Measurements of responses to sudden start of motion, and to gusts, reveal many interesting phenomena. Due to experimental difficulty, most of the results available so far are qualitative in nature. In the following, whenever comparison with theoretical results is mentioned, it is meant
that the results given by the two-dimensional linearized theory for infinitely thin wings, as exposed in the preceding chapters, are being compared. All the experiments quoted below are performed at low air speeds; hence, only the theory of incompressible flow are checked.

Change of Circulation due to Sudden Start of Motion. Figure 15.1


t, no. of semichord lengths traveled after start of motion Fig. 15.1. Growth of circulation and lift after a sudden start of motion. The theoretical values and Cj№ are for two-dimensional flat plate of zero thickness. The steady-state circulation and lift of the experimental airfoils are smaller than rmTh and CLm, respectively. If the ratio of the experi­mental instantaneous circulation to the corresponding experimental steady – state value (shown as asymptote) were plotted as a function of time, the experimental curves will appear to be in better agreement with the theory. (From data given by Walker, Ref. 15.119, and Francis, Ref. 15.113. Cam­bridge Aeronautical Laboratory. Wing chord 4 in. Reynolds number

1.4 x 105 (water). Span 6 in. between plane walls.)

shows the results obtained by Walker15119 and Francis15113 by photo­graphing the flow patterns in a water tunnel. The airfoil was suddenly moved with constant velocity in still water. In one case the angle of attack was so small (a = 7.5°) that the flow remained unseparated In the other case, however, the angle of attack was so large (a — 27.5° measured from the zero-lift line) that the flow began to separate after a

In Fig. 15.1 the measured values



(b) Airfoil: Clark-YH, Effective aspect ratio oo, Reynolds number 1.2 x 105.

From Farren, R. & M. 1648 (1935). Ref. 15.111. Airfoil angle of attack varied. Direction of flow constant

Fig. 15.2. Effect of rate of change of angle of attack on the maximum lift
coefficient. (Courtesy of Dr.-Ing. H. Drescher of Max-Planck-Institut fiir

of circulation Г and lift coefficient CL are compared with Wagner’s theoretical values (§ 15.1), Гтї11 and CLaa being the theoretical limiting value of Г and CL, respectively, as time / -»■ oo. The limiting values of the experimental curves were determined from steady lift measurements.

It is seen that the experimental values of Г are always lower than the theoretical ones. In the small-angle-of-attack case the shape of the experimental curve resembles closely that of the theory. In the large – angle-of-attack case the shape differs considerably.

Increase of the Maximum Lift Coefficient—Kramer’s Effect. The
maximum lift coefficient CLmax of a moving airfoil is different from that
of a stationary airfoil in a steady flow. Kramer15’115 first showed that

CL max increases when the angle of attack increases with time. In Fig. 15.2 are shown the results of Wieselsberger16,120 for a stationary airfoil situated in a flow the angle of which increases with time, and those of Farren15’111 when the angle of attack is first increased and then decreased. The rate of change of angle of attack dafdt is made nondimensional by multiplying with the wing chord c and dividing by the speed of flow V.

Fig. 15.3. Increase of C^max, over the stationary values with the rate of change of angle of attack. The curve marked К is given by Kramer, Ref. 15.115, for airfoils Go 398, Go 459, aspect ratio 5, Reynolds number 1.2 x 105 to 4.8 x 105. Curve N is given by Silverstein, Katzhoff, and Hootman, Ref. 15.118, for a high-wing monoplane Fairchild 22 (airfoil profile NACA 2 Rj 12), Reynolds number 10 x 105 to 30 x 105. Curve E is Ehrhardt’s result given in Ref. 15.117, Reynolds number 105 to 2.8 x 105. The points marked W are given by Wieselsberger, Ref. 15.120. Points F are given by Farren, Ref. 15.111. (Courtesy of Dr.-Ing. H. Drescher.)

