An investigation on the effect of second-order additional thickness distributions to the upper surface of an NACA 64 sub 1-212 airfoil

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s a AN INVESTIGATION ON THE EFFECT OF SECOND-ORDER ADDITIONAL THICKNESS DISTRIBUTIONS TO i1fl" UPPER SURFACE OF AN NACA 64 1 -212 AIRFOIL JANUARY 1975 CONTRACT NAS 2-8599

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DONALD S. HAGUE ANTONY l4. MERZ Prepared by AEROPHYSICS RESEARCH CORPORATION Bellevue, Washington 98009 y a • r c"'' ^,a°^FC.- tea' era Originally Published as Aerophysics Research Corporation TN-194 N75-24674 AN INVFSTIC'eT10N Olr1 THE (NASA-CE-13'770 1) EFFECT OF SECOND-OFDEF A)DITIONAL TEICKKESS DISTFIBUTIONS Tie"rH% UPPEF SUiF'AC'E OF AN Unclas NACA 64 SUB 1-212 AIFFOIL (Aerophysics G3/02 2+427 47 p HC Research Corp., Y3ellevua, Warh.)

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TABLE OF CONTENTS n n (I) Page _ L .........................:.................................

SUMMARY J^ .......................^.............................

INTRODUCTION A 4i MATHEMATICAL MODELS PROFILE REPRESENTATION ....................''................

AIRFOIL ...........................................

PRESENTA31ONOF RESULTS B SYSTEMATIC VARIATION OF AIRFOIL SHAPING PARAMETERS ...- .............

9, CONCLUSION ........................................................

TABLE , I (Convergence for C L Maximization) .........................

TABLE II (Optimal Airfoil Shaping Results) ........................

d ........................................................

REFERENCES ii

LIST OF ILLUSTRATIONS L^ Description Figure i Biquadratic Additional Thickness Distributions 3 (a) - 3 (v) Lift Coefficient, Biquadratic Modifications to 641-212 Moment Coefficient, Biquadratic Modifications to 641-212 k r.

Lift and Moment Variations Biquadratic Modifications to 64 212 1- Biquadratic Modifications to 641-212 Additional Thickness Distribution to Minimize Peak Pressure Minimum Peak Pressure Obtainable with Biquadratic Modifications Cs^ tL„ g

4w ABSTRACT n This report describes a series of low speed airfoil designs based on modifications to the NACA 64 1 -212 airfoil. Designs are based on potential flow theory. This report describes one of a series of airfoil modifications carried out under Contract NAS 2-8599, Application of Multivariable Search Techniques to Optimal Wing Design in Non-Linear Flow Fields. Mr. Raymond Hicks of National Aeronautics and Space %dministration's Aeronautical Division, Ames Research Center, served {fs contract monitor for the present study.

.

y AN INVESTIGATION ON THE EFFECT OF SECOND-ORDER ADDITIONAL THICKNESS DISTRIBUTIONS TO L" THE UPPER SURFACE OF AN NACA 64 212 AIRFOIL 1- by Donald S. Hague and Antony W. Merz L Aerophysics Research Corporation SUMMARY 4s An investigation has been conducted on the Lawrence Radiation Center, Berkeley, CDC 7600 digital computer to determine the effects of additional thickness distributions to the up per surface of an NACA 64 1 -212 airfoil.

Additional thickness distributions employed were in the form of two second- order polynomial arcs which have a specified thickness, y, at a given chordwise location, x. The forward arc disappears at the airfoil leading edge, the aft arc disappears at the airfoil trailing edge. At the l:s juncture of the two arcs, x = x, continuity of slope is maintained.

r The effect of varying the maximum additional thickness and its chordwise location on airfoil lift coeeficient, pitching moment, and pressure dis- tribution was investigated. Rovults were obtained at a Mach number of 212 airfoil. All 0.2 with an angle-of-attack of 6 0 on the basic NACA 64 1- calculations employ the full potential flow equations for two dimensional flow. The relaxation method of Jameson is employed for solution of the potential flow equations.

ti Increases in the rearward location of the maximum additional thickness and increases in the magnitude of the additional thickness both produce increases in the airfoil lift coefficient. Conversely moving the location of maximum thickness forward or decreasing the maximum thickness both ''reduce the magnitude of the quarter chord pitching moment. The magnitude of the largest pressure peak varies in a complicated manner with maximum additional thickness and its chordwise location. For maximum thickness locations forward of the 2/3 chord additional thickness initially produces G a reduction in pressure peak with a lift coefficient increase. With larger amounts of additionalthickness pressure peak value and lift

coefficient rise together. For maximum thickness locations aft of the 2/3 chord location additional thickness produces a monotonic rise in both lift coefficient and pressure peak magnitude. A consequence of this behavior is that for a given lift coefficient value the peak pressure can be minimized by careful selection of the location of maximum thickness and its magnitude. Generally as the lift coefficient rises the maximum r.

thickness location moves aft. For a C of 1.2 the optimal location for L maximum thickness is at the quarter chord. For a C of 1.8 the optimal L location is approximately at the half chord.

