The Linearized Wing Theory of the Supersonic Flow with the Karman's Fourier Integral Method

Citation

Chang, Chieh-Chien (1950) The Linearized Wing Theory of the Supersonic Flow with the Karman's Fourier Integral Method. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/67DX-1V06. https://resolver.caltech.edu/CaltechETD:etd-03042009-110100

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Part I gives a short introduction and some physical interpretation of von armAn's Fourier integral method applied to the supersonic wing theory. A short review of other current contributions to the Linearized supersonic wing theory is also given. Part II presents the general formulation of the von [...] method from the view-point of the elementary harmonic sources and doublets. First, the disturbance potential and the velocity components of a general flat body with symmetrical airfoil are derived. Next, the disturbance potential of the lifting surface is presented. In contrast to the well-known conical flow method, the von [...] Fourier integral method can treat a complicated plan-form as a whole, without considering the detailed geometry, as long as the airfoil sections are similar. Part III applies the method to the investigation of the wave drag of the non-lifting wing in supersonic flight. A general solution of the wave drag is obtained for the wing with a diamond- shaped airfoil. This solution allows a free choice of a number of the important geometrical parameters. For instance, the wing may be swept forward or backward, tapered or reversed tapered to any ratio. A number of the limiting cases are also investigated. For the practical aerodynamic problems, two useful families of wing plan-form with the fixed taper ratios 0.2 and 0.5, any swept angle, aspect ratio and Mach number are shown in the graphs. A particular application is demonstrated. The reversed flow theorem on wave drag as shown by von [...] and Hayes checks well with the consequence of the general solution. This method shows a certain elegance as no conical flow assumption is needed, and the mathematics is powerful enough to obtain a general solution covering all possible geometrical arrangements without detailed considerations. While in recent years, the direct problem of finding the lift distribution with given angle of attack on the wing has been well solved by the method of conical flow and others, the present treatment in Part IV, on the other hand, investigates the inverse problem, i.e., to find the downwash distribution in the plane of the wing with a pre- assigned lift distribution. This is particularly favorable with the present method. The general solution of the downwash of the tapered swept wings is derived for the case that a constant lift distribution on the wing is pre-assigned. Of course, the method may be applied to any lift or pressure distribution along the wing chord and span. The corresponding angle of attack on the wing and the downwash can be determined everywhere in the plane of the wing. To demonstrate the downwash distribution as given by the general solution, graphs are given to show the downwash of a number of wings including a swept-back tapered wing with supersonic trailing edge and a delta wing.

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