Aerodynamics - Thin Airfoil Theory
M&AE 305
October 3, 2006
Thin Airfoil Theory
D. A. Caughey Sibley School of Mechanical & Aerospace Engineering Cornell University Ithaca, New York 14853-7501 These notes provide the background needed to implement a simple vortex-lattice numerical method to determine the properties of thin airfoils. This material is covered in Lecture, but is not in the textbook [5]. A summary of results from the analytical theory also is provided, as well as a comparison of the thin-airfoil results with those of a complete inviscid theory that accounts for thickness effects.
1
The Vortex Lattice Method
We here describe the implementation of the vortex lattice method for two-dimensional flows past thin airfoils. The method is even more useful for three-dimensional wings, i.e., for the flow past wings of finite span, but that problem is not considered here. Instead, the reader is referred to standard aerodynamics texts, e.g., [2]. In this numerical procedure to solve the thin-airfoil problem, we place a finite number of discrete vortices along the chord line, with the boundary condition that the induced vertical velocity dyc v= − α, (1) dx be enforced at selected control points to determine the vortex strengths. Equation (1) simply says that the net velocity vector, comprised of components due to the free stream, at angle of attack α to the chord line, plus that induced by the point vortices, is tangent to the camber line whose slope is dyc / dx; the magnitude of the free stream velocity is taken to be unity. Thus, we discretize the chord line into a finite number N of segments, or panels, as illustrated in Fig. 1 (a). On each panel we place a point vortex and a control point, as illustrated in Fig. 1 (b). The most accurate results are obtained by locating the vortex one-quarter of the panel length, and the control point three-quarters of the panel length, aft of the leading edge of the panel. (This strategy can be shown to reproduce the exact results of analytical
October 3, 2006
Thin Airfoil Theory
D. A. Caughey Sibley School of Mechanical & Aerospace Engineering Cornell University Ithaca, New York 14853-7501 These notes provide the background needed to implement a simple vortex-lattice numerical method to determine the properties of thin airfoils. This material is covered in Lecture, but is not in the textbook [5]. A summary of results from the analytical theory also is provided, as well as a comparison of the thin-airfoil results with those of a complete inviscid theory that accounts for thickness effects.
1
The Vortex Lattice Method
We here describe the implementation of the vortex lattice method for two-dimensional flows past thin airfoils. The method is even more useful for three-dimensional wings, i.e., for the flow past wings of finite span, but that problem is not considered here. Instead, the reader is referred to standard aerodynamics texts, e.g., [2]. In this numerical procedure to solve the thin-airfoil problem, we place a finite number of discrete vortices along the chord line, with the boundary condition that the induced vertical velocity dyc v= − α, (1) dx be enforced at selected control points to determine the vortex strengths. Equation (1) simply says that the net velocity vector, comprised of components due to the free stream, at angle of attack α to the chord line, plus that induced by the point vortices, is tangent to the camber line whose slope is dyc / dx; the magnitude of the free stream velocity is taken to be unity. Thus, we discretize the chord line into a finite number N of segments, or panels, as illustrated in Fig. 1 (a). On each panel we place a point vortex and a control point, as illustrated in Fig. 1 (b). The most accurate results are obtained by locating the vortex one-quarter of the panel length, and the control point three-quarters of the panel length, aft of the leading edge of the panel. (This strategy can be shown to reproduce the exact results of analytical