Generalized Multi Point Inverse Airfoil Design

8/8/2019 Generalized Multi Point Inverse Airfoil Design

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AIAA JOURNALVol. 30. No. II . November 1992

C"'oF(z)IKKHKsP(q,)pQ(q,)ss;Siti cVv

Vi

V(q,)ii;(q;; )

W(q,)WF(q,)

WS(q,)Ww(q,)z

Generalized Multipoint Inverse Airfoil Design

Michael S. Selig and Mark D. MaughmertPennsylvania State University, University Park, Pennsylvania 16802

la a rather aeaeral sense, ia"erse airfoU desila ca a be takea to meaa the problem of speclfyinl a desired setof airfoU characteristics, sudl as tbe airfoU maxima . tblekaess ratio, pltchlal moment, part of tbe "e1ocltydistributioa, or bouadary-layer deve lop_t . From tills laformatloa, the correspoadinl airfoU shape is determiaed. This paper presents a metbod tbat approacha tbe desiga problem from this penpectlve. In particular,the airfoU is dl"ided Into seameats aloBl wbleb, toaetber witb tbe desila conditions, either tbe "e1ocitydistribution or boundary-layer developmeat may be presaibed. la additioa to tbese local desired distribut ions,sinlle parameten like the amoU tblekaess ca a be specified. Del_ iaa t ioa of tbe airfoU sbape is accompllsbedby coupUnl aa iacompressible poteatialDow l a n n e alrfoU desip metbod witb a direct lateanal bouadary-layeraaalysis method. Tb e resultia l system of aoallaear eqUtioDl Is soh-ed by a multidimensional N ewtoa itenatloatechaiqae. Aa eomple airfaU desila, DOt latended fo r practical appllcatioa, Is preseated to Hlustrate some ofthe capabUlties of the metbod. As tbls eomple Hlustnates, the desila methodoloty presented pro"ldes a meansof dealina simultaneously w i t ~tbe myriad requirements aad coastnaiats tbat can be speciDed in the desiaa of anairfoil.

Nomenclature= Fourier series coefficients= airfoil chord= pitching-moment coefficient for a given angle

of attack= pitching-moment coefficient at zero lift= complex potential function= total number of airfoil segments= main recovery parameter= closure recovery parameter= trailing-edge thickness parameter= harmonic function on circle= value of a desired generic parameter= conjugate harmonic function on circle= arc length about airfoil= arc length for segment i= relative arc length for segment i= maximum thickness ratio of airfoil= freestream velocity (V = 1)= airfoil velocity distribu tion nor malized by

freest ream velocity= velocity level for segment i= mUltipoint design velocity distribution= relative design velocity distribution as a function

of q;; for segment i= relative design velocity distribution as a function

of Si for segment i= total recovery function= trailing-edge recovery function

= closure recovery function= main recovery function= physical-plane complex coordinate (x + iy )

aaO L

a*(q,)E

r8*(q,)I '

r(q,)q,q,;q,;LEq,Fq,s

q,wq;;

= angle of attack from zero-lift angle= zero-lift angle of attack relative to airfoil

chordline= mUltipoint design angle-of-attack distribution= railing-edge included angle parameter= circle-plane complex coordinate ( ~ + i.,,)= local direction of flow about airfoil= main recovery parameter= step function= arc limit in circle plane= arc limit for segment i= leading-edge arc limit= trailing-edge recovery ar c limit= closure recovery arc limit

= main recovery arc limit= elative arc limit for segment iBoundary-Layer Variables

CD = dissipation coefficientcf = skin-friction coefficientH 11 H) l = shape factors. 0 / ~ an d 0)1 ~n = linear stability amplification factorR = airfoil chord Reynolds number. Veil 'R ~ = Reynolds number. V 021 "0., ~ . 0) = displacement. momentum. an d energy thicknessesII = kinematic viscosity

Subscript

= value for segment or at ar c limit q,;

Superscripts

= relative to beginning of segment= lower-surface quantity= Newton node index

Presented as Paper 913333 at the AIAA 9th Applied AerodynamicsConference. Baltimore. MD, Sept. 23-2S. 1991; received Oct. 17.1991; revision received April 30. 1992; accepted for publication May6. 1992. Copyright 1991 by the American Institute of Aeronauticsand Astronautics. Inc. Al l rights reserved.

