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 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 d2X/dW2 = 0 @ T-edge• X(W) incl. “shape param.” 0 ≤ g ≤ 0.15• X = 1 - (1-g) cos(pW/2) – g cos(3pW/2)• L-edge rad.(R), max (X,Z), aft slope (b)• Inputs, each surface: R, X, Z, b, g • Init. g=0.10 ; vary (g) to fine tune shape• R, [d2X/dW2]W=0 together set [dZ/dW]W=0

• b, [dX/dW]W=1 together set [dZ/dW]W=1

• Solve 5 eqns. for poly. coefficients (a):• Z = a1W1 + a2W2 + a3W3 + ... a5W5 • Optimum combo., control & smoothness

J. Philip Barnes www.HowFliesTheAlbatross.com

0 W 1

1

X

0

g

W

W

U

0 W 1

Z+0

-

3 Pelican Aero Group

b'cond W U X Z dZ/dW i a(i) U X Z W F(i,j) b(i) target foil X_t Z_t circle_X circle_Z W dZ/dW d2Z/dW2Parametric 1 0.00000 0.50000 0.00000 0.00000 0.15708 1 1.5708E-01 0.00000 1.00000 0.00300 1.00000 1.0000 -0.0030 0.02000 0.00000 0 0.1570795 1.94456861Polynomial 2 0.51522 0.24239 0.31000 0.10200 0.00000 2 9.7228E-01 0.02632 0.91742 0.02231 0.94737 0.9664 0.0043 0.01946 0.00325 0.04761905 0.23187286 1.21765714

Airfoil 3 0.51522 0.24239 0.31000 0.10200 0.00000 3 -2.7665E+00 0.05263 0.83540 0.03934 0.89474 0.9058 0.0075 0.01789 0.00614 0.0952381 0.2749602 0.611286644 1.00000 0.00000 1.00000 0.00300 -0.39164 4 2.3946E+00 0.07895 0.75451 0.05438 0.84211 0.8201 0.0024 0.01547 0.00837 0.14285714 0.29184882 0.115682215 1.00000 0.00000 1.00000 0.00300 -0.39164 5 -7.5438E-01 0.10526 0.67530 0.06756 0.78947 0.7211 -0.0125 0.01245 0.00969 0.19047619 0.28758058 -0.278931

0.13158 0.59830 0.07886 0.73684 0.5218 -0.0463 0.00917 0.00997 0.23809524 0.26673184 -0.5823279

3/24/2015 12:47 1 0.00000 0.50000 0.00000 0.00000 -0.19110 1 -1.9110E-01 0.15789 0.52405 0.08816 0.68421 0.3233 -0.0553 0.00598 0.00916 0.28571429 0.2334135 -0.80428342 0.44870 0.72435 0.31500 -0.05750 0.00000 2 1.2849E-01 0.18421 0.45305 0.09526 0.63158 0.1171 -0.0436 0.00323 0.00736 0.33333333 0.19127098 -0.95457243 0.44870 0.72435 0.31500 -0.05750 0.00000 3 -4.9142E-01 0.21053 0.38579 0.09993 0.57895 0.0196 -0.0188 0.00121 0.00476 0.38095238 0.14348425 -1.04296974 1.00000 1.00000 1.00000 -0.00300 -0.29765 4 1.6444E+00 0.23684 0.32272 0.10194 0.52632 0.0000 0.0000 0.00014 0.00165 0.42857143 0.09276779 -1.07925025 1.00000 1.00000 1.00000 -0.00300 -0.29765 5 -1.0934E+00 0.26316 0.26428 0.10109 0.47368 0.0139 0.0257 0.00014 -0.00165 0.47619048 0.04137061 -1.0731888

0.28947 0.21086 0.09728 0.42105 0.0438 0.0487 0.00121 -0.00476 0.52380952 -0.0089237 -1.03456040.31579 0.16283 0.09049 0.36842 0.0894 0.0698 0.00323 -0.00736 0.57142857 -0.0567972 -0.97313980.34211 0.12053 0.08088 0.31579 0.2619 0.1025 0.00598 -0.00916 0.61904762 -0.1013971 -0.898702

Upp. gu "shape" (0.00-0.15) 0.36842 0.08423 0.06878 0.26316 0.4917 0.0918 0.00917 -0.00997 0.66666667 -0.1423364 -0.82102190.0000 0.39474 0.05418 0.05474 0.21053 0.6916 0.0630 0.01245 -0.00969 0.71428571 -0.1796933 -0.7498742

Low. gL "shape" (0.00-0.15) 0.42105 0.03060 0.03957 0.15789 0.8636 0.0330 0.01547 -0.00837 0.76190476 -0.2140118 -0.6950340.0600 0.44737 0.01364 0.02437 0.10526 0.9522 0.0145 0.01789 -0.00614 0.80952381 -0.2463011 -0.666276

0.47368 0.00342 0.01058 0.05263 1.0000 0.0030 0.01946 -0.00325 0.85714286 -0.2780358 -0.6733752Upp. L.E. rad., Ru=r/c 0.50000 0.00000 0.00000 0.00000 0.02000 0.00000 0.9047619 -0.3111563 -0.7261064

0.0100 0.95238095 -0.3480683 -0.8342446Low. L.E. rad., RL=r/c 0.50000 0.00000 0.00000 0.00000 0.02000 0.00000 1 -0.3916428 -1.0075645

