Computer Aided Geometric Design 4 (1987) 17-22 17 North-Holland

B6zier presentation of airfoils

Wolfgang B O E H M *

Angewandte Geometrie und Geometrische Datenoerarbeitun~ Technische Universitaet Braunschwei~ D-3300, Fed Rep. Germany

Presented at Wolfenbuettel 24 June 1986 Received 5 August 1986; revised 2 October 1986

Abstract. B6zier presentations and cubic spline presentations of NACA-4-digit airfoils are developed and their errors are discussed. The given representations may help to use the Bemstein-B6zier technique in the design process of wings.

Keywords. Airfoils, B6zier curves, spline curves, wings.

I n t r o d u c t i o n

M o s t of wing sections in c o m m o n use are either N A C A airfoils or have been s t rongly inf luenced b y the N A C A investigations. [Abbot t '59, Wetzel '86]. The wing sections under cons idera t ion here are obta ined b y combining a thickness dis tr ibut ion z(u) with a meanl ine m (u) , as shown in Fig. 1,

=(.) =,.(u) +_.(u). z(u), where re(u) is the mean line unit normal vector at re(u) and u ~ [0, 1] is a parameter, e.g. cor responding to the chordwise posi t ion x, , of m. But the applied methods may easily be t ransferred to o ther wing sections.

Let ' + ' or ' - ' indicate the upper or lower side of the airfoil and let u = u_ ~ [ - 1, 0] be used to descr ibe the lower side. Then the parametr ic airfoil representa t ion x(u) m a y be

0

X_

Xm

m e a n ~ chord line

NACA 6420

Fig. 1. Airfoil description.

X

* Present address: Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A.

0167-8396/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-HoUand)

18 W. Boehm / B~zier presentation of airfoils

X x

_ X+(U)/

R

t :~ - C

-1

O

U_ U

Fig. 2. Composition of the non-parametric components.

c o n s t r u c t e d f r o m b o t h n o n - p a r a m e t r i c c o m p o n e n t s x ( u ) , y (u ) as s h o w n in Fig. 2. Le t R a n d p

be the radi i o f c u r v a t u r e of x (u ) and x (u ) at u = 0, respec t ive ly , a n d let t an a b e the s lope o f

y(u) at u = 0, t h e n

p + = R + tan2a +, p_ = R _ t an 2 a_ .

NACA-4-digit airfoils

In the special case o f a N A C A - 4 - d i g i t a i r foi l us ing the ' n a t u r a l ' p a r a m e t r i z a t i o n t = V~- the

th ickness d i s t r i bu t ion z (x( t ) ) is o f o r d e r 8 in t [see A b b o t t '59]. I ts B6zier o r d i n a t e s bi a re

s h o w n in T a b l e 1, t oge the r w i th the 8 th o r d e r B6zier absc issae xi o f x - t 2. T h e d i s t r ibu t ion

Table 1 B~zier points of the symmetric NACA-4-digit airfoil, q* = 1.

i = 0 1 2 3 4 5 6 7 8

x i = 0.00000 0.00000 0.03571 0.10714 0.21429 0.35714 0.53571 0.75000 1.0000 bi = 0.00000 0.18556 0.34863 0.48919 0.58214 0.55724 0.44929 0.30281 0.01050

z,

0 t

Fig. 3. B ~ e r presentation of thickness as a function of t, qr = 30.

W. Boehm / B~zier presentation of airfoils

Table 2 B~zier points of the mean line re(x), m* =1.

19

j = 0 1 2 3 4

xj-- 0.0 p*/2 p* (p* + 1) / 2 1.0 pj --- 0.0 0.1 0.1 0.1 0.0

Y ~ 0.08

0 P* 1 x

Fig. 4. B6zier presentation of the mean line as a function of x, m* -- 0.8, p* = 0.4.

z (y ) is shown in Fig. 3. A factor q * = qr/lO0 will control its amplitude. The leading edge radius of this parametric B6zier curve is

p = 1.1019 q.2.

The mean line f a NACA-4-digit airfoil [see Abbott '59] is a quadratic V 1 spline with respect to x (!). It is defined by its B6zier points xj, pj., given in Table 2. The abscissa p* = p / 1 0 indicates the chordwise position of the break point. Fig. 4 shows this quadratic spline. A factor m* = m/ lO to the y-ordinates will control the camber.

Various m, p, qr indicate the family of NACA-4-mpqr airfoils, where the integers m, p, q, r correspond exactly to the numbering system.

Hermite interpolation

Sometimes, the high order, 8, of the thickness will be regarded as a disadvantage in the process of designing wings. Composite B6zier curves of lower degree will be used. Table 3 shows the defining B6zier ordinates b k of a simple cubic Hermite interpolation of the thickness y(t) at t, - i/4. The resulting composite curve together with its 500-fold relative error is shown in Fig. 5. Note that b 3 - ½(b 2 + b4), etc.. The leading edge radius of this interpolant coincides with the exact value

p = 1.1019 q.2.

