J. Philip Barnes Practical parametric geometry for aircraft design 26 May, 2015 J. Philip Barnes “Regenosoar” regen-electric.

J. Philip Barnes www.HowFliesTheAlbatross.com

Practical parametric geometry for aircraft design 26 May, 2015 J. Philip Barnes

“Regenosoar” regen-electric aircraft rendered with Blender 3D open-source graphics software and 100% math modeled with parametric equations via Blender’s integrated Python programming language


Abstract Practical parametric geometry for aircraft design

J. Philip Barnes, Technical Fellow, Pelican Aero Group

Theory and application of practical methods for aircraft geometry parameterization and visualization are described. The methods, characterizing the surface geometry of complete aircraft, wings, fuselages, ducts, and new or existing airfoils, include fidelities ranging from “rapid visualization” to “high fidelity.” We apply two integrated programming and visualization platforms. The first is EXCEL and Visual Basic and the second is Blender 3D (open-source) with its resident Python programming language. In all cases, we characterize Cartesian coordinates (x,y,z) with parametric coordinates (u,v).

For “rapid synthesis,” we introduce modified trigonometric functions capable of quickly approximating an airfoil, wing, or fuselage with just a handful of parameters. We also introduce a “cubic quadrant” method for for fuselage cross section design. For “good fidelity” modeling of new or existing airfoils, we introduce a “parametric Fourier series” method satisfying specified leading edge radius, max&min vertical coordinates, upper&lower afterbody slopes, and aft-edge thickness. A “fine tuning” parameter allows further subtle adjustments. Upper and lower surfaces can also be modeled separately for greater control.

For “high fidelity,” we describe the theory and application of the cubic spline which is unique in its class by passing through, not just near, all specified points while preserving C2 continuity. Although the cubic spline not C3 continuous, we show that airfoil surface velocity distributions remain smooth with cubic-spline parameterization of the airfoil geometry. We also apply the cubic spline to characterize wings and fuselages. Core algorithms and code blocks are listed or otherwise made available to ensure ready access to the methods.


Presentation Contents ~ Practical parametric geometry

Objectives&Rationale

“Rapid vis”Trigonometric

EXCEL/VBBlender/Python

Airfoil Geom.Fourier-series

Cubic splineTheory & App.

Air vehicleApplications

Blender 3D rendering of python-programmed geometry


Python window Rendering window

C:\Users\Philip\graphics\Blender\regen_ne

Getting started: EXCEL as a scientific spreadsheet

• Purpose (typical):• Read input and/or data from spreadsheet• Edit & run algorithm; generate new data• Write to spreadsheet cells & plot results• Copy all data & plots as new sheet; re-run

• One-time setup:1) EXCEL Options ~ Formulas ~ R1C1 ...2) Trust Ctr. ~ settings ~ macro ~ enable & trust3) Toolbar ~ more... ~ all ... ~ Visual Basic ~ Add4) Set VB editor window to float on spreadsheet

• Typical operations:1) Type in the column headers, i.e. t, x, y, z2) VB ~ insert ~ module ~ Type: sub example 3) Enter or edit code ~ save file as *.xlsm4) Click run icon (note: module stays with the file) 5) Highlight applicable columns & plot the results6) New case: Copy sheet, revise inputs, repeat 4)

Microsoft Office Excel Macro-Enabled Wor

Powerful Parametrics for Airfoil Geometry

J. Philip Barnes June, 2015


W

W

U

Cubic Spline

Airfoil parametric geometry

• Objectives and Applications– Closely match/smooth existing airfoils– Geometric design of new airfoils– Option: modest-fidelity rapid vizualization

• Three methods herein– Trigonometric (“Rapid viz”)– Fourier Series (good fidelity)– Parametric cubic spline (high fidelity)

• Common approach– One or two parametric surfaces– Set LE radius, 1-to-3 midpoints, aft slope– X(W) parametric, 0 ≤ W ≤ 1, front to back– Z(U) Fourier, or Z(W) polynomial or spline– “Fine tuning” via one or more aux. params. – EXCEL files included herein, each method


“Rapid viz” airfoil shaping: Hybrid Cartesian & trig. functions


2. reshape X(u)

1. simple wave, Z(u)

5. lower negative cusp

4. add camber

3. add aftbody cusp 6. opposite-sign cusps

DZ = c sin (X3p)

