ChapterS

Systematic Airfoil Design Studies at Low Reynolds Numbers

MichaelS. Selig,* Ashok Gopalarathnam,t Philippe Giguere,f

and Christopher A. Lyon§ University of Illinois at Urbana-Champaign, Urbana, Illinois

Nomenclature cl = lift coefficient, chord = 1 H 12 = boundary-layer shape factor 8./82 a = airfoil angle of attack relative to chord line a* = segment design angle of attack relative to zero-lift line a~ = airfoil angle of attack relative to zero-lift line a,1 airfoil zero-lift angle of attack 81 = boundary-layer displacement thickness ~ = boundary-layer momentum thickness ¢ = segment arc limit used in conformal mapping

Subscripts

x, = chordwise laminar bubble reattachment location Xs = chordwi e laminar-separation location Xu = chordwise transition location

I. Introduction

F OR over 100 years, airfoil design has continued to capture the interest of practitioners of applied aerodynamics. The field is fueled by the ever-growing

Copyright © 200 I by Michael S. Selig. Ashok Gopalaralhnam, Philippe Giguere. and Christopher A. Lyon. Published by the American Institute of Aeronautics and Astronautics, lnc .. with permission.

• Associate Professor, Department of Aeronautical and Astronautical Engineering. Senior Member AIAA.

tGraduate Research AssistanL Department of Aeronautical and Astronautical Engineering: currently Assistant Professof'. Department of Mechanical and Aerospace Engineering, North Carolina State University. Member AlAA.

t Graduate Research Assistant, Department of Aeronautical and Astronautical Engineering; currently Senior Aerodynamicist, Enron Wind Corp. Member AlAA.

' Graduate Research Ass istant, Department of Aeronautical and Astronautical Engineering; currently Software and Aerospace Engineer, Frasca International. Member AlAA.

144 M.S. SELIG ET AL.

co~bi~ation ~f airfoil de ign requirements for unique applications. and this state of affru;s •s ~ot _hkel~ to change. When one considers all possible permutations of the myn~d atrfotl de •gn ~equirements, it quickly becomes apparent that the number o~ umque sets of requirements far exceeds the collection of existing airfoils. For th1s reason, _the adv~cement and use of methods for airfoil design continues to be the econoirucal solutwn. In contrast, the enrichment of airfoil "catalogs" for their own sake is felt to be of limited value.

The objective of this chapter derives from two topics. First, the alternative to our great l~gacy of airfoil design by geometric means guided by empirical study is to use an m~erse ~e~od, and there are certain advantages to be had by adopting the latter _while ~ahzmg that often geometric constraints must still be achieved. By adoptmg an mverse approach, the degree to which the aerodynamic performance can be controlled has reached a high level of sophistication, and this can be illus­trated clearly by examples. Second, inverse design in the classic sense involves s~ifyin~ a de ired velocity distribution based on boundary-layer considerations. Taking ~s. on: step further by directly prescribing the desired boundary-layer charactensttcs IS a st:p clos~r to controlling the desired outcome-the perfor­m~ce. Thus, employmg an rnverse boundary-layer-like approach can give the destgner tremendous power in achieving the performance goals in the face of all the tradeoffs that one mu t consider in the process of airfoil design. These two aspects of a_ mod~m inverse airfoil design methodology form the subject of this chapter: destgn v1a boundary-layer considerations in an inverse sense.

To illustrate this approach, three series of low Reynolds number airfoils are presented. In each case, state-of-the-art tools for airfoil design 1.2 and analysisl-5 were use?. Although these airfoils were each designed for specific applications, the systemattc and parametric studies show useful performance trends and tradeoffs in airfoil design at l_ow Reynolds n~mbers. As will be shown, the overall design process has been valtdated through wmd tunnel tests, and the e results are presented together with the predictions.

II. Design Process

As an overview, the de ign process proceeded as follows. PROFQIT..I.2 was first used for ra~id ~d interactive design. A new airfoil that appeared to meet the performance ObJectives was then screened through experience and analysis using the Eppler code3

·4 and/or XFOIT..5 to obtain the lift, draa, and pitchina moment

ch~cteris~cs ?f ~e airfoi l over a range of angles of a~ack. If at an; state the candidate at~OJI fatle~ to_ meet the design goals, that additional experience was ~ed _to redesign the airfoil to more closely match the desired performance. This Iterattv~ process c?ntinued _until a successful airfoil was designed, at which point the destgn was bmlt and wmd-tunnel tested to evaluate its performance. A more detailed description of each of these elements of the process follows.

A. PROFOIL

The PROFOIT.. codeu embodies an inverse airfoil design method and an inte­gral boundary:layer !llethod for rapid analysis at the design points. The method draws on the p10neenng :vork of~pple~.4·6·7 in inverse airfoil design and analysis through conformal mappmg and mtegral boundary-layer techniques, re pectively.

AIRFOIL DESIGN STUDIES 145

PROFOIL differs from the Eppler code in that laminar and turbulent boundary­layer developments can be directly prescribed through iteration on the velocity distribution. The method also allows for control over certain geometric constraints, such as the local geometry, maximum thickness, and thickness distribution. Addi­tional differences are discussed in Refs. 1, 2, and 8. Both the boundary layer and thickness-constraint capabilities are used in the examples here. More details of the method will be given with the design examples to follow. A Web-based version of PROFOIL and further discussion is available online at httpJ/www.uiuc.edu/ph/ www/m-selig (cited September 200 I).

B. Eppler Code

As mentioned, the Eppler code3 was used for first-stage screening of candidate airfoils. Thus. only the analysis mode is discussed. In the analysis mode, the inviscid velocity distributions are determined by an accurate third-order panel method. Performance is then determined through the use of an imegral boundary­layer method using the inviscid velocity distribution, whkh makes the analysis exceptionally fast (0.03-s elapse time on a 600-MHz PC per polar). In the version of the code used for this work, the drag caused by a laminar separation bubble is not calculated. Although the magnitude of the bubble drag is not determined, the method is invaluable when the user has had experience comparing its predictions with experiments. For example, the code bas been used to design several successful low Reynolds number airfoils that can be found in the literature.'>-15

C. XFOIL

XFOIL5 is a design and analysis method for subcritical airfoils. In the data pre ented here, XFOlL has been used as a postdesign viscous/inviscid analysis tool. A linear-vorticity second~order-accurate panel method is used for inviscid analysis in XFOIL. This panel method is coupled with an integral boundary­layer method and an approximate e" -type transition amplification formulation using a global Newton method to compute the inviscid/viscous coupling, requiring approximately 15 s of elapse time per polar on a 600-MHz PC. XFOlL has proven to be well suited for the analysis of subcritical airfoils even in the presence of significant laminar-separation bubbles.