Introduction

T RADITIONALL Y. inverse airfoil design is considered asthe problem of finding the airfoil shape corresponding toa specified velocity distribution. To more directly control theairfoil characteristics. the contemporary view of inverse airfoildesign follows principally along two lines of thought. First.emphasis is being placed on solving multipoint design problems either by inverse fo rmulationsl .l or numerical optimization.)' Second, there is continued interest in prescribing quan

tities other than just the velocity distribution. Fo r instance. it

Research Assistant. Department of Aerospace Engineering. 233Hammond Building; currently Assistant Professor. Department ofAeronautical and Astronautical EnaineeriDI. University of Illinois atUrbana-Champaign. Urbana, IL 61801. Member AIAA.

tAssociate Professor. Depertmeot of Aerospace EoIiJJee:rinI. 233Hammond Buildinl. Senior Member AIAA. 2618


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SELIG AND MAUGHMER: MULTIPO INT INVERSE AIRFOIL DESIGN 2619

is often the case that the designer wishes to specify local flowphysics such as the boundary-layer shape factor'-IO or a Stratford turbulent pressure recovery. I In other cases, it may bedesirable to control the geometry over a segment of the airfoil. 12 Besides specification of these desired local distributions,it is usually necessary to simultaneously achieve global!Jarameters such as the thickness ratio an d pitching moment.rhus, in broad terms, a modern airfoil design methodology.;hould allow for multipoint design, as well as for th e selectionand specification of the independent design variables, be they

the velocity distribution, boundary-layer development, or single design parameters.This paper presents an approach to solving this general

problem by coupling a boundary-layer analysis procedure withan inverse airfoil design method developed earlier. 213 Then,through a convenient parameterization of the desired designvariables and appropriately selected dependent variables, thenonlinear system is solved via Newton iteration. The method is

ighly flexible. For instance, it is possible to specify the airfoildocity distribution on the upper surface at one angle of

attack and some boundary-layer development on the lowersurface at a different angle of attack, while at the same timeachieving a desired airfoil thickness ratio and pitching moment. Of course, these prescribed conditions are subject to theconstraints that the airfoil shape must be closed an d uncrossedand that the velocity distribution must be continuous.

Formulation of tbe ProblemAlthough a numb er of schemes have been devised to achievedesired velocity distribution, 1 ~ . 1 1 . 1 2 . 1 4 - 2 1there are only two,mmon approaches for the achievement of a desired

boundary-layer development. One approach 6711 is to use aninverse boundary-layer method to determine the velocity distribution that yields the desired boundary-layer development,typically the shape-factor or skin-friction distribution. Theresulting velocity dist ribution is then used as input to a potential-flow inverse airfoil method that provides the corresponding airfoil shape.

The disadvantage of the method is that only single-pointdesign problems can be handled directly. Whether or not the~ ,ulting airfoil meets the multipoint design requirements is

ermined through postdesign analysis. If discrepancies doe,\lst, pa n of the velocity distribution is modified judiciouslyuntil the desired goals are eventually achieved. Another difficulty arises when the boundary-layer equations and the auxiliary equations may not be expressed in inverse form.

The other approach, which may be employed using almostany inverse airfoil method, entirely dispenses with the inverseboundary-layer solution as a driver to the inverse airfoilmethod. In an interactive and iterative fashion, all of thedesign goals are achieved by carefully adjusting the velocityd',fribution provided as input to the inverse method. Based on

iback from successive analyses and with some experience,t. _ velocity distribution may be changed in the direction necessary to bring the airfoil closer to the desired goals.

II is instructive to illustrate this iterative technique withinthe framework of the inverse airfoil design method describedin Ref. 2. The method uses conformal mapping to transformthe flow about the circle into tha t about an airfoil. A review ofthe theory is presented later. The circle is divided into a desired

5,~. 53 s.

Clrde dhicled IDIO fI"e R l m n t s aDd _ p p e d to aD airfoU.

2

v

a)

2

v v.

0b) 0 s. 53 5.

4 (e)

H,.

:3 H.3

20 5 . 53c)

Fig. 2 Velocity dlstributiODS fo r tbe tblrd aDd fourtb segments atI I '"' 5 d q plotted as a fanction of a) .; , b) s, and c) tbe correspondingsbape-factor dlstributloD.

