A Brief Summary of Aero-Hydrodynamics - Summary Pages | AOE 3014, Study notes of Aerospace Engineering

Download A Brief Summary of Aero-Hydrodynamics - Summary Pages | AOE 3014 and more Study notes Aerospace Engineering in PDF only on Docsity! Chapter 4 Summary Pages A vortex sheet can be used to synthesize the inviscid, incompressible flow over an airfoil. If the distance along the sheet is given by s and the strength of the sheet per unit length is y (s), then the velocity potential induced at point (x, y) by a vortex sheet that extends from point a to point b is 1 f° d(x, y= / Ay(s)ds (4.3) 2m Sa The circulation associated with this vortex sheet is rb T= | y(s)ds (4.4) Across the vortex sheet, there is a tangential velocity discontinuity, where y=m—u (4.8) The Kutta condition is an observation that for a lifting airfoil of given shape ata given angle of attack, nature adopts that particular value of circulation around the airfoil which results in the flow leaving smoothly at the trailing edge. If the trailing-edge angle is finite, then the trailing edge is a stagnation point. If the trailing edge is cusped, then the velocities leaving the top and bottom surfaces at the trailing edge are finite and equal in magnitude and direction. In either case, y(TE) =0 (4.10) Thin airfoil theory is predicated on the replacement of the airfoil by the mean camber line. A vortex sheet is placed along the chord line, and its strength adjusted such that, in conjunction with the uniform freestream, the camber line becomes a streamline of the flow while at the same time satisfying the Kutta condition. The strength of such a vortex sheet is obtained from the fundamental equation of thin airfoil theory: Jf r@d _ yy (« =) (4.18) In jo x —€ ~ dx