1、National Aeronautics and Space AdministrationLangley Research Center Hampton, Virginia 23681-0001NASA Technical Memorandum 4741Computer Program To Obtain Ordinates forNACA AirfoilsCharles L. Ladson, Cuyler W. Brooks, Jr., and Acquilla S. HillLangley Research Center Hampton, VirginiaDarrell W. Sprole
2、sComputer Sciences Corporation Hampton, VirginiaDecember 1996Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Printed copies available from the following:NASA Center for AeroSpace Information National Technical Information Service (NTIS)800 Elkridge L
3、anding Road 5285 Port Royal RoadLinthicum Heights, MD 21090-2934 Springfield, VA 22161-2171(301) 621-0390 (703) 487-4650The use of trademarks or names of manufacturers in this report is foraccurate reporting and does not constitute an official endorsement,either expressed or implied, of such product
4、s or manufacturers by theNational Aeronautics and Space Administration.Available electronically at the following URL address: http:/techreports.larc.nasa.gov/ltrs/ltrs.htmlComputer program is available electronically from the Langley Software ServerProvided by IHSNot for ResaleNo reproduction or net
5、working permitted without license from IHS-,-,-IntroductionAlthough modern high-speed aircraft generally makeuse of advanced NASA supercritical airfoil sections,there is still a demand for information on the NACAseries of airfoil sections, which were developed over50 years ago. Computer programs wer
6、e developed in theearly 1970s to produce the ordinates for airfoils of anythickness, thickness distribution, or camber in the NACAairfoil series. These programs are published in refer-ences 1 and 2. These programs, however, were written inthe Langley Research Center version of FORTRAN IVand are not
7、easily portable to other computers. The pur-pose of this paper is to describe an updated version ofthese programs. The goal was to combine both programsinto a single program that could be executed on a widevariety of personal computers and workstations as wellas mainframes. The analytical design equ
8、ations for bothsymmetrical and cambered airfoils in the NACA 4-digit-series, 4-digit-modified-series, 5-digit-series, 5-digit-modified-series, 16-series, 6-series, and 6A-series airfoilfamilies have been implemented. The camber-line desig-nations available are the 2-digit, 3-digit, 3-digit-reflex,6-
9、series, and 6A-series. The program achieves portabilityby limiting machine-specific code. An effort was madeto make all inputs to the program as simple as possible touse and to lead the user through the process by means ofa menu.SymbolsThe symbols in parentheses are the ones used in thecomputer prog
10、ram and in the computer-generated listings(.rpt file).A camber-line designation, fraction ofchord from leading edge over whichdesign load is uniformaiconstants in airfoil equation, i = 0, 1, ., 4biconstants in camber-line equation, i = 0,1, 2c (C, CHD) airfoil chordcl,i(CLI) design section lift coef
11、ficientdiconstants in airfoil equation, i = 0, 1, 2, 3dx derivative of x; also basic selectableinterval in profile generationd(x/c), dy, d derivatives of x/c, y, and I leading-edge radius index numberk1, k2constantsm chordwise location for maximum ordi-nate of airfoil or camber linep maximum ordinat
12、e of 2-digit camber lineR radius of curvatureRle(RLE) leading-edge radiusr chordwise location for zero value of sec-ond derivative of 3-digit or 3-digit-reflexcamber-line equationt thicknessx (X) distance along chordy (Y) airfoil ordinate normal to chord, positiveabove chordz complex variable in cir
13、cle planez complex variable in near-circle plane local inclination of camber line airfoil parameter, complex variable in airfoil plane angular coordinate of z angular coordinate of z airfoil parameter determining radial coor-dinate of zAbstractComputer programs to produce the ordinates for airfoils
14、of any thickness, thick-ness distribution, or camber in the NACA airfoil series were developed in the early1970s and are published as NASA TM X-3069 and TM X-3284. For analytic airfoils,the ordinates are exact. For the 6-series and all but the leading edge of the 6A-seriesairfoils, agreement between
15、 the ordinates obtained from the program and previouslypublished ordinates is generally within 5 105chord. Since the publication of theseprograms, the use of personal computers and individual workstations has prolifer-ated. This report describes a computer program that combines the capabilities of t
16、hepreviously published versions. This program is written in ANSI FORTRAN 77 and canbe compiled to run on DOS, UNIX, and VMS based personal computers and work-stations as well as mainframes. An effort was made to make all inputs to the programas simple as possible to use and to lead the user through