2

v

O ~ ~ - - ~ - - - - - 7 - - - - - - - - - - ~ ~ ~o .3 ~ 2 ~

4

2 L - ~ - - - - - - - - - - - - - - - - - - - - - - ~ ~ o 5

Fig. 3 Cbenges in the sbape-fact or distribution as a mull of ,..rylngthe slope of tbe nlocity distribution on tbe tblrd segment.

number of segments along each of which the airfoil velocitydistribution as a function of the circle arc limit q, is prescribedfor a specified angle of attack corresponding to the segment.For this example, five segments ar e used as depicted in Fig. 1.Attention, however, is focused only on the third and fourthsegments (on the lower surface) along which the velocity isprescribed for a = S deg.

The velocity distribution along any intermediate segment atthe design angle of attack is made up of a constant level anda velocity distribution relative to this constant level; i.e.,Vi + Vi(-ii), where -ii is the arc limit relative to the beginning ofthe segment i. By definition and without any loss in generality,it is taken that Vi(-ii = 0) = O. These relative design velocitydistributions are indicated in Fig. 2a for the third and fourthsegments. The origin of the relative coordinate system is positioned for each segment at the constant velocity level Vi . Therelative velocity is then measured from this origin. After specifying the airfoil design velocity distribution and angle of at-


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2620 SELIG AND MAUGHMER: MULTlPOIIIIT INVERSE AIRFOIL DESIGN

2

v

O L - ~ r - - ~ - - - - ~ - - - - - - - - ~ - - ~ ~ o 4>, 4>. 211'

4

3 -- ' ~ " " : ' - ~ : : : ' . : : : : : : - . . : " " " " oS. -----0

S.

2 L - ~ - - - - - - - - - - - - - - - - - - - - - - ~ o s

Fla. 4 Cbanaes in tbe sbape-fa ctor distribution as a result of nry in ltbe velocit, distribution alonl tbe fou,.11 seament.

,1----.-/["L----


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SELIG AND MAUGHMER: MUL TJPOINT INVERSE AIRFOIL DESIGN 2621

mains to determine the derivative of the mapping functionthrough Eq. (2) in order to obtain the corresponding airfoilshape. The derivative of the transformation is assumed to beof the form

dz ( 1)1-' ...dr = 1 - t exp l ~ / a m+ ibm)r-m Ir l ~ I (6)

. 1at on the boundary of the circle becomes

(dz) = (1 - e - ; . )1- . exp[P(ct + iQ(ct) (7)d r ,'0Substitution of Eqs. (3). (5), and (7) into Eq. (2) leads to

[2 sin ct>12-'v*(ct J

P(ct = -t. .21 cos[ct>/2 - Q*(ct]1

(8 )

Consideration of airfoil closure and compatibility betweene specified velocity distribut ion and the freestream leads to

the three integral constraints given by

(9)

Consequently the airfoil velocity distribution v(ct and angleof attack distribution Q*(ct may not be specified arbitrarilyt-;!l must satisfy these three equations. In addition, as seen

ough Eq. (8), the function P(ct must be continuous inL,,'der that the velocity distribution be continuous at any singleangle of attack. Having specified P(ct through v*(ct anda(ct subject to these three integral constraints and the condition that P(ct be continuous, the conjugate harmonic function Q(ct is determined through application of Poisson's integral. The functions P(ct and Q(ct define the derivative of themapping function which may be integrated about the circle togive the airfoil coordinates, :(ct = x(ct + iy(ct.

'IImrrical Formulation-he formulation of the numerical solution begins with the

'. jfication of the design velocity and angle-of-attack distributions. As mentioned, the design angle-of-attack distributionis specified in a piecewise manner by assigning a constantangle of attack to each segment of the airfoil. Likewise, v(ctis prescribed as a piecewise function. Its form is selected on thebasis that it must facilitate the numerical solution and permitthe design of practical airfoils. For each segment other thanthe first or the last, the design velocity distribution is writtenin the form

v(ct = V; + v;(q;;), (10)

w0 5

Fig.6 COmpollenl rec:OYrfY f.JKtioas u d Iotal rKOYrry fullCtionfor . , . . = 00, .s = 20 deg, ,. 0 : 0.3, KH _ 0.1, f - 0, aad K - 1 corrapondinllo an airfoil willi a cuped lnIiIi . . edae.

