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Airfoils and Wings

# Airfoils and Wings - Airfoils and Wings Eugene M Cliff 1...

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Airfoils and Wings Eugene M. Cliff February 21, 2000 1 Introduction The primary purpose of these notes is to supplement the text material re- lated to aerodynamic forces. We are mainly interested in the forces on wings and complete aircraft, including an understanding of drag and related nomeclature. 2 Airfoil Properties 2.1 Equivalent Force Systems In some cases it’s convenient to decompose the forces acting on an airfoil into components along the chord (chordwise) and normal to it. These forces are related to lift and drag through the geometry shown in Figure 1. From the figure we have c ( α ) = c n ( α ) cos α c c ( α ) sin α c d ( α ) = c n ( α ) sin α + c c ( α ) cos α Figure 1: Force Systems 1

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Figure 2: Flow Decomposition Obviously, we can also express the normal and chordwise forces in terms of section lift and drag. 2.2 Circulation Theory of Lift A typical ﬂow about a lift-producing airfoil can be decomposed into a sum of two ﬂows, as shown in Figure 2. The first ﬂow (a) is ’symmetric’ ﬂow and so produces no lift. The ’circulatory’ ﬂow (b) is responsible for the net higher speed (and hence lower pressure) on the top of the airfoil (the suction side). This can be quantified by introducing the following line integral Γ C = C −→ u · d −→ s This is the circulation of the ﬂow about the path C . It turns out that as long as C surrounds the airfoil (and doesn’t get too close to it), the the value of Γ is independent of C . The circulation is a property of the airfoil (at the given angle of attack). Such a circulation can be produced by imagining a cyclone like ﬂow with circular streamlines. The speed along any streamline varies inversely with radial distance r from the center. This last feature will make Γ the same along any streamline, and, it turns out, along any contour that simply encircles the center of the cyclone. We use the term vortex to describe such a ﬂow. The net result of this view is that we can reproduce the lift properties of the airfoil by replacing it with a vortex at the center of pressure. Additional analysis shows that the lift (per unit span) is related to the circulation by L = ρ V Γ , 2
Figure 3: Horseshoe Vortex System where ρ , V are the free-stream values of air density and velocity, respec- tively. 3 Three-dimensional Aerodynamics The typical geometry of wings has been introduced earlier. Here we explore the implications of the circulation theory introduced above. Since the 3-D wing can be thought of as a distribution of 2-D sections, when we replace each section by its vortex we end up with a line of vorticies. At the wing tip(s) something has to happen, because the vortex line cannot simply end.

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Airfoils and Wings - Airfoils and Wings Eugene M Cliff 1...

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