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icA mi 1390 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1340 LIFT Oi A BENT, FLAT PLATE* By F. Keune For the theoretical treatment of fin and rudder, or of a wing with flap, the profile is simplified in such a way that a bent flat plate may be considered. The aerodynamic forces on it can be calculated by means of known approximations due to Munk (ref. 1) or Glauert (ref. 2). The problem was recently treated exactly by the Russian work of Chapligin and Arjanikov (ref. 3). The calculations below will be carried out fol lowing the latter work, and the difference between them and the usual approximations will be determined.1 OUTLINE 1. Theory 2. Results 3. Summary 4. References 1. THEORY The lift coefficient of a profile at angle of attack a can be written in the form Ca = c sin (a ao) (1) The problem is to determine the liftcurve slope c and the angle of zero lift so. For the flat plate, the theoretical liftcurve slope is c = 2n; for other profiles, it deviates only a little from this value. The angle of zero lift for the flat plate is, moreover, 0o = 00. With the deflec tion of the flap, both values vary as a function of the chord ratio tR/t and the deflection angle B. *"Auftrieb einer geknickten ebenen Platte." (Bericht der Aero dynamischer Versuchsanstalt G6ttingen.) Luftfahrtforschung, March 20, 1936, Annual Volume, pp. 8587. 11 am especially grateful to Dr. I. Lotz for many suggestions in carrying out the analysis. 2 NACA TM 1340 In order to calculate c and ao we must determine the flow around the bent plate and in particular the circulation needed for smooth flow at the trailing edge B (fig. 1). With the aid of conformal mapping, the region outside of the profile (fig. 1) in the z plane is mapped, according to the SchwarzChristoffel procedure (ref. 4), onto the lower half of the A plane (fig. 2b), so that the border of the profile (see fig. 2a) on the h axis and the point at infinity in the z plane transform to the point = ik The mapping function has for this case the form z = c f ) (A )n d (2) 0 (2 + k2)2 (A 2)n where the exponent n is related to the deflection angle 0 of the flap (see fig. 1) by Snn (3) In order to be able to evaluate the integral (see eq. (2)) one carries out an appropriate linear transformation, by which the ? plane goes into the plane. The p axis is thereby mapped onto the axis. If we consider now the pointbypoint matching of the z plane with the t plane (fig. 3), there results from this transformation: the leading edge A (fig. 1) and the point A* (C = 1i); the location of the bend on the suction side and the zero point. 0*; the trailing edge B and the point B* ( = r2); the location of the bend on the pressure side of the profile and the infinitely distant point on the 5 axis. The point at infinity in the z plane falls at the point (3 = 1 i5. For these positions of the designated points the integral of the SchwarzChristoffel formula (2) can be calculated. If we introduce the abbreviation K2 = 1 + 82 (1 n2), we obtain the mapping function ( "n n K 1 + n z = 2(t tR) (4) 1 n2 1 + n ( 1)2 + 52 The position of the point t3 = 1 i6, the image of the point at infinity of the z plane, depends on the ratio of the flap chord to the forward part of the wing and on the deflection of the flap. The position is determined by the equation t = o = K 1( + n) (5) t tR K + 1 K n NACA TM 1340 In the z plane a parallel stream prevails at infinity; on this stream (on account of the circulation around the plate) is superposed a vortex whose strength depends on the angle of attack. To the parallel flow of the z plane there corresponds a flow in the t plane which is produced by a dipole at the point %3 = 1 ib; and to the circulation flow there corresponds a vortex flow around the same point. The strength of the circulation is given by the condition that in the z plane the flow is smooth at B; thus at B* the velocity is zero. If we give the position of the stagnation point t4 = 4(a), then the velocity must have the form d 2(t t(vK, l 1 + n n 2)(t + t4) d ( 1 n 2 ,1 + n [( 1)2 + 2]2 (6 In this equation the quantity N is a constant which is essentially dependent on the angle of attack a. We determine the circulation from the residue of the velocity func tion w (see eq. (6)). One obtains this by resolution into partial frac tions. According to lengthier calculation, which can be seen in Chapligin and Arjanikov (pp. 57, ref. 4), one has for the lift Ca = (1 + o)(< + 1) 1 + n,1 ,n. ca = (1i )(K + + 1) ( j ) (+ n'(n) sin(o ) (7) where the angle of zero lift (pp. 7 and 8, ref. 4) is calculated as +0o = arctan n nr + n arctan j2 (8) KI F n2 n/ By equating formulas (1) and (7) there is obtained for the liftcurve slope ( 1t + ) + (i+n)/2 K ( n)/ = ( ( (9) (1 + + 1) n, 1 n, In the evaluation one next determines, for a prescribed deflection angle 0 and ratio a = tR/(t tR), the value K according to NACA TM 1340 equation (5).2 With this equation the calculation of the angle of zero lift a. (eq. (8)), of the liftcurve slope c (eq. (9)) and of the lift ca (eq. (1)) are feasible.3 2. RESULTS The dependence of ao, the angle of zero lift, on the ratio tR/t for fixeddeflection angle B is seen in figure 5 (eq. (8)). Since for values n = p/i > 0.1 the curves for tR/t > 0.5 become approximately pro portional to one another, the curves for 1 = 450 