Upwash Interference onwings of Finite Span in a Rectangular Wind Tunnel with Closed Side Walls and Porous-Slotted Floor and Roof

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1 ,, = R. & M. No MNSTRY OF AVATON AERONAUTCAL RESEARCH COUNCL REPORTS AND MEMORANDA Upwash nterference onwings of Finite Span in a Rectangular Wind Tunnel with Closed Side Walls and Porous-Slotted Floor and Roof By D. R. HOLDER, M.Sc. LONDON: HER MAJESTY'S STATONERY OFFCE 965 PRC~ 4 s. 6d. N~,T

2 - 1qR Upwash nterference onwings of Finite Span in a Rectangular Wind Tunnel with Closed Side Walls and Porous-Slotted Floor and Roof By D. R. HOLDER, M.Sc. Reports and Memoranda No. 3395* November, ~9@ Summary. An exact solution is obtained for the upwash interference on wings of finite span in a rectangular wind tunnel with dosed side walls and porous-slotted floor and roof. A rdsumd of previous results concerning two-dimensional aerofoils and small-span wings is included for completeness. 1. ntroduction. The upwash interference on a two-dimensional aerofoil which spans a rectangular wind tunnel, and on a small-span wing, have been obtained by the present author in Ref. 1. Viscous flow through the slotted floor and roof is represented by the homogeneous boundary condition 868x + K 82 8xSn + (1R)O On = 0, which was originally derived by Baldwin, Turner and Knechtel. A r6sum6 of these results is given below, and the analysis is extended to provide an exact solution for wings of finite span. 2. Analysis Rdsumd of Previous Analysis. The velocity potential of a two-dimensional aerofoil in free air is taken as ~ tan - (zx) and the influence of the tunnel floor and roof is given by the interference velocity potential where and P ~= 2~ o {A(q) sin qx + B(q) cos qx} sinh qz dq e -q~. ~(sinh qh+ Kq cosh qh)(1-kq) - (lr) ~ cosh qhl" ~ A(q) = - -- q [_ (sinh qh+kq cosh qh) ~ + {(lr) cosh qh} ~ B(q) = (sinh qh+ Kq cosh qh) ~ + {(lr) cosh qh} ~ (1) (2) e Replaces Hawker Siddeley Aviation Ltd., Wind Tunnel Report No A.R.C Published with the agreement of Hawker Siddeley Aviation Ltd.

3 - F For a small-span wing the free-air velocity potential is taken as 1 = 27"~ 1 -'1- (X 2+F2_}_z2)12 y2_~_z2 and the boundary condition at the closed side walls is satisfied by the method of images, which, taking into account the free-air contribution yields a velocity potential where Let us write - Fs ~ (~rebm'~m f~ Ze-~ ) rmry s~- 27r,~0 j= + 0 ~sinqxdq cos--b (3) t (m= 0) r e = -- +qzandj = [1(m4 0)" ~ = fly, ~) + gx, y, ~). f we represent the influence of the tunnel floor and roof by the velocity potential 0 = fdy, z) + gdx, y, z), the boundary condition may be written a~ (A. + fo.) = o (O a ~ 1 0) Yx +- K o-x~-z +--R N (g~+ gq) = o z = _+ h. (4) t follows that fo represents the interference which would arise in the plane x = 0 from a closed floor and roof, and in Ref. 1 it is shown that where and gq = ~ Y', 2j {A,~(q) sin qx -t- Bin(q) cos qx} sinh rz dq cos b (5) Am(q) = B~(q) = - e ":~ (sinh rh+kr cosh rh) (1 -Kr) - (rlqr) ~ cosh rh q (sinh rh + Kr cosh rh) ~ + {(rlqr) cosh rh} 2 - ~lqn q[(sinh rh + Kr cosh rh) ~ + {(rlqr) cosh rh}~] " (6) l 2.2. Wings of Finite Span. A precedure introduced by Polhamus, and outlined in Ref. 4, may be used to extend the analysis to deal with the wings of finite span. A swept or unswept wing of arbitary loading may be represented as illustrated in Fig. 1. The velocity field is represented by a spanwise distribution of vortex doublets whose strength is proportional to the local loading. For a uniformly loaded unswept wing, the velocity potential of the wing and the images in the tunnel side walls is easily shown to be ~- 4~ --~ m~--o j.~1b + o--~-sinqx ~ 2e-r~ dq cos mzr (y b_~) de ) m=,( = 2~m ~== ~- sin qx dq cos--g- sin--b-i ~. (7)

