Technology & Engineering
Kutta Joukowski Theorem
The Kutta-Joukowski theorem in fluid dynamics relates the lift generated by a moving body to the circulation around it. It states that the lift per unit span of a lifting body in two-dimensional flow is proportional to the density of the fluid, the velocity of the flow, and the circulation around the body. This theorem is fundamental in understanding the principles of aerodynamics.
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4 Key excerpts on "Kutta Joukowski Theorem"
eBook - ePub- Ethirajan Rathakrishnan(Author)
- 2013(Publication Date)
- Wiley(Publisher)
The main problem associated with this transformation is finding the best transformation function (formula) to perform the required operation. Even though a large number of mathematical functions can be envisaged for a specific transformation.A transformation, which generates a family of aerofoil shaped curves, along with their associated flow patterns, by applying a certain transformation to consolidate the theory presented in the previous sections, is the Kutta−Joukowski transformation .Kutta−Joukowski transformation is the simplest of all transformations developed for generating aerofoil shaped contours. Kutta used this transformation to study circular-arc wing sections, while Joukowski showed how this transformation could be extended to produce wing sections with thickness t as well as camber.In our discussion on Kutta−Joukowski transformation, it is important to note the following:- The circle considered, in the physical plane, is a specific streamline. Essentially the circle is the stagnation streamline of the flow in the original plane 1 (z -plane).
- The transformation can be applied to the circle and all other streamlines, around the circle, to generate the aerofoil and the corresponding streamlines in plane 2 (ζ -plane) or the transformed plane. That is, the transformation can result in the desired aerofoil shape and the streamlines of the flow around the aerofoil.
It is convenient to use polar coordinates in the z -plane and Cartesian coordinates in ζ -plane. The Kutta−Joukowski transformation function
eBook - PDF- Pijush K. Kundu, Ira M. Cohen(Authors)
- 2010(Publication Date)
- Academic Press(Publisher)
This is called the Kutta condition , sometimes also called the Zhukhovsky hypothesis . At the beginning of the twentieth century it was merely an experimentally observed fact. Justification for this empirical rule became clear after the boundary layer concepts were understood. In the following section we shall see why a real flow should satisfy the Kutta condition. Historical Notes According to von Karman (1954, p. 34), the connection between the lift of airplane wings and the circulation around them was recognized and developed by three per-sons. One of them was the Englishman Frederick Lanchester (1887–1946). He was a multisided and imaginative person, a practical engineer as well as an amateur mathe-matician. His trade was automobile building; in fact, he was the chief engineer and general manager of the Lanchester Motor Company. He once took von Karman for a ride around Cambridge in an automobile that he built himself, but von Karman “felt a little uneasy discussing aerodynamics at such rather frightening speed.” The second person is the German mathematician Wilhelm Kutta (1867–1944), well-known for the Runge–Kutta scheme used in the numerical integration of ordinary differential equations. He started out as a pure mathematician, but later became interested in aerodynamics. The third person is the Russian physicist Nikolai Zhukhovsky, who developed the mathematical foundations of the theory of lift for wings of infinite span, independently of Lanchester and Kutta. An excellent book on the history of flight and the science of aerodynamics was recently authored by Anderson (1998). 6. Generation of Circulation 687 6. Generation of Circulation We shall now discuss why a real flow around an airfoil should satisfy the Kutta condition. The explanation lies in the frictional and boundary layer nature of a real flow. Consider an airfoil starting from rest in a real fluid.
eBook - PDF- Ronald L. Panton(Author)
- 2013(Publication Date)
- Wiley(Publisher)
As a first approximation it gives a remarkably good estimate of the lift as long as separation does not occur. 18.13 FLOW OVER A JOUKOWSKI AIRFOIL: AIRFOIL LIFT The geometry of a Joukowski airfoil is specified by the angle of attack α, the camber ratio h/, and the thickness ratio t/. We have no control over the distributions of camber and thickness, as all Joukowski airfoils have the same distributions. In practice this is not a critical simplification, as the events that produce lift are fairly insensitive to these distributions. This is especially true at modest angles of attack. The flow around a Joukowski airfoil is found using the ideas of Sections 18.10 to 18.12. In Section 18.10 the idea of conformally transforming one flow field into a much simpler flow field was introduced. Section 18.11 gave us a specific transformation, the Joukowski transformation, that transforms an airfoil shape in the z-plane into a circular cylinder in the ζ -plane. Since in Section 18.7 we already have in hand the solution for flow over a cylinder, the only remaining step is to reinterpret this flow after it is transformed back into the z-plane for the airfoil. The obstacle to this procedure is the fact that there are an infinite number of ideal flows over a circular cylinder and we must pick one. This difficulty is overcome by invoking the Kutta condition to select the flow that leaves the trailing edge smoothly with a finite velocity. The cylinder in the ζ -plane is shown in Fig. 18.22. Recall that the trailing edge of the airfoil maps to the point ζ = c on the circle and that the thickness and camber determine the center position of the circle, denoted by ζ 0 . To apply the Kutta condition, we must arrange the circulation constant so that the flow leaves the circle at the point marked TE; this point will be the rear stagnation point for the flow in the ζ -plane. Before we can write down the complex potential, we need one more detail.
eBook - PDF- Ira M. Cohen, Pijush K. Kundu(Authors)
- 2004(Publication Date)
- Academic Press(Publisher)
This is called the Kutta condition , sometimes also called the Zhukhovsky hypothesis . At the beginning of the twentieth century it was merely an experimentally observed fact. Justification for this empirical rule became clear after the boundary layer concepts were understood. In the following section we shall see why a real flow should satisfy the Kutta condition. Historical Notes According to von Karman (1954, p. 34), the connection between the lift of airplane wings and the circulation around them was recognized and developed by three per-sons. One of them was the Englishman Frederick Lanchester (1887–1946). He was a multisided and imaginative person, a practical engineer as well as an amateur math-ematician. His trade was automobile building; in fact, he was the chief engineer and general manager of the Lanchester Motor Company. He once took von Karman for a ride around Cambridge in an automobile that he built himself, but von Karman “felt a little uneasy discussing aerodynamics at such rather frightening speed.” The second person is the German mathematicianWilhelm Kutta (1867–1944), well-known for the Runge–Kutta scheme used in the numerical integration of ordinary differen-tial equations. He started out as a pure mathematician, but later became interested in aerodynamics. The third person is the Russian physicist Nikolai Zhukhovsky, who developed the mathematical foundations of the theory of lift for wings of infinite span, independently of Lanchester and Kutta. An excellent book on the his-tory of flight and the science of aerodynamics was recently authored by Anderson (1998). 6. Generation of Circulation We shall now discuss why a real flow around an airfoil should satisfy the Kutta condition. The explanation lies in the frictional and boundary layer nature of a real flow. Consider an airfoil starting from rest in a real fluid. The flow immediately after
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