, 3 1176 01306 6213 I 

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NASA-CR-176403 

19860003832 

A Reproduced Copy---- ------

OF 

Reproduced for NASA 

by the 

NASA Scientific and Technical Information Facility 

FFNo 672 Aug 65 

liBRARY COpy 

II'!.!' / f} ~f __ r) 

\.t\NGLEY RESEARCH ClN' lit 

LIBRARY. NASA 

LANGLEY STAl'Ui'I 

HAMPTON. VIRGINIA 

111111111111111111111111111111111111111111111 

NF00458 

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(UASA-Cb-17b~03) & DIR£CT-INvtRSE nET80D N86-IJJO) 

POB TB1HSOHI~ l~D S~PABA1En fLU.~ ABUUT 

AluFOILS f~nal Report (TeX4S A&A Un~ •• ) dO p He A05/KP A01 CSCL 011 G3/u2 Unclds O,,9dl 

College of Engineering 

Texas A&M University 

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A DIRECT-INVERSE HETHOD FOR TRANSOtHC AND SEPARATED 

FLOWS ABOUT AIRFOILS 

TAHRF Report No. 3224-85-12 

Dtclmblr 190:S 

Fin&l R.port 

NASA GrAnt NSG-U74 

Leland A. CArllDn 

Prof 0 ncr 

AerOGp&ee Enguw.rtng Dep:.rtment 

Texas A&H UOlv,rltty 

College Statton. TX 77843-3141 

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A DIRBCT-INWRSE HETHOD FOR TRANSONIC AND SEPARATED 

FLOWS ABOUT AIRFOILS 

AerospAce Engina.ring DePArtment 

Tex .. A&H University 

SUHHARY 

A d1rect-inv ..... t.chni~ and comput ... program call.d TAHSSP that CAn b. 

us,d for the analys.s of th' flow about ail'foU. at subsonic And low tran.onic 

frentream Vftlodtlltl is pt'eunt.d. Th. mttthod is based upon A direct-invers' 

noncon •• rvativ, full pot.nU&l invilScid m,thod, a Thw&it .. lAminar boundary layltr 

technlqul, And th. Barnw.ll tutbul.nt momentum 1nt.gr41 ,ch,me. and it ic 

formulated using CArtu1a" coordinat... Sinc, the mltthod utilulS inv ..... 

bouMary conditions in regions of IIparated flow, It i. SUitable for predlctmg the 

flowUeld about airfOIls havmg trailing Idg •• eparated flow under high lift 

conditIons. ComparIsons WIth ,xperlm@ntd data indlcatlP thAt th. m.thod should 

be a us.ful tool for apphed aerodynamIC analyses. 
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SYMBOLS 

is.ntropic spud of sDU'WS 

pr.ssur. coeHici.nt 

Hach numb.r 

v.l.xity 

velocity component in th. x or y direction r .. ~ctiv.ly 

transformtd v.locity at bcx.mary lay.r tdgt 

v.locity in the boundary laytr 

law-of-th.-wall and law-of-th.-w&~ v.locity pAramltt.ra 

Cartt.ian coordlnatls 

angll of attAck 

raUo of aptclflc hlats, Issum.d to b. 1.4 

circulation 

boundary lay.r thlckr .... 

polar coordlnate 

computational coordlnAt .... 

pot.ntlal function 

perturb" bon potlntlal 

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SYHBOL~ (Continued) 

Sublcripts: 

I , 01' f ..... t,. ... m condition 

b body 

• boundary laYI,. Idg_ 

i,J grid location 

LK ll&ding Idgl 

TB trailing Idgl 

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I S ,{,x,)' diffl,.entia bon 

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GKNERAL DESCRIPTION 

INTRODUCTION 

have be.n developed And aucenafully used for the deaign &nd anAlY8il of 

subsonic and tranaonic Airfoils At And ntar crui .. conditionl. Hown.r, in the 

AnAlYlil of high p.rformAnc. Airfoils, &lrodynAmiciltl would aleo 11J<e to b. Able 

to prldict Airfoil prnsure diltrlbutions And .. radynamic cC4ffici.nts &t high lift, 

high angll-of-attAcK conditions. Sinci sueh situatIons Are frequtntly 

charAC terized by ngions of Slp&r&tld flow on the ~r Ilurfaco And are 

dominated by strong viscoul interactIon effects, inviscid methods alone Are not 

applicable. Furthermore, subsonic-transomc an&lysis ml!thod~3,4 WhICh couple 

lnviscld and boundAry layer solutions tYPically only include tho effectl of weal< 

VISCOUS Interaction and generally fall to gIVe accurate rnults wh.n sepArated 

flow exists on the upper surfaci. 

However, it hu betn d.m;:)flstrat.dS- 8 that the dlrtCt-lnVerse techmQue 

coupled to A !SUItable boundary layer method can be 6uCCIPBsfully appl1ed to low 

speed flows About alrfollts having maSSlV, s.parAtion. In addItion, Barnw.1l9, 

DvoraK and Choi lO, ~nd Taverna l1 have dcpvrt)oped slmJ1ar m.thods for transonIc 

flows. Barnwell's method, however, is limited 1n apphetltlon In that it utllizes 

for its lnviscid solv.r th. trAnlomc small perturbll.t&on equation. Further, 

Referenci. 9 and 11 only lnclude the effects of VISCOUS lnteractlon due to a 

turbulent boundArY layer. On mOlt urfol11, partIcularly on thl lower surface At 

hlgh angles of attacK, ext~mave re910ns of lamlnar flow l)(l .. t. 

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This report dncrlb .... flow model ~nd computer progr .. m, caUed TAHSEP, 

which can be uud to predict tho flowfiold about. ISingle olemont transonic airfoil 

at high angle of AttAck high lift conditions with trAmng Idge separAtion. Sine. 

the m.'hod i .. baud upon thlt TRANDES4 and TRANSEp6 codn, it can als", b. 

us.d for subsonic-trAnsonic Inalys •• not involving "'par&tion. 

HETHOD OF APPROACH 

the TRANDES and TRANSEP program .. and thl ability of this m.thad to USI either 

the displac.m.nt .. uN£CI (drfoil ordinate plus diQPlacomllnt thicl<neu) or 

pr ... uro as th. 1111'4011 boundary condition. For the high 4ngl.-of-attatl< tlSt. 

the airfoil low.r .. urface only ctx~rilncn wuk visccu. int.raction and 

frequently has .. long lammAr run bRfore trlnsltionmg to fully turbulent flow. 

Thus, the prelent mod.l includu In inl'Hal laminar boundary layer calculation In 

its viscous mbrcUon slctlon. On thl upp.r surfac. the boundAry layer lSi dID 

inti&l1y laminar, but lt qUickly btcomn turbulent in char&eter followltd In many 

casel by boundary layer separation and A uparAted zone which can extend over a. 

