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1、Chapter 4,Incompressible Flow Over Airfoils,Of the many problems now engaging attention, the following are considered of immediate importance and will be considered by the committee as rapidly as funds can be secured for the purpose. The evolution of the more efficient wing sections of practical for
2、m, embodying suitable dimension for an economical structure, with moderate travel of the center-of-pressure and still affording a large range of angle-of-attack combined with efficient action. From the first annual report of the NACA, 1915,4.1 Introduction,Ludiwig Prandtl and his colleagues at Gttin
3、gen, Germany, showed that the aerodynamic consideration of wings could be split into two parts. (1) the study of the section of a wingan airfoil. And (2) the modification of such airfoil properties to account for the complete, finite wing.,Definition of airfoil,The purpose of this chapter:,Present t
4、heoretical methods for the calculation of airfoil aerodynamic properties.,Road map of this chapter,4.2 Airfoil nomenclature,Leading edge: 前缘 trailing edge: 后缘 Chord line: 弦线 chord length: 弦长 Thickness: 厚度 camber: 弯度 Mean chamber line: 中弧线,NACA “four digit” series airfoil,NACA2412: The first digit: m
5、aximum camber in hundredths; The second digit: the location of maximum camber along the chord from leading edge in tenth of the chord; The last two digits: maximum thickness in hundredth of the chord.,NACA0012: symmetrical airfoil,4.3 Airfoil characteristics(experiment),Special definitions,lift coef
6、ficient,angle of attack,lift slope,Maximum lift coefficient,zero-lift attack angle,Consequence of the flow separation,It is impossible for we to calculate with inviscid flow approximation!,Experiment results for NACA2412,Source of drag,profile drag,Aerodynamic center,4.4 Philosophy of theoretical so
7、lutions for low-speed flow over airfoils: The Vortex Sheet,Schematic figure of a point vortex,Schematic figure of a vortex filament,Point vortex is simply a section of a straight vortex filament,Construction of a vortex sheet,Try to remember the analogous situation for the construction of a source s
8、heet.,Infinite number of vortex filaments,The strength of each vortex filament is infinitesimally small.,Velocity induced by at point P,Definition of the strength of vortex sheet,Velocity potential induced by at point P,Difference of the superposition between the velocity vectors and velocity potent
9、ial.,The circulation around the vortex sheet is the sum of the strengths of the elemental vortices.,There is a discontinuity change in the tangential component of velocity across the sheet.,Let to be the circulation along the dashed line.,or,as,so,As the top and bottom of the dashed line approach th
10、e vortex sheet, , become the velocity components tangential to the vortex sheet immediately above and below the sheet.,or,The local jump in the tangential velocity across the vortex sheet is equal to the local sheet strength.,Philosophy of airfoil theory for inviscid, incompressible flows.,Step 1. R
11、eplace the airfoil surface with a vortex sheet of strength,Step 2. Find a suitable distribution of such that the wall boundary condition can be satisfied. That is, the combination of the free stream flow and the vortex sheet will make the vortex sheet(the surface of the airfoil) a streamline of the
12、flow.,Step 3. Calculate the circulation around the airfoil, and then get the lift by Kutta-Joukowski theorem,Note 1. There are no general analytical solution for an airfoil with arbitrary shape and thickness. This should be solved numerically with suitable digital computers. Vortex panel method (Sec
13、. 4.9),Note 2. Physical significance of the vortex sheet which has been used to replace the surface of the airfoil surface. Boundary layer is a highly viscous region, the vorticity inside the boundary layer is finite.,Step 4. Approximation for a thin airfoil, shift the vortex sheet from the airfoil
14、surface to the camber line of the airfoil. The upper and lower part of the vortex sheet are coincide together.,This time, Find a suitable distribution of such that the wall boundary condition can be satisfied. That is, the combination of the free stream flow and the vortex sheet will make the vortex
15、 sheet(camber line of the airfoil) a streamline of the flow.,Note 3. After the thin airfoil approximation, it is possible to give a closed-form analytical solution of .,4.5 The Kutta Condition,For potential flows, different choice of gives different lifting flow around circular cylinder. And it is t
16、he same to the situation of airfoils.,Two different flows around a same airfoil at the same attack angle,The nature knows how to pick a right solution. We need an additional condition that fixes for a given airfoil at a given attack angle.,Experimental results for the development of the flow field around an airfoil which is set into motion from an initial state of rest.,(a),(b),(c),Experimental results demonstrate that the flow is s