1.5

10

0. 5

00 .,,/2 ."

Fig. 7 Componenl rKOYrfY functions and total rKOYrfY functionfo r.w = )00, .s = 20, .F

'"

10 deg,,, .. 0.3, KH '" 0.1, f ' " 1/18, andK = 1 corresponding to an airfoU wilh a lO-deg lraillnl-Ngr anglr.

as discussed in the previous example. For the first segment, thedesign velocity distribution is expressed as

( l Ia )

while for the last

v*(ct = VfW(ct, ct>/-1 :S ct>:S 2 r ( l i b )

The recovery function w(ct for the upper surface is defined by

(12)

where

(COS ct>- cos ct>w)

ww(ct = 1 + K ,1+ cosct>w

Osct>sct>w (13a)

[

I - 0.36(cOS ct>- cos ct>S)2, O s t>s ct>sws(ct = I - cos ct>s

1, ct>s:sct>:Sct>w( l3b)

(l3c)

where q,w - q,1' The recovery function W(q,) for the lowersurface is of the same form except that ww(q,), ws(ct, WF(ctand the defining parameters p., KH , K, ct>w, ct>s. and q,F arereplaced by ww(q,), ws(ct, WF(ct, p., kH' k. ~ w- q,1-1. ~ s .and ~ F '

For a typical airfoil design. it is usually desirable to havect>,..>ct>s >ct>F. for instance, ct>,.. = 100, ct>s = 30, and ct>F= 15deg. In this case, the first factor w w ~ ( c t in Eq. (12) controlsthe main part of the recovery. The second factor WfH(q,)controls to a great extent the velocity distribution in the vicinity of the trailing edge. I f E ;o! 0, the last factor wF{ct is activeand results in the proper behavior of the velocity distributionup to and including the trailing-edge stagnation point. Toillustrate the effect of these component functions given in Eqs.(I3a-c) on the total recovery function of Eq.(I2), the following values are used: p. = 0.3, KH = 0.1, E = 0, and K = I. Withthese assigned values and those for the arc limits given previously, the total recovery function and the component functions are shown in Fig. 6. Since E = 0 in this case, an airfoilusing this total recovery function would have a cusped trailingedge. If a lO-deg trailing-edge angle is desired, it is requiredthat E = 1/18. The corresponding total recovery function andthe component functions are shown in Fig. 7 where, as compared with Fig. 6, only the value E has been changed.

The equations to be satisfied by an airfoil with I segmentsincludes the three integral constraints together with I continu-


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2622 SELIG AND MAUGHMER: MULTIPOIN T INVERSE AIRFOIL DESIGN

ity equations on P(tP), one coming from each junction between any two adjacent segments. Thus, in order to satisfythese ( I + 3) equations, it is necessary to identify ( I + 3) unknowns. Owing to the linearity of the equations, it is convenient to select 1', jJ., KH , and I 1.515,

HI2 = - 14.9375 + 12.5 H]2

+"'156.25 (1.195 - H] v 2 - 16

(16a)

(16b)

An attempt to solve the integral boundary-layer equationsbeyond the point of separation with a boundary-layer edgevelocity given by inviscid theory yields a shape factorH32< 1.515, which is not within the bounds of the correlations.

To circumvent this difficulty an d to integrate in the directmode beyond the point of laminar separation, the presentmethod replaces Eq. (16b) for separated flow with a fictitiousshape-factor relation given by

(17)

for H32< 1.515. This equation merely serves as a means tocontinue in the direct mode beyond laminar separation without having to resort to an inverse boundary-layer method. Ofcourse, the solution beyond the point of laminar separation isno longer a valid boundary-layer development. Nevertheless,the final solution after iteration yields the desired (i.e., prescribed) attached boundary-layer development for which thecorrelations are still perfectly valid.

Transition from laminar to turbulent flow is predicted by asimplified e"-method based on linear stability theory as discussed in Ref. 26 or the (H - R ) method of Eppler"

Multidimensional Newton IterationAs illustrated in the example shown in Figs. 1-4, the pre

scribed velocity distribution defined by the inverse designparameters Vi, all of the ai , tPi, etc., will no t necessarily resultin an airfoil having all of the specified properties. With experience and painstaking manipulation of the inverse designparameters, the desired properties can be obtained providedthat they are realistically achievable an d compatible.

