NACA four-digit airfoil section generation using cubic parametric curve segments and the golden ...

286 Pages·2016·4.38 MB·English

by  William T. Scarbrough

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Theses
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William T. Scarbrough
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Scarbrough, William T., "NACA four-digit airfoil section generation using cubic parametric curve segments
and the golden section" (1992). Thesis. Rochester Institute of Technology. Accessed from
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NACA FOUR-DIGIT AIRFOIL SECTION GENERATION
USING CUBIC PARAMETRIC CURVE SEGMENTS AND
THE GOLDEN SECTION
William T. Scarbrough
Department ofMechanical Engineering
Rochester Institute of Technology
One Lomb Memorial Drive
Rochester, New York
Submitted in Partial Fulfillment of the Requirements for the Degree of
Master of Science of Mechanical Engineering
28 APRIL 1992
Dr. Panchapakesan Venkataraman, Graduate Thesis Advisor
Dr. Hany Ghoneim, Professor of Mechanical Engineering
Dr. Amitabha Ghosh, Professor ofMechanical Engineering
Dr. Joseph Torok, Professor ofMechanical Engineering
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Table Contents
of
Abstract
v
Acknowledgements
vi
List of Symbols vii
List of Figures ix
1. Introduction i
1.1 A BriefHistory of Airfoils and 1
Aeronautical Development
1.2 Direction of Research 4
1.3 The Golden Section 4
2. NACA Four-Digit Airfoils 5
2.1 Symmetric Airfoils 5
2.1.1 Nomenclature 5
2.2.1 Thickness Distribution 5
2.2.2 Leading Edge Radius 6
2.2.3 Trailing Edge Angle 6
2.2 Cambered Airfoils 7
2.2.1 Mean Line 7
2.2.2 Method ofCombining Thickness 8
Distribution and Mean Line
2.3 NACA Designation Scheme 9
2.4 Method of Manual Layout 9
3. Approximation Techniques n
3.1 Polynomial Approximation 11
3.1.1 Cubic Spline Interpolation 11
3.1.2 Discrete Least-Squares 11
Approximations
3.2. Parametric Curves 17
3.2.1 Bezier Curves 19
3.2.2 Matrix Formulation for Bezier Curve 21
3.2.3 Derivatives ofBezier Curve 22
4. Results of Analysis 24
4.1 Symmetric Airfoils 24
4.1.1 Cubic Spline interpolation 24
4.1.1.1 Natural Cubic Splines 24
4.1.1.2 Clamped Cubic Splines 26
4.1.1.3 Overcoming the Problem of 30
Specifying Zero Slope
4.1.2 Least-Squares Polynomial 30
Approximations
u
4.1.3 Parametric Bezier Curves 30
4.1.3.1 Leading Edge Surface 30
4.1.3.2 Trailing Edge Surface 62
4.1.3.3 Assembling the Pieces 71
4.1.3.4 Error Analysis 71
4.2 Cambered Airfoils 72
4.2.1 Review 72
4.2.2 Detennining Points ofZero Slope 73
4.2.2.1 Linear Fit of Upper Surface 92
Ordinate
4.2.2.2 Polynomial Fits 93
4.2.2.3 The Arc Method 94
4.2.2.4 The Curve Method 95
4.2.2.5 Areas ofTriangles 95
4.2.2.6 Dimensionless Parameter 98
Combinations
4.2.2.7 The Linear Correction Factor 99
4.2.3 Utilization ofProperties at Point of 121
Maximum Camber
4.2.3.1 Further Review 121
4.2.3.2 Enclosing Tangent Triangle 122
4.2.4 Conventional Cambered Airfoil 150
Generation
5. Conclusions 152
m
Appendices
A. The Golden Section A-l
B. Cubic Spline Derivation B1
C. References c_1
D. Comparative Graphs of NACA D1
Four-Digit Symmetric Airfoils
and Bezier Curve Emulations
E. Comparative Graphs of NACA E_1
Four-Digit Cambered Airfoils
and Bezier Curve Emulations
F. FORTRAN Computer Programs F1
G. Pertinent Data Files G-l
IV
ABSTRACT
A simple, elegant and modern method of geometric description of NACA Four-digit airfoil
shapes is presented. Results are found to closely match conventionally described NACA Four
Digit airfoil shapes. The method developed allows user flexibility, and is easily adaptable to
manufacturing processes.
Acknowledgements
The author gratefully acknowledges the following people, withoutwhose help thisproject might
never have occurred.
Dr. Charles W. Haines, Dr. Panchapakesan Venkataraman, Dr. Mark H. Kempski, Dr. Chris
Nilsen (whounknowingly led meon thepath to theBezier curve), Mr. GeorgeKomorowskiand
David Hathaway, all faculty and staff of the Mechanical Engineering Department at RTT.
Mr. Ralph Culliton, my supervisor at AC Rochester, for his patience and understanding during
the last months of this project.
My daughters, Kelly and Megan, who never questioned my absence during the many long nights
away from home.
Lastly, but most importantly, my wife Coleen, who gave me moral support and took control of
all family matters during the course of my education.
This manuscript was prepared on an IBM PS/2with WordPerfectVersion 5.1; imbedded figures
were generated using DrawPerfect Version 1.1. The majority oftheplots were generated using
theProfessional Graphics Facility (PGF)onanIBM mainframe. Theremainder oftheplotswere
produced with DIS8, a plotting package available on the Digital Equipment VAX at RIT.
vi
List Symbols
of
ah bt, c^ d{ cubic spline interpolant coefficients
Bernfx) Bernstein basis polynomial
B, Bezier curve defining polygon vertex
c chordlength
E Young's modulus of elasticity
f,g arbitrary functions
Fn nth number in Fibonacci sequence
I moment of inertia
Jn,i ith nth-order Bernstein basis function
m maximum camber
M(x) bending moment
P chordwise position of maximum camber
PC*') parametric description of a point
Pn(x) generic polynomial
r leading edge radius
S.(x) cubic spline interpolant
t thickness ratio/distribution
x chordwise position
Xint(r) chordwise position of intersection ofleading
edge radius and thickness distribution;
^int chordwise position of intersection of trailing
edge angle and the line y=tl2
chordwise position of maximum thickness
(symmetric airfoils)
ordinate of camber line
yuu ordinate of intersection of leading edge radius
and thickness distribution
yi lower surface ordinate (cambered airfoils)
y, ordinate of thickness distribution
yu upper surface ordinate (cambered airfoils)
vu