17、the process by means of amenu.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20average value of ,Subscripts:cam camberedl (L) lower surfaceN forward portion of camber lineT aft portion of camber linet thicknessu (U) upper surfacex derivative with re
18、spect to xComputer Listing SymbolsFor reasons having to do with code portability, thecomputer-generated listing (.rpt file) will always havethe alphabetic characters in upper case. The followinglist is intended to eliminate any confusion.A camber-line designation, fraction of chordfrom leading edge
19、over which design loadis uniformA0,.,A4 constants in airfoil equationCHD, C airfoil chordCLI design section lift coefficientCMB maximum camber in chord length,CMBNMR number of camber lines to be combined in6- and 6A-series multiple camber-lineoptionCMY m, location of maximum camberCRAT cumulative sc
20、aling of EPS, PSI,D0,.,D3 constants in airfoil equationDX basic selectable interval in profilegenerationDY/DX first derivative of y with respect to x,D2Y/DX2 second derivative of y with respect to x,EPS airfoil parameter, = IT number of iterations to converge 6-seriesprofileK1, K2 3-digit-reflex cam
21、ber parameters, k1and k2PHI , angular coordinate of zPSI , airfoil parameter determining radialcoordinate of zRAT(I) ith iterative scaling of , RC radius of curvature at maximum thicknessfor 4-digit modified profileRK2OK1 3-digit reflex camber parameter ratio,RLE leading-edge radiusRNP radius of cur
22、vature at originSF ratio of input t/c to converged t/c afterscalingTOC thickness-chord ratioX distance along chordXMT m, location of maximum thickness for4-digit modified profileXT(12), location and slope of ellipse nose fairingYT(12), for 6- and 6A-series thicknessYTP(12) profilesXU, XL upper and l
23、ower surface locations of xXTP x/c location of slope sign change for6- and 6A-series thickness profileXYM m, location of maximum thickness for6- and 6A-series profiles or chordwiselocation for maximum ordinate of airfoilor camber lineY airfoil ordinate normal to chord, positiveabove chordYM y/c loca
24、tion of slope sign change for 6- and6A-series thickness profileYMAX y(m), maximum ordinate of thicknessdistributionYU, YL upper and lower surface y ordinateAnalysisThickness Distribution Equations for AnalyticAirfoilsThe design equations for the analytic NACA airfoilsand camber lines have been prese
25、nted in references 3to 7. They are repeated herein to provide a better under-standing of the computer program and indicate the use ofdifferent design variables. A summary of some of thedesign equations and ordinates for many airfoils fromthese families is also presented in references 8 to 10.The tra
26、ditional NACA airfoil designations are short-hand codes representing the essential elements (such asthickness-chord ratio, camber, design lift coefficient)controlling the shape of a profile generated within agiven airfoil type. Thus, for example the NACA 4-digit-series airfoil is specified by a 4-di
27、git code of the formpmxx, where p and m represent positions reserved for12pi- d02pipy m( )c-= RAT(I), I = 1 ITdydx-d2ydx2-k2k1-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-3specification of the camber and xx allows for specifica-tion of the thickn
28、ess-chord ratio as a percentage, that is,“pm12” designates a 12-percent-thick (t/c = 0.12) 4-digitairfoil.NACA 4-digit-series airfoils. Symmetric airfoils inthe 4-digit-series family are designated by a 4-digit num-ber of the form NACA 00xx. The first two digits indicatea symmetric airfoil; the seco
29、nd two, the thickness-chordratio. Ordinates for the NACA 4-digit airfoil family(ref. 2) are described by an equation of the form:The constants in the equation (for t/c = 0.20) weredetermined from the following boundary conditions:Maximum ordinate:Ordinate at trailing edge:Magnitude of trailing-edge
30、angle:Nose shape:The following coefficients were determined to meetthese constraints very closely:To obtain ordinates for airfoils in the family with athickness other than 20 percent, the ordinates for themodel with a thickness-chord ratio of 0.20 are multipliedby the ratio (t/c)/0.20. The leading-e
31、dge radius ofthis family is defined as the radius of curvature of thebasic equation evaluated at x/c = 0. Because of the terma0(x/c)1/2in the equation, the radius of curvature is finiteat x/c = 0 and can be shown to be (see appendix)by taking the limit as x approaches zero of the standardexpression
32、for radius of curvature:To define an airfoil in this family, the only input neces-sary to the computer program is the desired thickness-chord ratio.One might expect that this leading-edge radius R(0),found in the limit as to depend only on the a0term of the defining equation, would also be the mini-