Significant portion of tho alrfoll surface. In the preunt moC:.l, this separAtl"d 

r.glon is trIA ted inversely In that the pressure distribution dong the effectlvlP 

dlsplacment surfacit 5trftlmlin~ lS detlrmme" it.rat1v.ly &S p~rt of the solution 

and used as the .urfol1 boundary condltion. Consequently, tho presllnt method has 

To obbm the lnVlfOCld portlon of the flowueld, the full potential equation ior 

two-dimenSional compreS5lblit flow IS used 1M nonconservatlve form &s 

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whtNJ the subscripts denote partial diff.rlntiation. By dllfinin; & p.rturbation 

pot.ntial. f1. such that 

wh.re the v.lacity components are giv.n by 

U:: 11 : 9.( co$fi ttP .. ) 

(2) 

(3) 

th" governing .quation in terms of the a:.erturbation potential can be written &Si 

(4) 

with 

(5) 

The nonconSlervAtive form of thl ~.)tent1al equatIon WAY BIlleted for thl'l 

prn.nt problem b.CAUSI! for two-dImensIonal flow5 result5 obtained With it 

agree bItter with Eule" tlolution!S than those obtained udn~ the fully 

cooSRrvativlt form of the equAtion.12 In addition, the conservative formulatIon 

appeArs to breaK down in two-dlmenl51onal caS.9 shortly after the onset of 

sup.reriticA} f10w.13 

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In the ~ .. nt mod.l, EquAtions (3-5) ar. linit. diff.r.nced using a roat.d 

diff,r.f1C. sch.m. And solv.d iteratively using column reluaticn in a str.tch.d 

Ctlt'tniln grid which maps the infinite domain to a linitlt computational box. The 

appropriate boundary condition lot infinity is 

f' -, tP = -;;. TOlt (( ,-m;," (9 - r#,») (6) 

wh.r. " is the polar angl., and" is the circulation. which is d.t.rmined by the 

chang. in pot.ntial across the Kuita-Joukowsl<1 cui at the trailing .dg. of the 

airfol1. 

liJ<.wiSl, the appropriate aIrfoil boundary condition in the dir.ct r.gions 

Cr.gions without Sltparitlon having only WlAk viscous interaction) is the flow 

tangency condition given by the ardinatn of the airfoil displlC.m.nt surfac., i .•• 

m 

In the inverse ar separated flow region the pressure distribution along ther 

.ff.ctiv. dlsplace-m.nt surfaclt streambne is considered specified and used as the 

boundary condihon. As shown in Ref.rence 2, thIS apprOAch leads to a. derivative 

boundary condition for the invetrs. region of the form 

(8) 

Complete details concerning the finite difference schema, the stretched Carte!uan 

grid system, And the treAtment of the boundary condlt10ns are given In 

References 1,2, and 4. 

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To include viscaus effect., the basic &pproach is to c&lculate A boundary layer 

displacement thickn .. s far the weal< interaction reglans and to use it to corrut 

the location of the displacement slJl'fa.ce U.e., airfoil ordinAte plus d1spl~cament 

thickness). Far the str~ly intar&cting .'par&tltd zone on the ~r surfI.clt, the 

pressure is determined from the inter&ctlon solution and thl location of the 

displacemltnt surface is computed by inteorating thlt surface tangency condltion, 

Eq. (7), with the initial conditions specified by the displacem.nt surfac. 

ordinatn at the nparatio., point, which is the interface between the two regions. 

The locahon and 9lopes of the displacement surfac •• are updated regularly 

throughout the it.rative solut10n. 

In the pr~ .. nt method, the laminar portion of the bounda.ry lAyer is ccmputad 

using a compruslble Thwait.. method siml1ar to that used pre\tl0ut,ly in 

TRANSEP.6 Thct trlnsibcr'. loc&t1on lS determined from a Granvllle tl pe 

correla.tionl4 bued upon th. diff.r.ncl' betwlten the local momentum thlckness 

Reynolds number l.nd the value at the lamlnar lnstab1lity point combined With the 

pressure gradient history. Som.t1mn, part1cuh.rly on the upper surface- at high 

angl.s of attacl<, laminar separat10n IS predIcted upstream of the ~rt1nslhon 

point. In then cnn, the local mom.ntum th1cknns Reynolds number 15 compar~d 

to a.n empirlcal correlatlon in order to determine if the l&mlOa!' bubble 15 long or 

~ort. If the bubble is short, Its length 15 ASsumed to bl 001 hOi'lZonta.l dllh.-I< 

grld wldth And the turbulent flow computAtlon 15 initiAted at the n2xt do\vnstream 

grid pomt. If the estima.te lOd1Cltu tha.t the bubbllP 15 long, the calculAtion 

proceeds, but a warmng 15 prmted Which Ind1ca.tes that the results are probAbly 

1n error. 

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Aftlr transition, the turbul.nt boundAry later is computed using the 

limplifiad Kuhn And liiol .. n mlthod (SKAN) AS developed by BarnwIU in Reflrlnc" 

9. This mlthod WAS .. llctld bltcaus. it is Iffidlnt, Ntliabllh and yillds Ixclllint 

predictions of displacement thicknes .. s and siparation point location. The SKAN 

tlrbulent boundary layer method solvn the intl~r&1 ~o=:'m. of the momentum 

lquation, moment of momentum equation, and the derivative of the Cole.' 

law-of-th.-wa1l1aw-of-th. waM relationship applied at thw boundary iayer edg •• 

After considlrabll effort9, thn. equ.ltiOM can b. transform.d into a set of 

.1multAntlous ordina.ry dlfflr.ntialequa.tions, i ••• 

(9) 

which can be salved for th. wall frIctIon velOCIty, u*, thl wak. parAmeter up 

and the boundary lAyer thicKness ( , uSIng a second-order predictor-corrlcter 

tlchniqult. The remaimng quantIties of interlt!St such AS displacmtnt thid:ness 

and momentum thlc1<ness are then det!lrmined from these variables. The numerIcal 

inttgra.tlon IS tltrmmAted At the separatIon POint, whltrl the wall frictIon 

velocity, u*. vamshts. 

The method uses a. two-layer eddY-VIscosity model, whIch Ignores the VISCOUS 

sublAyer terms, conslstmg of an mner layer Prandtl mixing length model and an 

outer layer Clauser model; and thl intermittlncy factor as WIll as .ever&! 

denSIty ratios appearmg m the fundamental equatlons are approxImAted. In 

addltlon, the method assumes an adIabatIC wall. 