X = 1 - (1-g) sin(pu) + gsin(3pu)

DZ = c sin2(2pu)X = 1 - sin(pu) ; Z = c sin(2pu)

DZ = c sin (X3p)

DZ = c sin (X3p)

“Higher” fidelity ~ Parametric Fourier-series airfoil

• Fourier Series terms z(u)• Best used for one curve Z(U), not two Z(W)• Add 8 sinusoidal terms plus aft-edge width• Single L.E. rad.(R), max/min (X,Z) , two aft (b)• Use upper & lower fine-tune parameters (g)• Continuous in all derivatives• Solve eight eqns. for Fourier amplitudes• Satisfy end slopes (dW/dZ) & max/min• Compact “airfoil-sharing” formula• Airfoil construction sequence:• U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1)• g = gb + (gt - gb) cos2 (pU/2)• X = 1 - (1-g) cos(pW/2) – g cos(3pW/2)• Z = S m=1 to 8 {am sin(mpU)} + Za(1-2U)


0 W 1

1

X

0

g“fine-tune”parameter

Parameterizationfor X(W)

U

Z

0

First 4 termsof the series

Microsoft Excel Macro-Enabled Worksheet

W

W

U

Fourier Series


Parametric Fourier-Series airfoil ~ NLF(1)-0416 ~ matchFwd fine tuning, g inputs: 0.35000 0.17245 0.09366 0.30000

Upper gu 0.37500 0.12403 0.08208 0.25000

0.070 0.40000 0.08190 0.06715 0.20000

Lower gL 0.42500 0.04734 0.05008 0.15000

0.130 0.45000 0.02153 0.03232 0.10000

0.47500 0.00548 0.01527 0.05000

L.E. rad., R = r/c 0.50000 0.00000 0.00000 0.00000

0.0180 0.52500 0.00560 -0.01290 0.05000

0.55000 0.02243 -0.02337 0.10000

0.57500 0.05031 -0.03177 0.15000

0.60000 0.08864 -0.03863 0.20000

Upp. max. position, Xu 0.62500 0.13645 -0.04443 0.25000

0.3000 0.65000 0.19246 -0.04935 0.30000

Low. min. position, XL 0.67500 0.25506 -0.05316 0.35000

0.3500 0.70000 0.32250 -0.05529 0.40000

0.72500 0.39288 -0.05499 0.45000

Upp. max. elevation, Zu 0.75000 0.46431 -0.05163 0.50000

0.1045 0.77500 0.53503 -0.04499 0.55000

Low. min. elevation, ZL 0.80000 0.60345 -0.03545 0.60000

-0.0555 0.82500 0.66831 -0.02405 0.65000

0.85000 0.72871 -0.01231 0.70000

Upp. aft slope, bu, deg 0.87500 0.78419 -0.00201 0.75000

13.00 0.90000 0.83469 0.00529 0.80000

Low. aft slope, bL, deg 0.92500 0.88060 0.00857 0.85000

-10.00 0.95000 0.92269 0.00767 0.90000

0.97500 0.96206 0.00331 0.95000

Half trailing-edge, Za 1.00000 1.00000 -0.00300 1.00000

0.0030

Fourier Coefficients2.2794E-026.9443E-021.8405E-02

-1.6762E-021.3916E-03

-4.7482E-035.7805E-03

-8.0577E-05

-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)

NLF(1)-0416

0.0

0.2

0.4

0.6

0.8

1.0

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

0.14

0.16

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(X)Fourier Series

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) specificationsFourier Series

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

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

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

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

Trailing edge notesNo airfoil can have zero trailing-edge thickness;nor should it. Assume0.001 aft-edge thickness,unless input otherwise


Parametric Fourier-Series airfoil ~ PCS-001 ~ new designFwd fine tuning, g inputs: 0.35000 0.23585 0.12542 0.30000

