A COMPUTER PROGRAM FOR THE ANALYSIS OF MULTIELEMENT

AIRFOILS IN TWO-DIMENSIONAL SUBSONIC, VISCOUS FLOW

By Harry L. Morgan, Jr.NASA Langley Research Center

SUMMARY

A computerized analytical model which computes the performance characteristicsof multielement airfoils in subsonic, viscous flow has been developed under NASA con-tract to the Lockheed-Georgia Company. The model computes the viscous pressure dis-tributions, lift, moments, and local boundary-layer properties on each element of anarbitrarily arranged slotted airfoil in attached flow. The final viscous solution isobtained by an iterative technique for successively combining an inviscid solution withboundary-layer displacement thicknesses. The surface of each airfoil element is approx-imated as a closed polygon with segments represented by distributed vortex singularities.The ordinary boundary-layer solution is comprised of mathematical models representingstate-of-the-art technology for laminar, transition, and turbulent boundary layers. Anadditional boundary-layer model has been incorporated to compute the characteristics ofa confluent boundary layer which reflects the merging of the upper-surf ace boundarylayer with the slot efflux.

This computer program has been used extensively at Langley and throughout theindustrial and academic communities for both the design and the analysis of airfoils.Presented in this paper are summary descriptions of the general operation and capabil-ities of this program and a detailed description of the major improvements that havebeen made to the program since its initial formulation. Sample comparisons betweentheoretical predictions and experimental data are presented for several types of multi-element airfoils. Areas of agreement and disagreement are discussed with recommen-dations for areas of needed program improvement.

INTRODUCTION

During the initial design phase of an airfoil the effects of various modifications canbe easily evaluated by describing the potential (inviscid) flow around the airfoil. Manymethods are available to compute the potential flow and most generally require rathersmall computer storage and execution times, which make them very desirable during theinitial trial-and-error design phase. However, during the final design phase a more

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accurate assessment of the selected modification can be evaluated by describing the vis-cous (viscid) flow around the airfoil, which means simply that the boundary-layer prop-erties have been included in the analysis.

No general, mathematically closed form solution presently exists which describesthe viscous flow around an airfoil. Until very recently, airfoil design has relied mainlyon potential-flow theory to obtain the theoretical pressure distribution from which theboundary-layer properties were approximated, with the interrelationship between the twobeing largely ignored. However, the use of iterative techniques has provided a practicalsolution for defining the interrelationship between the potential and viscous flows aroundan airfoil. State-of-the-art technology in the areas of potential flow and boundary layers,plus the availability of high-speed large-capacity computers, has provided the backgroundcapability essential to the formulation of computer codes to compute accurately the per-formance of an airfoil in viscous flow.

One such computer code used extensively at the'Langley Research Center andthroughout the industrial and academic communities was developed by the Lockheed-Georgia Company under NASA Contract NAS 1-9143 in 1969. This computer program,entitled "2-D Subsonic Multi-Element Airfoil Program," was formulated to handle single-element and multielement airfoils (a maximum of four elements) in subsonic, viscousflow. A complete description of this original program is given in detail in. reference 1.Since the initial formulation of this program, there has been a continuing effort toimprove and modify the program to handle an ever increasing range of airfoil geometries.The initial debugging and general program maintenance and improvements have been per-formed through in-house efforts at Langley. The only other major contribution to theprogram improvement has been the work done by Delbert C. Summey and Neill S. Smithunder NASA Grant NGR 34-022-179 to North Carolina State University. Their work wasconcerned primarily with the single-element version of the program and consisted of pro-gram modifications to improve the lift and drag predictions, to reduce .the computer stor-age requirement, and to reduce the computer execution time. A detailed description oftheir work is presented in reference 2. . .

A summary description of the general operation and capabilities of the airfoil pro-gram and detailed descriptions of the major changes to it are presented in this paper.Sample comparisons between theoretical predictions and experimental data are also pre-sented for several airfoil geometries. Areas of agreement and disagreement are dis-cussed with recommendations for areas of needed program improvement.

SYMBOLS

al»a2>a3 coefficients of quadratic equation of the form f(x) = ajx2 + a£X + a.%

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AH aerodynamic, influence coefficient

c chord of airfoil, cm (in.)

