/ ARR Mo. L5H18 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGlNALLy ISSUED Septeniber ip'*-^ as Advance Restricted Report L5HI8 HUMEa^ICAL EVALUATION BY HARMONIC ANALYSIS OF THE € -FUNCTION OF THE THEODORSEN ARBITRARY -AIRFOIL POTENTIAL THEORY By Irven Nalman Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA ' WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. 153 DOCUMENTS DEPARTMENf Digitized by tlie Internet Arcliive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/numericalevaluatang 3^ ^^ NACA ARR No. L5Hl8 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT NUMERICAL EVALUATION BY HARMONIC ANALYSIS OF THE e -FUNCTION OP THE TKECDORSEN ARBITRARY- AIRFOIL POTENTIAL THEORY By Irven Nalraan SUMMARY A finite trigonometric series is fitted by harmonic analysis as an approximation function to the ^Jz-function of the Theodorsen arbitrary-airfoil potential theory. By harmonic synthesis the corresponding conjugate trigo- nometric series Is used as an approximation to the e-functlon. h set of coefficients of particularly simple form is obtained algebraically for direct calculation of the e--/alues from the corresponding set of \|f-values. The formula for e Is k=l where the surniTiation is for odd values of k only and K = ^K^ + —) INTRODUCTION In the determination of the flovv' about an arbitrary airfoil (references 1 and 2) the problem arises of trans- forming a curve, nearly circular, into a circle. This transformation, a basic problem, in conformal m.apping, is further reduced to the determination of the following two conjugate Fourier series: NAG A ARR No. L^HlS ~N \ ^' = aQ + y' (hjj, cos mcp + b^^^ sin mcp) > - \ in=l {b^ sin m(p - b,^ cos mo^ ^ (1) (See references 1 and 2 for significance of notation.) The followlnp; integral relations are equivalent to series relations (1 ) : n2TT -!/(o) = 2tt ?^ TT \|/ ( cp ' ) dCD • + r— c:Tr e(cp' ) cot — ■ — ^ d'Cp' > (2) PStt €(;o) = - 2Tr 'C/(cp' ) cot ^e-:^ dcp. -^ It is con'v'enlent to introduce a new variable s = q;f _ cp in relations (2). Because of the cyclical nature of these functions, the Units of integration may be Yi'ritten -rr to ir. V.'hen the Integral is broken into tn'o parts, -TT to and to rr, and -s is substituted for s in the first part, the folloring relations are obtained: 'i;(cp) p2n _^ P^ _ - e( '^ \'-V - sj I cot p ds e(c?) = ~ O TT > (3) It'i'X) - s) - \!;(q) + s jj cot 5" ds J Thus, by use of relations (1), (2), or (3), e may be determined if it is known or \j/ may be determined if e is known. MCA ARR Fo. L5Hl8 In the airfoil problem \{/ Is specified as a function-'- of cp by means of a curve and e Is to be deterrdined. In theory the Fourier coefficients may be determined in relations (1) but in practice, because of the unknovm analytic nature of the curve, it is neces- sary to resort to some type of numerical approximation. In references 1 and 2 an ap-prcximate method of han- dling the integrals of relations (2) is presented. In reference 5 a refinement of this method is given for the saxae Integrals. An alternative procedure is to approxi- mate relations (1) by a finite trigonometric series and then to determine the coefficients by harmonic analysis. A development of this method is now given. HARMONIC ANALYSIS The \l/-f unction is t o be approximated by a finite trigonometric series given by \!;(cci) = A,-; •*- A-, cos (i) + . . . + ^-,_i cos (n - 1 )cp -I- ixj^ cos nq) + 3]_ sin CD -I- ... + 3j^_]_ sin (n - l)cp n-1 = Ap. -♦- y {f\^ cos mcp + B^^. sin m.cp J + A^ cos ncp m^' If ''I' is specified at 2n equally spaced intervals in the ranp-:e (2n - 1)tt the range < cp < 2Tr - that is, 0, — , — , . . . , _ _ ' ' n n - then n A 2n-l in v=0 In practice, 'ii is given as a function of = cp - c and therefore '^ is t al-ren as a function of cd as a first approximation. An iteration process is neces- sary to determ.ine both \]; and e correctly as func- tions of cp. ::aca arr no. l5hiS - — : -^ cos n — r=J _ 1"^ ^ zz ~ -> xj sLn ^ - r. / = sow £(c) = / (-^ sin r.Z - B^ ccs x") + k^ sin ng = — ^ (sin. r:Cp^ \{fj, cos :n -^ - cos siT \ tj/j, sin n— | + j^ -in nc> (-1)^ ^/^ 1 ^T \ / Tr\ i t r .'.en. the order of sussiation is interchanged. 2^-1 "-^ 2^-2. rvr.j=,- . .'^ > sir. r-i- - — ) + r— sm ncp / .-±/ r. / - / ^ V n / £m / r-=j -'::=i p--, Nov.' ix c is evai-aated at xhe sane roints r st tji c^ - t; - « - --i the values of ■;!? ■r-. ! TV C = = , the variable cecoEies , -— — - ^"j - - / 1 : a. • r-. and the last tem be cones zero, lifter this is "c-erJ^cr^e ci . and 2n-l r--l €(p) = - -y V.^ > sin kr The suinnation with respect to k is alsc rrcs D to 2r: - 1 tecause of the periodic itT. But; 'shen k is odd. \ . l-TT ^ kTT y- s-^z n — = eot zr- n ^n iri=i ar.5 w^nen a is even. \ kr / s :Ln ~ — — O e(G) = -- ^ '.-. cot ~ NACA ARR No. L5Hl8 cv n '='> = n-z:e-K-\)-*g Finally, then, n ^Cf) => =i.(*-,c- \)