It should be noted that viscous effects are neglected in the present analysis. At the higher lift coefficients the effect of viscosity could be significant. Further investigations incorpor*ting a viscous flow model are therefore desirable.

INTRODUCTION The National Aeronautics and Space Administration and others are n, currently conducting a. series of theoretical and experimental studies to define airfoil sections having improved performance from the aspects of lift,, drag, pitching moment or pressure distribution characteristics, i refs 1 and 2. Analytic investigations ;sing airfoil surface repre- sentations based ou high-order polynomials may result in impractical profi?es, for example, very thin trailing edge thickness distributions or severe reflexes in the profile. The present study employs low-order - -- polynomial arcs of second - order whose Characteristics are selected to avoid such problems. 'Optimization studies using multivariable search techniques, reference 3, generally indicate that shspe changes which ; provide increased lift produce unfavorable changes in moment characteristics. Conversely profile changes which improve the moment characteristics decrease the lift y3i ( coefficient. With the low - order model of the present investigation a } systematic examination on the effect of _profile changes " can be carried a nd tre nds revealed b optimization studies wereconfirmedishAend nterestingb the systematic y product of the

investigation of profile changes is that a gain in lift coefficient can be produced while reducing the peak negative pressures. This tends to decrease the pressure gradient and hence holds promise for the development of practical single component high lift coefficient airfoils.

MATHEMATICAL MODELS Potential Flow Equation Potential flow analysis is based on solution of the two-dimensional potential flow cTiation + Oyy-2uv`'0xy = ( a 2 -u2) ( a2 -v2 ) 0 x x is the velocity potential, u and v are the velocity components where U = 0x , V-^: 0y and a is the local speed of sound determined from the energy !aquation`and y: the stagnation speed of sound a 2 (^2 1 = v2) a o 2 - ) ( n2 ii Solutions are obtained by Jameson's finite difference scheme, reforenc `e 4," AIRFOIL PROFILE REPRESENTATION BasicAirfoil -212 airfoil were approximated by Ordinates for the basic NACA 64 1 four cubic chain polynomials in the manner of Hicks + a2.x2 + a 3. x 3 ; j = 1,2,3,4 e a o. F 1 + a l. x yj = ^ J J J .

Cot,' ,'ueients in the four polynomial arcs are selected on the following C

Are represents forward portion of upper surface i = l. - Fl=V^ Are represents aft portion of upper surface 1 = 2. - F 2 1 Are represents forward portion of lower surface i = 3 - F3 VrX_

=

i`_a - Arc represents aft portion F4=1 The coefficients a are determined by introducing four boundary conditions on the .ui °9°oil profile in each of the four airfoil arcs. Crout's method for trajngulaeication and back substitution of the resulting systems of linear simultaneous equations. Note that if four points are specified on the r aft portion (i = 2 oi'4), a discontinuity in slope occurs where the poly- numials join. This produces a small ripple in the pressure distribution at the juncture point. However, since the juncture occurs at a region of small slope (x = .5) the effect is not significant. The approximate NACA 64 1-212 airfoil developed by this method is presented in Figure 1.

Additional Thickness In the present study additional thickness is limited to the upper airfoil surface. The additional thickness has the form

x

; Ay (x) - y I - f X %x) 2] 1 \ /

r /

A =YI1-/ x=x x Y(x)

LL ` 1-x I

J

These functions are of second-order ranging parabolically with 4 _ ix-xl.

Additional thickness is zero at the ending edge (

x =0) and trailing edge (x=1)

at x=x. Additional thickness and slope of6ithe and has a maximum of Ay=y additional thickness are continuous throughout the interval o < n <1. Second derivative of the additional thickness distribution is constant in the forward and aft airfoil arcs but has a discontinuity at the ,arc junction, x=x. It follows that a continuous polynomial representation of the additional n

C, thickness distributions, valid in the interval o <n <1 would be in the form of an infinite series. This type of additional thickness distribution P, is referred to as a "biquadratic" function in recognition of the above characteristics. A sequence of biquadratic arcs having varying maximum thickness positions are presented in Figure 2.

ra PRESENTATION OF RESULTS Additional Thickness Optimization Lift Coefficient Maximization Maximization of lift coefficient has the form = Max (C L ] where CL = iAp(x)dx and the integration is around the airfoil contour. Since the airfoil contour is completely described in terms of the two parameters x and y p = Max [C L ] = Max [C L (" Y)] where xL< x5 xM YL5 Y:5 YM This two variable multivariablr search problem was solved by a combination of directed random-ray and pattern searches, Ref. 3. Table I presents the results of 30 iterations using these search procedures. Lift gains are produced at 27 of the 30 iterations and continue to be made at the compu- tati_on termination.