C(j section profile-drag coefficient

Cf skin-friction coefficient r i

GI section lift coefficient

cm . section quarter-chord, pitching-moment coefficient

cn section .normal-force coefficient

Cp . local static-pressure coefficient, Pstatf? " P°°

H boundary-layer form factor, d*/0

K local curvature, cm~* (in-1)

M Mach number

N number of corner points for polygon approximation of airfoil

p , -static pressure, N/m2 (lb/ft2)

q^ • free-stream dynamic pressure, N/m .(lb/ft ) .

RQ • Reynolds number based on momentum thickness, ^

s surface distance along airfoil contour, cm (in.)

Ue velocity at edge of boundary layer, m/sec (ft/sec)

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Uoo free-stream velocity, m/sec (ft/sec)

v local velocity on airfoil surface, m/sec (ft/sec)

x airfoil abscissa, cm (in.)

z airfoil ordinate, cm (in.)

a angle of attack of airfoil, deg

y strength of vortex singularity, m2/sec (ft^/sec)

6 boundary-layer thickness, cm (in.)

5f s or v angular deflection of flap, slat, or vane, deg

6* , , . . - • boundary-layer displacement thickness, cm (in.).

9 . . boundary-layer momentum thickness^ cm (in.)

v • kinematic viscosity, mVsec (ft^/sec) -

0 cosine distribution angle, 0i = |?, deg

i// stream function, m^/sec (ft^/sec)

Subscripts:

c control point . - . • • . • • .

1 matrix row

j "' ' matrix column -

1 lower• • - - - ' -1 '

te trailing edge

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u upper

00 free stream

Abbreviations:

B. L. boundary layer

NC number of components

TE trailing edge

PROGRAM OPERATION AND CAPABILITIES

The original airfoil program developed by the Lockheed-Georgia Company wasdesigned to handle single-element and multielement airfoils. As many as four elementscan be handled which typically consist of an airfoil with a leading-edge slat device and adouble-slotted trailing-edge flap system. The airfoil program is composed of three mainparts: (1) geometry specification, (2) potential flow, and (3) boundary layer. The non-overlay version of the program requires a computer storage of approximately200 000 octal locations (CDC 6600) and an execution time of approximately 200 CPU sec-onds for a typical four-element airfoil. An overlay version has reduced the requiredstorage to approximately 65 000 octal locations with only a slight increase in executiontime. The single-element version developed by North Carolina State University requiresa computer storage of approximately 110 000 nonoverlay octal locations and 53 000 over-lay octal locations with execution times of the order of 30 CPU seconds for a single case.

A flow chart of the single-element version of the airfoil program is presented infigure 1. After data input and geometry specification (subroutine READIT and GEOM),the program enters an iterative cycle which involves the determination of interrelation-ship between the potential flow and the boundary layer (subroutines MAIN2 and MAINS).After each iteration a convergence check is made which consists of a simple comparisonof the computed normal-force coefficients. Experience has shown that only five iterationsare necessary to obtain a converged solution. This rapid convergence is possible becauseof the unique method used to combine the physical airfoil geometry and the computed dis-placement thicknesses from the boundary-layer computations in order to obtain the nextiteration geometry. This convergence method will be discussed further in a followingportion of this paper.

The flow chart for the multielement version is very similar to that for the single-element version, the exceptions being.an expanded geometry routine to handle the posi-

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tioning of the elements relative to one another- and an additional boundary-layer routine-to handle the merging of boundary layers between fore and aft elements (confluent bound-ary layers). A program routine was available in the original multielement version to :

correct the pressures in the slot regions between elements to account for the high fric-tional and surface curvature effects. This routine employed a marching-type.calculationprocedure which required an accurate definition of the slot inlet conditions. An empiri-cal formula was derived to estimate the inlet conditions from the potential-flow solution.Experience has shown that this slot-analysis routine produces erroneous results and ithas, therefore, been omitted from the more recent program version.

Since the inception of the airfoil program, the program input has been kept as sim-ple as possible to make it more user oriented. The coordinates of each element of amultielement airfoil can be input with respect to a separate coordinate system and easilypositioned relative to other elements by specifying the pivot point location and deflectionof the element. The boundary-layer transition location (transition from laminar to tur-bulent) on each surface of an element can be input as either fixed or free. The total number of calculation points at which the pressures are desired can also be input and will beautomatically allocated using the formula

= 2 (1)

where Nj is the number of points allocated to the ith element, NSp is the total num.- .„• iber of calculation points, Nc is number of components, ci is the chord of a given ele-;i"ment, and c<p is the summation of the values of c^. During a single machine pass, theangle of attack and Mach number can be varied for a constant Reynolds number, Prandtlnumber, and stagnation temperature. To represent the effects of compressibility, thewell-known Karman-Tsien pressure correction law is employed which, therefore, limitsthe input Mach number to that producing critical flow. The laminar and turbulentboundary-layer routines contain methods to.predict boundary-layer separation, but do notcontain methods to model the flow after separation. Therefore, the angle of attack shouldbe limited to that producing only minor separation (less than one percent of the surface).

