f02_sp - Fluids Lecture 2 Notes 1 Airfoil Vortex Sheet...

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Fluids – Lecture 2 Notes 1. Airfoil Vortex Sheet Models 2. Thin-Airfoil Analysis Problem Reading: Anderson 4.4, 4.7 Airfoil Vortex Sheet Models Surface Vortex Sheet Model An accurate means of representing the flow about an airfoil in a uniform flow is to place a vortex sheet on the airfoil surface. The total velocity V ( x, z ), which is the vector sum of the freestream velocity and the vortex-sheet velocity, can be forced parallel to the airfoil surface γ ( s γ (s) V V V by suitably setting the sheet strength distribution ). A panel method is normally used to numerically compute γ ( s ). By using a sufficient number of panels, this result can be made as accurate as needed. The main drawback of such numerical calculations is that they give limited insight into how the flow is influenced by changes in the angle of attack or the airfoil geometry. Such insight, which is important for effective aerodynamic design and engineering, is much better provided by simple approximate analytic solutions. The panel method can still be used for accuracy when it’s needed. Single Vortex Sheet Model In order to simplify the problem sufficiently to allow analytic solution, we make the following assumptions and approximations: 1) The airfoil is assumed to be thin , with small maximum camber and thickness relative to the chord, and is assumed to operate at a small angle of attack, α 1. 2) The upper and lower vortex sheets are superimposed together into a single vortex sheet γ = γ u + γ , which is placed on the x axis rather than on the curved mean camber line Z = ( Z u + Z ) / 2. z z z γ (x) γ (x) γ (x) γ (x) l x x x Z(x) Z (x) u u Z (x) l Z(x) n 3) The flow-tangency condition ˆ V · n = 0 is applied on the x -axis at z = 0, rather than on the camber line at z = Z . But the normal vector ˆ n is normal to the actual camber line shape, as shown in the figure.
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