This sophisticated trial-and-error approach through the inverse method may be automated by solving the problemthrough multidimensional Newton iteration. Thus, controlover the inverse design parameters is selectively given up; i.e.,the values are determined through Newton iteration, in favorof matching desired airfoil characteristics that are not explicitly given as input to the potential-flow inverse airfoil designproblem.

As illustrated in the example of Fig. 4, the shape factor HI2at the beginning of the fourth segment is specified as 3. This

value is obtained by the adjustment of the slope d I V d ~ l .Through Newton iteration, d V l / d ~ lbecomes the unknown inorder to satisfy the Newton equation

(18)

where the notation .. = > " means that this inverse design parameter has a first-order effect on the corresponding Newtonequation. I t further serves as an aid in keeping an equalnumber of equations as unknowns.

For the fourth segment, the relative design velocity distribution V 4 ( ~ 4 )is adjusted such that 8 12.($4) = 0, that is, throughNewton iteration

(19)

The numerical problem, however, must be discretized forincorporation into the Newton system. The design velocity


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SELIG AND MAUGHMER: MULTIPOINT INVERSE AIRFOIL DESIGN 2623

distribution v . ( ~ . )is defined by a desired number of moveablespline supports as shown in Fig. 5. Fo r the three nodes shown,the following three equations must be satisfied:

- - -IV.(tP4 = 41.) o = H I2.(5. = ~ l ) (20a)V 4 ( ~ 4= ~ : ) => 0= H l2.(54 = 411) (20b)- - -3V4(tP4 = 414) => o = H l 2.(54 = ~ 1 > (2Oc)

where the superscripts indicate the index of the nodes in termsof the arc limit ~ 4 an d the corresponding nodes in 54'Practically any conceivable desired airfoil property can be

incorporated into the Newton system with iteration on someinverse design parameter. As in the companion paper, 2 thepitching moment at a given angle of attack may be specified bythe adjustment of the specified velocity level V; as

Only specified V; 0= Cm - P (21)

where p is the value of the generic desired parameter, in thiscase the pitching moment.

An arc limit 41; between two segments can be iterated tocorrespond to a specified x;lc or s;lc location as

41; 0= x;lc - p (22a)

or

41, ... O = s ; l c - p (22b)

e generally, the arc limit ma y be adjusted so that a specified boundary-layer pro perty is reached at that location. Forexample. q,; may be iterated to correspond to the point wherethe linear stability amplification factor n is a value of 9 for agiven operating condition, i.e.,

q,; - 0= n(s,) - 9 (23)The basic inverse formulatio n t hrough the specification of

v*(tP) and a*(q,) can lead to an airfoil that is crossed. Fortu!"' .' "Iy, this problem can be remedied through appropriate

ton iteration. By empirical obseryation, the trailing-edge\ ",,)city ratio of uncrossed airfoils is always less than unity.Moreover, it can be shown that the thicker the airfoil, thelower the trailing-edge velocity ratioY Many inverse methodsmake use of this fact and allow for the adjustment of aninverse design paramete r in order to match a specified trailingedge velocity ratio, which is somewhat less than unity. Ashortcoming of this approach is that the airfoil thickness is notknown a priori, thereby making it difficult to preassign theproper trailing-edge velocity ratio. Also specifying the trailingeC7e velocity ratio is not a viable option for the design of

lils having a finite trailing-edge angle in which case theL "ding-edge velocity is always zero.