The XFOIL analyses in this work were used primarily to study tradeoffs and effects of systematic variations in airfoils and for selecting the most suitable can­didates before wind tunnel testing. For this purpose, the value Tlcrit = 9 was found to be quite suitable based on prior experience with comparisons between XFOIL results and the University of Jllinois open-return subsonic wind tunnel data. This value llcrit = 9 was used for transition prediction in the data presented here.

D. Wind-Thnnel Tests

This section describes the wind-tunnel experiments. Because details of the method can be found in Refs. 16, 17, and 18, only a summary is given here. The experiments were performed in the University of Ulinois open-return subsonic wind tunnel. The rectangular test-section dimensions are approximately 2.8 x 4ft in cross section and 8-ft-long. To ensure good flow quaJity in the test section, the

146 M. S. SELIG ET AL

tunnel settling chamber contains a 4-in.-thick honeycomb and four anti turbulence screens, resulting in a turbulence level of less than 0.1% over the Reynolds num­ber range tested. 16 The SA 703x series models were made from hot-wire cut foam cores covered with fiberglass and resin under a mylar vacuum bag. The SG604x and S607x series models were from polyurethane RenShape®. milled using a numerically-controlled machine, then sanded and painted.

To isolate the ends of the airfoil model from the tunnel side-wall boundary layers and the outer support hardware, the airfoil models were mounted horizontally between two 3/8-in.-thick. 6-ft-long Plexiglas® splitter plates. Gaps between the model and splitter plates were nominally 0.05 in. All models had a 12 in. chord and 33 5/ 8-in. span. One side of the model was free to pivot. At this location, the angle of attack was measured using a potentiometer. The other side of the model was free to move vertically on a precision ground shaft, but it was not free to rotate. A loadcell restrained the motion of the model and measured the lift force. Linear and spherical ball bearings within the lift carriage helped to minimize any frictional effects.

The drag was obtained from the momentum method To ensure that the wake had relaxed to tunnel static pressure, the wake measurements were performed 14.8 in. (approximately 1.25 chord lengths) downstream of the model trailing edge. Each vertical wake traverse consisted of between 20 and 80 total-head pressure measure­ments (depending on the wake thickness) with points nominally spaced 0.08 in. apart. Owing to spanwise wake nonuniformities, 19

•20 wake profile measurements were taken at four spanwise locations spaced 4 in. apart over the center 12 in. of the model span. The resulting four drag coefficients were then averaged to obtain the drag at a given angle of attack.

The airfoil pitching moment was measured via a loadcell connected between two lever arms, one metric with the model and the other fixed to the lift carriage angle-of-attack adjustment plate.

The lift, drag, moment, and angle-of-attack measurements were corrected to ac­count for the effects of solid blockage, wake blockage, and streamline curvature.21

The velocity was not only corrected for solid and wake blockage but also for a "circulation effect" that is unique to setups that make use of splitter plates. For the current tests, the freestrearn velocity rather than being measured far upstream was measured between the splitter plates for higher accuracy. Because the pitot-static probe that was used to measure the freestream velocity was located fairly close to the model, the probe measurements were therefore corrected for airfoil circulation effects so as to obtain the true freestream test section speed. The details of this correction procedure can be found in Ref. 22.

Overall uncertainty in the lift coefficient is estimated to be 1.5%. The drag measurement error comes from three sources: accuracy of the data acquisition instruments, repeatability of the measurements, and the locations of the particular four wake profiles used to determine the average drag coefficient. Based partly on the error analysis method presented in Refs. 23 and 24, the uncertainty due to the instruments and measurement repeatability are less than I% and 1.5%, respectively. Based on a statistical analysis (for a 95% confidence interval) of the spanwise drag results for the E374 airfoi1 19 at a = 4 deg, the uncert.ainties due to the spanwise variations were estimated to be approximately I .5% at and above Re = 200,000. The current airfoils are expected to have approximately the same uncertainties. A more detailed discussion of this topic is presented in Ref. 20. For the angle-of-attack sensor, the uncertainty is estimated to be 0.08 deg.

AIRFOIL DESIGN STUDIES 147

To determine the accuracy of airfoil profiles, each model was digitized with a Brown and Sharpe coordinate measuring machine. Approximately 80 points were taken around each airfoil, and the spacing between points was approximately proportional to the local curvature. Thus, near the leading and trailing edge , the spacing was relatively smaJI, whereas over the rnidchord it was no ~realer t~an 0.7 in. These measured coordinates were compared with the truecoordmates usmg a two-dimensional least-squares approach (rotation and vertical translation), which yielded an average difference of approximately 0.010 in. or less for all airfoils discussed in this chapter.

Data taken on the E387 model for Re = 200,000 and 460,000 are presented in Ref. 16 and compares well with data taken in the NASA Langley low-turbulence pressure tunnel (LTPT).23 Moreover, surface oil-flow visualization taken to_ de­termine the laminar separation and oil-accumulation lines showed that the hnes agreed with NASA Langley LTPT data to within 1-2% of chord.25 This good agreement serves to validate the current experiments.

m. Parametric Studies in Airfoil Design

In this section, three airfoil series are discussed with special emphasis given to the design process involving the aforementioned tools.

A. SA 703x Series This first series is for application to radio-controlled (RIC) model aircraft, specif­

ically RIC soaring. The interest in designing such a series was motivated by the results of a survey taken at the 1996 AMA Nationals/Unlimited Thermal Soaring Competition in which flight duration and landing precision were emphasized. The survey showed that of the 101 responses (from nearly all of the participants) 40 of the pilots used the SD7037 airfoil14 (shown in Fig. 1 together with its _veloc­ity distribution). Two other airfoils (S3021 and RG15) each numbered 8 muse, followed by the fourth (SD7080) with 6, and then 24 different airfoils were used on the remainder. Despite the overwhelming popularity of the SD7037, it was obvious that this airfoil could not be the optimum for a wide range of sailplanes with different sizes, wing loadings, and weather conditions. For example, in some situations pilots would benefit by having a faster version of the SD7037 Oower lift), and in others a slower version (higher lift) might be preferable. Thus, the SD7037 became the baseline for this first series.