Upper gu 0.37500 0.16765 0.10526 0.25000

0.200 0.40000 0.10905 0.08159 0.20000

Lower gL 0.42500 0.06190 0.05708 0.15000

0.100 0.45000 0.02757 0.03422 0.10000

0.47500 0.00686 0.01486 0.05000

L.E. rad., R = r/c 0.50000 0.00000 0.00000 0.00000

0.0115 0.52500 0.00667 -0.01022 0.05000

0.55000 0.02606 -0.01639 0.10000

0.57500 0.05695 -0.01953 0.15000

0.60000 0.09782 -0.02075 0.20000

Upp. max. position, Xu 0.62500 0.14694 -0.02101 0.25000

0.4140 0.65000 0.20250 -0.02098 0.30000

Low. min. position, XL 0.67500 0.26270 -0.02096 0.35000

0.3500 0.70000 0.32583 -0.02099 0.40000

0.72500 0.39037 -0.02097 0.45000

Upp. max. elevation, Zu 0.75000 0.45502 -0.02076 0.50000

0.1465 0.77500 0.51875 -0.02025 0.55000

Low. min. elevation, ZL 0.80000 0.58079 -0.01941 0.60000

-0.0210 0.82500 0.64064 -0.01826 0.65000

0.85000 0.69803 -0.01680 0.70000

Upp. aft slope, bu, deg 0.87500 0.75295 -0.01501 0.75000

7.00 0.90000 0.80552 -0.01286 0.80000

Low. aft slope, bL, deg 0.92500 0.85606 -0.01030 0.85000

4.30 0.95000 0.90498 -0.00732 0.90000

0.97500 0.95278 -0.00401 0.95000


0.0005

-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)

PCS-001

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X(u)

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(X)Fourier Series

Target Airfoil

specifications

RUN

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(u) specificationsFourier Series

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Cubic spline ~ Parametric u(t) or Cartesian y(x)

• Get smooth curve passing through (1_to_n) points• VB array dim. (n) elements: 0_to_n ~ ignore 0th elem.• 1st & 2nd derivative Continuity (3rd is not continuous)• Independently control L/R-end slope or 2nd derivative • Internal-node continuity yields tri-diagonal system• End constraints are applied in first and last rows• Parametric x(t) ; v “velocity”; a “acceleration”

t

x

12

n

3

• Set ends; Solve linear EQs. for internal-knot accelerations (a)

t

t

t

i+1ii-1

a ≡ d2x/dt2

vdx/dt

x

cubic

parabolic

linear

0

:

:

0

aa:aaa:

aa

.........

...:...p ...

:...p

...p...

n

1-n

1i

i

1-i

2

1

1-n

2-n

i

3

2

1

2

3

2

11

22

33

22

10000000

0000

0000

0000

0000

000

0000001

n

n

i

nn

nn

ii

s

s

s

s

s

rqp

rqp

rq

rq

rq

Microsoft Office Word Document

Parametric cubic spline ~ Various end constraints

“Stiff” ends “Flexible” ends “Flat” endsMicrosoft Office Excel Macro-Enabled Wor

Parametric cubic spline airfoil

• Cubic spline(s) pass through all set points• Wider design space including “unusual”• Match 0th, 1st, 2nd derivatives, ea. node• Discontinuous 3rd derivative• Input LE rad.(R), 3 pts. (X,Z) , aft slope (b)• g can be varied but is normally fixed (0.1)• 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”• U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1)• X = 1 - (1-g) cos(pW/2) – g cos(3pW/2)• EXCEL solves for cubic splines, Z(W) • Package: sol’n data block & interpolator


0 W 1

1

X

0

g

0 W 1

Z+0

-

Microsoft Excel Macro-Enabled Worksheet W

W

U

Cubic Spline


Parametric cubic spline airfoil Sample Gauss-Seidel convergence

Parametric cubic spline airfoil ~ 13-point match


Parametric cubic spline(blue) closely matchestarget (white points)



Parametric cubic-spline airfoil ~ NLF(1)-0416 ~ 9-pt matchUpp. L.E. rad., Ru=r/c 0.14286 0.70700 0.06143 0.71429