PROGRAM THEORY AND MODIFICATIONS

Airfoil Geometry Specification

The user inputs to the airfoil program are the upper- and lower-surface coordinatesof the airfoil shape. The airfoil is modeled within the program as a polygon approxima-tion which is illustrated in figure 2. This polygon consists of N number of cornerpoints with N - 1 number of straight line segments. Previous experience has shownthat an N equal to 65 is sufficient to obtain an accurate viscous flow solution for a

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single-element airfoil. Of the 65 total points, 32 points are distributed, on the upper sur-face, 32 on the lower surface, and 1 at the leading-edge nose location. In most instances,the user desires to input more than 65 coordinates and does not want to be concernedabout whether or not the input spacing will affect the computed results. For this reason,an automatic collocation method has been included in the airfoil program.

The original version of the airfoil program distributed corner points with thecosine formula

. (2)6

where 0.j.= ^-, i = 1, 2, . . ., N. This procedure is illustrated in figure 3 and a sam-ple distribution shown in figure 4(a). This method, in effect, closely spaces points nearthe leading and trailing edges of the airfoil. A close spacing near the leading edge isvery desirable because of the high curvature and high velocities occurring in that region.Experience has shown, however, that closely spacing points near the relatively thintrailing-edge region can result in extreme oscillations in the computed velocities. Thereason for these oscillations is discussed further in the potential-flow portion of thispaper. To overcome this problem, a new collocation method has been formulated andincorporated into the airfoil program.

The basis of the new method is that points are spaced relative to the local curva-ture. This method will closely space points only in regions of high curvature. To uti-lize this method the curvature at each user input coordinate is computed with the formula

,2

(3)3/2

where - a^, a.%, and a3 are taken from a curve fit of the airfoil points of the form

z2 = ajx2 + a2X + a3 (0 s x < o.8c) (4)

and

z = ajx2 + a2x + a3 ' (0.8c S x s c) (5)

A curvature summation is then computed from the following equation and stored for back-ward interpolation:

Is (6)

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where

/ / T 5 / \ ~ 2 •si = si-l + \l(xi ~ xi-l) + (zi ~ zi-l)

The maximum value of K is divided into N equal portions and the s value .corre-spond ing to each portion is then determined by backward interpolation between the Sjand Ki arrays. Additional backward interpolations are then made between the. si, 24,and Zi arrays to determine the new distributed airfoil coordinates. A sample of theresults of this collocation method is presented in figure 4(b). Note the improvement inthe distribution of points as compared with the cosine method.

Caution by the user should be exercised regarding the number and relative posi-tions of the input coordinates. If more input points are given, the computation of thecurvature summation array will be more accurate and, therefore, the distribution ofpoints more accurate. The'curve fit formula of equation (4), which is used in the nose. ,region, is designed to approximate an infinite slope and a better estimation of the highcurvature in that region can be obtained if more points are input. , .

' IPotential-Flow Solution

The potential-flow methods used to determine the velocity at specified locations onthe surface of an airfoil generally fall into two categories. One category consists of con-formal transformation methods and the second, singularity distribution methods. Con- .formal transformation methods have not been greatly utilized because of the difficultiesencountered when trying to obtain transformation equations for airfoils of arbitrary shapeand because of the inability of this method to handle blunt-base airfoils. Singularity dis-tribution methods have been widely utilized since the advent of the high-speed, high-capacity digital computers which are needed to solve the large systems of simultaneousequations characteristic of these methods. These methods can handle arbitrarily shapedairfoils at any orientation relative to the free stream.

