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;(,Q i7 ('jt I NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1310 CORRECTION FACTORS FOR WIND TUNNELS OF ELLIPTIC SECTION WITH PARTLY OPEN AND PARTLY CLOSED TEST SECTION* By F. Riegels SUMMARY A wind tunnel of elliptic section with partly open and partly closed test section contains a wing with rectangular lift distribu tion. The additional flow caused by the wall interference is determined by conformal representation. The correction factor for an elliptic jet of 1:/2 axial ratio is plotted for several spanchannel width ratios and several included angles, that is, the angles which the lines connecting the end points of the solid part of the tunnel boundary form with the center of the ellipse. (Compare figs. 1 and 5.) As on the circular jet (refer ences 5 and 6), it is found that the correction for angle of attack. and drag becomes zero at a certain included angle. This angle varies with the wing span. The theory is so applied that it can be utilized also for elliptic tunnels with different axial ratio. An extension to include wings suspended over the median plane of the tunnel is likewise easily possible. I. INTRODUCTION The finite boundary of jets causes additional velocities in the flow, especially at the wing. As a consequence, the angle of attack and the drag coefficient measured in the jet at a given lift must be corrected. Calculations dealing with the effect of the jet boundaries are numerous. Open and closed jets have been explored and also jets *"Korrekturfaktoren fMr Windkanlle elliptischen Querschnitts mit teilveise offener und teilweise geschlossener Messstrecke." Luftfahrtforschung Bd. 16, Lfg. 1, 1939, pp. 2630. NACA TM 1310 whose boundaries consist partly of solid walls and partly of free jet boundaries (references 1 to 6). The last arrangements have the advantage that they make it easy to conduct a greater number of tests and also permit choosing a boundary in such a way that the mean additional velocity at the wing, and hence the correction factor, becomes exactly zero for the measure ments. If the wing model is suspended with the suction side downward, jets with fixed boundaries produce additional downward velocities, those with free jet boundaries, upward velocities. Therefore, it can be expected that, with suitable distribution of fixed and free boundaries over the section, no additional velocity is produced at all. The present article deals with the effect of a partly open and partly closed jet of elliptic section. 2. ELLIPTIC JET WITH ONE SOLID WALL (FIG. 1) With the assumption of small additional velocities whose squares are negligible, and a nondeformed jet boundary, the determination of the additional flow can be reduced to the consideration of the flow condition in a section infinitely far downstream, where an additional velocity exists which is twice as great as that at the wing (reference 1). Suppose that the jet has the elliptic section with the axes 2a' and 2b' shown in figure 1; the wing of span b = 2s and rectangular lift distribution is mounted in the center of the jet section. The chosen system of coordinates (x,y) has its origin in the center of the ellipse, so that the shed vortices of the wing push through the section at the points x = ts. Now, the problem is to define an additional flow in such a way that the boundary conditions are satisfied. They are: for the solid part, disappearance of the normal velocity component; for the free jet boundary, constant pressure, which for the assumed smallness of the velocity is identical with the requirement that the tangential velocity component shall disappear (reference 1). But, as such an additional flow is difficult to define in the zplane, it is attempted to find a plane by conformal representation in which the potential of the required additional flow is easily obtainable. Now, reference 5 cites a report by K. Kondo which treats the corresponding problem for circular jet. The mapping function z" = c tan t is used which