More can be deduced from the character of the trailing-edgevelocity distribution than from the value of the trailing-edgevelocity ratio. Figure 8 shows the trailing-edge velocity distribution an d the corresponding trailing-edge shape for three,symmetric, 8070 thick, cusped airfoils at a 5-deg angle of attack. Only the last 25070 of chord is shown, a nd the vertical y I c

v a) b) c)

:ty/c ===-., ==- ====-I

0 0 2 5 [ ~ ~ ~-0025 ............... ............... ...............x/c

0.75 I

FIR.S Impact 01 tM tralllal-edle nlodt , . dillrilMltioa oa tb e . . . . pe01 tM trallial edle: a) moclmateiy tbick tml i a l e d p , b) tbla trallialedRe, aa d c) crossed trailinl edle.

scale has been expanded to five times that of the x l c scale. Thetrend is that the larger the drop in velocity (i.e., pressurerecovery) at the trailing edge, the thicker the airfoil in thevicinity of the trailing edge (e.g., case a) . I f here is no drop inthe velocity, the trailing edge is very thin (e.g., case b). I f thevelocity shows an increase, th e airfoil is usually crossed (e.g.,case c) . Although these comments are specific to symmetricairfoils such as those shown in Fig. 8, the same trends areobserved for non-symmetric airfoils as long as the net velocitydrop is considered. For example, if the velocity decreases on

the upper surface by the same amount that it increases on thelower surface, there is zero net velocity drop. In such aninstance, the airfoil will be thin at the trailing edge.

The trend just discussed must be translated into an equationif typical trailing-edge shapes are to result from the designmethod. The functions given in Eqs. (12) an d (13) have beenderived so that normal trailing-edge velocity distributions canbe produced. In particular, the closure contributions, wtH(tP)and wfH(tP) have a dominan t effect on the trailing-edge velocity distributions. By prescribing the sum (K H + KH ) = Ks to bein the range 0 to 0.8, normal trailing-edge velocity distributions result and give rise to uncrossed airfoils-the smaller Ksin this range, the thinner the airfoil in the vicinity of thetrailing edge. Negative values for Ks usually produce crossedairfoils. To achieve the desired value of Ks , the leading-edgearc limit may be used for iteration, that is,

0 = K s - p (24)

For a finite trailing-edge angle (E -F- 0), the functions wHtP) andwF(tP) are active and produce a zero trailing-edge velocity. Itis still necessary, however, to specify Ks in order to havecontrol over the thickness in the vicinity of the trailing edge.By these means, the designer has great control over the airfoilgeometry in the region of the trailing edge.

Iteration on the design angles of attack can easily be used tocontrol the usable lift range of the airfoil. By adding anincrement to each a i, the polar will be shifted upwards which.in turn, will decrease the zero-lift angle of attack. This shift inthe polar can be controlled through a prescribed zero-lift angleof attack as

Q; 0 = QO L - P (25)

By adding an increment to the upper surface a; and subtracting an equal increment from the lower surface a; , the width ofthe polar will be increased. This has the effect of thickeningthe airfoil and may be controlled through

O = t l c - p (26)

where means to adjust the a; in opposite fashion.The relative design velocity distribution for a segment may

be used to control the relative boundary-layer shape-factordistribution as previously mentioned, or the relative velocitydistribution in 5,,2 or anyone of other desired distributions.

For example, either the local airfoil geometry may be specified, the n-development, or a curve in the H-R diagram usedby Eppler for predicting transition. Through iteration on thepreceding segment, the initial value is set as previously ex-plained for the specification of the shape-factor distribution.

In the design of a new airfoil, the iterative process is takenin stages. In a typical case, Ks is first satisfied in order for theairfoil to be uncrossed. Then Cm is sometimes specified tobring the airfoil into a normal range. After this, any desiredinitial segment conditions are added to the Newton systembefore iterating on the segments that have a specified distribution of some type. I f only a few small changes are made to theinverse design parameters of a converged solution, it usually ispossible to iterate on the full Newton system from the outset.The convergence of the solution can be disrupted if the New-ton scheme attempts to take a step that is too large. To prevent


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2624 SELIG AND MAUGHMER: MULTIP OINT INVERSE AIRFOIL DESIGN

this from happening, a maximum step size for any of theunknowns can be preset. If any of the predicted step sizes atany point in the iteration exceeds a preset maximum, then afractional step is taken to avoid exceeding a maximum. Because the method incorporates an incompressible inverse airfoil design method and an integral boundary-layer method,the convergence is quite rapid an d easily allows for the interactive design of a wide range of specialized airfoils.