and 600 are therefore not plotted any further. If one plots the angle ao of zero lift for different chord ratios against the deflection angle 3 (fig. 6), one obtains almost straight lines. The dashed lines representing Glauert's approximation deviate from these curves only for small chord ratio and there one has somewhat smaller values. Figure 7 gives the ratio of the liftcurve slope c of the plate with flap deflection to the flat plate value of 2n. The lift coefficient ca is referred to the original wing chord t. For the actual chord of the wing t', which becomes smaller with increasing flap deflection,4 we obtain somewhat larger deviations from c/2n = 1 in the reversed sense (c/2n > 1). For all curves 3, they are the largest with chord ratio tR't = 0.5 in both cases (eq. (9)). Glauert assumed that the quantity c/2n, referred to the original chord t, varies as the ratio t'/t. This assumption results in values that are too small. Figures 8 and 9 show the dependence of the lift coefficient ca on the ratio tR t for fixed deflection 1 with an angle of attack of the profile of a = 5 or a = 100 (eq. (1)). Figure 10 is a cross plot of figure 8 and points out that the depend ence of lift on the deflection angle B for values 0 > 150 does not remain linear any longer. The linear dependence assumed by Glauert thus holds only for deflections to 150. Glauert has replaced the sine of the angle of attack, referred to the zerolift direction, with a ao itself. The lift becomes too large, therefore, for large a ao. Up to deflection of 1 = 150 this error is counterbalanced in practice by the toosmall value of the liftcurve slope. 2For calculation of the quantities c and ao for chord ratios tp/t > 0.5, one employs the values 0 < tR/t < 0.5, while imagining the wing surface and flap exchanged, as shown in figures ha and b, and also the potential correspondingly changed. 3The specified formulas hold for positive and negative deflections; only the sign of ao and therefore the lift ca are changed. 4t'2 = (t tR)2 + tR2 + 2(t tR)tR cos P, (see fig. 1 and the sketch in fig. 11). NACA TM 1340 Figure 11 gives the circulation distribution for the chord ratio tR/t = 0.5 and the deflection angle 0 = 300 with the angle of attack a = 70; Glauert's approximations are shown by dashed lines. For this case the lift, referred to the original chord, is ca = 3.25 and accord ing to Glauert ca = 3.33, referred to the same chord. 3. SUMMARY The lift on a bent, flat plate is calculated exactly by use of con formal mapping, and the results for the angle ao of zero lift, the liftcurve slope c, the lift coefficient ca and the circulation dis 1 dr tribution are compared with those obtained by Glauert's approxi vo dx mation. This approximation suffices for deflection angles of the flap p < 150, when the angle of attack a is so large that the small inexact itudes of the value of ao of Glauert's approximation in the term (a cro) can be neglected: otherwise, one obtains values somewhat too small in this region. The results for the liftcurve slope are practi cally the same for 0 < 9. Translated by Paul F. Byrd National Advisory Committee for Aeronautics Ames Aeronautical Laboratory Moffett Field, California 4. REFERENCES 1. Munk, M.: Elements of the Wing Section Theory and of the Wing Theory. TNACA Rep. 191, 1924. 2. Clauert, H.: Theoretical Relationship for an Aerofoil with Hinged Flap. ARC Rep. and Mem. lo. 1095, 1927. 3. Chapligin, S. A., and Arjanikov, N. S.: On the Forward Aerofoil and Flap Slot Theory. Trans. Centr. AeroHydrodyn. Inst., l1o. 105, 1'31. 4. Courant, R., and Hurwitz, A.: Funktionentheorie, SpringerVerlag, Berlin, 1922. NACA TM 1340 NACA TM 1340 0 ..x Figure 1. The bent or hinged flat plate. 0 6 a) r I 5 3 A 1,. Figure 2. Mapping of the outer region of the bent, flat plate onto a half plane. NACA TM 1340 B" Figure 3. Correspondence of the z plane (fig. 1) to the 5 plane. 6) S (t t) b) ,J . Figure h. Exchange of the wing surface and flap. NACA TM 1340 Figure 5. Angle of zero lift a,,, as a function of the chord ratio for various flap deflections. NACA TM 1340 30 0 o P0,75 200 AV 20 I//f,/.// 10 0O 15O 30 Figure 6. Zero lift angle as a function of the flap deflection 3 for various chord ratios. Glauert's approximation is shown by dashed lines. NACA TM 1340 100 0 7_\ _ / 0,98   q98 \IN *12 q96 q94z o q 5 1o Figure 7. Ratio of liftcurve slope c to flat plate as a function of the chord ratio for various flap deflections. NACA TM 1340 4 ^ 93 .o5 1 0 Figure 8. Dependence of the lift coefficient ca on chord ratio for various flap deflections at an angle of attack a = +5o. NACA TM 1340 2 . gO 30 00 tRtt 0 0,5 ,0 Figure 9. Dependence of lift coefficient ca on chord ratio for various flap deflections at angle of attack a = +100 NACA TM 1340 40 S  315 / 0o5 40 2,0 0 150 300 o 600 Figure 10. Lift coefficient ca as a function of flap deflection for various chord ratios at angle of attack sr = +50. NACA TM 1340 0 0,1 q2 0,3 0,4 0,5 0,6 0,7 0,8 9 1,0 Figure 11. Circulation distribution with tR/t = 0.5, a. = 7, 3 = 300. The dashed lines give Glauert's approximation. NACALangley 22155 1000 CD co S C s uU IV roi E . j :3.< g a C., i t on 0 < ,4 .4rm . "' a. cS ai /o S S<^S vr b csZ <2 11 B i ca 4B .0 .g~ 0 ri i B B. g  s,( O> 5 a . cc r. 4G ) _ St k 4U (0 ( D.. >~ 0 C:(.o ) 4 1 (U05.  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