The increment of the maximum lift coefficient, ACimax over the stationary c da.

values is plotted against — — in Fig. 15.3, where the results of other

authors are also presented. The values found by Kramer are represented by the straight line

AC£raax = 21.7^ (1)

which is seen to be invalid for small values of angular speed parameter. c dcx.

For — — <0.21 x 10”3, the NACA data obtained by Silverstein,

/T Л* 7 J

Katzhoff, and Hootman15118 gives a straight line whose slope is about 17 times higher than that of Kramer’s Eq. 1. In the intermediate range, Ehrhardt’s15117 test results show a curious transition from the NACA curve to Kramer’s curve.

Motion of a Wing Encountering a Gust. By dropping a wing model through the jet of an open-section wind tunnel, and photographing the trajectories of two little lamps attached to the. wing leading and trailing edges, Kiissner15116 obtained a good qualitative agreement with his theory of gust response. The theory predicts that the lift created by a gust is due to circulation and that the resultant force acts through the forward aerodynamic center. In Kilssner’s tests a rectangular wing (chord 0.204 meter, span 0.415 meter), with and without tip plates, fell (at zero angle of attack) into the horizontal free jet of the wind tunnel. When it reached the geometric jet boundary, its velocity was of order 6.13 meters per second. The ratio of the wind velocity w to the falling velocity of the airfoil U could amount to w/U — 0.359 without separation of the flow. A ratio w/U = 0.359 corresponds to an angle of attack a = 19.5°, at which separation will occur in a steady flow. This delay of separation is another revelation of Kramer’s effect. In experiments with higher wind speed, the wing turned (pitched) to the wind, which means that the center of pressure moved backward because of the separation of flow.

For tests in which wjU < 0.359 the wing with its center of gravity at 0.236 chord length behind the leading edge did not turn (pitch) when passing the jet boundary. Assuming the aerodynamic center of this rather short wing to be located at 0.236 chord (the theoretical value is 0.25 for a two-dimensional wing without thickness), the experiments may be taken as confirming the theoretical location of the center of pressure.

From measurements of the curvature of the trajectories of the airfoil, the lift force acting on the wing can be calculated. However, the accuracy of such measurements was rather low. Within the experimental error, no significant difference between the theory and experiment has been found with respect to the transient lift force, provided that separation did not occur.

Flap Motion—Adherent Flow. An example of the measurements of transient pressure distribution over an airfoil due to a sudden deflection of a flap is shown in Fig. 15.4, which is given by Drescher15110 from experiments in a water tunnel. The pressure distribution was measured by a multiple manometer. At the top of this figure is a curve of the normal force coefficient Cn vs. time, Cn being the force (normal to wing chord) divided by q x (wing area). Next is a curve of the flap force coefficient CF vs. time; CF, like Cn, is referred to the main-wing area (including flap) and is the component of force acting on the flap in the

Fig. 15.4. Transient normal force coefficient, flap force coefficient, and pressure distribution over an airfoil following a sudden deflection of flap. (Courtesy of Dr.-Ing. H. Drescher.)

direction normal to the main wing chord. In the lower part of Fig. 15.4 are plots of pressure distribution over the airfoil and the flap. The pressure coefficient Cv is defined as the actual pressure jump across the airfoil divided by the dynamic pressure q — pU2. The angular speed of flap £2 was a constant in each experiment, and was expressed in non – dimensional parameter Q. c/2U. The wing model had a symmetric profile Go-409; it spanned wall to wall in the tunnel, so that in the mid-span section, where the measuring holes were arranged, the flow

/3 = 0° 15“

— Ec-~

•*—————————- C—————————– *-

Fig. 15.5. Transient normal force and flap force coefficients for a higher
angular speed of flap deflection. Reynolds number 6 x 106. a = — 5°.
ft = 0° -> 15°. Qcj2U = 0.356. (Courtesy of Dr.-Ing. H. Drescher.)

approximated well a two-dimensional one. The flap was mounted to the main wing without a slot.

