# Three-Dimensional Oscillating Wings

[1] See Notation List for meanings of symbols which are not defined locally in the text. In the following, D/Dt is the substantial derivative or rate of change following a fluid particle..

[2] It should be noted that more parameters are introduced when one takes account of the dependence on state of viscosity coefficient, conductivity, and specific heats.

[3] The bar over any symbol will be employed to designate a complex conjugate for the remainder of this chapter.

[4] For further details on this procedure, see Theodorsen (1931), Theodorsen and Garrick (1933), and Abbott, von Doenhoff, and Stivers (1945).

[5] See also Lagerstrom and Cole (1955) and Friedrichs (1953, 1954).

[6] An illuminating discussion appears in Section A.3 of Jones and Cohen (1960). See also the extended development in Chapter 10 of Liepmann and Roshko

[7] This scheme is associated with the names of Multhopp (1941) and Vandrey (1938).

[8] The Biplane. Let us consider two supersonic wings, with associated diaphragm regions, separated by a distance d in the f-direction (Fig. 11-7). Because there is a certain artificiality in the use of sources to represent the flow over the upper surface of each of these wings, some care must be

[9] These are bodies of similar shape, but stretched differently in the z-direction, or у-direction, or both. Thin airfoils of different thickness ratios constitute one class of affine bodies.

[10] This notation is commonly used in connection with mechanical or electrical vibrations and implies that the real (or imaginary) part of the right-hand side

. ft

must be taken in order to recover the physical quantity of interest. Here h is a complex function of position and allows for phase shifts between displacements of different points.

[11] If dimensionless x-variables are adopted in (13-28), based on reference length l = c/2, it is clear how the aforementioned к = шс/2С/„ will arise as one parameter of the problem.

The general planar wing problem, (13—16)—(13—17), has stimulated some imaginative research in applied mathematics. For M > 1, there are many analytical solutions appropriate to particular wing planform shapes, such as rectangular or delta, and all details have been worked through for elementary modes of vibration like plunging and pitching. Miles (1959) constitutes a compendium of such supersonic information, as does Landahl (1961) for the vicinity of M = 1. In the range 0 < M < 1 the only available exact linearized results pertain to the two-dimensional airfoil, whereas in constant-density fluid a complete and correct analysis has been published for a wing of circular planform.

Since the advent of high-speed computers, numerical methods have been elaborated to cover very general wing geometry and arbitrary continuous deflection shapes. The approach for subsonic speed has been through superposition of acceleration-potential doublets, culminating in complete lifting surface theories which generalize the steady-flow results of Section 7-6. The definitive works are those of Watkins et al. (1955, 1959).

The influence-coefficient methods mentioned in Chapters 8 and 11 have proved adaptable to supersonic wings, although there are some details of the treatment of singularities that have apparently been resolved only very recently. Nonplanar wings and interfering systems represent an extension that is likely to be mechanized successfully within a short period of time.

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