Optimization has moved the position of maximum thickness to the most rearward position allowed, x = 0.9. At termination, lift is increasing monotonically with increasing thickness, y. Based on this isolated result lift is maximized for additional thickness of the form assumed by moving the position of maximum additional thickness as far aft as allowed and ^2 introducing as much additional thickness as allowed.

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Moment Minimization Minimization of the momor!,4,coefFicient has form A = Min [CM^ where ( u , "m ({^ (x - J^) ep(x)dx n;ten'ui^i aY "son rosolted in a solution directly opposed to lift .,f Al ,..]^i. The psi=ition of maximum thickness moved to the forward and the amount of additional thickness was minimized. That is, the basic NACA 64,-212 airfoil has less adverse moment than any airfoil generated by addition of biquadratic thickness to the upper surface of the airfoil.

Lift vs. Moment Tr a de Preliminary work using other airfoil thickness representations has indicated that the requirements of lift maximization and moment minimization oppose each other. This is confirmed by the results reported above.

It has been found as a re,ult of previous studies that it is very difficult to produce an airfoil for which C M <'.177 - .22 CL This function has been used to define airfoil which have favorable lift/ moment dyinracteristics by solution of the problem j p = Min 1.177 - .22 C L - CM I - Solution of this problem by directed random-ray and pattern search I indicates that additional thickness should be added as far forward as possible and that maximum amount of additional thickness should be employed.

Optimization Strtttiiary Three optimal airfoil results have been obtained consistent with the class of airfoil profiles considered here. These results are summarized in Table II. It can be seen that in all cases the position of maximum thickness, z, is either at the extreme forward or rearward position allowed.

Similarly, depending on problem specification, the amount of additional"„ OF ^ P 0^^j QU GE

AID

I _..,icss is either minimized c —,, maximized. The low dimensionality of

this problem (two parameters, x and y) permit a ready mapping of these

results as a function of x and y. This is done in the following section.

it SYSTEMATIC VARIATION OF AIRFOIL SHAPING PARAMETERS II A systematic investigation on the effect of variations in the airfoil shaping parameters x and y was undertaken. The resulting airfoils and q calculated pressure distributions are presented in Figures 3(a) to 3(v).

It should be noted that the-airfoils are not drawn to scale in Figure 3.

G c toristics the vertical scale is exagerated.

To emphasize profile chari, The pressure signatures vary in a radical manner with R and P. The basic airfoil exhibits a sharp pressure peak at the leading edge. The magnitude i of the plak pressure is reduced by introducing additional thickness in j a forward location, x = .1, and the peak position moves aft. However, if the amount of additional thickness is increased the pressure peak sagnitude This peak is well aft of the leading edge. This effect again increases.

persiiits vnt,il rearward locations of x are encountered. For example, introducin,.^ ;additional thickness at x = .8 results in a rearward "hump" The increased circulation produced by this in the pressure distribution.

hump results in an increased leading edge peak in the c.'irfoil pressure {{ Flow separation would probably be encountered with these distribution.

II rearward additional thickness distributions unless devices such as ^j rotating cylinders or blowing were employed.

Figure 4 illustrates the effect of varying position of maximum It can be seen that thickness and maximum thickness ° on lift coefficient.

lift coefficient is maximized by increasing both x and y. This con'irms i Since the additional thick- optimization studies in the previous section.

u, ( ness and the basic 12% airfoil thickness are additive Figure 3 presents To first-order the airfoil lift coefficient as a function of thickness.

thickness required is 12% + y k^ t/c ORIG .^^ P,`i^^ L 01.1 POOR RUE ly

As it moves to the extremes of the range the actual airfoil thickness is less than this amount as the positions of maximum thickness on the ;asic and additional thickness distributions are significantly different.

Moment coefficient variation with i and y is presented in Figure S.

It can be soon that the increased lift available from additional thickness ' is accompanied by an increase in undesirable pitching moment coefficient.

'rhe conclusion of the previous section that moment coefficient is minimized by moving z forward and diminishing is borne out by Figure 5, again confirming the_ optimization study results.

A final verification of the optimization procedures employed is provided_by Figure 6. Here the variation of C M and CL with i and y is presented together with the line function 22 CL - CM = 0 .177 - .

It can be _seen that based on this function the most favorable C M - CL trade involves moving i forward and introducing the maximum y.

Figure presents the relationship between pressure peak and lift coefficient for 'a range of i and y values. For each value of y (maximum additional thickness) there is a point at which the pressure peak magnitude is minimized. Cross plotting the peak pressures as a function of CLin Figure 8 reveals the minimum peak pressures as <a function of CL.

