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The circulation theory of lift is an alternative way of thinking about the generation of lift on an aerodynamic body. However, let us keep things in perspective. Its development at the turn of the twentieth century created a breakthrough in aerodynamics.
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The approach we have discussed above-the definition of circulation and the use of Equation (3.140) to obtain the lift-is the essence of the circulation theory of lift in aerodynamics. rĪt this stage, let us pause and assess our thoughts. The important point here is that, in the Kutta-Joukowski theorem, the value of Г used in Equation (3.140) must be evaluated around a closed curve that encloses the body, the curve can be otherwise arbitrary, but it must have the body inside it. These vortices have the usual singularities in V x V, and therefore, if we choose a curve that encloses the airfoil (such as curve A in Figure 3.37), Equation (2.137) yields a finite value of Г, equal to the sum of the vortex strengths distributed on or inside the airfoil. On the other hand, we also show in Chapter 4 that the flow over an airfoil is synthesized by distributing vortices either on the surface or inside the airfoil. As we show in Chapter 4, the flow outside the airfoil is irrotational, and the circulation around any closed curve not enclosing the airfoil (such as curve В in Figure 3.37) is consequently zero. The same can be said about the flow over the airfoil in Figure 3.37. It is only when we choose a curve that encloses the origin, where V x V is infinite, that Equation (2.137) yields a finite Г, equal to the strength of the vortex. If we take the circulation around any curve not enclosing the origin, we obtain from Equation (2.137) the result that Г = 0. Irrotational at every point except at the origin. Therefore, the lifting flow over a cylinder as shown in Figure 3.33 isįigure 3.37 Circulation around a lifting airfoil. Recall that all three elementary flows are irrotational at all points, except for the vortex, which has infinite vorticity at the origin. In Section 3.15, the lifting flow over a circular cylinder was synthesized by superimposing a uniform flow, a doublet, and a vortex. See Reference 9 for a particularly lucid treatment of inviscid, incompressible flow at a more advanced level.) Hence, more advanced treatments of such flows utilize the mathematics of complex variables as an important tool. (It can be shown that arbitrary functions of complex variables are general solutions of Laplace’s equation, which in turn governs incompressible potential flow. Such mathematics is beyond the scope of this book. The general derivation of Equation (3.140) for bodies of arbitrary cross section can be carried out using the method of complex variables. Indeed, the concept of circulation is so important at this stage of our discussion that you should reread Section 2.13 before proceeding further. The Kutta-Joukowski theorem states that lift per unit span on a two-dimensional body is directly proportional to the circulation around the body. This result underscores the importance of the concept of circulation, defined in Section 2.13. In turn, the lift per unit span L’ on the airfoil will be given by the Kutta – Joukowski theorem, as embodied in Equation (3.140): If the airfoil is producing lift, the velocity field around the airfoil will be such that the line integral of velocity around A will be finite, that is, the circulation Let curve A be any curve in the flow enclosing the airfoil. For example,Ĭonsider the incompressible flow over an airfoil section, as sketched in Figure 3.37.
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