airfoil profile design by reverse engineering bezier curve

410

Int. J. Mech. Eng. & Rob. Res. 2012 Rama Krishna N Parasaram and T N Charyulu, 2012

AIRFOIL PROFILE DESIGN BY REVERSEENGINEERING BEZIER CURVE

Rama Krishna N Parasaram1* and T N Charyulu1

*Corresponding Author: Rama Krishna N Parasaram, [email protected]

Though it is easier to model and create an airfoil profile in CAD environment using camber cloudof points, after the creation of vane profile it is very difficult to change the shape of profile foranalysis or optimization purpose by using cloud of points. In this paper, we investigate anddescribed the creation of airfoil profile in CAD (CATIA) environment using the control point of thecamber profile. By means of changing the values of control points, the shape of the profile canbe easily changed and also the design of the cambered airfoil was established without affectingthe basic airfoil geometry. In this paper, the Quintic Reverse Engineering of Bezier curve formulawas used to find the camber control points from the existing camber cloud of points.

Keywords: Airfoil, Quintic reverse engineering bezier curve, Camber, NACA-4 digit series,CATIA

INTRODUCTIONEach airfoil comprises an inner airfoil surfaceand outer airfoil surface. The inner airfoil andouter airfoil surface define a vane airfoilthickness (Joel, 2003). Airfoil includes aleading edge positioned along a first inner andouter airfoil along a first airfoil surface junction,a trailing edge positioned along a secondinner and outer surface junction. The airfoilinner and outer airfoil surface are speciallyconfigured to provide a airfoil camber line,positioned between the two airfoil profile andextending along the length of the airfoil that is

ISSN 2278 – 0149 www.ijmerr.comVol. 1, No. 3, October 2012

© 2012 IJMERR. All Rights Reserved

Int. J. Mech. Eng. & Rob. Res. 2012

1 Nova College of Engineering and Technology, Jangareddygudam, Vegavaram, Andhra Pradesh, India.

substantially length of the camber line

(Richard, 1990).

The “NACA airfoils” are airfoil shapes for

aircraft wings developed by the “National

Advisory Committee for Aeronautics (NACA)”

(Jacobs et al., 1933). The shape of the NACA

airfoils is described using a series of digits

following the word “NACA.” The parameters

in the numerical code can be entered into

equations to precisely generate the cross-

section of the airfoil and calculate its

properties.

Research Paper

411


The NACA has different series of digit airfoilare (1) 4 digit series airfoil; (2) 5 digit seriesairfoil; (3) 7 digit series airfoil; and (4) 8 digitseries airfoil.

We are taking the general example creationof standard airfoil like the NACA four-digitairfoils sections are airfoil shapes for aircraft.The NACA four-digit wing sections define theprofile by one digit describing maximumcamber as percentage of the chord (Moran,2003) NACA 2412.

• One digit describing the distance ofmaximum camber from the airfoil leadingedge in tens of percents of chord.

• Two digits describing maximum thicknessof the airfoil as percent of the chord.

For example, the NACA 2412 airfoil(Abbott and Von, 1959) has a maximumcamber of 2% located 40% (0.4 chords)from the leading edge with a maximumthickness of 12% of the chord. Four-digitseries airfoils by default have maximumthickness at 30% of the chord (0.3 chords)from the leading edge.

DEFINITION OF NACA-0011AIRFOILThe NACA 0011 (Abbott and Von, 1959) airfoilis symmetrical, the 00 indicating that it has nocamber. The 15 indicates that the airfoil has a15% thickness to chord length ratio: it is 15%as thick as it is long. The thickness and camberformula (1 and 4) using the creation NACA-0011 airfoil.

The Thickness formula (1) for the shape ofa NACA 0011 foil, with “11” being replaced bythe percentage of thickness to chord is 0.11.

Using the NACA 0011 aerofoil profile extractthe cloud of points from NACA four-digit SeriesProfile Generator by giving the proper input(Richard, 1990). The Input Variables are:

Maximum Camber: 0

Maximum Camber Position: 0

Thickness: 11

No of Stream wise Points: 40

Points Distribution: 1

Points Size: 4

OUTPUT OF THE NACA 4-SERIES PROFILEGENERATOROutput of the NACA 4-series (Jens andRobert) Profile Generator-0011(Figure 2)and profile cloud points were calculated forChord Length value is 1. After getting theCloud of points Data, we multiply with ouroriginal Chord 40 mm and multiply with theparsed cloud points data. We got the realscaled Cloud points shown in Tables 1 and 2.In addition to that we need to divide the value1 into 40 division as shown in the Table 1 tdivision column.

