linear thzory offers a marked improvement on the linear theory of Ref Concluding Remarks Theoretical and experimental considerations point to large effects of mean incidence on oscillatory pitching derivatives for wings of moderate to low aspect ratio. FOP incompressible flow, a significant advance on linear liftingsurface theory has been made by means of Gersten's simplified model of separated flow, although it is recognised that this model is unrealistic and unreliable for the prediction of load distribution. The use of a theory for slow pitching oscillations is not too restrictive, since the experiments indicate that frequency effects are small. Although the calculated values of -z 6 for the gothic wing leave scope for further improvement, the other three pitching derivatives show a fairly good agreement between experiment and nonlinear theory. It is intended to examine the consequences of modifying the elementary vortex sheets of Gersten's model to represent rolled-up vortex elements. 6. Acknowledgement The authors acknowledge the assistance of Mrs. S. Lucas, who carried out the theoretical calculations and helped to prepare the diagrams. JER( 267)

11 - IO - No a Author(s) J. G. YWight and Miss A. virilkinson J. G. Wright H. C. Garner W. E. A. Acum and H. C. Garner K. Gersten H. C, Garner and Doris E. Lehrian H. lhulthopp H. C. Garner and Doris E. Lehrian References Title, etc. Low speed wind-tunnel measurement of oscillatory derivatives for a family ol" slender wings. Part II, Damping in pitch. Bristol Aircraft Ltd. Report W.T.348. May, Low speed wind tunnel measurements of the oscillatory longitudinal derivatives of a gothic wing of aspect ratio MoA S&T Memo l/62. A.R.C.24,357. April, Multhopp's subsonic lifting-surface theory of wings in slow pitching oscillations. A.R.C. R.& M July, The estima%ion of oscillatory wing and control derivatives. A.R C. c.p.623. March, Nlchtlineare Tragflgchentheorie insbesondere f&r Tragfltigel mit kleinem Seitenverhgltnis. Ingen-Arch., vol 30, pp , Non-linear theory of steady forces on wings with leading-edge flow separation. NPL Aero Report No A.R.C.24,523. February, Methods for calculating the lift distribution of wings. (Subsonic lifting-surface theory). A.R.C. R.& M January, Non-linear theory of wings In slow pitching oscillation with leadingedge flow separation, Report in preparation.

12 I *2* I -0 Lifting-rurfaca theory (Raf. 3) - - Slander-wing thaory (Eqn. 17) + Exparimant (Raf. I) 8.2% thick a Exparimant (Raf. I) 90% thick X Exparimant (Raf. 2) 90% thick -"(j 0 4, o h T.E. I.6 Pitching damping against axis position for gothic wings (A = 0.75) at a = 0

13 FIG. 2 Wing Section ofl wake Vortex model of steady flow over a wing with leadina-edae senaration

14 FIG. 3 O- a I- -A Linear theory (Ref.7) - Non-linear theory (Ref. 6) 3- q Previous tests (Ref. 2) x W. T. balance (Ref I- CL * I 1-I X x F LX- -Cm A x O-02,04 Steady lift against incidence and pitching moment (M-724)

15 FIG Linear theory (Ref. 3) Non-linear theory (Ref. 8) + Experiment (Ref. I) 82% thick 0 Experiment (Ret /o thick X Ex riment (Ref.2 5.O h thick - - -m. 8 I.A, I I c h I Pitching damping against axis position for gothic wings (AaO.75) at a:=zo

16 FIG Non-linear theory (Ref. 8) Slender-wing theory(eqn. If) X Experiment (Ref. 2) IO0 a IO0 a Lift derivatives against mean incidence for pitching ostgiiations about h

17 FIG m Non-linear theory (Ref. 8) 0 Slender-wing thcory(eqn.17) X Experiment (Ref. 2) 1 I 10 a Dwect pitching derivatives against mean incidence for oscillations about h = 0.724

18 _- A.R.C. G.P. No.695 February, 1963 Garner, H. C. and Lehnsn, MJXS D. E. PITCHING DERIVATIVES FOR A GOTHIC WING OSCILLATINGABODTAMEAN INCIDENCE This note briefly introduces a non-linear theory of wings in slow pitohmg oscillation with leading-edge separatzon in incompressible flow. The oscillatory ldt and pitohmg moment become linear functions of mean incidence. Comparisons mth measured pitching aer1vatlves are made for a gothic alng of aspect ratio A.R.C. C.P. No.695 February, 1963 Garner, H. C. and Lehrian, Miss D. E. PITCHING DERIVATIVES FOR A GOTHIC WING OSCILLA~GABOUT AMEAN ll'icide?jce This note br~ef'ly introduces a non-linear theory of wings in slow pdchmg oscdlat~on with leadmg-edge separation m incompressible flow. The oscdlatory lift and pitchlng moment become linear functions of mean inc1aence. Compar~ons mth measured pltchlng derivatives are made for a gotkuc wmg of aspect ratio A.R.C. C.P. No.695 February, 1963 Garner, H. C. and Lehrian, Miss D. E. PITCHINGDERJJATIVES FORAGOTHIC WING OSCILLATINGABOUTAMEBNINCID~NCE This note briefly introduces a non-linear theory of wings in slow pitching oscillation with leadmg-edge separation in inoompressible flow. The oscillatory lift and pitching moment become linear functions of mean inoidenoe. Comparisons with measured pitching derivatives are made for a gothic wing of aspect ratio 0.75.

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