For singularity distribution methods, either source, sink, or vortex singularitiesare distributed on the surface of the airfoil and integral equations formulated to deter - .mine the velocity induced at a point by the singularity. By dividing the airfoil surfaceinto N segments and specifying either zero-normal or a tangential flow boundary condi-tion for each segment, the integral equations can be approximated by a correspondingsystem of N - 1 simultaneous equations. By satisfying the Kutta condition at the trail-ing edge of the airfoil, the Nth equation can be formulated and then the singularitystrengths determined with any one of a number of matrix-inversion techniques. The

• ' *" * •

Kutta condition usually employed is that the velocities at the upper- and lower-surfacetrailing edge be tangent to the surface and equal in magnitude. - ' • • , _ . _ _ _

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\I

The singularity distribution method used in the original version of the airfoil pro-gram consisted of a distribution of vortices at the corners of the polygon approximationof the airfoil, with an additional constraint that the vortex strength vary linearly alongthe surface segments. A control point was selected at the midpoint of each segment anda boundary condition of no flow normal to the surface was applied. This resulted in Nunknown vortex strengths and N - 1 boundary conditions and equations. An additionalequation was obtained by satisfying the Kutta condition at the trailing edge as . ,.

because, for a vortex singularity, the magnitude of the tangential surface velocity is equalto the vortex strength at the corner point.

Several problems were encountered with the original singularity distributionmethod. For airfoils with a trailing -edge cusp, the upper- and lower -surf ace vortices'near the trailing edge tend to become identical and, thereby, generate an almost singularmatrix. For other types of airfoils, a too close spacing of vortices near the trailing, edgeresults in extreme oscillations of the vortex strengths. This method could not handleairfoils with an open or blunt -base characteristic of the recently developed Langleysupercritical or "shockless" airfoils. To overcome these handicaps an improved singu-larity distribution method has been incorporated into the airfoil program.

The new singularity distribution method was first formulated by H. J. Oellers to,compute the pressure distribution on the surface of airfoils in cascade, and is describedin reference 3. Instead of working with induced velocities, characteristic of the previousmethod, Oellers' method employs stream functions. The stream function for a uniformfree stream plus that of the vortex sheet is set to be a constant on the airfoil surface.This is represented mathematically by the Fredholm integral equation .

_ L C2?r Jf

In r(s,£) d£ = Ilexes) cos (a) - O^zfe) sin (a) . (8)0

where . if is the unknown stream function constant, r(s,£) is the distance between twopoints on the airfoil surface, x(s) and z(s) are coordinates of a point on the surface,and y(£) is vortex strength at a point. By dividing the surface into N segments andassuming C9nstant vortex strength for each segment the above equation becomes

' ; - - N . . • . . - . . . •

i// - } Ayyj = Uoofxi cos (a) - zi sin (a)J (9)

where the influence coefficient Ay is

! « " ' • ' ' (10)

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Si

By specifying a control point at the midpoint of each segment (denoted by a "c" subscript),the influence coefficient becomes

At, - it, in (r2) - tl in (r - £

and

where

(i = j) (12)

As = sj+1

= (xj+l - xc,i) 2 - z 2,i)

n _ ^^__^___^___^__^_____^__^___^_________^_

t, _ (XJ - Xc»l13

-As

.As J

(13)

To determine the vortex strength (y) at the intersection of two segments, the following"interpolation formula is used:

(j ..* 1 or N) (14)

For this method, an additional equation is needed to obtain an N by N system ofequations. (The unknowns are N-l number of y's and «//.) A new method of apply-ing the Kutta condition has been formulated to reduce the oscillations of the vortexstrengths caused by a too close spacing of points near the trailing edge. This new Kuttacondition simply requires that the vortex strengths (y) vary quadratically for the last foursegment corners near the upper and lower surface of the trailing edge and that at thetrailing edge

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This quadratic variation is expressed as .

y(s) = ajs^ + a2s + a3 (16)

By combining equations (14) and (16), the equation for the vortex strength at the lower-surface trailing edge becomes ">

(17)

where

Ji =_ S4s2(s3 - s4)

D

Jo =_ 84 (S3 " S2)(S3 + S2 " S4)

D

J, =

S2(S4 -S3) s4

S3(S2•5-

-S3)"83

D

_. _ J . =" S3s2(s2 - s3) (s5 - s4)= -

D(s5 - s3)

and

D = s22(s3 - 84) + s3

2(s4 - s2) + s42(s2 - s3)

A similar expression can be obtained for the upper-surface trailing-edge vortex byreplacing the subscripts in equation (17) as follows:

(i = 1, 2, . . ., 5)si = SN - sN+1_t

Ji = JN-i= l ,2 ,3 ,4)

Combining equations (16) and (17) with equation (15) yields the needed Nth equation in theform .