Figure 1: Profile Geometry

Note: 1 – Zero lift line; 2 – Leading edge; 3 – Nose circle;4 – Camber; 5 – Max thickness; 6 – Upper (outer) surface;7 – Trailing edge; 8 – Camber mean-line; 9 – Lower (inner)surface; 10 – C-Chord length.

412


Table 1: Upper Camber Cloud of Pointsand ‘t’ Value

X Y Z t1/40

0 0 0 0

0.246 0.495 0 0.025

0.552 0.728 0 0.051

0.978 0.949 0 0.076

1.522 1.157 0 0.102

2.179 1.351 0 0.128

2.947 1.529 0 0.153

3.819 1.688 0 0.179

4.791 1.828 0 0.205

5.857 1.946 0 0.230

7.011 2.042 0 0.256

8.244 2.116 0 0.282

9.550 2.166 0 0.307

10.92 2.194 0 0.333

12.34 2.200 0 0.350

13.81 2.184 0 0.384

15.33 2.149 0 0.410

Table 1 (Cont.)

X Y Z t1/40

16.87 2.095 0 0.435

18.43 2.025 0 0.461

20 1.941 0 0.487

21.56 1.844 0 0.512

23.12 1.737 0 0.538

24.66 1.622 0 0.564

26.18 1.500 0 0.589

27.65 1.374 0 0.615

29.07 1.245 0 0.641

30.45 1.115 0 0.666

31.75 0.986 0 0.692

32.98 0.859 0 0.717

34.14 0.737 0 0.743

35.20 0.620 0 0.769

36.18 0.510 0 0.794

37.05 0.408 0 0.820

37.82 0.317 0 0.846

38.47 0.237 0 0.871

Figure 2: NACA 4 Digits Series Profile Generator

413


Total number of generating points in theoutput of the generator is only 80 points,Leading Edge created by using nose circle(2) and Trailing edge created by using angle.We need to find the Control points fromcalculated Cloud of points refer (Tables 1 and2). For Quintic Reverse Engineering of BezierCurve (Prautzsch et al., 2002) formula, weneed to pick only Six Cloud of point atEquidistance out of forty points of uppercamber profile, similarly on lower camberprofile please refer (Tables 1 and 2) picked

Table 1 (Cont.)

X Y Z t1/40

39.02 0.170 0 0.897

39.44 0.116 0 0.923

39.75 0.077 0 0.948

39.93 0.054 0 0.974

40.00 0 0 1

Table 2: Lower Camber Cloudof Point’s Value

X Y Z

0 0 0

0.246 –0.49 0

0.552 –0.72 0

0.978 –0.94 0

1.522 –1.15 0

2.179 –1.35 0

2.947 –1.52 0

3.819 –1.68 0

4.791 –1.82 0

5.857 –1.94 0

7.011 –2.04 0

8.244 –2.11 0

9.55 –2.16 0

10.92 –2.19 0

12.34 –2.2 0

13.81 –2.18 0

15.33 –2.14 0

16.87 –2.09 0

18.43 –2.02 0

20.00 –1.94 0

21.56 –1.84 0

23.12 –1.73 0

24.66 –1.62 0

26.18 –1.50 0

27.65 –1.37 0

29.07 –1.24 0

30.45 –1.11 0

Table 2 (Cont.)

X Y Z

31.75 –0.98 0

32.98 –0.85 0

34.14 –0.73 0

35.20 –0.62 0

36.18 –0.51 0

37.05 –0.40 0

37.82 –0.31 0

38.47 –0.23 0

39.02 –0.17 0

39.44 –0.11 0

39.75 –0.07 0

39.93 –0.05 0

40.00 0 0

c0 0 0 0 0

c1 = f 3.8197 1.6885 0 0.1795 = u

c2 = g 13.819 2.1844 0 0.3846 = v

c3 = h 26.180 1.5003 0 0.5897 = w

c4 = j 36.180 0.5102 0 0.7949 = r

c5 40 0.0462 0 1

Table 3: Picked Six Cloud of Pointsof Upper Camber and ‘t’ Division

Upper Camber Points

Des X Y Zt

414


points marked as BOLD letter and picked pointplotted in the Tables 3 and 4.

out the control points from original cambercloud of points. We equally divide the 40 pointsCamber points’ value into six Divisions refer(Tables 1 to 4). Using the Six Original camberpoint value, the Six Control point of the CamberCurve using formula (6) was find out.