For the series, the objective was to produce a range of similar performing airfoils that differed with respect to their lift range. One approach would be to make simple camber chanoes to the SD7037 to arrive at the new airfoils, but this process was not used becauS:it is more attractive to control the performance by using the inverse capabilities in PRO FOIL. In particular, PRO FOIL was used to set the lower comer of the polar by specifying that the laminar boundary layer on the lower surface be close to separating at the lift coefficient at the lower comer of the polar. The shape of the polar above this lift coefficient was then obtained by tailoring the aerodynamics of the upper surface, which is discussed later. .

Figure 2 shows the particular laminar boundary-layer development presc':'be_d for the lower comer of the polar. This boundary-layer shape parameter H12 dJstn­bution which is close to laminar separation. was specified to be achieved at the design' angles of attack of 1.75, 2.15, 2.55, and 2.95 deg relative to the zero-lift

148

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M. S. SELIG ET AL.

0.5

a(deg) C1

5.30 0.24 7.30 0.47 9.30 0.71

11.30 0.94 13.30 1.17

x/c

-Cso7037

Fig. 1 SD7037 airfoil and inviscid velocity distributions.

line to produce each respective airfoil of the SA 703x series shown in Fig. 3. (The SA7037 mimics the baseline SD7037.) In the figure, the velocity distributions are plotted at angle of attack increments equal to those that separate the aforemen­tioned lower surface design angles of attack. The corresponding boundary-layer shape parameter developments as predicted by XFOTI.. for the lower surface design condition are shown in Fig. 4, and they are practically identical to the prescription shown in Fig. 2.

It is worth noting that the laminar boundary-layer H12 developments are plotted for a Reynolds number of200,000; however, the laminar development to transition is independent of the Reynolds number, as discussed in Ref. 4. Also, although the

~ -·~~o-

17 18 5

Critical H12

for laminar separation 4~------~~-------------------

17 18 .

- H12prescription

0 ~--------------L-------------__J 0 0.5 x/c

Fig. 2 Lower-surface laminar boundary-layer development prescribed for tbe SA 703x airfoiJ series for tbe lower comer of tbe drag polar.

1.5

VN

1.0

0.5

AIRFOIL DESIGN STUDIES

-. ----~·--

SA7038, az = 6.2 deg SA7037, az = 5.8 deg SA7036, az = 5.4 deg

SA7035, az = 5.0 deg

0.5 x/c 0

_csA7o38 -------==-=--=-- -

-CsA7o37

-CsA703s

-CsA7035

Fig. 3 SA 703x airfoils and inviscid velocity distributions.

149

design specification was for there to be laminar flow on ~hi s segment at the. design condition, in application the Reynolds number could qmte well be off-dest~n ~d cause transition before the end of this segment when operating at the destgn hft coefficient of the lower urface. Neverthele s, the prescription serve to define the lower-surface velocity distribution and corresponding geometry.

When each airfoil i operated below the respective design angle of anac~ for the lower surface, laminar separation and subsequent transition in the lammar­separation bubble quickly move forward and lead to higher drag at the end of the

17 - H

12development (XFOIL)

_ H12

prescription

o L-------------~------------~ 0 0.5 x/c

Fig. 4 Laminar boundary-layer development achieved for the SA 703x airfoils and the agreement witb tbe prescription.

150

1. 5

SR7038 SR7037 SR7036 SR703S

I I I I -r -,- .... -,

I I f I -.- -,- -.-, I I I I I I f I

Re • 200lOO Re • 20CXMXl Re • 200000 Re • 20o:XJO

-:--:- ~- ~ -

M. S. SELIG ET AL.

"" • 0. 000 "". 0. 000 "" • 0.000 "" • 0.000

I I I I -,-,.-r-r-' I I I -,- r - i-r-' I I I -.-i-.-.--

- ! -~- ~ -~ -

Ncrll. • 9~ 1)(X)

Ncrlt. • 9~000

Nc:r-il • 9 . 1))0 Nc,.n. • 9 . 000

1. 5

a

#OIL w 6.8

f' I I I I • ,.,-,-r , -r rT

xv lc

Fig. 5 XFOIL predictions for the SA 703x airfoil series.

low-drag range. This resuJt is found in the predictions of XFOIL shown in Fig. 5 and validated by the wind tunnel test results shown in Fig. 6. (In place of the SA 7037, the baseline SD7037 was tested.) AdditionaJ data for these airfoils over the Reynolds number range from I 00,000 to 300,000 can be found in Ref. 18, and tabulated data are available online at http://www.uiuc.edu/ph/www/m-selig (cited September 2001).

Of criticaJ importance in the design of low Reynolds number airfoils is the upper-surface pressure distribution. The tendency of the flow to form a laminar­separation bubble can lead to a significant degradation in performance owing to

A SA7038 ~ SD7037 a SA7036

1.5 0 SA7035 Re = 200,000

I I

1.0

c, I

0.5 I

t-

0.0

-o.5 0.00

'

lA-"' , ' I I til

#P ~

IIi: " ' l'>.... t-1:' . )'.,

:s..

! 0.01 0.02

ca

1.5 I T f .l

·>--I I I

I I I I C:>t'" I

! 1.0 I

I c, ~ I

0.5

.

0.0

-o.s 0.03 - 10

c-+-~~ ~-

IJ\l 1i'

:_..j~ l ~'i"" I ~ I ~

fl

:JJ~- I 1-1--~-

I iil9 ~-

I

I

0 10 20 a (deg)

Fig. 6 Measured drag polars for the SA 703x airfoiJ series.

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AIRFOIL DESIGN STUDIES

Airfoil A, az = 6 deg

Airfoil B, az = 6 deg

0.5

-c==

151

Fig. 7 Inviscid velocity distributions for airfoils A and B to study the different effects on drag.

the high bubble drag. To mitigate these adverse effects, a transition ramp in the pre sure distribution is often employed to gradually bring the flow to tran ition in a thin bubble without a large pressure rise and high drag associated with an otherwise thick bubble. A general discussion of transition ramps can be found in Refs. 4 and 26, and additional details specific to low Reynolds number airfoils are discussed in Refs. 13, 14, 27, and 28.