0.0100 0.52632 0.00505 -0.00976 0.05263 0.01946 0.003250.55263 0.02005 -0.01908 0.10526 0.01789 0.00614 0 -0.1910955 0.25698633

Upp. max. position, Xu 0.57895 0.04462 -0.02799 0.15789 0.01547 0.00837 0.04761905 -0.1815189 0.158964320.3100 0.60526 0.07811 -0.03634 0.21053 0.01245 0.00969 0.0952381 -0.1747604 0.13626573

Low. min. position, XL 0.63158 0.11969 -0.04384 0.26316 0.00917 0.00997 0.14285714 -0.1675705 0.174723160.3150 0.65789 0.16834 -0.05009 0.31579 0.00598 0.00916 0.19047619 -0.1573742 0.2601692

0.68421 0.22294 -0.05466 0.36842 0.00323 0.00736 0.23809524 -0.1422715 0.37843641Upp. max. elevation, Zu 0.71053 0.28231 -0.05715 0.42105 0.00121 0.00476 0.28571429 -0.1210366 0.51535739

0.1020 0.73684 0.34527 -0.05721 0.47368 0.00014 0.00165 0.33333333 -0.0931186 0.65676473Low. min. elevation, ZL 0.76316 0.41070 -0.05461 0.52632 0.00014 -0.00165 0.38095238 -0.0586412 0.788491

-0.0575 0.78947 0.47759 -0.04930 0.57895 0.00121 -0.00476 0.42857143 -0.0184028 0.89636880.81579 0.54505 -0.04147 0.63158 0.00323 -0.00736 0.47619048 0.0261238 0.9662307

Upp. aft slope, bu, deg 0.84211 0.61240 -0.03157 0.68421 0.00598 -0.00916 0.52380952 0.07279089 0.9839092814.00 0.86842 0.67915 -0.02039 0.73684 0.00917 -0.00997 0.57142857 0.11877626 0.93523715

Low. aft slope, bL, deg 0.89474 0.74501 -0.00911 0.78947 0.01245 -0.00969 0.61904762 0.16058303 0.80604687-14.00 0.92105 0.80988 0.00065 0.84211 0.01547 -0.00837 0.66666667 0.19403968 0.58217103

0.94737 0.87384 0.00679 0.89474 0.01789 -0.00614 0.71428571 0.21430006 0.24944222Half trailing-edge, Za 0.97368 0.93710 0.00666 0.94737 0.01946 -0.00325 0.76190476 0.21584338 -0.206307

0.0030 1.00000 1.00000 -0.00300 1.00000 0.02000 0.00000 0.80952381 0.1924742 -0.7992440.85714286 0.13732244 -1.54353620.9047619 0.0428434 -2.45335110.95238095 -0.0991823 -3.542856

1 -0.2976486 -4.8262184

Pub lic DomainJ. Philip BarnesPhil's web site

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(X)

0.00

0.25

0.50

0.75

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X(u)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(X)

Polynomials

Target Airfoil

specifications

RUN

-0.10

-0.05

0.00

0.05

0.10

0.15

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(u) specifications

Polynomials

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Upper and Lower 2nd Derivatives, d2Z/dW2

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Upper & Lower 1st Derivatives, dZ/dW Vs. W

Instructions Parametric Polynomial AirfoilDeclare upper & lower inputs in column 1Initialize (g)=0.10 (typ. range 0.00 to 0.15)(g) affects both forebody & aftbody; effectdepends on maximum thickness location(R) fattens forebody; (b) sets aftbody slopeParametric coord. 0 ≤ W ≤ 1, fwd-to-aft,RUN gets two nth-order polynomials Z(W),with coefficients (a) in column 9, where:Z = a1W + a2W2 + a3W3 + ... + anWn with:X = 1 - (1 - g) cos (pW/2) - g cos (3pW/2)U (graphics coord.) goes counter clockwise------ SAVE BETWEEN ITERATIONS -----------

J. Philip Barnes www.HowFliesTheAlbatross.com

Parametric polynomial airfoil: design/match examplePurpose: show that this method can design various “classes” of airfoil (Cubic spline method is best suited to precise match of a given airfoil)

Inputs DirectionsConfirm smooth

Confirm match

Target

shapePolynomial

coefficientsBoundary

conditions

NLF(1)-0416

Confirm L.E. continuity

4 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com

Parametric polynomial airfoil ~ Design space examples

Difficulty with “unusual” designs, such as LNV109AMethod is well suited to a wide range of “usual” airfoils

7 cases

Microsoft Excel Macro-Enabled Worksheet

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 0th, 1st, 2nd derivatives, ea. node• Discontinuous 3rd derivative• Solves 5 eqns. spline-knot 2nd 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-g) cos(pW/2) – g cos(3pW/2)• L-E rad.(R), 3 points (X,Z) , aft slope (b)• g can be varied but is normally fixed (0.1)• Package: sol’n data block & interpolator

J. Philip Barnes www.HowFliesTheAlbatross.com

0 W 1

1

X

0

g

W

W

U

0 W 1

Z+0

-

6 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com

Parametric spline airfoil ~ Design space examples

Comprehensive design space

Microsoft Excel Macro-Enabled Worksheet

7 Pelican Aero Group J. Philip Barnes www.HowFliesTheAlbatross.com

Parametric cubic spline airfoil Sample Gauss-Seidel convergence