UNSTEADY MOTIONS IN GENERAL. EXPERIMENTS

The aerodynamics of unsteady motions of airfoils with arbitrary time history is described in § 15.1 as a simple generalization, involving one integration, of the harmonic-oscillation case. In particular, the indicia! admittance (response to a unit-step function) is derived from the oscillating case. This does not imply that there are no shorter methods of deriving the indicial admittance, but rather emphasizes the reciprocal relation between the admittance and the indicial admittance. The effects of finite span is briefly mentioned in § 15.2.

The ultimate test of a theory lies in experiment. Some general con­siderations in experiments are given in § 15.3. Some of the established experimental results are outlined in § 15.4. Owing to the limitation of space, the experiments cannot be described in detail. Moreover, since the numerical uncertainty of the experimental results available at present is such that a scheme of empirical correction of the theoretical results is not yet generally acceptable, the aim of our discussion will be to point out the conditions under which the linearized theory is inapplicable and to describe some of the phenomena yet unexplained by theory.

15.1 UNSTEADY MOTIONS OF A TWO-DIMENSIONAL AIRFOIL

The aerodynamic response of an airfoil performing an arbitrary un­steady motion about a mean uniform rectilinear translation can be cal­culated from that of an oscillating airfoil by means of a Fourier analysis. In § 8.1 it is shown that, if the response?/ to a forcing function F= F„eimt is

(1) then the response to a periodic function F(t)

CO

F(t) = ^ Cn einmt

n~~ CO

IS

These results can be applied directly to the airfoil problem if the linearized theory is accepted. The airfoil displacements (change of angle of attack, velocity of translation, aileron angle, etc.) may be considered as the forcing function and the induced lift, moment, or pressure distribution as the response. The admittance l/Z(iw) is given by the theory of har­monically oscillating airfoils, as applied to у (x, t) =f (x) ewt The procedure can be expressed also in Laplace transformation. In Eqs. 5 and 6, put

im = s (7)

and assume F(t) — 0 for t < 0; then

(9)

Hence, formally, V2n^( — is) is the Laplace transformation of F(t), and Eq. 8 shows that y(t) is the inverse Laplace transformation of JSf{F}/Z(s), i. e.:

These formal steps can be mathematically justified for suitable classes of forcing functions F(t). The Laplace transformation of the response is equal to the Laplace transformation of the forcing function multiplied by 1 IZ(s), which is obtained by replacing /со by s in the admittance to harmonic oscillation.

When F(t) is a unit-step function,

J?{1(0} = – (11)

s

the response, called indicial admittance and denoted by A(t), is given by

m <12»>

or

"«-•НтЫ <12‘>

in agreement with Eq. 32 of § 8.1.

Note that the downwash distribution over the airfoil must be the same in the unsteady motion as in the harmonic-oscillation case in order that the above formulas be applicable.

There are other methods of deriving the indicial admittance. For special problems special methods may be devised that are much shorter than the Fourier-transformation method mentioned above. Recently, great advance has been made in the calculation of indicial admittance with respect to a sudden motion of the airfoil or to a sharp-edged gust, for both the two-dimensional case and the finite-aspect ratio case. The success is remarkable, particularly at high subsonic Mach numbers and at M = 1 (linearized theory). On the other hand, the second – and higher-order theories are investigated in the supersonic case, and significant corrections to the first-order linearized theory (which is presented in the last chapter) are revealed for Mach numbers near and below Vl. See bibliography, and in particular, Lomax,15 84 and van Dyke.14-45

Example. Wagner’s Problem. Consider an airfoil moving recti – linearly in a fluid with a relative speed U which is so small that the fluid may be considered incompressible. Let the angle of attack be suddenly increased by an amount a. Owing to this change the relative velocity of the fluid will have a component normal to the airfoil. This normal velocity is uniformly distributed along the chord and is a unit-step function of time:

v(x, t)= – U*l(t) (- 1 < я < 1) (13)

The lift and moment induced by this upwash can be found by the Laplace transformation. Note that the same problem arises if an airfoil suddenly starts to move in a stationary fluid with a constant velocity U and angle of attack a.

In § 13.4 it is shown that, if

v(x, t) = v0 eimt = iwy0 eimt

and the stalling moment about the mid-chord point is

M/a = – 7TPU*iy0k C(k) 4*

C(k) is Theodorsen’s function.

Replacing tk by s, we obtain, for the lift force,

■ TrpbUv0 8(t) — 2-rrpbUv0ЛЄ

The first term gives an impulse function. The total impulse is obtained by an integration over an infinitesimal time interval. If we remember the scale factor b/U (Eq. 15) in changing from т to t, the magnitude of the total impulse is seen to be – прЬЪ0. This is the total impulse that is needed to move abruptly a mass тгрЬ2 to a velocity v0. The quantity npb’2 is the apparent mass of the fluid associated with the vertical motion (cf. § 6.7, Eq. 9).

The second term gives the lift due to circulation. Define a special function Ф(т):

Ф(т) = &-1 {^“7^} (R1 * > 0) (19)

Then the circulatory lift can be written as

L,(r) = – 2ттрЬ Шй Ф(т) (20)

The function Ф(т) is the Wagner’s function defined in § 6.7, and is shown graphically in Fig. 6.6.

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