In the experiment of Fig. 15.4, the angle of attack was a = — 5°, and the flap was deflected from /3 = 0° to /3 = 15° at constant angular speed Q. In general, Cn(t), CF(t), and Cv{x) curves all agree fairly well with theoretical values (dotted curves) based on infinitely thin two-dimensional plate. The airfoil profile shape causes a discrepancy in the pressure distribution, particularly at the leading edge, where a sharp edge is assumed in the theory.

In Fig. 15.5 is plotted another result by Drescher15110 for a higher angular speed of the flap motion. A periodic oscillation in Cn(t), which gradually dies out, appears after the flap has ceased to move. This is connected with the flow picture of Fig. 15.6. The vortex surface deflected by the flap motion is followed by periodic vortices of decreasing intensity.

Such periodic vortices had been observed after sudden starting or sudden stopping of the flap motion. One may conclude that the circulation about the wing cannot instantaneously assume the value which is given by the kinematic conditions, but that it oscillates about the prescribed value with appreciable amplitude.

Flap Motion—Separated Flow. Experimental pressure distribution on an airfoil with flow separation is also reported by Drescher.15110 The beginning of the separation process depends on the Reynolds number. Low Reynolds number will favor the separation. But the most important factor influencing the separation process is the angular velocity with which the flap moves. If the main-wing angle of attack is small, and the flap

(в) Ф) (c)

Fig. 15.6. Flow following a sudden deflection of flap, (a) Immediately after stopping the flap motion. (6) Next instant, (c) Approaching steady state. (Courtesy of Dr. Ing. H. Drescher.)

is deflected to a large angle, the flow separates in the flap region, and in later stages a Karman vortex street is developed. For Go-409 airfoil, a = 0, /3 moves from 0 to 60°, separation occurs at /3 = 12° when Oc/2£/ is 0.014, but it occurs after /3 = 60° when Qc/2U is Г.16. In the latter case very high suction is obtained at the flap nose, and the CL and CF values may exceed the corresponding theoretical values at the instant before separation occurs. Very large oscillations in CL(t) and CF(t) curves are often observed after separation.

If the main-wing angle of attack a is small and /3 decreases from 60° to 0, the separated flow becomes adherent again, but often after an ap­preciable time delay, which is required to scavenge away the dead water accumulated behind the wing. More complicated motions of the flap and the main wing induce more complicated responses; but such responses can generally be understood on the basis of the facts mentioned above: oscillation in circulation before it reaches a steady value, and time delay required for the scavenging process.


Harmonic Oscillations. Results of oscillating-wing experiments have been published by many authors. (See bibliography.) For a wing without a flap, the experimental results in general agree with the theo­retical. There are, however, some small quantitative difference between theory and experiment, and among various authors. Typical results obtained by Halfman15129 are shown in Figs. 15.7 and 15.8. The airfoil (NACA 0012 section, chord 1 ft, span 2 ft) was tested in a 5 X 7-ft wind tunnel with plates shielding the wing tips so that two-dimensional flow condition was assured. Oscillations in two degrees of freedom were imparted to the airfoil: h and a. The expressions of aerodynamic force and moment corresponding to the translational motion h = h0eimt (h0 real) are, respectively,

Ь – = + IL,/ e«»*+*4 <f, LT = tan-1 Ь*

Aqb Rlt

Мж. — /~r a _i_ r 2 жм+ф„.) j ~ Imil


Similarly, those corresponding to pure pitch a — a0eto* (a0 real) are given by the same formulas except that the subscript T is replaced by P. The force L, as well as the displacement h, is taken positive downward in these figures. Both the magnitude and phase angle are plotted in Figs. 15.7 and 15.8. The solid curves are theoretical values.

More serious differences between theoretical lift and moment coefficients and experimental ones are found for oscillating flaps. But test results are meager. See papers by Drescher15126 and Walter.15-135,15136 Experimental results for oscillating wings having a mean angle of attack near or greater than the static stalling angle have been reviewed in § 9.4.

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