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Thus, on the lower 5urfaclt the flow is comput.d u5ing direct boundary 

conditions (airfoil specified' including the effocts of wlf&k VISCOU. inte,':tlon. On 

the upper surface, the flowUeld is &150 tomputed directly with viscous 

interaction up to the slparAbon point, which is determined &II part of the 

boundary layer .olution. Downstream of •• paration, lnVlru boundary condition. 

are utihzed, And the pre.sure must be specified. Fortunately, if thlt skln friction 

at the WAll i. auumed to bl zlro in the Slparatltd zone, the SKAN formulatIon can 

bl used to obtain a clo.ed fOl'm .olutlon for the vlfloClty, and hence ~he pressure, 

at the out.r Itdge of th •• Iparated zone. The re,altant analytic Ixpressions for 

the velocity and prl.sure are 

&lcz _ L ~,. ('DS" (X,. - ](",.) '# ,pT. -4 .. p ) 

~ - " {(H c.J* (x rl- JC,t .. )] u, -'I [ II· cl (JC -~.Irp)] ./& no) 

un 

As can be seen, the separated pressure dapends upon the flowheld solution 

via thlt lnV15Cld perturbat10n potent1als at the separation pOint and the tr.uling 

edge, the size of the separated zone, and thru ca* thi' boundlu'y layer solutlon at 

the separation POtnt. In addItion, th1S closed form solution predlcts a variable 

pressure dlstrlbutlon for the separat.d region. At low freestream Hach numbers 

thiS variation is extremely small and 15 essentlally constant. nOwever, at 

freestream Hach numbers of 0.3 And Above, the VAriation becomes slgnf1cant and 

tnfluences the resultant flowf1elr solution. Th1S trend and separated pressure 

VA"Jatlon 1S In accord Wlt~ exper1men'tl.l observations and IS a slgmficant 

improvement over prevIous methods whlch assumed constant pressure In 'the 

separatl!d zone regArdlt's5 of flow conditions. Howev,r, since at low speeds 'the 

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nparattd pr .. sure 1\\ ustntillly constar.t and th. cl'mpl,xlty introduced by 

Equations (10-1 U ~ay not be wlrranted, the preSltnt method contains the option 

of eUher using a constlnt prtuur. in the s'pArated r'gion Dr the variat.le 

distribution dlPt.rr .. lnl'd by the closed fc.rm .·;,lutton given above. 

In principle, the separ .. trd r'gion and the waKe shoL.ld be aC!:Uf'ately mod.lad 

with rllpltt to physi,:i.l phen"m.na and lnterr"al details, lind thIS approach has 

betn taKen by other mvnhgatorsSO.15-16. In thQ prn.nt model, ho .... '.v.r. the 

wal<e region contains v.ry ffhol cO:llputation&l pC'!nts since the cDordinatl' sY5tem 

rapidly stretches to infmity. Thus, thcr \'lake is auumed to be inviscid WIth a 

constlnt pressure tral11ng edge Tormed by the upper and lower dusplacam'fnt 

surfaces. Fortunately, extensIve numerical ,xperiments; wiih the prltSt>nt and 

prevlous7 models mdlcate that the pr"ssure distributIon and aercdynamlC: 

coeffIcients are ~rlmi.rlly de~~"dent upcn obtammg accurate predictlons for the 

location of thlt separation point tlnd the mlgnitudIP and varIatIon of the separAted 

pressure. Apparently, the dl. taIls of the wak~ regIon are of secondary 

lmporttlnce. Smce the presvnt method obtams the separation pomt locatIon 

dIrectly from the soluhon foro the wall frlchon velOCIty, ut:-, and the pressure 

varIatIon from a soluhon whIch couples the mV1SCld and VISCOUS par~5, It should 

INTERACTION AND ITERATION PROCEDURE 

Th~ itertltlon and mieractl0f'l procedurt> used 1n TAHSEP 15 51mlla.' to that 

used 1M the low-speed program TRAr~SEP6, and It 15 ouilIned In schematl~ form on 

FIgure 2. The program f1rst reads all necessary mput and Imtlahzes the 

perturbatIon potential at all gr1d po~nt:. to zero. Next It computes 

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transformation factors and coorcUNt •• &nodat.d with the .tretctwd CArtnl.n 

grid for the miti&1 grid sptcWld by tho input datA. Includltd in thics prIX ... 1 • 

the computation of all airfoil Ot'dinAt .. and ,lope. roquired on the computaUCt'I&l 

grid. 

Sinee the initiAl grid aa noc"mally vlry coa .... with a default .1ze of S3 M 7. only 

fifty invicC1d it .lrative cycl.. are comp)Uted on this grid. The calculation 

.ucc.ssive column OVlr-rol&xation acheme U1I.d in TRANDESS.4. Th11 iHnited 

numblr of cycles •• rvcrs to rapidly croate an approximate stl.rtlno co~ut1on fOt" 

MJCc. .. ding grids. Aft.r thIS' aro compl.t.d, th. grid ts th!n h&l\'~d and tho 

solution lntarpolated onto & new gr~:1, \,t,lch h&1J a dtfault '.i~~~ of 25 :: :? 

Aftlr obtaanlng all Mcnury coorchnatcts, strotchmg factors. &lrfoll 

ordlnatn and sloplS, ~tc •• ior thus Sllcond grid, thl method then ptrforms fifty 

invlscid it.rativ. cycln b.fo,.. consldlPrlng any typ. of VllCOUS interaction. 

Experience hAS shown that at Is important to perform & hmlted number of lnviscid 

cyellS At th~ b'ginmrlg of nch n.w grid In order to eliminate any "problemG" 

introduced by grId h41vlng and inttrpolatl0n. 

After the •• lnltld Ittr.ltloos, thlP program thIn chICks to au whltther or not 

th~ user deslr'l Vl"5COUll lnteraction to t. includltd by "xammlng thIP value of th. 

varabl. ITACT. If VISCOUS inttractlon IS desired, whsch 15 speciflf:a by thlt 

ITACT defAult value of one, the program th!n checKs to 5U if an lnat1d l.amlntlr 

boundary IAYDr 1!J to be lncludl'd UlAH-1) or 1f the V1_COUIl calcul.&hon5 ~rlt to bE' 

for" turbulent boundary lay!?r With user speCified trAns1tlon POints UlAH-O>. 

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Upon complltt~ the boundAry laYlr computation. for th. current flowfilld 

aoluUan, thl program thin caJculaitl th' ordinatll and .lopta of thl ~r and 

lower dilplaclmlnt slr'acl •• Stnce it only involvn \lful< vilcous int.raction, the 

lOWl,. surf act computattona art from the Il&dlng .dOI or th. JOWl,. aurfac. 

stagnation point. which.vlr i. furthlr Aft, to thl tr"il1~ Idgl. HOWIvor, on thl 

LIPPII' IU'f"cl thlY "rt only from the 1.lding "dOt to thl up"r"tion po1nt or to 

tnl trailing Idgl, whichevlr ,. 11S1. This proclSVl involvn smoothing of the 

displac.mlnt thiclO'lIss vllulS, pt"O~rly adding them to thl liNcn ordin&tos, and 

splint fitting tht resulting points. 

detlcted en the upp,r surface. If .lfpu&tion don not eXIst prior to the lut grid 

point en thl "lrfoll upptr surfacl, additIonal invllcid cycl~1 al'l ptrform.d 

blforl I'Itturmng to the VISCOUS interaction loop. Howevor, if "'paration is 

prldlct.d, thin thl method must determln, thl p~ssurt distribution and the' 

location of the displacement surface in the uepar&ttd zon •• 

El<actly how thl .. parat,d pressure dIstrIbution 15 d.termined d,p.nds upon 

thl uSlr speclfled VAriable, I<SEP. If I{SEP 15 zero, the pressure is assum.d to be 

constant 1n the uparated ID~ anCi 11 computed by 

(12) 

Wt'llll! thIS expressIon 15 a smAll pl!rturba.hon 4pprO)(lmahon for CP1st'P, lh. usage 

has been found to be accura.te and tld'Guatl! for low spel!d tncOmpres151ble flows. 