maps the inside of a circle in the z"plane on the inside NACA TM 1310 of a strip in the splane in such a way that a part of the circumference is changed in the one, the remaining part in the other of the straight lines bounding the strip (fig. 3). But in the plane of the strip, an additional potential that satisfies the boundary conditions is easily indicated and, if it succeeds in mapping the inside of the ellipse on the inside of a circle, the aforementioned representation is fundamentally accomplished. The procedure is developed step by step. With the aid of the function z = k sn(2 arc sin ) (1) the inside of the ellipse in the zplane is mapped on the unit circle in the z'plane (fig. 2), where 2K is the half real period, K the modulus of the Jakobian elliptic function sn Z and e = a'2 b'2 half the distance of the foci of the ellipse. As to the theory of this mapping function, the reader is referred to the report by de Haller (reference 7) and the "Schwarzschen Abhandlungen" (reference 8). Next, the plane z' is rotated about n/2 and followed by a translation z" = I c2 iz' (2) as a result of which the axis of the ordinates of the new z"plane exactly separates the fixed part of the circumference from the free part. (Compare figs. 2 and 3.) Both parts meet in the points z" = tic. The subsequent transformation z" = c tan ( (3) projects these two points to infinity, where the arcs of the circle change into straight lines of distance n/2 (fig. 3). Combining these 4 NACA TM 1310 transformations produces finally the mapping function of the zplane on the (plane S= arc tan [1 c2 irk en ? arc sin (4) c Zn e Since the derivative of this function is used later on, it is given here i.? 1k ccnZ dn Z d 1 rt (5) dz z2(1 ksn2Z + 2i k(l c2)snz) where, for abbreviation, ?K arc sin M = Z is introduced. it e The boundary conditions, disappearance of the normal component on the fixed part and disappearance of the tangential velocity on the free part of the circumference of the ellipse, can be satisfied now in the splane by repeated reflection of the original vortex doublet at the boundaries. The result is the vortex system represented in figure 4. The potential of this flow is readily defined, since the vortex rows can be added up in horizontal direction (reference 5): tan ill' tan + 2 + V + in' tan tan Pi 2 2 FO = log (6) t' + iq' tan + 2o + El i' tan 2tan2 2 2 with I' + iq' and i' ir' representing the points corresponding to the vortex points z = Ts in the splane. The potential of the additional flow Is obtained by subtracting the potential of the original flow in the zplane from the potential Fo tan ( iT)' tan 4 + +21o1 log + + 2t + V in)' 2 2 2 F = T3 log  l log z + s ( ) 2rt ( 6 + IT!' ( + 2io + * T* 2n z s tan tan 2 2 NACA TM 1310 cci co (U O 3 0) pa N +J 3 I. 4) 0 CO 0 3 ID 0 4 41 H 0 0) r) *~ 191 5 +3 4r) U 0 a4 .4 0 II 3 4' +J e 0 .4 0' F .1 i a a 4 U U 0 E uj4) Cc +3 0O 0 a5 u 4 alP 5 +' co r 0 P oo *r4l U t ^ 0 + Is 0 4' + l 1:31> o fl) II 42j  II 4' 3 Nto p N p  0 0 41 > (U 0 E0 3 cy  s 4 3 3 rf SII d 1> 0 *0 U; 0 c Ei CMj + +3 + rI o CM 0 0 +\ Cuj r_ 42  Mr + + +r N 5^ 03 a.' +J 0 *JO 5 5 42 +  UJ1 + c 5 + ~ ' + 0 N " NM  ^ . 5 N 4 U, 11o IL rz.l 0 t C! P11DI .c c u o r4 c > d 0c 0 L) U 14 L L o 4ri 42 0 4) 0 + o o L 4. 613 0 U  0 rC. =S I +. U 0 o o 4 tWU c3 uU 0' U 0 ,1 NACA TM 1310 For the calculation, the following relations are added. For points of the major axis, the mapping function (equation (1)) changes to x' = /k sn( arc sin A; k) V IT e (12) for 0 < x < e, and x' = Ln arc cosh ; k' n e (13) for e < x with k' denoting The points z = is equations (12) and then at the points t' = arc tan 1' = arc tanh For the extreme case of becomes = a'b's 4(a,2 b'2) the modulus k' = f1 k2 in which the vortices in (13) to z' = s,; in the complementary to k. the zplane lie change tplane, the vortices 1 2c2 + s 12 + 1V + s12(2 4c2 + 512) 2+11 c2 1 + B12 + /l + sl2(2 +12 12) 2cs a wing with zero span, formula (11) fK k 4 c + 2c2 + C vbwhere k (K22 k 4K2 6 1 2 + k2) 6 b1.2 A2 a,2L 