0.0130 0.15306 0.68289 0.06538 0.69388

Low. L.E. rad., RL=r/c 0.16327 0.65830 0.06926 0.67347

0.0130 0.17347 0.63324 0.07304 0.65306

0.18367 0.60774 0.07671 0.63265

fwd upper, Xfu 0.19388 0.58182 0.08025 0.61224

0.1500 0.20408 0.55555 0.08363 0.59184

fwd lower, XfL 0.21429 0.52896 0.08683 0.57143

0.1500 0.22449 0.50211 0.08984 0.55102

0.23469 0.47507 0.09263 0.53061

fwd upper, Zfu 0.24490 0.44791 0.09519 0.51020

0.0900 0.25510 0.42072 0.09747 0.48980

fwd lower, ZfL 0.26531 0.39356 0.09944 0.46939

-0.0480 0.27551 0.36654 0.10104 0.44898

0.28571 0.33974 0.10219 0.42857

mid upper, Xmu 0.29592 0.31326 0.10286 0.40816

0.4500 0.30612 0.28721 0.10298 0.38776

mid lower, XmL 0.31633 0.26168 0.10249 0.36735

0.4500 0.32653 0.23678 0.10133 0.34694

0.33673 0.21261 0.09945 0.32653

mid upper, Zmu 0.34694 0.18927 0.09679 0.30612

0.0950 0.35714 0.16688 0.09330 0.28571

mid lower, ZmL 0.36735 0.14552 0.08891 0.26531

-0.0520 0.37755 0.12530 0.08362 0.24490

0.38776 0.10631 0.07757 0.22449

aft upper, Xau 0.39796 0.08864 0.07088 0.20408

0.8000 0.40816 0.07238 0.06371 0.18367

aft lower, XaL 0.41837 0.05760 0.05618 0.16327

0.8000 0.42857 0.04438 0.04844 0.14286

0.43878 0.03279 0.04063 0.12245

aft upper, Zau 0.44898 0.02288 0.03288 0.10204

0.0450 0.45918 0.01470 0.02534 0.08163

aft lower, ZaL 0.46939 0.00829 0.01814 0.06122

0.0000 0.47959 0.00369 0.01142 0.04082

0.48980 0.00092 0.00533 0.02041

Upp. aft slope, bu, deg 0.50000 0.00000 0.00000 0.00000

14.00Low. aft slope, bL, deg 0.50000 0.00000 0.00000 0.00000

-14.00 0.51020 0.00092 -0.00481 0.02041

0.52041 0.00369 -0.00945 0.04082


0.0030 0.54082 0.01470 -0.01815 0.08163

0.55102 0.02288 -0.02221 0.10204

0.56122 0.03279 -0.02609 0.12245

0.57143 0.04438 -0.02976 0.14286

Public Domain 0.58163 0.05760 -0.03323 0.16327

J. Philip Barnes 0.59184 0.07238 -0.03650 0.18367

Phil's web site 0.60204 0.08864 -0.03956 0.20408

0.61224 0.10631 -0.04241 0.22449

0.62245 0.12530 -0.04505 0.24490

0.63265 0.14552 -0.04747 0.26531

0.64286 0.16688 -0.04967 0.28571

0.65306 0.18927 -0.05163 0.30612

0.66327 0.21261 -0.05333 0.32653

0.67347 0.23678 -0.05473 0.34694

0.68367 0.26168 -0.05580 0.36735

0.69388 0.28721 -0.05654 0.38776

-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)

NLF(1)-0416

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)

spline

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

spline

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


1 W 0 W 1

Upper Lower

Aft upper Aft lower

W

W

0 U 1


Parametric cubic-spline airfoil ~ PCS-001 ~ new designUpp. L.E. rad., Ru=r/c 0.14286 0.70700 0.09871 0.71429