At frustrum HACh numbers of l'.3 and hlgh,r, howav,r, th~ v&ruble separated 

pressure optIon. specIfled by a KSEP valu~ of ono, should bll used. In thIS cue, 

thlP preuur~ dI5trlbuhon &long thll dlUpl&cem"nt Itrfamhn. an the Sflplrltfd 

13 
,..gion is dltt.rmintd by Bquation. (to) and U U abovlt. Not. that both approachu 

by condition. &t both thlt Hparation point and at thlt traiUng ItdOIt and not ~t on 

conditions in thlt vicinity of .. paration. This .... ult i. in agrltemrmt with thlt 

conclu.lon of Grout7 that conditions at thlt downstrum Itnd of the uparation 

zone partially influence the .eparation preSSUN levltl. 

After dlttwrmining thlt Iltparatttd prUIU'1t distribution corrnponding to thlt 

prltditttd Slpar4tton point and th. c:urrttnt potltnUal flow solution, the 

COM'ltIponding upptr SuNate displACltment surfaclt must allo btt computttd for thtt 

.ltparated zone. When GoparAtton exiats, the prltViOU5 mClthod of adding thlt 

computltd dlsplacltmltnt thiCi'NU to thlt orlgin&l airfoil ardinatn 11 inappt'opt'iatlt 

lince the valun for dlsplacamlint thicl<nns prcrdlcted by the SKAN mltthod .... 

probably inaccuratlt In .eparatod r'glon.. In.te.d, the prltG~nt approach IS to 

lolvlt, using the current potential flow solution, the differentiAl equ&tlon 

siflc:s. .,. g~ ~'I.' 

C".tct i ~ (1., u:n 

for thrt y-ordm&tes of tho up.tra.ted dispb.cement surf,c. n a function of x. 

Bued upon previous studlU2,6, EqUAtIon (13) is solved uSing the Runge-UuttA 

method of order four And thlP di!placement surfAce ordmate at the separation 

point loS the mitial condlbon. In addition, m thlt j:)roces. of solVing th15 equAtion 

IA .. and <?t .. mur.t be evduAttd by f!mte dlfferencu. Whll~ !Several formul.ttlons 

ar. pOG!nbl.2, numlrlcal studi.n mdicA t~ that accurate d15pIAc~mt'nt surfaCts Arl' 

cbtatnl!d using tha followinr;, 

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• .... fo.CIt. Its yalu. is dlt~rm1ned AS part of the inven. pruur. boundary 

condition. 

Sinelt thlt prlt .. nt proelSs II It.ratiVIt and the potential lolutlon usn thlt 

IlparAted prlt .. ur. distribution AS an lnv.rse boundary condlttan, the solution of 

Eql. (13) And (14) Ihould yield upon convergonc. a Itparatld zont duplac.mltnt 

surfAce or free strnmhn. that is compAtIble wIth the pressure dlstrlbubon lind 

potlntll.l flow solution. 

At thiS point In the lttration-mtlract1on proeedurit A checK 15 madl to lie If 

thl solution has convergld or if the malumum number of Ulrabons for" glvfm 

grid SlZe has bun .)(cud.d. If ntither sItuAtIon 15 true, tIn more inviscld cycles 

With the new displAcement surfaces and separated prlssure dustrlbutlon are 

performed pr10r to rep.atlng the VISCOUS interactIon loop. If. however. eith.r 

condition us satisfIed and the fmest grid speCified by the user has not been used, 

the gr1d 15 refmed and the enhrl procns shown on Figure 2 is repu.tlPd. If the 

lut grId solution has been obtuned, tt-en " find output is pr1nted and the 

soluhon 15 f1mshld. 

.\! 
i 

I 

'I 

It should bl notld that thl calculations an a given grid ar. stopped and 

auumld to bl convtrgtd whln thl mad mum plrturbation potltnUal changl is Ie .. 

than som. Ullr sp.ciflld valu •• Howlvlr, whltn separation is pre.,nt it ts usual 

for th. calculation. on Qach grid to bcr termlnat.d dUi to th. numb.r of it.rattv. 

cyellS elCcllding a malCimum usar spec1fied value (particularly on computar. which 

only retain "vln 11gnificant digits). In tho .. c .... , th. ed.t.ne. or degr .. of 

converg.nce can b. dltlrmined by examining th. variatlM in th. numb.r of 

Suptrsonic points, th. location of separation, and the trailir.g edge ordinitl of 

the upper ditlplac.mtnt tlurfaCl. All th •• e values are printld out .very tan 

itarattvcr cycln. If they siabiliz. prior to the .nd of th. computation an a oivtn 

grid, than the rnult. can be .. sumed to bit converged. Normally, it ts tluffic1ent 

to plrform 800 cycln on th. coar., gr1d (2Sx13)' 400 on the mldlum grid (49)(25), 

and 400 on the fine grid «97)(49), although occnionaly mort may be needed. In 

dlttrmining convtrgenccp, it should be remembered that the prtlent method 1tl 

f'uppoud to obtain a steady state solution. At ang1n of attacK above maximum 

lift, thl Actual flowfield about an airfoil is usually unstudy 17,18. In those 

cues. thl present method probably wlll nat converge and may enter some type of 

osclllatory behavlor WhlCh appurs to rClpresent an un5tudy flow patt,frn. 

How.v.r. the present method is not "tlme-accurate" and such result!:l should only 

be viewed as andlcativi of the prlsence of slgmflcant unst.Ady pl'enom.na. 

16 

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.. 

"~~:'l:~c~ ~~~ .~~ ~- -. '-"'. - ........ :..~,~-~ .. '". 

USER INSTRUCTIONS 

CODa DESCRIPTION 

~--,--...- ------

Thlt TAHSEP cod. consists of a main program and .i;htun subroutlnos. The 

subroutin .. and their relationships are Ilhown in & subrouUne tr.e on Figure 3. 

Th. submutiM nAm .. and th.ir f\M'\ctions are as follows: 

FOIL - Ruds in Initial airfoil shape a.nd d.tlrmines Drtfinatn and slop .. at 

I:omputational points. 