The effect of the transition ramp is demonstrated using two example airfoils A and B. Figure 7 shows a comparison of the geometries and inviscid velocity distributions. These airfoils were designed using PROFOIL to each have a differ­ent transition ramp that is reflected in a different shape for the transition curve (C1-xufc curve) on the upper urface. The two airfoils were analyzed using XFOIL, and Fig. 8 shows the drag polars and upper-surface transition curves for a Reynolds number of 200,000. For the sake of discussion, the transition ramp is defined here as the region over which the bubble moves gradually as defined by the transition curve. (In this context. the tran ition ramp might be more aptJy caJled a "bubble ramp."14

)

From the figure, it can be seen that airfoil A has lower drag than airfoil Bat lift coefficients from around 0.3 to around 0. 7, above which value airfoil B has lower drag. Also noticeable is the correlation between the drag polar and the shape of the upper-surface transition curve. For the C1 range from 0.3 to 0 .7, where airfoil A has lower drag, the transition curve for airfoil A is shallower than that for airfoil B; that is, there is a larger change in the vaJue of x1,jc for airfoil A than for B. For values of C1 from 0.7 to 1.2 where airfoil B has lower drag, the transition curve for airfoiJ B is shaJiower than for A. This figure shows that the steepness of the transition curve is a direct indication of the bubble drag. By adjusting the shape of this curve, it is therefore possible to tailor the drag polar of an airfoil at low Reynolds numbers.

Figure 8 also includes an overlay of the variation of bubble size (x, - x,) with C1• The size of the bubble for each C1 was obtained by determining the chordwise extent over which the skin friction C 1 , as predicted by XFOIL. was less than

152

--Airfoil A - -Airfoil B

1.5

IT,-1-l

// // 1.0 '/

i f I 7

c,

l't ~ f-..:-.

0.5

f-.< 0.0

0.00 O.Dl

M. S. SELIG ET AL.

XFOIL v6.8 Re = 200,000 Airfoil A

/

I c,

I

H- i i

0.5

I

0.0 0.02 0 .03 0 0.5

cd x /c S.IJ.r

Airfoil B 1.5 ....---.---..,

c,

0.0 1.0 0 0.5 1.0

x /c s,tr,r

Fig. 8 XFOIL predictions for airfoils A and B to illustrate the effects of changes in the transition ramp on drag.

or equaJ to zero. Studying the bubble-size variation for the two airfoils further illustrates the connection between the shape of the transition curve and the bubble drag. The bubble is larger when the transition curve is steeper.

Figure 9 shows the inviscid velocity distributions for airfoil A at C1 vaJues of 0.5 and 1.0 with the upper-surface bubble location marked in bold. A similar plot for airfoil B is shown in Fig. 10. Comparing the velocity drops across the bubble for the four cases, it can be seen that aJthough airfoil A bas a smaller velocity

2.0

1.5

0 .5

Airfoil A

c,= o.5, 1.0

O+-r--.--.---.-.--,--.--,--.---, 0 0.5 xlc

Fig. 9 lnviscid velocity distributions for airfoil A witb the locations of the bubble marked.

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AIRFOIL DESIGN STUDIES

Airfoil B

C1= 0.5 , 1.0

0.5 xlc

153

Fig. 10 Inviscid ,·elocity distributions for airfoil B with the locations of the bubble marked.

drop than airfoil B at C1 = 0.5, the situation is reversed for c, = 1.0. Because the pressure drag caused by the bubble increases with increasing velocity drop across the bubble, airfoil A has smaller bubble drag at the low C1 and larger bubble drag at the higher C1• Thus, a steeper transition curve results in a larger bubble and also larger velocity drop across the bubble causing an increase in bubble drag.

Understanding the connection between the transition ramp in the velocity dis­tribution, the c,-x, fc transition curve, and the performance is the first step to obtaining optimum low Reynolds number airfoi l design. The next step in design involves the implementation of these ideas. In studying the velocity distributions shown in Figs. 3 and 7 and the resulting behavior of the transition curves shown in Figs. 5 and 8, respectively. we see that the connection between the two is not straightforward owing to the subtle (yet important) differences. Moreover, it is worth adding that the differences in the airfoil shape are even less useful in guid­ing the de ign toward an optimum with re pect to the transition ramp and its impact on drag.

In coping with this problem. a useful approach derive from an inherent feature of the Eppler theory for inverse airfoil design. Briefly, in the Eppler method, the designer can specify for a segment of the airfoil a design angle attack a• (relative to the zero-lift line) over which the velocity is constant. For instance, the forward upper surface can be defined as one segment and given a design angle of attack of 10 deg (C1 = 2rra, "' 1). When the resulting airfoil is then operated at 10 deg with respect to the zero-lift angle of attack a:~, the velocity over that segment is then constant. For a higher angle of attack. the resulting pres ure gradient is adverse and vice versa. Further discussion can be found in Refs. 4 and 29.

Because I) the boundary layer responds to the pressure gradient, 2) the design angle of attack a* for a segment has a direct effect on the pressure gradient, and 3) many such egments can be used to define an airfoil, these three aspects can be

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1.?5 13

• Beginning of segment o End of segment

1.59 3.34 5.07 6.63 7.41 8.21

0 0.5 x/c

0.25

109 8 7 6 5 4 3 21

-~o-17 18 19

Fig. 11 SA 7035 inviscid velocity distributions showing the zero pressure-gradient segments and corresponding design angles or attack.

conn~ted to yield_ an elegant solution to having precise control over the C1-xufc transJtmn curve. F1gure II shows the velocity distributions for the SA7035 airfoil at several des ign angles of attack. As seen, for segments 8, 9, 10, 11, 12, and 13 the ~esign angles of attack a• are I .59, 3.34, 5.07, 6.63, 7.4 1, and 8.21 deg, respectively. When the airfoil operates at these values of a~, the velocity gradient over the respective segments is zero, as hown in the exploded portion of the figu~. ~en the~ design angles of attack a• are plotted vs the segment-endpoint arc limits if> us~ ~ the co~o~ mapping to generate the airfoil, the resulting curves _shown m F1g. 12 ITilllllc the corre ponding transition curves (duplicated from F1g. ~) over the re_ ~clive surface of the airfoil, thereby providing a means of controlling the transitiOn ramp and resulting drag as was done also with the example of Fig~. 7 and 8. For the SA 703x series, the a* -if> curve shown in Fig. 12 maps from leadmg edge back to segment-endpoint I or approximately 75% of the chord (s~e Fig. 11 ). It_ is over this region that the ramp is controiJed by the a* -4> curve, wtth th~ re~amder being controlled by the final trailing-edge pressure recovei?' seen m F1g. 11. The essence of this technique has been employed in the des1gn of mo t of the S*xxxx four-digit airfoils reported in the literature

AIRFOIL DESIGN STUDIES 155

10 1.5 SA7038 a: (deg) SA7037 c, SA7036

SA7035 1.0

0

0.5 -5

- 10 L----L.-----'-----' 0.0 '----'--~ 180 120 60 0 0 .0 0.5 1.0

41 (deg) xJc

Fig. 12 Linkage between the a •- if> and C1-x" cun•es for the SA 703x airfoil series.

and archived online at http://www.uiuc.edu/phlwww/m-selig (cited September 2001) and in Refs. 14, 16, 17, 18, and 30.