4 From the similarity of equations (3) and (7), it follows that CQ = fo(y, z) + go(x, y, z) where, as before, 0 represents the influence of a closed floor and roof and --~s~ f: ql tarry( mzrs]m~s gq -- 2~b me o 2j {Am(q) sin qx + Bin(q) cos qx} sinh rz d cos -if- sin ~--]~-- (8) with Am(q) and Bin(q) as defined in equations (6) Equations (5) and (8) become identical as S -+0, and it has been shown that a limiting process as 1R -+0 yields results consistent with those of Ref. 5. The irterference in a wind tunnel with uniformly porous floor and roof is obtained by putting K = 0. For a tunnel with completely open floor and roof K = 0 and 1R -> O Numerical Results. The mean incidence correction, obtained by integrating the interference over the span of the wing, is A~C 8 - CL~ - 1 f~ ~ bh 2~- ~_s ~Z ( + CQ -- 1) ~ dy with x = z = 0 = 8c- ~ ~--o -q (sinhrh+kr coshrh) 2 + {(rqr) coshrh} ~ ~sm-b-) b (9) where 3 c is the corresponding interference factor for a closed wind tunnel. We may write ~ + N 3~ sin m~rs~mtrsl2 (10) -U ] -U J and 8,~ is a function only of K, 1R and tunnel heightwidth ratio A. Once 3, is known and values of 8 m have been calculated, the interference factor may easily be obtained for any wing span and, with a simple extension of the analysis, for any symmetrical wing loading. The summation is rapidly convergent, and only the first three terms are required for practical purposes. t is easily shown that 3 o is the interference factor for a two-dimensional aerofoil, and that 3m(A ) = 32~(~2). Values of 3~, m = 0, 1 and 2 have been calculated for ~ = 1 and ~ = 0-5 (the latter for halfmodels on the side walls of a square tunnel) for various porosity and slot geometry parameters, see Fig. 2. ncidence correction factors have been calculated for spantunnel-width ratios ~ -- 0 and 0-5, Figs. 3 and 4. The parameters for zero incidence correction are shown in Fig. 5. Compressibility effects may be taken into account by div!ding the porosity parameter by ~(1-32). t is perhaps worth noting at this stage that, with a suitable choice of the two parameters, it is possible to obtain zero mean upwash interference and zero mean blockage interference at a given subsonic Mach number. 3. Concluding Remarks. t is hoped that the results of this investigation will prove useful for many tunnels, not only the correcting results, but also in obtaining improved testing conditions. 3

5 NOTATON X~ y~ 2: n z~ P el $ S b h C Cartesian co-ordinates with their origin at the centre of the working section: x + ve downstream and z + ve upwards Outward normal at tunnel wall Freestream velocity Vortex strength USCL A~C Mean incidence correction factor = -- cls Lift½o U2S Wing semi-span Wing area Tunnel semi-width Tunnel semi-height 4s Tunnel cross-sectional area = 4bh K Slot geometry parameter = d log cosec ~ 7r for a rectangular tunnel (c = Kh) a d R q Slot width Distance between slots pv Porosity parameter defined by Ap -- R ~n Variable of integration j( q~ + -b~- ], where m is an integer 4

6 No. Author(s) REFERENCES Title, etc. 1 D.R. Holder... B. S. Baldwin, J. B. Turne~ and E. D. Knechtel. 3 E.C. Polhamus E.W.E. Rogers... 5 D.D. Davies, Jr. and D. Moore Upwash interference in a rectangular wind tunnel with closed side wails and porous slotted floor and roof. A.R.C.R. & M April, Wall interference in wind tunnels with slotted and porous boundaries at subsonic speeds. N.A.C.A. Tech. Note May, Jet-boundary-induced-upwash velocities for swept reflection-plane models mounted vertically in a 7- x 10-foot, clos@, rectangular wind tunnel. N.A.C.A. Tech. Note November, Wall interference of closed rectangular wind tunnels. Thesis for M.Sc., University of London Analytical study of blockage- and lift-interference corrections for slotted tunnels obtained by the substituation of an equivalent homogeneous boundary for the discrete slots. N.A.C.A. Research Memo. L53E07b. June, (90848) A*

7 b~- --, Fic. 1. Representation of a finite-span wing. 6

8 R~o L\i \ i ~" (X=0"5) - 0"05 R= 0 "", "2 0 "4 0'6 l.l,c \ 0-8 '0-0'0,,,~\ R=, R=5 --0'15-0"005-0, O-Z 0 '4 l c \ \- ~= ~,, 0'6 0'8 '0 FG. 2. The incidence correction for a twodimensional aerofoil for various porosity and slot geometry parameters. -0'010-0" ,2 0" l-i-c Values of 81 and 82 for various porosity and slot geometry parameters. Fie. 2 (continued).

9 s \ P,=O 0"10 ~"- ~ ~ _..._~...._._ ~ +o.o~ ~,~-. R= T OdE 0'10 ~'0"0! ~..~ -0.c \\\ \ 3 \,, 00 \ \ -0"05 \!\ \- i '~ ""x,,,,, -o.zo \ --O.lO ~ -o.2s ~ 1 R= -0'15 0 FG ~ "4 0"6 0-8 l.o 0 0"2 0,4 1 0'6 O.B The incidence correction factor for a small-span wing, for various porosity and slot geometry parameters.

10 0.0 R=b "- 6 +o.o5 ~"~-. +0,05 "~ x~''~ ~ R= J_ 3 \ -0'0 \, \ -.,~ ~-_-~ -0,, ;~= 0 Fic. 4. 0, , Z B 1-0 lli+ l+c The incidence correction factor for a finite-span wing (or = 0.5) for various porosity and slot geometry parameters.

11 ' -C 0-~ 0,6 0"4 R O-Z = ~=:os, l.. ' i i! 0.? "~ '0 +c FC. 5. Values o the porosity and slot geometry parameters for zero incidence corrections. (90848) Wt (5 265 Hw. 10

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