0.0120 0.15306 0.68289 0.10702 0.69388

Low. L.E. rad., RL=r/c 0.16327 0.65830 0.11486 0.67347

0.0120 0.17347 0.63324 0.12209 0.65306

0.18367 0.60774 0.12856 0.63265

fwd upper, Xfu 0.19388 0.58182 0.13415 0.61224

0.1500 0.20408 0.55555 0.13872 0.59184

fwd lower, XfL 0.21429 0.52896 0.14227 0.57143

0.1500 0.22449 0.50211 0.14485 0.55102

0.23469 0.47507 0.14649 0.53061

fwd upper, Zfu 0.24490 0.44791 0.14722 0.51020

0.1000 0.25510 0.42072 0.14708 0.48980

fwd lower, ZfL 0.26531 0.39356 0.14610 0.46939

-0.0204 0.27551 0.36654 0.14433 0.44898

0.28571 0.33974 0.14179 0.42857

mid upper, Xmu 0.29592 0.31326 0.13852 0.40816

0.5800 0.30612 0.28721 0.13456 0.38776

mid lower, XmL 0.31633 0.26168 0.12995 0.36735

0.4500 0.32653 0.23678 0.12471 0.34694

0.33673 0.21261 0.11889 0.32653

mid upper, Zmu 0.34694 0.18927 0.11251 0.30612

0.1345 0.35714 0.16688 0.10563 0.28571

mid lower, ZmL 0.36735 0.14552 0.09826 0.26531

-0.0200 0.37755 0.12530 0.09047 0.24490

0.38776 0.10631 0.08236 0.22449

aft upper, Xau 0.39796 0.08864 0.07401 0.20408

0.8000 0.40816 0.07238 0.06554 0.18367

aft lower, XaL 0.41837 0.05760 0.05704 0.16327

0.8000 0.42857 0.04438 0.04861 0.14286

0.43878 0.03279 0.04035 0.12245

aft upper, Zau 0.44898 0.02288 0.03236 0.10204

0.0630 0.45918 0.01470 0.02475 0.08163

aft lower, ZaL 0.46939 0.00829 0.01760 0.06122

-0.0130 0.47959 0.00369 0.01103 0.04082

0.48980 0.00092 0.00513 0.02041

Upp. aft slope, bu, deg 0.50000 0.00000 0.00000 0.00000

8.00Low. aft slope, bL, deg 0.50000 0.00000 0.00000 0.00000

3.50 0.51020 0.00092 -0.00436 0.02041

0.52041 0.00369 -0.00805 0.04082


0.0010 0.54082 0.01470 -0.01365 0.08163

0.55102 0.02288 -0.01567 0.10204

0.56122 0.03279 -0.01724 0.12245

0.57143 0.04438 -0.01842 0.14286

Public Domain 0.58163 0.05760 -0.01927 0.16327

J. Philip Barnes 0.59184 0.07238 -0.01983 0.18367

Phil's web site 0.60204 0.08864 -0.02017 0.20408

0.61224 0.10631 -0.02034 0.22449

0.62245 0.12530 -0.02040 0.24490

0.63265 0.14552 -0.02040 0.26531

0.64286 0.16688 -0.02039 0.28571

0.65306 0.18927 -0.02039 0.30612

0.66327 0.21261 -0.02040 0.32653

0.67347 0.23678 -0.02040 0.34694

0.68367 0.26168 -0.02040 0.36735

0.69388 0.28721 -0.02039 0.38776

-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)

PCS-001

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)

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Z(X)spline

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

spline

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


X = 1 - (1 - g) cos (pW/2) -g cos (3pW/2)

1 W 0 W 1

Upper Lower

Aft upper Aft lower

W

W

f

Laminar airfoil study ~ integrated geometric/aero design

Parametric cubic spline

• Pressure coefficient

Discontinuous 3rd-deriv.of cubic spline does notdisrupt smooth airflow

• Velocity ratio

Theodorsen Angle (f)


Parametric fuselage section: “cubic quadrant” method

Four parameters per quadrant

One cubic curve per quadrant


Parametric Fuselage – cubic spline & trig. compared

cubic-spline basis

Trig. functions provide 99% desiredresult with just 1% of computation




Parametric wing: cubic spline throughout (EXCEL/VB)

symbol description station, v = 0:1 → v → Ev0↓ 0.0000 0.2000 0.4000 0.7500 1.0000 Ev1↓ u x y z xp zp v c b0 b1 u1 z1 u2 z2 u3 z3 u4c local chord chord, c 0 1.0000 0.6600 0.3800 0.1800 0.0200 1 0 1 7 13 0 0.001 0.25 0.1 0.5 0 0.75

bo0 upper tr. edge boattail angle bo0 0 7.0000 9.0000 9.9000 10.0000 10.0000 1 0 0.5 0 0.0015 0.427531 0.210985 0.048455 0.968519 7.200982 12.79902 0 0.001 0.25 0.1 0.5 0 0.75bo1 lower tr. edge boattail angle bo1 0 13.0000 11.0000 10.1000 10.0000 10.0000 1 0.049734 0.399053 0 0.023936 0.351654 0.190995 0.094122 0.894321 7.663301 12.3367 0 0.001 0.25 0.1 0.5 0 0.75

u c'clockwise from upper t.e., 0:1 u1 0 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.103766 0.283219 0 0.05198 0.258874 0.168472 0.137865 0.800195 8.227776 11.77222 0 0.001 0.25 0.1 0.5 0 0.75Z=z/c foil vertical coord. (local) Z1 0 0.001 0.001 0.001 0.001 0.001 1 0.165539 0.137789 0 0.081196 0.131747 0.132356 0.180979 0.701419 8.784027 11.21597 0 0.001 0.25 0.1 0.5 0 0.75