B. SG604x Series

The SG604x series of airfoils31 as shown in Fig. 13 was designed for small variable-speed horizontal-axis wind turbines having a rated power of l- 5 kW. The operational Reynolds number for uch machine is typically below I 06

• The focus here will be on the performance at a Reynolds number of 300,000, which represents rotors on the lower end of the range. In ideal conditions, variable- peed wind turbines operate at a constant tip-speed-ratio (Q RJ V00), which leads to the airfoil operating at a single angle of attack over a wide range of wind speeds. As a result, for optimum aerodynamic performance during variable-speed operation, the low-drag lift range (drag bucket) can be reduced in favor of having greater lift­to-drag ratios. However, to account for possible variations in the tip-speed-ratio caused by atmospheric turbulence and operational considerations, the best lift-to­drag ratio conditions should occur over a range of lift coefficients centered about the design lift coefficient. In rotor design, another factor deals with the tradeoff between the blade solidity and the design lift coefficient. With all else being equal, a high-solidity rotor requires an airfoil with lower lift than that required for a low-solidity rotor. Therefore, given the range of rotor designs, a family of airfoils covering a range of lift coefficients is desirable. These general considerations and others were taken into account in setting the design requirements, as detailed more thoroughly in Ref. 31. The current discussion is mainly concerned with the details of the design approach.

Again PRO FOIL was used to prescribe the de ired aerodynamic characteristics. In a manner similar to that used in the previously di cussed SA 703x series, the boundary-layer shape parameter was prescribed on the lower surface to control the lower comer of the low-drag range of the polar. In this respect, there were some minor differences in the prescriptions, but generally a similar behavior was obtained. The transition ramp on the upper surface differs from that of the SA 703x series in much the same way that airfoil B differs from airfoi l A in Fig. 7. Since the objective was to achieve a high lift-to-drag ratio with le emphasis being given to operation over a wide range, the ramp on the upper urface was made more shallow

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SG6043, o.z = 9.0 deg SG6042, o.z = 6.5 deg

SG6041 , o.z = 4.0 deg

0 0.5 xlc

-~ SG6043

-c= SG6042

-C SG6041

---

Fig. 13 SG604x airfoils and inviscid velocity distributions.

at the specific design lift coefficients of 0.6, 0.9, and 1.2 for the SG6041 , SG6042, and SG6043, respectively. As before, the transition curve was controlled through the shape of the a*-l/J curve shown in Fig. 14. Additional constraints included the pitching moment, which increased with tbe design lift coefficient, and also the airfoil thickness of 10%. It should be added that for ease of construction a finite trailing-edge angle was used, producing a zero trailing-edge velocity as seen in Fig. 13.

Performance predictions at a Reynolds number of 300,000 are shown in Fig. 15 and compare relatively well with the experimental results of Fig. 16. Most im­portantly, the trends in the predictions agree weiJ with experiment, and also the behavior of the a• -l/J curve is reflected in the transition curve.

Figure 17 shows the resul ting experimentally determined lift-to-drag ratios at a Reynolds number of 300,000 compared with those for many previously existing airfoils. The SG6040 shown in the figure is the thicker companion root airfoil for the eries of tip airfoils discussed here. Clearly, the objective of achieving high lift-to-drag ratios has been achieved. Given the high level of performance, these airfoils wiU likely find their way into applications beyond wind energy.

C. S607x Series

ln contrast to the first two series, the S607x series is composed of three series of three airfoils each. The effort finally Jed to an airfoil for its intended application

I

I

~

AIRFOIL DESIGN STUDIES 157

- SG6043 - · SG6042 ·- SG6041

a.· (deg)

-15L-----~-----L----_J

180 120 60 0 ¢ (deg)

Fig. 14 SG604x airfoil series o • -4> cun·es for control over the C,-x, curves.

SG604 3 Re • 300000 SG604 2 Re • 300000 SG6041 Ro • 300000

"" • 0.000 Ku~t • 9.000 "" • 0.000 Ku~t • 9 .000 "" • 0.000 Nul< • 9 .000

l.S

-·- ~--~-~-. l. 0 1--~~...;_"'7""'--t----;

0.0 ~....;,._.,.....;;-'i::...;_~....;.....-..~. ~ • ........,,

I I I I -.- -,- -.-·~ -1 I 'f I -.- -.- -,- -~ -

I • I t .. ' . -~ -:--:- ~-

-., -r-r·r­' I I I ... ,- r- ;·r-

-7 - ~-~-:--- { · ~-L-~ --~ -:--:- ~ -

-o. 5 L...o......._.._._J oo.L..__._ ........ ~z.l.oo......_.._.._.__.300

10~ • Co

-6 -4l 2 I

l q 6 8 10

a

-0.5

.. ... -.... ... '.' .... ""' ' /! ,-.-r ,-r

I I I I

T .-.-r .-.-I I I I .. -.-. '' ~~-:-~ ~-:-

0.0 0.5

xu.lc

Fig. 15 XFOIL predjctions for the SG604x airfoil series.

:t t

1.0

158 M. S. SELIG ET AL.

A SG6043 ~ SG6042 B SG6041 Re = 300,000

1.5

I 1.0 I

c,

0 .5 -+-j

0.0

--{).5 0.00

I

I

I .A _.-Cr I

1.5

~ (! )..(:: I I->

~- ..--1" ,......;:r 1l:1' .!.-" I

I 1.0

. 'cf I

A I ; c,

'lo.

"' 0.5

"' I

-n+= !'-.... ~ I

" ---~ 0.0

I

"' ! -:'-f-

0.01 0.02 --{),5

0 .03 -10

cd

I

H d.

J{

--+- f if 'i" ir .. 'ild

If'-! .,

..CI g

' I

!F f!1 &>

-~ ";A 41 .. .} I

41. <-l. ~dr:;! I

A. ~-.){(;!

ii'l I p

r.{ I I I

I i-I I

! I I

.1 J I ,_____.___

0 10 20 a (deg)

Fig. 16 Measured drag polars for the SG604x airfoil series.

Re = 300,000 SGmc3lbt:lll"h

• SG604x Ad)il Family _,,•

100 ° Previoosty existing airfoils ___ ,,,,-" FX 63-137 (lic=13~)'o

SG&lC! (lie-lOS)' /', .,.., .....