= 0 (i = 1, 2, 3, 4, N-4, N-3, N-2, N-l) (18)

The Oellers1 method with the modified Kutta condition has been incorporated intothe current version of the airfoil program. This new potential-flow method combined

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with the new geometry specification method has successfully overcome the problemsencountered with t h e original methods. - . . . - ' - .

Boundary-Layer Solution . .

The pressure coefficients computed in the potential-flow portion of the airfoil pro-gram are corrected to account for compressibility with the well-known Karman-Tsiencorrection law. Using the isentropic flow relations, the local Mach number is computed-and input to the boundary-layer portion of the program. The boundary-layer develop-ment on a typical multielement airfoil is illustrated in figure 5. The boundary layer con-sists of an ordinary boundary layer (nonmerging boundary layer) and a confluent boundarylayer (merging boundary layer). The ordinary boundary layer is composed of laminar,transition, and turbulent regions. The confluent boundary layer is composed of core,main I, main II, and turbulent regions as shown in figure 6. The confluent boundary- "layer model was developed by Sure sh H. Goradia from the Lockheed-Georgia Companyand is one of the unique features of this program. The meaningful parameters outputfrom the boundary-layer portion of the program are (1) the,displacement thickness 6*,(2) the momentun thickness 0, (3) the form factor H, and (4) the skin-friction coefficientCf. The theoretical development of the boundary-layer methods used in this program arequite lengthy and, therefore, only a brief description will be presented in this paper.

A flat-plate boundary-layer analysis is performed on each surface of an airfoilelement, and the leading-edge stagnation point is the plate leading edge. A flow chart ofthe boundary-layer computations is presented in figure 7. An initial laminar boundary-layer region exists from the stagnation point to the point of transition from laminar toturbulent. The laminar boundary-layer model used is the method of Cohen and Reshotkoas presented in reference 4. After computing the laminar boundary-layer characteristicsat a discrete point, routine BLTRAN is called to check for transition and, if transition hasoccurred, to check for the formation of a long or short transition bubble and for laminarstall. The sequence of calculations within BLTRAN is presented in figure 8; An initialcheck is made to determine if the laminar boundary layer is stable or unstable based onthe instability criterion established by Schlicting and Ulrich as presented in reference 5.If the boundary layer is unstable, a transition check is then made based on an empiricallyderived transition prediction curve. 'If transition1 has occurred, the initial quantitiesneeded to start the turbulent calculations are computed. If transition has not occurred,the formation of either a long bubble with corresponding laminar stall or a short bubblewith corresponding'reattachment is determined. The user .can input a fixed transitionlocation and a check will be -made at the beginning of BLTRAN to determine whether ornot the fixed location has been reached. . .

After computing the transition location and corresponding initial boundary-layerproperties, the turbulent boundary-layer calculations are made. The original version of

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the program used a modified Truckenbrodt method derived by Gdradia to compute theturbulent boundary-layer properties. The Goradia method was less sensitive to localpressure oscillations and was capable of traversing localized separated/reattaching flowregions. These local pressure oscillations were caused primarily by the numericalinacccuracies associated with the potential-flow method. The Goradia method underesti-mated the boundary-layer displacement thicknesses which, therefore, resulted in anunderestimation of the lift,and pitching-moment coefficients. After incorporatingOellers' potential-flow method, the pressure oscillations were reduced which meant thatthe original Truckenbrodt method described in reference 6 could be used to improve theturbulent boundary-layer computations. The original Truckenbrodt. method was codedand incorporated into ,the recent program version and has resulted in an imprpvement inthe estimation of the lift and pitching-moment coefficients.

If a slot exiting plane is reached during the turbulent boundary-layer computations,the confluent boundary-layer computation is initiated. The confluent boundary layer is aresult of the mixing between the slot efflux and the wake from the forward element, andcan exist from the slot exit to the trailing edge of the element depending upon the pres-sure distribution. The confluent boundary-layer model was formulated by Goradia andconsists of various regions and layers as illustrated in figure 6. The model is based onthe assumption that-the merging of fore-and-aft element boundary layers will have"similar" velocity profiles if nondimensipnalized in a way analogous to that for a free-jetflow. By utilizing this assumption, the governing partial differential equations were,reduced to a set of.ordinary differential equations which could be easily solved with avail-able numerical techniques. .Several empirical constants were needed to establish thesimilar velocity profiles and were obtained from experimental tests performed byGoradia as reported in reference 7. The only improvement made to the original conflu-ent boundaryTlayer routines has been, minor adjustments in the values of these empiricalconstants. The model formulated by Goradia assumed that the core velocity exiting theslot is greater than the velocity at the upper edge of the wake layer at the slot exit.Experience has shown that this velocity relationship is not always true and, therefore,erroneous performance predictions can occur. It is generally believed that this velocityrelationship should be true for an efficient design and, therefore, no additional work hasbeen done in this area.