To define six original picked points andhave a Bezier curve passing through all sixorigin points. Basically, given the upper profilesix original points c0, c1, c2, c3, c4 and c5,we will used find upper six control points ofp0, p1, p2, p3, p4 and p5 such that the Beziercurve calculated using points p(i), will passthrough the points c(i).

Figure 3 and Table 3 shows that when ‘t’ iszero, the equation effectively collapses into justp0. When ‘t’ is one, there is no change inequation gives p5. When t is between zero andone, the resulting point lies on the curve itself,so iterating t from zero to one will give theBezier curve. Since we know the curve willpass through p0 and p5, we need to find p1,p2, p3 and p4.

c6 0 0 0 0

c7 3.819 –1.6885 0 0.1795 = u

c8 13.819 –2.1844 0 0.3846 = v

c9 26.180 –1.5003 0 0.5897 = w

c10 36.180 –0.5102 0 0.7949 = r

c11 40 –0.0462 0 1

Table 4: Picked Six Cloud of Pointsof Lower Camber, ‘t’ Division

Lower Camber Points

Des X Y Zt

QUINTIC BEZIER CURVEFORMULAWe created all input data for our QuinticReverse Engineering Bezier Curve formula. InBezier, the Control Points are the driving pointsto create a curve (Farin, 1997). We needcontrol points of the camber curve to createan airfoil in CATIA. We develop the QuinticReverse Engineering of Bezier formula (6)from Quintic Bezier curve formula (5) to find

Figure 3: Example Quintic Bezier Curve Figure

415


Suppose we want the curve to pass throughp0 when t = 0, f when t = u, g when t = v, h whent = w, j when t = r and p6 when t =1, where f, g,h and j are the points to be passed through.Next, we make sure that 0 < u, r < 1 and u notequal to r. These conditions will ensure asolution can be found. Next, we substitute thedesired points into the equation.

AIRFOIL DESIGN IN CATIAAfter find out the control point of the bothCamber profile using Quintic ReverseEngineering Bezier Curve formula, Using theSix Upper and Six Lower Camber Control

Figure 4: Upper Camber Profile 3D CurveCreation

Figure 5: Lower Camber Profile 3D

Figure 6: Final 2D Airfoil Profile Design in CATIA

416


Points create a airfoil profile in CATIA. TheControl Points are plotted in CATIA (Richard,2006) using points coordinate option andafter using the 3D The Control Points areplotted in CATIA (Richard, 2006) using pointscoordinate option and after using the 3Dcurve option in free style shaper or QuickSurface Reconstruction Module in CATIAcreate a upper Camber and Lower Camber

3D Curve as shown in the Figures 4 and 5.Finally, created an airfoil profile design inCATIA using control points see Figure 6.

VALIDATE THE AIRFOILDESIGNAfter created the airfoil profile in CATIA tovalidate the airfoil design with NACA 4-digitsprofile Generator output data by means of read

Figure 7: Airfoil Created in CATIA Using Cloud of Points

the cloud of points output data (Tables 1 and2) using digitized shape editor module and plotthe cloud of points value in CATIA as shown inthe Figure 7.

Figure 7 is the base design created usingthe formula of the NACA profile Generator and

coincide with our airfoil design which is createdusing the control points (refer Figure 6) tovalidate the airfoil design. If it coincidesperfectly, the design is validate. The Figure 8shows that both the airfoil profile designscoincide perfectly.

Figure 8: Coincide the 2D Airfoil with Cloud of Points

417


RELATION BETWEENCONTROL POINTSBy assigning the Y axis length differenceformula relation (Sham and Deepak, 2006)between the upper and lower cambercontrol points to control the profile easily(refer Figures 9 and 10). By maintainingconstant Y axis length difference between

the upper control point p2 and the lowercamber control point p8 was shown in theFigures 9 and 10 and similarly maintainsthe constant Y axis length differencebetween p1 and p7, p3 and p9, p4 and p10.For example constant Y axis lengthdifference between the p2 and p8 is 7.308as shown in the Table 7.

