The Effect of Wind Tunnel Wall Interference on the Stalling Characteristics of Wings

In this paper it is shown that the distortion of the lift distribution of a wing in a wind tunnel caused by the wind tunnel wall interference can be considered as an induced twist in the wing. The magnitude of this twist is calculated for a wing in a circular closed wind tunnel. In numerical examples and an experimental check it is seen that this induced twist (washin for a closed wind tunnel working section) is large enough to cause very marked changes in the stalling characteristics of a wing unless the wing span is less than eighty per cent of the wind tunnel diameter. T ALMOST universal use of highly tapered and highly loaded monoplane wings has made the problem of stalling characteristics one of great importance. A wing with a tendency toward tip-stall will almost invariably have undesirable lateral stability and control characteristics near the stall. The normal induction effect, which can be calculated by the Prandtl wing theory, causes the effective angle of attack of the tip sections of an untwisted highly tapered wing to be considerably greater than that of the central portion of the wing; consequently the stalling angle is first reached at the wing tips as the angle of attack is increased; and a tip-stall is produced. This effect may be counteracted either by twisting the wing so that the geometrical angle of attack of the tip sections is less than that of the center or by changing the airfoil section so that the tip sections have more camber than the central portion of the wing. A combination of these two methods is normally used to prevent tip-stall. As both of these methods increase the wing drag, the optimum design uses only sufficient twist or camber to prevent tip-stall. The theoretical methods of solving this problem are generally satisfactory; however, it is advisable to obtain an experimental check of the results in a model test. Wind tunnel investigations of the stalling of tapered wings, utilizing both force measurements and floss tufts to show the flow over the wing, have been carried out and described by Nazir and by C. B. Millikan, and such tests are now generally included in any wind tunnel test program. In all such tests which the author has observed, the distortion of the lift distribution due to the interference of the wind tunnel walls has been neglected. Although this distortion is generally of little significance below the stall, it will be shown in the present paper to be of considerable importance in connection with the spanwise development of the stall. The pertinent calculation will be carried out for the case of a closed wind tunnel having a circular cross-section; similar results would be found for any other shape of closed wind tunnel working section. The general problem of the lift distribution for a wing of arbitrary plan form in a circular wind tunnel has been solved by C. B. Millikan. His method is very similar to that developed by Lotz for determining the lift on a wing in an unbounded fluid. For the present investigation it will, however, be sufficient to consider the much simpler inverse problem of determining the plan form necessary to produce a given lift distribution. In the analysis, the following notations will be used : T = Circulation about the wing (in the wind tunnel) T0 = Circulation about the center of the wing (in the wind tunnel) CL = Lift coefficient (in the wind tunnel) U = Velocity of airstream w' = Down wash velocity due to wind tunnel wall interference S = Wing area b = Wing span AR = b/S = Wing aspect ratio r = Wind tunnel radius x = Distance along the wing span from the center of the wing, positive to the right a = Angle of attack corrected for wind tunnel wall interference a i = CL/r AR = Self-induced angle of attack a i = w'/U = Induced angle of attack due to wind tunnel wall interference Aa'j = Induced washin of wing due to wind tunnel wall interference If a wing in a wind tunnel has an elliptic lift distribution as in Fig. 1, the circulation is r = r 0 \ / i (2x/b) 2 The downwash at the wing due to the wind tunnel wall interference can be calculated from the image vortex system and is