None of the boundary-layer methods used in the airfoil program include curvatureeffects; All the methods used are basically integral methods which are generally lessaccurate than finite-difference methods but require considerably less computer time.More accurate infinite-difference methods are available, as described in references 8, 9,and.10, and can be easily incorporated into the existing program.

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Equivalent Airfoil Geometry

The airfoil program uses an iterative procedure to obtain the viscous solution andthe basic steps are as follows:

(1) Compute a potential-flow solution for the basic airfoil.

(2) Compute boundary-layer properties based on the potential-flow solution.

... (3) Construct a modified airfoil by adding the boundary-layer displacement thick-ness to the original airfoil.

(4) Compute the aerodynamic performance coefficients.

(5) Repeat steps (1) through (4) until convergence of the performance coefficients isobtained.

The most important step in the iterative procedure is step (3) which involves selecting ageometry modification method that will insure convergence for almost any input airfoilshape in a reasonable number of iterations. The method developed by Lockheed has beenhighly successful and has provided converged solutions after only four or five iterations.

The method developed by Lockheed to modify the airfoil shape is based on theassumption that the effect of the boundary layer on the basic thickness and basic cambercan be considered separately and then superimposed to determine the net effect. Theaddition of the boundary layer has an uncambering effect near the trailing edge whichcauses a reduction in the effective angle of attack and lift coefficient, and it has a thicken-ing effect along the airfoil which causes an increase in the local surface velocities andlift coefficient. .Experience has shown that the thickness effects are of secondary impor-tance for multielement airfoils and are, therefore, omitted in the multielement programversion. Thickness effects are, however, included in the single-element program versionto improve the overall accuracy of the performance predictions. The camber change isgiven as the difference in the magnitude of the upper- and lower-surf ace displacementthicknesses as illustrated in figure 9(a). The thickness change is given as the differencein two thickness solutions as shown in figure 9(b).. The first thickness solution is for a .symmetric airfoil at a 0° angle of attack with the same thickness distribution as the orig-inal input airfoil/; The second thickness, solution.,is also, for a> symmetric airfoil at a 0°angle of attack with the thickness distribution of the original input airfoil plus the sum of,the upper- and lower-surface displacement thicknesses. . .

' Applying superposition and a proportioning technique to prevent over-correction dur-ing the initial iterations, the velocity distribution becomes

(vtotal)i =

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where BT stands for the basic thickness distribution and

kt = 1 (i * 2)

k i = f . (1 = 2)

The addition of the displacement thickness to obtain the thickness solution often producesa symmetric airfoil with a very thick trailing edge. In actual flow the wake acts as anafterbody with a rapidly decreasing thickness distribution. Based on the work of Powellas reported in reference 11 and from observations of available experimental data,Summey and Smith from the North Carolina State University formulated the followinganalytical expression to represent the afterbody shape:

= ±I[(zte -Zoo)e-6.9xx+ZooJ(i _xx) . . (20)

where xx = £ - 1 for c = x § 2c and z^, =-i GJ c. The Squire and Young drag form-. c . 2 u ° o

ula from reference 5 is used to compute the drag coefficient at infinity and is given as

Hu+5 H1+5:

.,Slk»Y 2 .'jOt* 2

After each boundary -layer solution, the displacement thickness distribution issmoothed three times by using a standard least-squares smoothing technique. To preventover -correction during the initial iterations, a proportioning technique similar to that.used for the velocities is used to compute the effective displacement thickness distribu-tion and is given as

where BL stands for the present boundary-layer solution.

COMPARISONS BETWEEN EXPERIMENT AND THEORY

Both the single -element and multielement versions of the airfoil program have beenwidely distributed and utilized by the industrial and academic communities. The programis widely used within Langley for the design and analysis of new subsonic and transonicairfoils for application to helicopters, general aviation aircraft, and transport aircraft.Shown in figure 10 is a comparison between the theoretical and experimental performancecharacteristics for the recently developed NASA GA(W)-1 airfoil that was designed byRichard T. Whitcomb especially for application to general aviation aircraft. .(See ref. 12.)