1  is a constant solely dependent on the axial ratio of the elliptic tunnel section. (14) (15) (16) NACA TM 1310 3. ELLIPTIC JET WITH TWO SOLID WALLS (FIG. 5) In this case, it is appropriate to map the inside of the elliptic section on the inside of a rectangle (fig. 5). The fixed walls correspond then to two opposite sides of the rectangle and the free jet boundaries to the other two sides. The first step is the same as before, namely, the inside of the ellipse is mapped on the inside of the unit circle with the aid of the mapping function z' = k sn arc sin z; k) (17) given by equation (1). The inside of the unit circle becomes the inside of a rectangle by means of the function ( i=' 2z' sn; cos ) = 2 (18) 2 1 + z'2 if C represents the coordinate of the plane of the rectangle and cos is the modulus of the elliptic function sn (compare reference 6, p. 170), where c, is an angle specifying the tunnel section opening ratio in the plane of the circle. (Corpare fig. 5.) The connection between the splane and the original zplane is therefore given by the mapping function cos = 2/k sn (Z; k) sn(; cos (19) 2) 1 + ksn2(Z.; k) 2K z where, for abbreviation, arc sin = Z, as before. The modulus k Ir e is again dependent on the axial ratio of the elliptic section. The square of the derivative of this function Is given by 16K2 2 6K 2 kcn 2Z dn2 Z (dt 2 A2 (20) S= (e2 z2)(1 2k cos sn2Z + k2snZ) 8 NACA TM 1310 The conditions at the jet boundaries are satisfied by repeated reflection of the vortex doublet on the sides of the rectangle, while the sense of rotation of the vortices is inverse for reflection at the sides that correspond to the fixed walls and remains the same for reflection on the sides of the rectangle that correspond to the free jet boundaries. The potential of such a system of vortices is given by (compare reference 6) sn c n F0 i 2 2 dshSLk^ Fo 2# g g 2 s 4p enF + log (21) dnrF 2 where T6' is the location of the original vortex in the tplane (in the zplane, the wing is in the center of the jet, hence z = Ts). Subtracting from this expression the potential of the original vortex in the zplane ri z + s F1 = r log z (22) S2i z s leaves the potential of the lookedfor additional flow F = FO F1 (23) Since F = 0 + it, the stream function can be deduced, and so yields the mean downwash over the span of a wing suspended in the center of the section NACA TM 1310 1 *(s) *(s) 2 2s Iir dt\ cnfl w log s 4ns Fdz z = s snt'dnt' With CnVt r =a 2 and from the relation for the additional angle of attack  c v a F V 8 F0 (24) (25) the correction factor 5 follows as 2KcnZ dn Z(l k2snZ) , a'b' x F = 2s2 log e2 s2snZ(l 2k cos esn2Z + k2s n Z) . (26) where k is the modulus of the present elliptic function. For the extreme case of a wing with disappearing span, this equation simplifies to 5 = alb' 1 K2 (1 6k cos 3 + k2) 6(a'2 b' 2 (27) NACA TM 1310 4. RESULT For an elliptic tunnel of l: 2 axial ratio, the correction factors 3 were computed by equations (11) or (16) and (26) or (27) for wings with rectangular lift distribution for the spanmajor ellipse axis ratios b = 0, 0.2, 0.4, 0.6, and 0.8 and plotted in 2a' figures 6 and 7 against an angle p or I which is decisive for the ratio of fixed wall and free jet boundary. It is noted that the correction factors for rp = 0 or S = it and p = 2v or b = 0, that is, for completely open and completely closed jets, assume the values given by Sanuki and Tani (reference 9). The diagrams indicate the distance which the solid part of the jet boundary must reach to produce zero correction. The amount of this favorable coverage varies with the span of the wing, as seen from figure 8 where the angles for which 5 = 0 are plotted against the spantunnel width ratio. Clearer than the plotting of the angles is the representation of the ratio d/b', where d is the distance between the major axis and the point at which solid and free boundary meet (fig. 9). In general, the span of a model wing for a tunnel of l:V2 axial ratio amounts to about b = 0.7 x 2a'. So in