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INTRODUCTION
In recent years a large amount of work has been expended on the study of aerodynamic forces on oscillating wings. These forces are needed for the investigation of the dynamic or aeroelastic stability of an airplane. Such problems have received increased attention with the advent of transonic and supersonic airplanes.
There are two reasons why the phenomenon of flutter has become more important as airplanes have surpassed sonic speed. Firstly, of course, the more slender shapes required for supersonic flight have made the airplanes more flexible and therefore more prone to aeroelastic instabilities. Secondly, the aerodynamic forces at transonic speeds are such as to favor the occurrence of flutter, so that the critical speed usually has a minimum at or near M = 1. For an excellent discussion of transonic flutter problems the reader is referred to a paper by Garrick (Ref. 18).
The affinity to flutter in the transonic speed range may be explained from well-known aerodynamic properties of transonic flow. The lifting pressures due to a given amount of deflection are known to be at maximum at or near M = 1 (cf. the lift curve slope). This must lower the flutter velocity since an increase of all aerodynamic derivatives by the same amount has the same effect as, for example, increasing the air density. However, an effect which is probably even more important is that due to the large phase lags between motion and unsteady air pressures that occur at transonic speeds. When an object travels at a speed near that of sound the flow perturbations created move forward at about the same speed as the object itself. Hence there will be a slow accumulation of disturbances and, if the flow is given sufficient time to build up, the well-known typical transonic non-linearities will occur. Since a pressure wave set up at a point will spend a long time before it travels off the object, it is evident that large and possibly destabilizing phase differences between motion and pressure can easily be created. These are directly responsible for one-degree-of-freedom flutter of control surfaces (control surface "buzz") and also for the low or negative damping in pitch sometimes encountered by tailless aircraft of high or medium aspect ratios.
Because the computation of aerodynamic forces on oscillating three-dimensional wings is so complicated, even on the basis of linearized theory, most flutter calculations in industry today are made by use of aerodynamic derivatives obtained from two-dimensional (strip theory) analysis . For high supersonic Mach numbers or for large-aspect-ratio wings in subsonic flow this procedure may be justified . At transonic speeds, however, cross-flow effects are always very large as is well known in the case of steady flow. Therefore, the use of strip theory can lead to large errors in the computed flutter speed near M = 1. For example, a strip-theory flutter calculation of a configuration involving a control surface will always show one-degree-of-freedom flutter of the control surface at transonic speeds unless the hinge stiffness is very high or artificial damping is provided. As shown in Chapter 7, however, the three-dimensional theory, on the contrary, gives positive hinge moment damping at M = 1 for rectangular control surfaces of aspect ratio less than 3.5.
Since there is no prospect of integrating the full non-linear transonic equations of fluid motion, any three-dimensional lifting-surface theory would have to be based on the linearized equations. For sub- or super-sonic flow linearized theory is known to hold well for thin wings. For transonic flow, however, the above-mentioned non-linear accumulation of disturbances precludes the use of linearized theory in the steady, non-lifting case no matter how thin the wing is. In the oscillating wing case the situation is somewhat better, though. Firstly, one is concerned with the lifting part of the flow. According to the transonic equivalence law, Ref. 54, linearized theory is capable of describing the steady lifting flow for wings of low aspect ratio, and wind tunnel experiments do, indeed, confirm fairly well the predictions of the theory at least for wings with swept leading edges. Secondly, if the wing oscillates rapidly the non-linear disturbance accumulation will not have time to develop and hence the linearized equations will be applicable. The conditions necessary for this to apply are discussed in Chapter I.
The main part of the present monograph is devoted to the study of lifting surface theories. Most of this is based on recent theoretical work by the writer (Refs. 30–39) but available investigations by other workers in the field have also been included for completeness. The writer's methods have been developed with the aim of covering the reduced frequencies of interest in flutter research. For the sake of simplicity, however, most numerical results given are those for stability derivatives, i.e. for rigid-body wing motions, but evaluated at frequencies of interest in flutter work. One exception to this is Chapter 5 on wing-body interference at sonic speed in which only results for stability derivatives at low reduced frequencies are given.
No comprehensive treatment of the present subject would be complete without reference to experimental results. In Chapter 11 material available in the open literature is collected on the subject for cases where direct comparisons with theory are possible.
The writer realizes that there are transonic unsteady-flow problems that merit attention other than the oscillating wing problem and which have been left out in this monograph. In principle, the oscillating case can be considered as the Fourier transform with respect to time of an arbitrary time dependent motion so that it is possible to make use of the given results for any type of motion. However, such a method is seldom practical. References dealing with transient phenomena at transonic speeds can be found, for example, in Miles' recent book (Ref. 49).
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Excerpted from "Unsteady Transonic Flow"
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Copyright © 2019 Mårten T. Landahl.
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