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Excellent agreement is demonstrated until an angle of attack is reached at which theboundary layer begins to separate near the upper-surf ace trailing edge. Currently,a flow separation model does not exist in the program; however, a research studyis being proposed by Langley in this area of boundary-layer research. Excellent agree-ment is also demonstrated for the drag prediction of the GA(W)-1 airfoil, but this is notgenerally true, for all airfoils. The method used in the program to compute drag con-sists of a simple integration of the pressure and shear (skin friction) forces. Animproved drag computation method is currently under development by Lockheed-Georgia,under NASA Contract NAS 1-12170, which computes the drag from computed downstreamwake characteristics. This improved drag method will be incorporated into the single-'element airfoil program after verification at Langley. '

' • • *•

A Fowler^type, _single-slotted flap was also designed for the GA(W)-1 airfoil and .later tested at Wichita State University under a NASA grant. , The results of the flappedairfoil tests are presented in reference 13, and a summary of the agreement between,experiment and theory is presented in figure 11. Excellent agreement is demonstrateduntil a 40° flap deflection is reached where separation occurs on the upper surface of theflap. Similar agreement was obtained from tests of a blunt-base airfoil with a single -slotted flap reported in reference 14. Typical agreement between the experimental andtheoretical confluent boundary-layer velocity profiles for this airfoil is shown in fig-ure 12. The multielement airfoil program can also be used to perform a gap optimiza-tion study of a given flap system. Presented in figure 13 are the results of a gap optimi-zation study for a 10° drooped-nose airfoil with a single-slotted flap as reported in ref-erence 15. The viscous and inviscid theory predictions are also presented in this figure,and although the agreement between viscous theory and experiment is fair, the viscoustheory does predict the correct optimum gap of 2 percent.

The agreement between theory and experiment for multielement airfoils with threeor more elements has not been generally as good as that for single- or two-element air-foils. This can usually be attributed to the fact that an airfoil with three'or. moreelements generally will have some separated flow on at least one element. Shown infigures 14(a) and: 14(b) is the agreement between viscous and inviscid theory and experi-ment for a typical three-element airfoil with a leading-edge slat and a single-slotted,trailing-edge flap. (See ref. 16.) Excellent agreement is demonstrated over an angle-of -attack range from -4° to 12° for the force and moment coefficients. A typicalpressure-distribution agreement is shown' in figure 14(b) for an angle of attack of 8°.Shown in figure 15 is the agreement between theory and experiment for a typical four-element airfoil with a leading-edge slat and a double-slotted, trailing-edge flap. (This ismodel C in ref. 14.) The low -5°/15° vane/flap deflection case showed poor agreementbecause at those deflections there were large overlap areas between the vane and flap andthe airfoil program does not correct the pressure distribution for slot effects. (Previous

728

discussion pointed out that the original slot flow correction method proposed by Lockheedwas erroneous and, therefore, has been dropped from the program.) The high 30°/65°deflection case showed very poor agreement because of the large amount of flow separa-tion on the upper surface of the flap element. The intermediate 14°/33° case, however,showed good agreement because there were no large overlap or separated flow areaspresent. Shown also in figure 15 is the extremely poor drag agreement which is typicalfor multielement airfoil cases.

The sample cases presented in this paper were selected to demonstrate some ofthe good as well as bad features of the airfoil program. The first major area of neededprogram improvement is the computation of profile drag. Many researchers use thewell-known Squire and Young drag formula (eq. (21)) to compute the profile drag; how-ever, this formula was derived based on a proven erroneous assumption about the varia-tion of the downstream boundary-layer form factor, H. The second major area ofimprovement is the computation of the characteristics of the flow region after boundary-'layer separation. The lack of experimental data for separated flow regions has greatlyhampered the development of a theoretical model. Experimental tests have recently beencompleted'at Wichita State University to map the velocities and pressures in the sepa-rated regions of the GA(W)-1 airfoil. The third major area of improvement is the com-putation of the pressure corrections in the slot areas between overlapping elements ofmultielement airfoils. This area has been of lesser importance because the designergenerally desires to have as much Fowler motion (increase in effective chord) as possi- -ble which results in relatively small overlap regions.