Figure 9 plots the position of maximum additional thickness as a function,'of C L . As C L increases i moves aft. The associated values of y required for the low peak pressure is also plotted in Figure 9. Finally, Figure 10 plots the minimum C attainable as a function of C L using the biquadratic additional thickness airfoil model.

CONCLUSION A numerical investigation into a class of modified airfoil shapes has been completed using full two-dimensional flow potential flow equations.

Airfoils studied were obtained by modifying-the NACA 64 1 -212 airfoil by additional thickness distributions based.on a biquadratic variation with OMGINAL OP POOR UAGE Q ALIZY b

c^ chordwise position. Free streamwise Mach number was held constant at M = 0.2 and the basic airfoil is hold at 6 o anglo-o£-attack. Results of the study may be summarized as follows: Significant changes in pressure distribution, lift and 1.

moment can ba introduced by the biquadratic pitching thickness modification.

The requirements for improving lift and moment co- 2.

efficient characteristics are directly opposed to each other. That is, increases in lift result in increases in adverse moment. Conversely, decreases in adverse moment peoduce deerenj es i< lift.

High lift airfoils require the addition of a thickness 3.

distribution biased to the rear of the foil and as much thickness addition as possible.

Low adverse moments require a thickness distribution 4.

biased to the front of the foil and as little additional thickness as possible. Therefore, the best airfoil based on moment considerations is the unmodified foil.

off characteristics are obtained S. Favorable lift / moment trade - by a thickness distribution biased to the front of the foil employing as much thickness as possible.

6. There exists a class of airfoil exhibiting low peak pressures for a given CL which require an intermediate location of maximum additional thickness and thickness amount. Generally, the position of maximum additional thickness moves to the rear with increasing C L , and the amount of additional thickness required increases with increasing CL.

ORIGINAL PAGE f w OF pOOB QUALM

g TABLE I C CONVERGENCE FOR C MAXIMIZATION L ^I

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x -CL Y i FUNCTN( 1) ?SUOA( 1) 1 'ALPHA( 2) ^0 .^ UUOOE-03 -;8,41 1 Ji j

U . X08 1" U36^t:-03 -.8942

3 .2583 1.1667E-03 -,5952 10 1 5 .2626 1r4155E-03 -,8968 10 1 6 .2753 1.8311E-03 -.8982 2 1 2 1 7 .2879 - 4 21166E-03 -09022 } .3132 3..0776E-03 +.9078 2 1 3 4..7398E-03 -,9196 2 1 9 .3638

10 2 10 .3688 4..7533E-03 -.9198

i0 2 11 8 502172E 03 ''-.9216

10 2 12 .3824 5..5611E-03 -.9258 2 2 13 .4010 6.3824E-03 -:9320 ' . 1 1196 7:2U37E-03 -,9385 2 2 14 2 15 .4567 8".8462E-03 -.9520 16• .5311 • 1.2131E-02 -;9826 „ 2 2 .5505 1,2761E-02 -.9863 10 3 17 10' 3 .5675 1;2812E.-02 -;9920

0 58.08 1.494E-02 -.9993

j 10 3 20 - .6305 2 3 21 1..4856E-02 -1.018

.6802

2 3 22 1.6219E-02 --1.041

1x89/1/IE-02 -1.105 2 3 23 .7746 19000 2 3 24 2.439/IE-02 10 11 25 .9000 2 "U-1.6E-02 -is313 ^^^ ^0 4 26 .9000 2.7157E-02 -1.350 It 10 4 27 ,9000 i; 3.0116E-02 -1:399 2 4 28 9000 -°" 3,5838E-02 -1':493°-

2 4 29 9

4r1560E-02 -1:586

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2 4 30 5.30.03E-02 -1,765

4 9000'"

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ORIGINAL PAGE IS OF POOR QUALITY

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TABLE II OPTIMAL AIRFOIL SHAPING RESULTS r ,, Maximum - Thickness Position Thickness Problem

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Min CM Forward Max.

Min CM / C L Trade - 0

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n j( REFERENCES ii Ll Hicks, R: M., Merman, E. M., and Vanderplaatz, G. N., "Assessment 1.

of Airfoil Design by Numerical Optimization," NASA TMX-3092, July 1974.

Liebeck, it. H., "A Class of Airfoils Designel..for High Lift in 2.

Incompressible Flow," J. Aircraft, Vol. 10, No. 10, October 1973.

Hague, D. S. and Glatt, C. R., "An Introduction to Multivariable 3.

Search "techniques for Parameter Optimization (and Program AESOP)," NASA CR73200, April 1968.

Jameson, A. "Transonic Flow Calculations for Airfoils and Bodies 4.

of Revolution," Grumman Aerodynamics Report 390-71, December 1971.

^L^ ft 1i,