this case with the use of one solid wall the covering would have to extend to d = 0.385 to b' produce zero correction. By choosing a partial covering of the jet boundaries with two solid walls, the correction would disappear for d4 = 0.64, so that the last arrangement seems more promising in many b* respects as far as experimental technique is concerned, but naturally calls for more elaborate test preparations. Translated by J. Vanier National Advisory Committee for Aeronautics NACA TM 1310 REFERENCES 1. Prandtl, L., and Betz, A.: Vier Abhandlungenr zur Hydrodynamik und Aerodynamik. Kaiser Wilhelm Instituts fUr Stromungsforschung, GCttingen, 1927. 2. Theodorsen, Theodore: The Theory of WindTunnel Wall Interference. NACA Rep. 410, 1931. 3. Theodorsen, Theodore, and Silverstein, Abe: Experimental Verifi cation of the Theory of WindTunnel Boundary Interference. NACA Rep. 478, 1934. 4. Tani I., and Taima, M.: Two Notes on the Boundary Influence of Wind Tunnels of Circular Cross Section. Rep. Aeronaut. Res. Inst. Tokyo, Nr. 121, 1935. 5. Kondo, K.: The Wall Interference of Wind Tunnels with Boundaries of Circular Arcs. Rep. No. 126 (vol. X, 8), Aero. Res. Inst. Tokyo Imperial Univ., Aug. 1935. 6. Kondo, K.: Boundary Interference of Partially Closed Wind Tunnels. Rep. No. 137 (vol. XI, 5), Aero. Res. Inst., Tokyo Imperial Univ., Mar. 1936. 7. de Haller, P.: L'Influence des Limites de la Veine fluide sur les characteristiques aerodynamiques d' une surface portante. Comm. de 1' institute d' Aerodynamique de 1' Ecole Polytechnique Federal, Zirich 1934. 8. Schwarz, H. A.: Gesammelte Math Abhandlungen, Bd.II, p. 102. 9. Sanuki, M. and Tani, J.: The Wall Interference of Wind Tunnel of Elliptic CrossSection. Proc. Phys. Math. Soc. Jap. III.s., Bd 14, 1932, p. 592. NACA TM 1310 \ / 2a'  Figure 1. The elliptic jet with one solid wall. / 1 I I  Figure 2. Mapping of zplane on the z'plane. Ile N NACA TM 1310   N. 0 0 Figure 3. Figure 3. 0 i;,1 0 b ___ / ^0c /Mapping of the z "plae on the plane. Mapping of the z "plarne on the s plane. I q q 0I Figure 4. The reflection system for compliance with the boundary conditions. C) 0 NACA TM 1310 Sj I Cd ri 0 u cd 0 *0 o ci~ *O 4 0 cd ( a) us Q. *!ii bc) .. C *I ml Cd 1, NACA TM 1310 / \ / r 2s Jos./ Figure 6. The correction factor 6 plotted against angle p one solid wall. 08 10 1 .4 NACA TM 1310 V~Y~71 1 If I II U.  4 I I 9 9 4i . & I i I ~I~i  I I 4 4 I ~ 4 ~~ I f 4 I I I~~ I 't 4 ~ I 1 I. 1 4 4 4 4 ~.~. 4t.&..1 4 /1 2a's L 2a  Figure 7. The correction factor plotted against angle a two solid walls. N \ 200 600 08 06 0.4 0.4 06 (28 10 1.2 1.4 140 k0 02 NACA TM 1310 180 150  120 90 60  300 ~ S a'  0 02 04 06 08 O Figure 8. The values of and d for zero correction factor plotted against the spantunnel width ratio. NACA TM 1310 0 0.2 0.4 06 08 1.0 Figure 9. for 6 = 0. L I *r4 4 V) 0 03 a aa OO kcd UO 0 0 a(U r I , I O 01\ 4 Ed zE U +3 HO OU 0+ rOO k+ OF. ed .0 r JO 0U *H k +3 MU 00 u tn O E e 00 0 H ,1 V N E in c d S ld M 0 0 SB Id cc i 0) u 0 p, rI M M l 0) 0 a a) c 0) rA ci A, 43 ca aj 8 0 Q) rd P '6W *SIO S0 04 0 9 O4 C1 r 3 3 O2 J' 14 0 0d 14 a) 140 ^r c m 'd 0} o N 4 0 o4 + + H +3 CD " n! C a ru k r ) S00 0 + *H U 14a 0 to 0 oT O U ilrl 4'C U *H r 4 0 " A O0 H +3 0 +O 'd +P 0 u P rH 'd u I 2 ) 14 >o3 0 02 0= 14 UPCO 'ddi fr r4 0 14 rI a N 4 U) d 0 bo 0 O'd S2 + R 0 (D (D 0 0 0 0 4) *c IQ H a 0) I H SI ) M CM 02 4+2 U w 0 0.0 >a 4 +) >02 04*' +) 0 0 02 0 14 to 4 Hi ca14 r42rI al .iH ii . 0, o < a > 'd 0 U ) rd ti $4 0 a u a P, S 4'M U) 4 ou I' a)  P4 U) a) O S 0) 4 0 I p 0 r 0 ab El 4Q fl U) 1 4 0. 0 O S.r 4' h + H a +3 4 ) O 4  14 1002 00S +3 (,+2 ( +3 Oi .0 ui HO0 0 + 0 'd U 2 Hl o L a r > o *C1 ,0 +4 I H 0*2 4 < U r 4 U 4O3 U OW dd SI $ u c u 0 fiL fi r1 a P4 0 *3 *d 4) 0 0 P. *r 'd U *H rH I 1 0 >a ri o k r4 aU k rI mw0 OM go (u adr o0 'd 00 0 'd > 0 0 0 ( 4+ +43 X: > 0 in 0 fl Cd 0 od 0j 41 u Cd a) 0 A Q) N +) 0 >0.4 P, +3 M 4' 0 P. o << ) UNIVERSITY OF FLORIDA 31262 08105 805 8 
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