CONCLUDING REMARKS

This paper has discussed in detail the theoretical and operational features and cap-abilities of the Langley "2-D Subsonic Multi-Element Airfoil Program." Several majormodifications have been made to improve the applicability and prediction accuracy of thisprogram and involve the use of:

(1) Curvature instead of cosine method to distribute segment corner points for'polygon representation of the airfoil •' '

(2) Oellers' stream function method to obtain the potential-flow solution

(3) Modified Kutta condition to reduce pressure oscillations at the trailing edge

(4) Truckenbrodt method to obtain the turbulent boundary-layer characteristics

Comparisons between experimental data and theoretical predictions indicate excel-lent agreement for single-element airfoils in attached flow and good agreement for multi-

729

element airfoils in attached flow with small overlap regions between elements. Threemajor areas of program improvement are needed and involve the computation of:

(1) Profile drag from downstream wake characteristics

(2) Flow characteristics after boundary-layer separation

.(3) Pressure corrections in the slot areas between overlapping elements of multi-element airfoils

The airfoil program has been demonstrated as an effective tool for the design and analy-sis of single-element and multielement airfoils in viscous flow, and has been widely dis-tributed to and utilized by both the academic and industrial communities.

730

' ' • • REFERENCES ' : '\ - , " , • " • • • • • - -

1. Stevens, W. A.; Goradia, S. H.; and Braden, J. A.: Mathematical Model for Two-Dimensional Multi-Component Airfoils in Viscous Flow. NASA CR-1843, 1971.

2. Smetana, Frederick O.; Summey, Delbert C.; Smith, Neill S.; and Garden, Ronald K.:. - • * ' » . • • - . - • • ,

Light Aircraft Lift, Drag, and Moment Prediction - A Review and Analysis. NASACR-2523, 1975.

3. Oellers, Heinz J.: Incompressible Potential Flow in a Plane Cascade Stage. NASATT F-13,982, 1971.

4. Cohen, Clarence B.; and Reshotko Eli: The Compressible Laminar Boundary LayerWith Heat Transfer and Arbitrary Pressure Gradient. NACA Rep. 1294, 1956.(Supersedes NACA TN 3326.)

5. Schlichting, Hermann (J. Kestin, trans!.): Boundary-Layer Theory. Sixth ed.,McGraw-Hill Book Co., Inc., 1968.

6. Truckenbrodt, E.: A Method of Quadrature for Calculation of the Laminar and Tur-bulent Boundary Layer in Case of Plane and Rotationally Symmetrical Flow. NACATM 1379, 1955.

7. Goradia, S. H.: Confluent Boundary Layer Flow Development With Arbitrary Pres-sure Distribution. Ph. D. Thesis, Georgia Inst. Technol., 1971.

8. Dvorak, F. A.; and Woodward, F. A.: A Viscous/Potential Flow Interaction AnalysisMethod for Multi-Element Infinite Swept Wings. Volume I. NASA CR-2476, 1974.

9. Callaghan, J. G.; and Beatty, T. D.: A Theoretical Method for the Analysis andDesign of Multi-Element Airfoils. AIAA Paper No. 72-3, Jan. 1972.

10. Bhateley, I. C.; and Bradley, R. G.: A Simplified Mathematical Model for the Analy-sis of Multi-Element Airfoils Near Stall. Fluid Dynamics of Aircraft Stalling.AGARD Conf. Pre-Print No. 102, Apr. 1972, pp. 12-1 - 12-12.

11. Powell, B. J.: The Calculation of the Pressure Distribution on a Thick CamberedAerofoil at Subsonic Speeds Including the Effects of the Boundary Layer. C.P.No.1005, British A.R.C., June 1967.

12. McGhee, Robert J.; and Beasley, William D.: Low-Speed Aerodynamic Characteris-tics of a 17-Percent-Thick Airfoil Section Designed for General Aviation Applica-tions. NASA f N D-7428, 1973.

13. Wentz, W. H., Jr.; and Seetharam, H. C.: Development of a Fowler Flap System fora High Performance General Aviation Airfoil. NASA CR-2443, 1974.

731

14. Omar, E.; Zierten, T.; and Mahal, A.: Two-Dimensional Wind-Tunnel Tests of aNASA Supercritical Airfoil With Various High-Lift Systems. Volume I - DataAnalysis. NASA'CR-2214, 1973.

15. Foster, D. N.; Irwin, H. P. A. H.; and Williams, B. R.: The Two-Dimensional FlowAround a Slotted Flap; R. & M. No. 3681, British A.R.C., 1971.

16. Harris, Thomas A.; and Lowry, John G.: Pressure Distribution Over an NACA 23012.Airfoil With a Fixed Slot and a Slotted Flap. NACA Rep. 732, 1942.

732

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