Figure 9: Constant Y Length Difference Relation Between Upper p2 (Red)and Lower p8 (Red) Camber Control Point

Figure 10: Constant Y Length Difference Relation Between Upper p2 (Red)and Lower p8 (Red) Camber Control Point

418


By means of changing the control point p1,p2, p3 and p4 it automatically update thelower camber control point p7, p8, p9 andp10 using formula relation as shown inFigures 9 and 10.

Y 3.654 –3.654 –3.654 = 3.654 + X – 3.645 – 3.654= –7.308 = X

Constant Y value length difference p8 is –3.654 = 3.654 – 7.308

Table 7: Constant Y Length Difference Between the p2 and p8

Axis P2-Upper Camber ControlPoint (CP)

P8-Lower CamberControl Point (CP)

Constant Y-Axis Length DifferenceFormula P8 = p2 + Difference (X)

RESULTSBy changing the Y-value of the upper control pointp1, p2, p3 and p4 change the airfoil profile andwithout affecting the basic airfoil geometry. To

create a camber airfoil, by changing upper

Figure 11: Control Point p1 (Shown in Red Color) Y-Value Changes Whichis Located 10% of the Length of the Airfoil

Figure 12: Control Point p2 (Red) Y-Value Changes Which is Located 40%of the Length of the Airfoil

419


camber control point Y-value which is situated

10%, 40% 80% and 95% of the chord length. It

shows how the airfoil profile modified in CATIA

as shown in Figures 11 to 14.

Figure 13: Control Point p3 (Red) Y-Value Changes Which is Located 80%of the Length of the Airfoil

Figure 14: Control Point p2 and p3 (Red) Y-Value Changes Which is Located 40%and 80% of the Length of the Airfoil

CONCLUSIONIn this study, we have investigated anddesigned an airfoil profile in CAD environmentusing control points. We have employed aQuintic Reverse Engineering Bezier curveformula to find out the control points of thecamber profile which is used to create an upperand lower camber profile. By using the control

points, we easily modify the shape of theprofile so that to produce the cambered airfoilshape without affecting basic airfoil geometry.The objective of this work is to find a simpleand accurate way to design the airfoil profilein CATIA using six camber control pointposition. However the present method wasapplied only for six camber control point

420


position in the airfoil. The future work can be

attempted to increase the number of camber

point positions by means of increasing the

degree ‘n’ value of the Bezier curve formula

and to employ a high order Reverse

Engineering Bezier curve formula to find out

the more camber control point locations on the

airfoil design.

REFERENCES1. Abbott and Von Doenhoff (1959), “Theory

of Wing Sections-Basics for All NACA

Airfoil Surface”.

2. Farin Gerald (1997), Curves and

Surfaces for Computer-Aided Geometric

Design, 4th Edition, Elsevier Science and

Technology Books.

3. Jacobs E N, Ward K E and Pinkerton R

M (1933), “The Characteristics of 78

Related Airfoil Sections from Tests in the

Variable-Density Wind Tunnel”, NACA

Report No. 460.

4. Jens Trapp and Robert Zores, availableat www.ppart.de/aerodynamics/profiles/NACA4.html

5. Joel P Castan (2003), “Cambered AirfoilUse in Turbocharger”, US Patent, Vol. 7,No. 771, p. 162.

6. Moran Jack (2003), “An Introductionto Theoretical and ComputationalAerodynamics”.

7. Prautzsch H, Boehm W and Paluszny M(2002), Bezier and B-Spline Techniques.

8. Richard Cozzens (2006), “AdvancedCATIA V5 Workbook: Knowledge Wareand Work Benches Release”, Vol. 16,SDC Publications.

9. Richard Eppler (1990), Airfoil Design andData.

10. Sham Tickoo and Deepak Maini (2006),CATIA V5R16 for Engineers andDesigners.

11. Strang Gilbert (2003), Introduction toLinear Algebra, 3rd Edition, p. 71.

airfoil profile design by reverse engineering bezier curve

Documents

Transcript of airfoil profile design by reverse engineering bezier curve