,/

---- E387\ ,-' r-J 0

/ S01032C\11>1Dl') o O""GO•n7a~

, ' o-AtB (\t-7~)

,,-' ~C181fo:·Y lbt:tt.?S)

~\ / ,\'506060 ,507037 (llc=9.2S)

SG604t/ (lA> 10..4"1 .,.,5822 (lA> 16'1') (\lc:IDl'l/o o, BW-3 (111>5")

AG15 ~) 0,.NACA2414 (I/<>8..K) {\'c-14"1

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.5 0.6 0.7 0.8 0.9 1.0 1.1 12 1.3 1.4 1.5

Uft coefficient lor maximum lifl·lo-drag ratio

Fig. 17 Maximum lift-to-drag ratio vs the corresponding lift coefficient for the SG604x airfoil series compa.red with several previously existing a irfoils.

AIRFOIL DESIGN STUDIES 159

that required a low pitching moment and optimum performance at a Reynolds number of 150,000 and lift coefficient near 1. Only data for a Reynolds number of 200,000 are presented here, however. The series evolved from a 9%-thick family to a 12%-thick one, and then permutations in the pitching moment were made while other improvements were incorporated. The series will be discussed in this order.

As in the prior examples, the lower surface of each airfoil was designed by pre­scribing the boundary-layer shape parameter development at its respective design angle of attack. For the first series S6071/2/3, the study centered on the upper­surface transition ramp. Figure 18 shows that small changes in the a • -¢ curve affect the transition ramp, making it more shallow in going from airfoil S6071 to S6073. These differences result in small changes to the resulting velocity distribu­tions shown in Fig. 19. As seen, the sbaHower a *-¢ curve for airfoil S6073 results in a shallower pressure gradient over the forward upper surface. The differences in the airfoil shapes are so minute that a magnification of the y axis is needed to highlight the differences, as shown in Fig. 20. These small changes to the ramp, however, do have a significant effect on the bubble, as may be deduced by the drag predictions shown in Fig. 21. Wind tunnel tests depicted in Figs. 22 and 23 confirm the predicted trends.

10

a· (deg) - 86071 - · S6072

5 - - S6073

0

-5

- 10

-15

-20

120 60 0 ¢ (deg)

Fig. 18 S6071/U3 airfoil series a:•-4> curves for control over the C1 -xu curves.

160 M. S. SELIG ET AL.

2.0

VN

1.5

1.0

0 .5

0

- C S6071

- C S6072

- C s6o73

S6071, c1 = o.6

S6072, C1

= 0.6 S6073, C

1= 0.6

0.5 x/c

------------ ----

----- -

Fig. 19 S6071/2/3 airfoiJs and inviscid \'elocity distr.ibutions.

The next perturbation included largely a change in the airfoil thickness from 9% to 12%. Figure 24 shows the two bounding airfoils S6074/6 and their corresponding velocity distributions. Again the differences appear minor; however, as seen in Fig. 25 the trends are similar to the prior series, aJbeit with there being higher drag owing to the higher airfoil thickness. XFOll.. predictions indicated similar increase in drag.

The wind runnel tests of the S6074/5/6 airfoiJs reveaJed an undesirabJe feature of the series- stall hysteresis as shown Fig. 26, which was aJso found on airfoiJ S6071 . Such hysteresis cannot be predicted by any method currently available. This type of hysteresis, however, has been found on other airfoils. As described in

- 56071 -- 56072 -- S6073

Fig. 20 S6071/1J3 airfoils witb the thickness magoi6ed to how the smaiJ differences.

AIRFOIL DESIGN STUDIES 161

56{)7J Ro • 2001100 llo • 0.000 llc•IL • 9. 000 56{)72 Re • 2001100 llo • 0.000 lk•IL • 9. 000 56073 Ro • 2001100 llo • 0. 000 lk• 11 • 9. 000

J.S llfOIL..!.,hl

Airroil - S6071 - 56072 -- S6013

0.5

-0.5 L----l-------300

-0. 5 tOO 200

. ·I 0.0 0.5 1.0

to•. Co

Fig. 21 XFOIL predictions for the S6071/1J3 airfoiJs.

Ref. 13, when the inviscid velocity distribution on the forward upper surface tends toward a concave shape, staJI hy teresis of the type found here can be reduced. Unfortunately, no computationaJ tool exists to quantify this effect, and the degree to which the pressure distribution (or velocity distribution) should tend in the concave direction has not been quantified. Nevertheless, the chief aim of the finaJ series was to eliminate this hysteresis.

Figure 27 shows the two bounding airfoils S6077/9 designed to avoid staiJ hysteresis while satisfying the other constraints. A secondary purpose of this finaJ

A S6071 0 56072

1.5 a S6073 Re= 200,000 I 1.5

I i ,_.j._

.A=-:

""" -f--+-t I ·I' d-:t

1.0 1.0

cl cl fi' I I I Df 0.5 0 .5

IP

14

0.0 I~ 0.0 ~ 11 ·~.

~ ~

--{).5 0 .00 0.01 0.02

-0.5 0.03 -10

cd

I i I I _l

J

I ~ I

l.r I ~ · r i r X

X_ I I X

'ir 1-

lT' I ~ J 1-,_L ~- f-4 i-f-.-

lA

!i'f I

_1i1 ' 0 10 20 n(deg)

Fig. 22 Measured drag polars for the S6071/1J3 airfoils.

162 M. S. SELIG ET AL.

56071 Re = 200,000 86073 Re = 200,000 A Increasing a A Increasing a

B Decreasing a s Decreasing a 1.5 l 1 I -0.3 1.5

~ f. t t+1++-+-+-...---.I I

1.0 r--.,--1r-----'l:fl-++-+-i -0.2 1.0 ~~ Gf

c/ ~ i::l

r;:]

I c/ I ~

-~- l I r;:]

I If- I lg> I I ~ ~~

ff I ...... . 1· - 0.1

' Ji' t-:--i-H-IJ::i~. ~'--+-tf--t-t-----f C I :f I I m

0.5 0.5

E I "" . :~ .,.,. ... . -· 0.0 ~:J!iA~~- . i . . if .. t;i; .. ij .. -it· .~t. ~=~ 0.0 0.0

'r~ S'" · . .

-+l li El

-0.5 p,;! I 0 .1 -0.5 ~ 1 - 10 0 10 20 -10 0 10

Ct (deg) a (deg)

Fig. 23 Measured lift cones for tbe S607113 airfoils.

2.0

VN

1.5

1.0

0.5

0

- C 56074

56074, c1

= o.6 S6076, c

1 = o.6

- C 5so7s ~-

Fig. 24 S6074/6 airfoils and inviscid velocity distributions.

