1. 1 Pelican Aero Group Powerful Parametrics for Airfoil Geometry J. Philip Barnes 30 Mar, 2015 J. Philip Barnes www.HowFliesTheAlbatross.com W W U

2. 2 Pelican Aero Group Parametric polynomial airfoil Design new airfoil or smooth existing foil Math, versus tabulated, characterization Parametric coordinates are (U and W) For graphics & mfg. use (0 ≤ U ≤ 1) Upper & lower polynomials: X(W), Z(W) U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1) X(W) shaped for dX/dW = 0 @ L-edge X(W) shaped for d 2 X/dW 2 = 0 @ T-edge X(W) incl. “shape param.” 0 ≤  ≤ 0.15 X = 1 - (1-  ) cos(  W/2) –  cos(3  W/2) L-edge rad.(R), max (X,Z), aft slope (  ) Inputs, each surface: R, X, Z, ,  Init.  =0.10 ; vary (  ) to fine tune shape R, [d 2 X/dW 2 ] W=0 together set [dZ/dW] W=0 , [dX/dW] W=1 together set [dZ/dW] W=1 Solve 5 eqns. for poly. coefficients (a): Z = a 1 W 1 + a 2 W 2 + a 3 W 3 +... a 5 W 5 Optimum combo., control & smoothness J. Philip Barnes www.HowFliesTheAlbatross.com 0 W 1 1X01X0  W W U Z+0-Z+0-

3. 3 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com Parametric polynomial airfoil: design/match example Purpose: show that this method can design various “classes” of airfoil (Cubic spline method is best suited to precise match of a given airfoil) Purpose: show that this method can design various “classes” of airfoil (Cubic spline method is best suited to precise match of a given airfoil) Inputs Directions Confirm smooth Confirm match Target shape Polynomial coefficients Polynomial coefficients Boundary conditions NLF(1)-0416 Confirm L.E. continuity

4. 4 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com Parametric polynomial airfoil ~ Design space examples Difficulty with “unusual” designs, such as LNV109A Method is well suited to a wide range of “usual” airfoils 7 cases

5. 5 Pelican Aero Group Parametric cubic spline airfoil Same as param. poly. airfoil except: Wider design space including “unusual” Cubic splines in lieu of polynomials Match 0 th, 1 st, 2 nd derivatives, ea. node Discontinuous 3 rd derivative Solves 5 eqns. spline-knot 2 nd derivatives Gauss-Seidel in lieu of Gaussian Diag. 3 midpoints versus single midpoint Any position, not necessarily max/min Less compact “airfoil-sharing package” Upper & lower cubic splines: Z(W) U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1) X = 1 - (1-  ) cos(  W/2) –  cos(3  W/2) L-E rad.(R), 3 points (X,Z), aft slope (  )  can be varied but is normally fixed (0.1) Package: sol’n data block & interpolator J. Philip Barnes www.HowFliesTheAlbatross.com 0 W 1 1X01X0  W W U Z+0-Z+0-

6. 6 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com Parametric spline airfoil ~ Design space examples Comprehensive design space

7. 7 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com Parametric cubic spline airfoil Sample Gauss-Seidel convergence