u2 0 0.25 0.25 0.25 0.25 0.25 1 0.236968 -0.04563 0 0.099866 -0.04623 0.065076 0.225 0.610098 9.238646 10.76135 0 0.001 0.25 0.1 0.5 0 0.75Ev_ spline start/end edge constraints Z2 0 0.1 0.1 0.1 0.1 0.1 1 0.318051 -0.25242 0 0.088465 -0.26756 -0.04973 0.271468 0.530646 9.557754 10.44225 0 0.001 0.25 0.1 0.5 0 0.75

v sparwise parameter, 0:1 u3 0 0.5 0.5 0.5 0.5 0.5 1 0.406877 -0.42884 0 0.047867 -0.46766 -0.1841 0.321684 0.461477 9.760373 10.23963 0 0.001 0.25 0.1 0.5 0 0.75Z3 0 0 0 0 0 0 1 0.5 -0.5 0 0.0005 -0.54374 -0.2664 0.376489 0.401712 9.871555 10.12844 0 0.001 0.25 0.1 0.5 0 0.75u4 0 0.75 0.75 0.75 0.75 0.75 1 0.593123 -0.42884 0 -0.02937 -0.45139 -0.24846 0.436094 0.350225 9.934743 10.06526 0 0.001 0.25 0.1 0.5 0 0.75Z4 0 -0.05 -0.05 -0.05 -0.05 -0.05 1 0.681949 -0.25242 0 -0.0435 -0.25246 -0.16176 0.5 0.306187 9.976475 10.02352 0 0.001 0.25 0.1 0.5 0 0.75u5 0 1 1 1 1 1 1 0.763032 -0.04564 0 -0.05047 -0.04341 -0.06294 0.566996 0.268707 9.998582 10.00142 0 0.001 0.25 0.1 0.5 0 0.75Z5 0 -0.001 -0.001 -0.001 -0.001 -0.001 1 0.834461 0.137787 0 -0.05302 0.124879 0.019427 0.635267 0.235607 10.00523 9.994773 0 0.001 0.25 0.1 0.5 0 0.75

wo washout (trailing-edge up) wo 0 0 1 3.2 6.1 8 1 0.896235 0.283217 0 -0.04648 0.249268 0.086731 0.702586 0.203931 10.00303 9.996974 0 0.001 0.25 0.1 0.5 0 0.75do dihedral (local x-rotation) do 0 0 0 0 0 0 1 0.950266 0.399052 0 -0.0293 0.344731 0.147184 0.766574 0.171026 9.999158 10.00084 0 0.001 0.25 0.1 0.5 0 0.75s spar chord sta. (x-xLE)/c (local) s 0 1 1 1 1 1 1 1 0.499999 0 -0.0005 0.427217 0.209342 0.825 0.136872 9.997489 10.00251 0 0.001 0.25 0.1 0.5 0 0.75

xb spar backbone x (global coord.) xb 0 0.5 0.35 0.35 0.46 0.54 1 0.876085 0.104408 9.99737 10.00263 0 0.001 0.25 0.1 0.5 0 0.75yb spar backbone y yb 1 0 0.2 0.4 0.75 1 1 0 0.483562 0.048455 0.004313 0.380554 0.232849 0.918767 0.075956 9.997956 10.00204 0 0.001 0.25 0.1 0.5 0 0.75zb spar backbone z zb 0 0 0.034 0.074 0.09 0.09 1 0.049734 0.385768 0.048455 0.026299 0.306532 0.214557 0.952901 0.052623 9.998722 10.00128 0 0.001 0.25 0.1 0.5 0 0.75g 0.0:0.10 option moves Zmax aft g 0 0.0900 0.0900 0.0900 0.0900 0.0900 1 0.103766 0.27355 0.048455 0.053419 0.216183 0.193818 0.979357 0.034331 9.999423 10.00058 0 0.001 0.25 0.1 0.5 0 0.75

TBD spare parameter 0.165539 0.132667 0.048455 0.081408 0.092947 0.160205 1.000001 0.02 10 10 0 0.001 0.25 0.1 0.5 0 0.75