I

~ ~

iM

0.1 20

AIRFOIL DESIGN STUDIES

A 56074 B 56076 Re = 200,000

- 0 .5 0.00

I

I ' I I I ~

·-:,; I" ..!!I

w gJ

I ~ rs ~-

1~ ' :"1· J ~

-~

O.Q1 0.02

cd

1.5

I .A-~

1.0

c/

0.5

0.0

-0.5 0.03 -10

I ' I 1

r, IJ: ~

Al A"

§I

I j£p> ~

l!i ~

F1 --7

~

' I I

I

0 10 a (deg)

Fig. 25 Measured drag polars for tbe S6074/6 airfoils.

163

20

series was to examine the effects of a change in the pitching moment, which was specified using the inverse capabilities of PROFOIL. In comparing the velocity distributions shown in Fig. 24 with those in Fig. 27 it is seen that this current series has a more concave velocity distribution on the forward upper surface, and this difference is reflected in the shapes of the S6074/6 vs S6077/9 shown in Fig. 28. As seen in Fig. 29, this change in the velocity distribution is enough to eliminate the staiJ hysteresis. Finally, the performance is shown in Fig. 30. One feature of

1.5

1.0

c/

0.5

0.0

-0.5 -10

56074 Re= 200,000 86076 Re = 200,000 A Increasing a A Increasing a B Decreasing a

- 0.3 s Decreasing IX

1. 5 t-.--,-...,.,,--''---;:-r-r-J "-rl -0.3

I I I ' .tt -+ !# ~

; .r-, I I

f-

I

I

' I

I ~

~

I . li' -0.2

~- L_ bfhf·~· 1.0 f----r-f-''-+.----li'it-. if-T-1+--~-Tilt'-i -0.2

c/ I Wl -~·· ·

1£ $

}$ fl rn;.

11 I II

· ·· ·· ... · ··· ·· I

I ~

0 10 a (deg)

.. ~ -0.1 0 .5 1-+--++--.&"1-+~-~ - 0.1

0.0

f- GJ I II 1*-4.-,! e_-H-+ +--H 11-t-++ T-t 0 . 1 -0.5 ...__,,__,___,_......_.__,_J........L __ ......l.......J

20 - 10 0 10 0 .1

20 IX (deg)

Fi . 26 Measured lift curves for the 8607416 airfoils.

164 M. S. SELIG ET AL.

2.0

VN

1.5

1.0

0.5

soon. c,= o .6

S6079, c,= o.6

0.5 xlc -----~-

0

-~6077

-Csoo79 ===--=-=====--Fig. 27 S6(}77/9 airfoils and inviscid velocity distributions.

using a more concave distribution is that the a* -<P curve (not shown) becomes more shallow. As a result, the laminar-separation bubble drag is reduced at the upper comer of the low-drag range, yielding lower drag than the S6074/6 but also lower maximum lift as a tradeoff.

IV. Summary and Conclusions

In this chapter, three series of airfoils were designed to illustrate the power of modem computational tools for low Reynolds number airfoil design and analysis. Emphasis was placed on the design of the airfoils based on boundary -layer consid­erations. More specifically, the parameterization of the design problem centered around prescribing desirable boundary-layer features directly through an inverse

S6074 S6076 S6077 S6079

Fig. 28 S6074/6/7/9 airfoils with the thickness magnified to show the small differences.

A

B 1.5

1.0

I

0.5

0.0 ~

4>

F

AIRFOIL DESIGN STUDIES 165

S6077 Re= 200,000 Increasing a Decreasing a

,1.

·i'J.: .. 1r fl ~--Ji: ~

~ gJ

~ I I ~ i I I

I I iT ... ~ ... ~ ~ .. l.

.... ··~

~ ~-H .;it

-0.3

-0.2

-0.1

c m

0.0

S6079 Re = 200,000 A Increasing a 13 Decreasing a

1.5 I -0.3

I j I I

lr.tl

1.0 . iQ;M

-0.2 if.J

c, I t ~ l,. .. 7"' I

0.51--L---,.-+-~!4-H--;....+--~~-. -0.1 ll'l .;y.

II I Cm

~ ~-~ aJ. 0.0 ~ ······· · · · ···

II f7 ' ~ r<+t-t-

0.0

-0.5 e"l I 0.1 -0.5 '---11.$7a:. ·,___._....__ _ ___.. __ ___J 0.1 -10 0 10 20 -10 0 10 20

(l (deg) (l (deg)

Fig. 29 Measured lift curves for the S6077/9 airfoils .•

method. Formulating the design problem in this way offers the designer consider­ably more power than one would otherwise have using more traditional methods of inverse design (based on a single-point velocity distribution) and design by geo­metric perturbation. The design approach and philosophy can be used successfully to assess design tradeoffs with a high degree of control. Finally, wind-tunnel testing of low Reynolds number airfoils is, however, stiJl needed to provide engineers with a necessary level of confidence required to malce important engineering decisions.

A S6077 e S6079 Re = 200,000

1.5 1.5

I ! i ..!t ! I L

~ i I

1.0 1.0

I ! I.~

A. 0 .5 0.5

i I

0.0 0 .0

A ~ I

----' ~ I .t:--:--0.5

0.00 0.01 0.02 -0.5

0.03. -10

cd

I I I I I I

I I ..r.L

!W' IS ~

IK IK !" I

.d ;r

I,/" I I

a J!.. I I

I ,,.,. I I /!{

~' 0 10

(X (deg)

Fi . 30 Measured dra Jars for the S6077/9 airfoils.

I l l

i

I I I

20

166 M. S. SELIG ET AL.

Acknowledgments

Funding for this work was provide through a variety of sources for which we are grateful. namely, private donations (see Refs. 16, 17, and 18 for individual1istings), the National Renewable Energy Laboratory. and Aero Vironment. Also, we thank all the model makers, Mark Allen, Allen Developments (SA 7035, SG604x); Tim Foster and Frank Carson (SA 7038): Jerry Robertson (SD7037); D' Anne Thompson (SA 7036); and Yvan Tmel, Tine! Technologies (S607x series), for their meticulous and skillful efforts in model construction. The authors also wish to thank Mark Orela for providing the XFOIL code used in this work.

References 1 Selig, M. S., and Maughmer, M. D., "'A Multi-Point Inverse Airfoil Design Method

Based on Conformal Mapping," AIM Journal, Vol. 30, o. 5, May 1992, pp. 1162-1170. 2Selig, M. S., and Maughmer, M. D., "Generalized Multipoint Inverse Airfoil Design."