VISACT -- Computes turubl.nt boundAry JiAy.r whltn vistous int.raction in:luded. 

THWAIT -- Computes tho laminar bcundary layer when viscous interAction 

included. 

FIT2 -- Curv. fit roubn. usctd by Thw&ites meihod. 

VALUE --lnitu.l1ze5 the flow field to zero perturbation potmntial. 

SOL VE -- S,ts up the matru( co,Hici,ntG used in the SLOR ~l&)(ation Ichlml. 

PRESS -- Computes the pressure dUltrlbubon on the aU'foll. 

COORD -- S.ts up thct coordinatn 1n th. computatu)fl41 and phYS1C&1 Qrid!& and 

computes tho stretChing fa.ctors. 

FLOWS -- Solves flo .. "fl.ld in .front of the airfoll. 

FLOW2 -- Solves nowfield in the dlrect rlglon &bove and below the airfoil. 

FLOW3 -- Solves flowfield in thl' inverse region. 

WAKE -- Solves the flowheld behmd the Alrfoll. 

SHAPE -- Computlt6 the shlp. of 1he &lrfOll dlsplacement 6urface in thlt 

separated zone. 

TRID -- Trldlagonal .qua hon solvor. 

HAL VE -- Doubles th. grld SlZe and lnterpola.in old v&luu to obtAm starhng 

values on thlt new grld. 

PLOT -- Crttaivs A prmter plot of th~ Cp and dl!;plACl!ment ~urhce. 

11 

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ARC -- J>.tlrmines thl AN: llngth of th, airfoil cooNfinatn and aplinn thl 

coordinate. vlrsus Arc llngth. 

SPLINE -- Computes a cubic splinl thl'u a .et of points. 

Th. TAHSEP cod. ls written in FORTRAN IV programming languag. and i. 

design.d for use on IBH, AHDAHL, CDC, DEC, and similar comput.rs. In 

nonav.rlAY mod. it requir .. less than 320,000 bytes an an Amdahl 470/V9. Using 

" FORTRAN H Itxttnd.d compUtr At the optimization l.vltl twa, it nuda Ibout S 7 

s.conds for compilation and obtains a solution on I 97)(49 grid in about 160 

atconds It " rat. of around 15,000 PDlnts/ •• cond. Sam. sUght modifica.tion to 

formats, .tc. mlY b. required to run the progrAm on diffctrent comp'Jter systems 

Dr und.r A FORTRAN 77 compiler. 

INPUT DESCRIPTION 

Th. f.nput to th. TAHSEP cad. II rlad in fight separat. blocks. The fi"5t ont 

contains A user supplied title, while th. second and third blocKs speCify all the 

floAting point and integer paramtters needed to run thlt progro1m. Thll. 

parameters are input via namell.ts and if not sptClfu!d are ,Uisigned default 

vAlun by thl program. Blocks four and s.ven are optional And &re only includ.d 

when the parameter IREAD IS one. They read & non-zero perturbation potential 

starting soluhon and &n Intl&l &irfoll dtscl'lphon. Blocks five and r 19: tare 

assoclt1ttd WIth input for the design option in the progrl1m And are or.l;, Includttd 

wh.n th. parameter INV IS one. FmAlly, blocK SIX contollns the dncrlptlon of the 

airfOil under cOnli1derAtlon. FOI' an analysIs computahon, only blocks one, two, 

threr dnd six , ... ould be 1nc1ud~d in th~ Input s'tream. 

18 

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DETAILBD INPUT DESCRIPTION 

Input BlocK 1: Titl. 

This blocK consisb of a single UM of input and is rud by the main program. 

NTITLB D.scription of c..... Up to 80 a.lphanum.ric chulet.rs. Appurs on the 

print.d output At the b'giming of the results Df nth grid. 

Input BlocK 2: Ii ~olting POlnt Parameters 

This block of input i& rud by the maln program via 4 nlm.list caUed FINP. 

M Freestr.am ~fach numb.r Creal varIable). Default 0.5 

W RelaxatIon 41c10r for subSOnIC points. Should be In th. range O<W<2.0. 

Default 1.7 

Xl X location whCire 'th. direct mod. cAlculation procedurft stops. In the 

analysls mode lt should be set to 0.5 (i.e. the traIling edge). In the inverse 

(deslgn) mod. it is usuilly set to shghtly less th .. n thlt thIrd point from thlt 

leadlng edge or larger. Default 0.5 

X2 End of the inverse reglon. For an1lY61 .. C~S.!5 srt to A large number. In 

the inverse Cdes1gn) case set to 0.5 ( 1.e. the tra111ng edge). Defl.ult 10000.0 

ALP Angle of attAcK In degrees. Default 0.0 

e:ps Subsonlc damplng fa.ctol' to mAtch dIHerence cquAhons a.t SOnlC hne If 

nuded. Has no eHect on the accuracy of the solutlon. Only affects stab1hty .. nd 

convergence rate. l~orma.lly 1t 15 not nceded. D~fault 0.0 

19 
"IF Zetrsp ._ 

EPSS SupersolOic dAmping fActor for i~erative stability. Has no affllet on the 

.. ccuracy of the converged .. olution, only on the ltability And convergence rAte. 

Should typicAlly be About M2 max - 1 ,whlre HmA)( il the maximum local Hach 

number. Default 0.4 I 

>:4 Tht positive X locationl whQre the coordinate strltching chang". It 

should be nur the airfoil trailing edge. Default 0.49 

S4 The posItive psi value in the computational plane where the Itr.tchi~ 

thangell. Default 2.0 

CONY Converglnct! cri1llria control v&lUlt. Iterations stop when the maximum 

change In the perturbAtion potentia,l bet\.,.een relaXAtion _/eles is 1e51 than CONE. 

Default t.E-OS 

Ai Stretchmg consbnt for the Y dIrection. It can be uSld to control the Y 

and eta. 5pa.cmg nur the horizontal aKis. It 15 usually beGt to have the psi and 

eta spacing equal nur the lu::iing edge of ttto ,irfoll. Dllfa.ult 0.246 

A2 First stretch1ng constant for the X-directton. It 1S "qulva.1ent to (2111' • 

(dx/dJ ) At l II J'I • Th~ valUe! of A2 det:trmln!5 the her110nbl step 5izt nur the 

leading and traIling ~dge!l. i.e. 