33、mum radius on the profile curve. This is not true in gen-eral; for the NACA 0020 airfoil, for example, a slightlysmaller radius (R = 0.0435 as compared to R(0) =0.044075) is found in the vicinity of x = 0.00025.NACA 4-digit-modified-series airfoils. The 4-digit-modified-series airfoils are designate
34、d by a 4-digit num-ber followed by a dash and a 2-digit number (such asNACA 0012-63). The first two digits are zero for a sym-metrical airfoil and the second two digits indicate thethickness-chord ratio. The first digit after the dash is aleading-edge-radius index number, and the second is thelocati
35、on of maximum thickness in tenths of chord aft ofthe leading edge.The design equation for the 4-digit-series airfoilfamily was modified (ref. 4) so that the same basic shapewas retained but variations in leading-edge radius andchordwise location of maximum thickness could bemade. Ordinates for these
36、 airfoils are determined fromthe following equations:from leading edge to maximum thickness, andfrom maximum thickness to trailing edge.The constants in these equations (for t/c = 0.20) canbe determined from the following boundary conditions:Maximum ordinate:Leading-edge radius:yc- a0xc- 1/2a1xc- a2
37、xc- 2a3xc- 3a4xc- 4+ + + +=xc- 0.30=yc- 0.10=dydx- 0=xc- 1.0=yc- 0.002=xc- 1.0=dydx-0.234=xc- 0.10=yc- 0.078=a00.2969= a10.1260=a20.3516= a30.2843=a40.1015=Rlea022-t/c0.20- 2=R1 dy/dx( )2+ 3/2d2y/dx2-=x 0yc- a0xc- 1/2a1xc- a2xc- 2a3xc- 3+ + +=yc- d0d11xc- d21xc- 2d31xc- 3+ + +=xc- m=yc- 0.10=dydx- 0
38、=xc- 0= Ra022-=Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4Radius of curvature at maximum thickness:Ordinate at trailing edge:Magnitude of trailing-edge angle:Thus, the maximum ordinate, slope, and radiusof curvature of the two portions of the s
39、urface match atx/c = m. The values of d1were chosen, as stated in refer-ence 4, to avoid reversals of curvature and are given inthe following table:By use of these constraints, equations were writtenfor each of the constants (except a0and d1) in the equa-tion for the airfoil family and are included
40、in the com-puter program. As in the 4-digit-series airfoil family,ordinates vary linearly with variations in thickness-chordratio and any desired thickness shape can be obtained byscaling the design ordinates by the ratio of the desiredthickness-chord ratio to the design thickness-chord ratio.The le
41、ading-edge index is an arbitrary numberassigned to the leading-edge radius in reference 4 and isproportional to a0. The relationship between leading-edge radius Rleand index number I is as follows:Thus, an index of 0 indicates a sharp leading edge(radius of zero) and an index of 6 corresponds toa0=
42、0.2969, the normal design value for the 20-percent-thick 4-digit airfoil. A value of leading-edge index of 9for a three times normal leading-edge radius was arbi-trarily assigned in reference 4, but I = 9 cannot be used inthis equation. In reference 4, the index I is not usedin computation. Instead,
43、 the index I = 9 was assigned toan airfoil where which(because ) thus has a three times normalleading-edge radius. The computer program is written sothat the desired value of leading-edge radius or the indexI is the input parameter. The value of a0is then computedin the program.NACA 16-series airfoi
44、ls. The NACA 16-series air-foil family is described in references 6 and 7. From theequation for the ordinates in reference 7, this series is aspecial case of the 4-digit-modified family although thisis not directly stated in the references. The 16-series air-foils are thus defined as having a leadin
45、g-edge index of 4and a location of maximum thickness at 0.50 chord. Thedesignation NACA 16-012 airfoil is equivalent to anNACA 0012-45.Thickness Distribution Equations for DerivedAirfoilsNACA 6-series airfoils. As described in references 9and 10, the basic symmetrical NACA 6-series airfoilswere deve
46、loped by means of conformal transformations.The use of these transformations to relate the flow aboutan arbitrary airfoil to that of a near circle and then to acircle had been developed earlier, and the results are pre-sented in reference 11. The basic airfoil parameters and are derived as a functio
47、n of , where isdefined as . Figure 1, taken from reference 10, showsthe relationship between these variables in the complexplane. These parameters are used to compute both theairfoil ordinates and the potential flow velocity distribu-tion around the airfoil. For the NACA 6-series airfoils,the shape
48、of the velocity distribution and the longitudinallocation of maximum velocity (or minimum pressure)were prescribed. The airfoil parameters and whichgive the desired velocity distribution were obtainedthrough an iterative process. Then the airfoil ordinatescan be calculated from these parameters by use of theequations presented in references 10 and 11. Thus, foreach prescribed velocity distribution, a set of basic airfoilparameters is obtained. However, as stated in refer-ence 10, it is possi