The error function shows the typical shape having zeros with horizontal tangents at the knots. Note that a correction of bll by -0.002 will decrease the maximal error, as dashed in Fig. 5.

Table 3 Thickness distribution, B ~ e r ordinates of a 4-segment Hermite interpolant.

k = 0 1 2 4 5 7 8 10 11 12

Pk ~ 0.0 0.12371 0.23625 0.41420 0.47758 0.51263 0.48371 0.32619 0.20538 0.01050

20 W. Boehm / B$zier presentation of airfoils

zA

b ~ , Ib~ 0 1/16 2/16 3/16 6/16

~ b ~ 2 ~/16 1 t

I [ =o.oob-~ . . . .

Fig. 5. Hermite interpolation of the thickness with 500-fold relative error, qr = 30.

Spline approximation

Some numerical experiments [Boehm '81] have shown that instead of the 'natural' parametri- zation by t = v~- a simple piecewise cubic parametrization u, defined by

x=½u213-ul,

will decrease the maximal error both in the case of a 4-segment cubic Hermite interpolant and in the case of a 4-segment cubic spline. Using this cubic parametrization a least square approximation of 15 equidistant inner points and fixed end points by a cubic spline with knots at 0, 0, 0, 7,a z,2 7,3 1, 1, 1 yields the control points a i, ci, di of x(u), y(u), z(u), respectively, shown in Table 4.

Fig. 6 shows the corresponding polygons and functions y(u), z(u) for m = 8, p = 4 and qr = 30 together with their relative errors. The error functions show the typical oscillation: Their least square approximation as above would yield the zero spline. The numerical experiments have shown also that knots 0, 0, 0, p*/2, p*, (p* + 1)/2, 1, 1, 1 would decrease the mean line error effectively, while the thickness error will not be strongly influenced.

From the control points ' . ' the BO.ier points 'o ' can easily be derived by simple subdividing the control polygons [Boehm, Gose, Kahmann '85], as sketched in Fig. 6. Note that the used parametrization x(u) is a cubic spline with knots - 1 , - 1 , - 1 , 0, 1, 1, 1, thus it will be reproduced exactly.

Table 4 Spline control values ai, ci, d i of parametrization x(u) , camber y ( u ) and thickness z(u), respectively.

i = left end 0 1 2 3 4 fight end

a i = 0.0 0.00000 0.06250 0.29688 0.62500 0.87500 1.0 Ci = 0.0 0.00000 0.34243 1.10260 0.95066 0.42964 0.0 di = 0.0 0.14953 0.41169 0.55036 0.38788 0.15563 0.01050

W. Boehm / Bdzier presentation of airfoils 21

y

e~

m e a n l ine

0.08_ c2 c3 .

Co e

[ • U2 •

error lO0 fold

u

0.0005

Z,

thickness z(u) 0.3 ,4 ~ L

-

1/12 1/12 1/12 Ul U 2 U 3 • U -~ ~ ! °.°°°~

er ro r 500 fold ]

Fig. 6. Sp]L~¢ app rox ima t i ons o f mean Un¢ and tl~ck3ess w i t h rc ]adv¢ errors, m = 8, p = 4, qr = 30.

X

i a4

a3

.a 2

/ I ~ / i1 ~

U ~

~/4 2/4 3/4 4/4 ¢2-d2

Fig. 7. Superposidon of the control polygons of a spline approximation of a NACA 6430 wing section.

22 W. Boehm / B$zier presentation of airfoils

Superposition and leading edge radius

In practice it is often sufficient to replace the normal n ( u ) of the mean line by the unit vector in y-direction. In this special case y- and z-ordinates are only to be added. Furthermore, if, as above, mean line and thickness correspond to the same knots one has simply to add the control point ordinates. Fig. 7 shows the corresponding superposition of the control polygon of a N A C A 6430 wing section spline approximation, it corresponds exactly to Fig. 2. Note that only a double knot at u = 0 is necessary to combine both parts of the airfoil.

Because x ( u ) is curvature continuous and y ( u ) is tangent cont inuous at u = 0 the paramet- ric airfoil above is curvature continuous at the leading edge. One gets immediately [see Boehm, Gose, Kahmann '85]

.2 p = 1.07324 q

Note that in the case of the superposition the airfoil tangent at the leading edge is vertical. The B6zier points ' o' can be constructed either f rom the control points ' . ' by subdividing

the control polygon as above or by superposition.

References

Abbott, I. and von Doenlaoff, A.E. (1959), Theory of Wing Sections, Dover, New York. Boehm, W. (1981) Some experiments with airfoils, Reehenzentrum der TU Braunschweig. Boehrn, W., Gose, G. and Kahmann, J. (1985), Einfiihrung in die Methoden der Numerischen Mathematik, Vieweg,

Braunschweig/Wiesbaden. Welzel, B. (1986), Verfahren zur Approximation von NACA Profilen durch B6zierkurven mit Hilfe vorgegebener

Formparameter, Diplomarbeit in Schiffsteclmik, TU Berlin.