See Appendix A of Reference 4. Dltfault 0.15 

A3 Second stretching consbn~ for the )(- dlrectlon. Ii determmes the 

physical 10ca1ion of the vertlcd grld 11ne ad,lACent to the grid Side edge. Default 

3.87 

20 

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RN Frentrum Reynolds number b .... d on chord length. Used only when 

viscous interaction UTACT-U included. Default 20.E+06 

XIBDLY The ie-locAtion &t which upper surfac. transition is nsumed to occur • 

Thtt turbulent boundary layer calculation starts at the next grid point. The 

relationship to percent chard IS: 

XIBDl Y • ('Xtchord - 50.0)/100.0 

U .. d only if viscous interaction included <ITACT-U And laminar boundary layer 

ignored (llAK-O). Default -0.44 

CIR Circulation about Airfoil. If an initial solution is input UREAD-U, It 

must be the corresponding value of CIrculation (CIRaCLI2.0). Default 0.0 

CDCORR CorrectIon to the wave drag coefflcllmt. Becaune of the lacl< of .. large 

number of POlnts In the leading and tra.lling edge regIons, the wave drag 

coefficiltnt has an error associated WIth grid $b., spacing, and lift cordflclent. 

The magmtude of CDCORR must be determIned by the user by empirIcal methods. 

Note that the correchon should be different for nch aIrfoil and grId comblnabon. 

Default 0.0 

RDEL R.laxatl0n paramlbr for the boundary layer dlsplcement thIckness. It 15 

used only whe VISCOUS lnterachon 15 lncluded UTACT"U and IMAX is less than or 

equal to 55. Default 0.25 

RDELFN Fme grId relaxatIon parameter for the boundary layer dIsplacement 

thIckness. It 15 used only when VISCOUS lntt!racbon III mcluded and IMAX 15 

greater than 5~. Default 0.125 

21 

_____ 0- '~ 
.. 

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SP MAximum value Allowed far the Nuh-MacdonAld S'pAration parAmeter 

wh.n x(XSEP. U •• d only in the dnign cu. UNY-H wh.n computing the boundary 

layer aver the design surfAce. Default 0.004 

XSEP X location Aft.r which the Nash-MAcdonald separAtion paramltter can 

.xc •• d SP. Used ol"ly In the design cu. C1NY-U when computing the boundary 

lay.r ov.r the design IJurfacl. D.fault 0.44 

RePB tJot used. Ignore. 

CPD Not us.~. Ignar'" 

XHON Nat used. Ignore. 

XlSEP Should always be initiabzed to 0.5. Default 0.5 

XPC Locatlon after which the lower surface displacedment thlcK"ess is 

requIred to continue decreasing once it h15 started to dec"t>a.~:. Upstrea.m· XPC 

the dusplacement thIckness 15 reqUIred to be monotonlcally Incroaslng. For most 

aft-cambered aIrfoIls It should be s.t to 0.1, al,d for convenbonal aIrfOIls It 

should be set to 0.5. Defa.ult 0.1 

XlBDLY The x-locatIon at whIch lower surface tranSItIon IS assumed to occur. 

Same relatIonshIp to chord as XlBDLY. Used only If VISCOUS Interactlon Included 

()TACT"O a.nd laminar boundary layer Ignored ()LAHcO). Default -0.44 

RLAX Relaxa.tl0n par~meter for the separated pressure levelm the constant 

upara.bd pressure optIon (f{SEP=O). Sometlmes needed to enhance convergence. 

Used only when tTACT=l, IHASS"l, and KSEP=O. Default 1.0 

22 I 

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RADt:J lea.din\r edge radius of the airfoil nondimensiona11zld by the chord. 

U •• d only if ITACT-' and IlAM-l. D~f&ult 0.0159 

Input Block 2: Inbglr Parameters 

Thifl blocK of input i. r.ad by the main program via a nam.l1st caUed UNP. 

IHAX Number of vertical grid linn in the horizontal direction on the firtst grid. 

1-' corresponds to up§trum Infinity and J-IHAX corrtlSlponds to downstrum 

infinity. For oath grid refinement IHAX i. increased such thllt the new JHAX 15 

two times the old value minus onl. The limit on JMAX 1599. Default 13 

JMAX Number of horlZontal grid l1nn in the vertlci.l direction on the fIrst grid. 

J-t corruponds to infinIty below the airfoil And J-JKAX 15 infinIty above the 

airfoil. The same formula and 11n11t that apply to IKAX also apply to JMAX. 

Default 7 

JKASE An Integer number ducribing the CASe bl!1ng computed. It IS hmltcd to a 

maximum of SIX digits and 115 printed a.t the beginnmg of the pressure printer plot 

for nch grId. D.fAult 100 

INV Par&mllbr determining the program mode. It should bii l2ro for analysll5 

cases And one fer inv .. rlll deslgn cases. Default 0 

MITE:R MaxImum number of interabons (con,plete r~lau.tlon cycles> allowed on 

the first grid. MITER 15 halved for each grId refmement. However, on the fourth 

grId, MITER is reset to 400. Defa.ult 1600 

NHAlF Number of grld rehnements. D~fault 2 

23 

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ITACT VISCOUS interaction control para.mlter. It should b~ lilt to zero for 

analysis clSn without lnbractlon and for dllign cun. It should bit onll for 

analysis CIS .. with int&raction. Default 0 

ISKP2 Airfoil update control paramltt.r for grid two. It tlhculd bl 21ro H an 

a1rfoll shape update is desired en grid two every t,n iterAtions. It should bl onl 

it an updatl 111 not dnired until grid two solution is completctd. Only uud In the 

lnves. design mode <INV-U. Default 0 

ISKP3 Same IS ISKP2 but for grid 3 

ISKP4 S~me a.s ISKP2 but for grid 4 

ITERP Interpolltton paramettr fer t!Vl deSIgn pressure d15trl·/I.!tlOr. on grid 

four. If In the deSign mede the Input pressure r' .strlbutlen for grid four i9 to be 

read as Input data. INTERP should be zero. If It IS de!ur,d to hnerarly 

Interpolate the prn5ure distribution of grld three, It should be one. Default 0 

IREAD Starting solution control par ameter. If IREAD lS zero, the Imtlal 

perturbatlon solutlon 15 assumed to be zero everywhere. If It IS one. an lmtlal 

soluhon 15 reAd as data. The latter would only normAlly occur when a user Wished 

to restart a solutlon which had preViously been saved. Default 0 

LP Relaxation cycle Interval at which boundary layer detillls are prlr~- i. For 

dlagnosbc purposes suggest 50 or 100. For normal information purposes, suggest 

a. value of 200. DefAult 1000 (no prmtout) 

24 

----", I 
I 

! 

I 

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. -, 

IHASS Hassiv. separAtion parAmeter. It ehculd bit one if the manlV. 

separation option i. dnired in analysis cue. and is active only if ITACT is one. 