AlAA Journal, Vol. 30, No. 11, ov. 1992, pp. 2618-2625. 3Eppler. R. and Somers, D. M., "A Computer Program for the De ign and Analysis of

Low-Speed Airfoils:• NASA TM 80210, Aug. 1980. ~Eppler, R., Airfoil Design and DaTa, Springer-Verlag, New York, 1990. 5Drela, M., "XFOll.: An Analysis and Design System for Low Reynolds Number Air­

foils." Low Reynolds Number Aerodynamics, edited by T. J. MueUer, Vol. 54 of Lecture Notes in Engineering, Springer-Verlag, ew York, June 1989, pp. 1-12.

6Eppler, R., "'Direkte Berechnungvon Tragflugelprofilen aus der Druck.veneilung;· lngenieur-Archive, Vol. 25, No. 1, 1957, pp. 32-57 (translated as "Direct Calculation of Airfoil from Pressure Distribution," NASA TT F-15, 417, 1974).

7Eppler. R., "Ergebnisse gemeinsarner Anwendung vo Grenzschicht- und Potentialthe­orie," Zeitschrift for Flugwissenschaften, Vol. 8, No. 9, 1960, pp. 247-260 (translated as "'Results of the Combined Application of Boundary Layer and Profile Theory," NASA TT F-15, 416, March 1974).

8Selig, M.S., " Multi-Point Inverse Design oflsolated Airfoils and Airfoils in Cascade in lncompres ible Flow," Ph.D. Thesis, Pennsylvania State Univ., State College, PA, May 1992.

9Thies, W., Eppler-Profile, MTB I, Verlag fur Technik. und Handwerk., Baden-Baden, Germany. 1981 (republished. Hepperle, M., MTB lfl, 1986).

10Thies, W., Eppler-Profile, MTB 2, Verlag fur Technik. und Handwerk, Baden-Baden, Germany, 1982 (republished. Hepperle, M .. MTB lfl, 1986).

11 Althaus, D .. Profilpolaren fiir den Modellflug- Windkanalmessung an Profilen im Kritischen Reynolds::tJhlbereich. Neckar-Verlag, Villingen-Schwenningen, Germany, 1980.

12Aithaus, D., Profilpolaren for den Modelljlug- Windkanalmessung a11 Profilen im Kritischen Reynolds::tJhlbereich. Band 2. Neckar-Verlag, Villingen-Schwenningen, Germany, 1985.

13Selig, M. S., "The Design of Airfoils at Low Reynolds umbers," AIAA Paper 85-0074, Jan. 1985.

14Selig, M. S., Donovan, J. F., and Fraser, D. B., Airfoils at Low Speeds, Soartech 8, SoarTech Publications, Virginia Beach, VA, 1989.

15Hepperle, M., Neue Profile for Nurjfiigelmodelle, FMT-Kolleg 8, Verlag fur Technik und Handwerk GmbH, Baden-Baden, Germany, 1988 (republished, Hepperle. M., MTB 1/2, 1986).

16Selig, M. S., Guglielmo. J. J .. Broeren. A. P., and Giguere, P., Summary of Low-Speed Airfoil Data, Vol. I. SoarTech Publications, Virginia Beach, VA, 1995.

AIRFOIL DESIGN STUDIES 167

17Selig, M.S .. Lyon, C. A., Giguere, P., Ninham, C. N., and Guglielmo, J. J .. Summary of Low-Speed Airfoil Data, Vol. 2, Soar Tech Publication , Virginia Beach, VA, 1996.

18Lyon, C. A .. Broeren, A. P., Giguere. P., Gopalarathnam, A .. and Selig, M.S., Summary of Low-Speed Airfoil Data, Vol. 3, SoarTech Publications. Vuginia Beach, VA, 1998.

19Guglielmo. J. J .. and Selig, M. S., ··spanwise Variations in Profile Drag for Airfoils at Low Reynolds umbers," Journal of Aircraft, Vol. 33, No. 4, July-Aug. 1996, pp. 699-707.

20Guglielmo, J. 1 .. "Spanwise Variations in Profile Drag for Airfoils at Low Reynolds Numbers," M.S. Thesis, Dept. of Aeronautical and Astronautical Engineering, Univ. of lllinois at Urbana-Champaign, Urbana, ll., 1995.

21 Rae, W. H., Jr. and Pope, A., Low-Speed Wind Tunnel Testing, Wiley, New York, 1984. 22Giguere, P., and Selig, M.S., "Freestream Velocity Measurements for Two-Dimensional

Testing with Splitter Plates," AJAA Journal, Vol. 35, No.7, July 1997, pp. 1195-1200. n McGhee, R. J., Walker, B.S., and Millard, B. F. , "Experimental Results for the Eppler

387 Airfoil at Low Reynolds Numbers in the Langley Low-Turbulence Pressure Tunnel." NASA TM 4062. Oct. 1988.

24Coleman, H. W., and Steele, \V. G., Jr .. Experimemation and Uncertainty Analysis for Engineers, Wiley, New York, 1989.

2SLyon, C. A., Selig, M. S., and Broeren, A. P., "Boundary Layer Trips on Airfoils at Low Reynolds Numbers," AIAA Paper97-0511 , Jan. 1997.

26\Vonmann, F. X .. "Progress in the Design of Low Drag Airfoils," Boundary Layer and Flow Control, edited by G. V. La.chmann, Pergamon, London, 1%1, pp. 748-770.

27Donovan. J. F., and Selig, M. S., "Low Reynolds Number Airfoil Design and Wind Thnnel Testing at Princeton University," Low Reynolds Number Aerodynamics. edited by T. J. Mueller, Vol. 54 of Lecture Notes in Engineering, Springer-Verlag, New York, June 1989, pp. 39-57.

28Drela, M .. "Low Reynolds-Number Airfoil Design for the M.I.T. Daedalus Prototype: A Case Study," Jormzal of Aircraft, Vol. 25, No.8, Aug. 1988, pp. 724-732.

29Gopalarathnam, A., and Selig, M. S., "Low-Speed Natural-Laminar-Flow Airfoils: Case Study in Inverse Airfoil Design," Joumal of Aircraft. Vol. 38, No. I, Jan.-Feb. 2001, pp. 57-Q3.

30Selig, M. S., The Design of Airfoils at Low Reynolds Numbers, Soar/eel! 3, SoarTech Publications, Vuginia Beach, VA, 1984.

31 Giguere. P. , and Selig, M.S., ""New Airfoils for Small Horizontal Axis Wmd Turbines ·· ASME Journal of Solar Energy Engineering, Vol. 120, May 1998, pp. 108-114. '