Edit columns 4-10, open VB editor and click the Run icon 0.236968 -0.045 0.048455 0.099023 -0.07843 0.097187Phil Barnes, 08 Mar 2015 0.318051 -0.24527 0.048455 0.08762 -0.28996 -0.01059

0.406877 -0.41609 0.048455 0.048145 -0.48009 -0.136950.5 -0.48496 0.048455 0.00225 -0.5521 -0.21505

0.593123 -0.416 0.048455 -0.02667 -0.46415 -0.199550.681949 -0.24512 0.048455 -0.0402 -0.27423 -0.119240.763032 -0.04484 0.048455 -0.04658 -0.07371 -0.026760.834461 0.132812 0.048455 -0.04857 0.088442 0.0508960.896235 0.273656 0.048455 -0.04193 0.208591 0.1146970.950266 0.385825 0.048455 -0.02524 0.300795 0.172157

1 0.483563 0.048455 0.002376 0.380288 0.231259

0 0.446655 0.094122 0.01115 0.321105 0.2487830.049734 0.356289 0.094122 0.031961 0.251539 0.2328260.103766 0.252592 0.094122 0.056873 0.16695 0.2142960.165539 0.122426 0.094122 0.082011 0.052472 0.1838130.236968 -0.04168 0.094122 0.097215 -0.10497 0.1267370.318051 -0.22657 0.094122 0.08584 -0.29704 0.0298790.406877 -0.38419 0.094122 0.048998 -0.46825 -0.0832

-1.25

-0.75

-0.25

0.25

0.75

1.25

-1 -0.5 0 0.5 1 1.5 2

zp(xp)

-0.25

0.00

0.25

0.50

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

z(x)

-0.25

0.00

0.25

0.50

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

z(y)

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

y(x)

SummaryThe table above represents one half wing. Half-wing geometry is parametric with (u,v) using cubic splines, airfoil c'clockwise Vs. u, sparwise Vs. v Input one column per wing "sparwise" station, including the local airfoil as a column (5-points for now)Spline-edge integer constraints are [not] used for the airfoil ; set the boattail slopes (+ for typical foil)x/c for the airfoil is an output: x/c = 1-sin(pu), given (u) as an input. x/c is optionally modified with g.Airfoil "spline Left and Right" (lower t.e., upper t.e.) edge slopes (dz/du) are then given by -p tanbSpline-edge constraints are used versus sparwise position for all other parameters, i.e. c(v), b(v),...Sparwise position (v) has an airfoil "backbone" point at xb,yb,zb (global)The spar backbone chordwise station s = (x-xLE)/c, nominally 0.25, is anywhere from 0.0-to-1.0The airfoil is first translated such that its backbone is anchored to the backbone global positionThe airfoil is then "washed out" (trailing edge up), rotating about a local y-axis thru the backbone pt.The airfoil is then rotated about a local x-axis thru the backbone point by the dihedral angle (d).

RUN


23Energy From an Atmosphere in Motion - Dynamic Soaring and Regen-electric Flight Compared J. Philip Barnes www.HowFliesTheAlbatross.com

Application: Dynamic soaring in the jet stream


Application: Regen of electrical power in ridge lift


About the Author

Phil Barnes has a Master’s Degree in Aerospace Engineering from Cal Poly Pomona and BSME from the University of Arizona. He is a Principal Engineer and 34-year veteran of air vehicle and subsystems performance analysis at Northrop Grumman, where he presently supports both mature and advanced tactical aircraft programs. Author of several SAE and AIAA technical papers, and often invited to lecture at various universities, Phil is presently leading several Northrop Grumman-sponsored university research projects including an autonomous thermal soaring demonstration, passive bleed-and-blow airfoil wind-tunnel test, and application of Blender 3D software for flight simulation. This presentation includes highlights of one such collaboration (public domain) using EXCEL/Visual Basic and Blender 3D with its resident Python programming language to parameterize and visualize aircraft geometry. Outside of work, Phil is a leading expert on dynamic soaring, and he is pioneering the science of regen-electric flight.

J. Philip Barnes Practical parametric geometry for aircraft design 26 May, 2015 J. Philip Barnes “Regenosoar” regen-electric.

Documents

airfoil geometry

cubicspline parameterization

philip barnes regenosoar

programmed geometry

aircraft design

fourierseries airfoil

parametric equations

new data