In invlrse design casn CINV-U It should be zero. DefAult 1 

ILAH Boundary layer param.ter. If zero, boundary IS computed as If all 

turbull"lt with transition at XIBDL Y and XLBDLY. If one, boundtlry layer 15 

consIdered lammar-turbulent with natural transition. Default 1 

IPRTI Print parameter. If one, perturbation potential values prlMted d the 

completion of nch grId. DefAult 0 

IPRT2 PrInt parameter. If one, )( I1nd y veloclt1u lot each grId pOInt prmbd At 

the completion of each grid. Default 0 

KSEP Separaied pressurfl distribution parameter. If zero, pressure 1n 

separated zone IS assumed to be constant. If one, pressure 1n separated zone 15 

conSIdered vl.rl"ble. Default 1 

Input Blocl< 4: Startmg Solubon (OpbonaU 

ThIS bloc\< of data 15 read by subroutin'! VALUE on:/lf the mteger parameter 

IAEAD has the value of one. 

P<I.J) Nondlmensl0nAl pertlJrbahon potentsal at pomt I,J. Reold by rows starting 

at J=JHAX down to J=l. Each row runs from 1=1 to I=lHAX and starts on a new 

11ne. Format SE1S.7 

25 
--------~ .... ~ ...... -.~ ." 

psm Nondimlnslonal perturbation potential at point 1 on tht yeO (1 ... J-JD) 

grid lint that ia usoeiatld with thl l~r gurfac. of tht airfoil. Rud from I-S 

to l-IHAX. Format :5EUS.7 

This s1ngle line of input 1s rud in subrout.nt COORD only when tht invlr •• 

dnign mode 1. actiY, UNV.U. 

XS,X2 Samlt deflrution AS in BlocK 3. However, when thl my.r .. dlugn modt is 

actiyo, these v .. lun AI'O rud prior to th. solution of eAch grid. Thl. bloc\( 

corrtspondll to the fir"t grid; and, thus, should Alv/ayt l'St Xl"O.~ :nd X2a lOOOO.O 

Format 21' 10.S 

Input Block 6: AIrfoil DescrIptIon 

ThlS blocK of data 1S read by subroutine FOIL And dltScrihts the 11rfOll ulOd in 

the an&lysui mode or th. start1ng 11rfol1 for the mver!!e des1gn mode. 

NI The number of coordmate pairs u5IPd to dncrlb. the upper surfAc" of the 

.urfOll. Haxlmum value hmlted to 99. Format IS 

XlIII,Yl(U CocrdlnAtlf pAIrs descrIbing the upper surface of thtr Airfoil. Th~ 

ludlng edge corresponds to XI=O.O and the tral11ng edglt 15 XIal.O. Thlt vertic"l 

ordinate, YI,lS nondlm.n510nahzrd by chord. Read "it"rtlng w1th I-I to I-tH. 

Format 81'10.4 

26 

----.:. 

I 

I 
DERIX, DERIY, DERFX, DERFY Param.ter. dncrib!ng th. luding And trailing 

.dOI of th. airfoil. DERIX is dx/d. of th. airfoil ~r sur-fu. at th.luding 

.dg •• It is usually z.ro. DERIY 1. dy/ds of th. airfoU upplr nurfac. At thlt 

luding .dg. and it is uSUl.lly S .0. DERFX is d3x/ds3 of thlt airfoil upptr surfAce 

d th. trailing .dg •• It i. usually suffidlntly a(curd. to un 0.0. DERFY Is 

d3Y/ds3 of th. airfoil ~r surfae. At thlt trAUing .dge. It is usually 

sufflci.ntly Accurd. to un 0.0. Format 4F 10.4 

NIB Thl numb.r of coordinAte pairs und to dncrib. th. low.r surfac. of thl 

airfoil. Maximum vAlu. limited to 99. Format 1:5 

XJB(l),YIBUJ Coord1nAtit pairs dncr1bmg thlt low.r surfaclt of the A1rfoil. ih~ 

leading .dge corresponds to Xl-O.O and th. trAlllng IdO. 1. XI-l.0. The vlrbeal 

or(l1nAt., YIB, is nondimltnsion&llZtcS by the chord. Rud atArtmg w1th IsS to 

I-NIB. Format 8Fl0.4 

DERIXB,DERIVB,DERFXB,DERFYB PArAmeters dncrlbmg the ludmg And trAlling 

edge of the A1rfoil. DERIXB 1S dy. Ids of the Alrfodlow.r surface at th~ leading 

Idg •• It is USUAlly zero. DER1VB 1S dylds of the AirfDillow." surface At thl 

lud1ng edo. And 1t 1S usually -1.0. DERFXB 1S d3K/ds3 of thl! alrf011lDw.r 

surfac. At the trai11no edge. It 1S usu&lly 5ufflcumtly accurate to use 0.0. 

DE:RFVB 15 d3Y/ds3 of thlt AlrfDlllow,r surface at the truhng edglt. It 15 usually 

sufhClltntly accurate to us. 0.0. Format 4F10.4 

21 

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Input BlocK 7: St&rti"9 Airfoil Description (Optional) 

Thi~ block of data i~ read from subroutine FOIL and i~ only rud if the 

integer input parameter IREAD hI one. It effectively overwrites the information 

from input blocK she. 

VUU),V! (J),SLUlU,SLLm V&1uu dncrlbing the airfoil on the ~t&rtino grid. YUIJ) 

and YL<U are the upper and lower surface ordinAtes, nondlrncnuor..l.l1zed by chord, 

At cnord locAtion xm. SLUm and SLUU art the up~r and lowu surface SlOP"l at 

chord location xm. The X(1) values depend upon the sizl And spaCing associated 

with the starting grid. Thlt group of four values 1. rud starting at tht I value 

corrnponding to the first p01nt dO'lmst"r af'\ !'" the lel:.c., :J edg~ n-ILE) and 

ending WIth the pomt JJst upstream ot UH~ tralhn~ eoge (I-ITE). :-~"mat 5E15.7 

DUPOLDII>,DLWOLDm Va.lues describing the bound&ry layer displacement 

thicKnnsn on the shrtlng grid. DUPOLD(J) And DLWOLDm are tht upper and 

lower surface displacement thIcKnesses correspondmg to the chord locatIon xm. 

These Art read 'ihrtlng at the 1 value correspondlng to the hrst p01nt 

downstream of the lUdlng edge lI=lLE) and er.dlng With the pomt JUst upstream of 

the tra111ng edge U"lTEU. Format 5£ 15.7 

Input Blocl< 8: De!ugn Pressure DistrIbution Unverse DeSIgn Hode only) 

ThiS blocK of dAta cOnfnsts of four sections which Are only Included m the 

inverse deSign mode <INV= 1>. In th&t mode only thi! lut three sections would 

u!>ually be Included. 

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Sletien I -- StArting solution dnign prl •• url distribution rl&d by subroutinl 

FOIL. Thil sletion would only bl lOc1ud.d if & dnign lolution we ... bling 

restArted h.I.INV-I. IR~t.D·l, ,\nd HHALF-U And it would only aUlet the first 

grid connid.rld. 