Demonstration of the MethodTo illustrate some of the capabilities of the method an

example airfoil is considered. It should be stated that thisairfoil only serves to illustrate the method an d is not intendedfor practical use. The main design goals. although defined inmathematical detail later. may be stated as follows. Along theforward part of the upper surface of the airfoil. the n -development is prescribed at the design angle of attack an d Reynoldsnumber. By prescribing the linear stability amplification factor n, the extent of laminar flow an d the location of transitionis controlled. This is advantageous when there is interest indesigning for low drag. Following this segment. a linear rampis introduced for a different angle of attack. On the lowersurface. at yet a different angle of attack. the boundary-layershape-factor distribution is prescribed much like the example

of Figs. 1-4. Specification of the boundary-layer shape factoris desirable in instances where flow separation is to be avoided.These characteristics described are obtained using seven

segments on the airfoil-four segments on the airfoil uppersurface and three on the lower. All of the inverse designparameters are listed in Table I, with the exception of thetrailing-edge angle parameter E. Many of the inverse designparameters listed in Table I are selected as unknowns in theNewton iteration in order to match the design goals. but forthose that are fixed the following values are used:

a l = a2 = 10 deg (27a)

a3 = a4 = IS deg (27b)

a5 = a6 = a7 = 8 deg (27c)

K = k = I (27d)

s= 15 deg, ~ s = 340 deg (27e)

F= 10 deg, ~ F= 350 deg (270

E = 1118 (27g)The value for E is selected to yield a IO-deg trailing-edge angleand the arc limits s, ~ s , F' an d ~ F are set to confine theclosure and the finite trailing-edge angle contributions ofthe pressure recovery to a small region near the trailing edge.The small values for K and k, which partly define the mainpressure recovery, will give a slight adverse pressure gradientat the beginning of the recovery on the upper and lowersurfaces at the corresponding design angles of attack, a l an d

a7, respectively.As discussed in the preceding section. the design goals arematched in stages. In this example case, the process is taken inthe order of increasing complexity. First, the leading-edge arclimit is iterated to match Ks . Afterward s. the leading-edge arc

Table 1 InYene design p.rameten f o r . seven-segment airfoil

a(


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SELIG AN D MAUGHMER: MULTIPOINT INVERSE AIRFOIL DESIGN 2625

2

v

O ~ ~ - - - - ~ - - - - ~o 0 5 x/c

fig. 11 Airfoil an d velocity distributions fo r a = 8, 10, an d 15 deg.

F, = 1 X 106 Th e initial value of n = 2 is set by adjusting theocity distribution o f the fourth segment, i.e.,

(32)

The velocity distribution for the third segment is adjusted togive the desired linearly increasing n-growth given by

(33)

where $3 is measured from 53 to 52, that is, $3 = 53 - 5.Since th e length o f the segment is 53 - 52 = 0.5 an d th e initial

\ -lue is n = 2, this gives n = 9 at the en d o f th e segment torespond to the point of transition. As shown in Fig. 10, this

desired n-growth, based on th e analysis method of Ref. 26, isachieved. Finally, th e airfoil shape an d th e velocity distributions at the design angles o f attack ar e shown in Fig. 11.

ConclusionsA hybrid-inverse airfoil design technique has been devel

oped by coupling a potential-flow, multipoint inverse airfoildesign method with a direct boundary-layer analysis method.The potential-flow inverse design parameters can be iteratively

; usted automatically through multidimensional Newton it-

tion in order to obtain desired airfoil characteristics that~ . ~ no t explicitly given as input to th e potential-flow inversemethod. This combined approach makes it possible to prescribe along segments o f the airfoil either the local geometryor, together with th e design conditions, the velocity distribution or some boundary-layer development. At the same time itis possible to specify single parameters such as thickness ratioor pitching moment. Although the current implementation ofthis approach makes use o f an incompressible inverse designmethod an d a direct integral boundary-layer method together" :!h an en-method for transition prediction, either component

dd be replaced by an alternative design or analysis method.j r instance, th e method could be extended to handle compressible airfoil design, multielement airfoil design, or allowfor th e design o f airfoils with regions o f separated flow. As itstands, fairly sophisticated airfoil design studies can now bemade with relative ease. This should ultimately lead to improvements in new airfoil designs.

AcknowledgmentTh e support o f the NASA Langley Research Center under

C an t NGT-S0341 is gratefully acknowledged.

ReferencesIEppler. R Airfoil Design an d Data, Springer-Verlag, Berlin,

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