CPU<U Upptr surfael inver .. rlgion prusur. cOlfflcflnt vAIUlI for th. dnign 

cu. starting with I-U, which i. thlt I1r.t grid POlOt aftlr Xl and ending with 

I-ITE, th. grid pllint JJst upsttrum of th. trailing Idgl. Format SEIO.3 

CPL(1) Lower .urf&c~ inver .. region pressure coefficient valulS for thl dealgn 

ClU starting with l-U" which is thl iirst grId point after Xl and ~ndmg with 

I-ITE, the grid POlOt ~st upstream of the tralling edge. Format SElO.3 

S.ction 2 -- ThiS nction reads in the startlOg and cPrldmg points of tht lOver ... 

rlglon from subroutine COORD and the Inverse deSign pre55url distribution from 

subroutlnt FOIL for the second grid. Used only 10 th. inv.rn dlSign uu 

CINVaU. 

XI,X2 Xl 15 tht loca. tl0n where th. direct calculation stops and th!' lOverSI 

calculation begms. TYP1Ca.lly, It 15 s11ghtly les5 th .. n thl! third pOint from the 

ludlng edill or larg.r. X2 is th. location wherl thl' InVer59 calculation stops. It 

should alw<1Ys be set to 0.5. Formi.t 2F 10.S 

cpum Upper surface mverse reg10n preS5UrIP cOllfhclent valu.s for 'tha deSign 

case stArting W1tt, I-U, which 15 the first grid POlOt after Xl and cndmg With 

I-IT£::, the grid pOint ),Jst upstrum of the tr.ll11ng edg!. Format 8E10.3 

29 

--_. '\:! 
CPLCU Lowtr ~ut'fact lnvtrst regIon prtuur. co.fhcient vAlue. for th. duign 

cas. starting with I-Ill which is th. first grid point after Xl and ending with 

I-ITE, th. grid POlnt !1st up~tr .. m of th. tra.ihny Idgt. Format SIUO.3 

Sletion 3 -- This sletion rl!ads in the .'artlng And endlng pOints of thl lnvern 

re;1on from subroutin. COORD and tht lnvtrs. dllign pressut'l di~tributicn froll' 

.ubroutine FOIL for the third grId. Used only in the inverse dni;n CAS. <INV-U. 

The input v&riabln and dncriptions arIP th. sam. loS Section 2 abov •• 

S.ction 4 -- ThIs .eetlon reads in th~ starting and ending pOints of th. inverSt 

region from ~ubroubne COORD olnd the invern dnign pres!-ut'e dIstribution from 

subroutine FOIL for the fourth grid. Used only In tnt lr.ver t. de cign case (J N V. U 

whln ITERP 15 nro. Note that In Rtfltrl,"ce~ (2) And (4), the un of grId four for 

inverse deSIgn 1S not recommended. The 1nput var1ables and descrIptions Are the 

s&m~ 45 Slebon 2 abovt. 

OUTPUT DESCRIPTION 

THe printed output When the program lS operattd In the lnverse desIgn mode 

is identical to that descr~bed 1n Reference (4). When thlt program 1S opera.ted In 

the analys1s modi \filth the massive separat10n optIon, the output for rrids two, 

threl, and four has th~ form shown below. Smce the f1rst gr1d 4sr.umes mVl!lCld 

flow only, lts prmtout only mcludes those portlons assoclated wlth a.n mVlsCld 

solution. 

30 

r 

, 
... 

, 

r, 

~ , I 

I . , , I 

., 

• . 1 ~ , 

.' 

• 

• r 

, 

\ 

I :-1 I 

-.. 

t. 

2. 

3. 

4. 

Hudlng (uur supplied) 

CAS'" Numbtr 

H loch number and .. ngll. of &ttac/( 

C .... typ. callouts, i ••• , 

INVISCID ANALYSIS CASE 

WITH LAHINAR TURBULENT VISCOUS INTERACTION 

AND HASSIVE SEPARATION 

AND VARIABLE PRBSSURE IN SEPARATED RBGION 

5. Input dAtI. in nAmlfUsts FINP .nd IINP 

6. Coordinate System for th~ current grid printed nIt xm followed by J t Y(J) 

7. Ordlnates of th" current airfoll dlliplacement surface 

x -- Horizontal ordlnatltt where -0.5 15 the ludlng edgL' and 0.5 15 the 

trailing edge! 

YU -- Upper dlsplac"mrnt surface ordlnate 

Yl-- Lower dIsplacement surface ordlnate 

UPPER SLOPE -- Slope of upptr d1!~plac9ment surface 

LOWER SLOPE -- Slop. of lower dIsplacement surfcaclt 

8. Iteration hl$tory at ten-cycle lntery"ls 

CIR -- Clrculatlon 

DPM -- Haximum VI correctIon (absolute va.lue) In the last relaxation cycle 

WIth the correspondmg (l,J) grId 10clI.tlon 

NSSP -- Numbl'r of supersonIc grId POlnts 

DEL TAY OR YUTE -- In the deSIgn case, the change In YU(lTE), the upper 

5urhce tralhng edge ordinate, !>lnle the last surface update. Should go to Zl"ro If 

31 

. '.-~., 

• 

I 

i 

I ' • 

I 

I, 

r 

I 

I ~ 

-
~ . I • , 

" ........ 

; 

e, 

t' 

. ~ 

I' 

, 

, - ~. t.: - . 

~ t" I 

rk_ 

• 0 

.. 

I 

f 

;1 

.IP 

conv'l'g!n;. In thl viscou. anllysis cas" it is th, curr.nt valul of YUUTE); and 

it should approach a constant valul if th. solution is converging. Only ching .. 

aft.1' th. f1rst fifty cyeln on each grid. 

SBP AT X -- Thl current x ordinatl valul for thl upper '1urfacl s.paration 

point ( x • -0.5 1s th. Juding Idgl and K • 0.5 is th. trAiling Idg.). 

9. Boundary Layer Information 

EVlry LP cyeln ruults of thfl currlnt boundary laYlr solution al'l print.d 

first for the lawlr surface of thlt airfoil and thin for th!p upp.r surfaci. In each 

can, thl lamInar solution ( if ILAM-S) is printed "lrst followld by thl tUl'bulC!nt 

solution. 

94. Laminar Bound~ry Laytr lnformaf:cn (Printed ever'l LP cycles) 

X -- Horizontal ordsnate 

MACH (4 -- LocAl "!4ch number 

CF -- Skln friction eoltHlClent (the O.iE 11 snltlal value is arbltl'll.ry .Rnd 

should bt ignored) 

D-STAR -- 6 " Ie, non-dImensIonal boundAry layer dlGplaeement thIcKness 

D-THETA -- ele, ncn-dlmem51onal momentum thIckness 

H -- f! Ie, shAp,," factor 

RE-THETA -- Local ReY,lold!5 number based on e 

RE-STAR -- Local Reynolds numbel' based on 6" 

TH -- e' (du/d:d/'V ,Pressure gradIent parameter 

9b. POSSIble Lammar Flow Messages 

a. SEPARATION OCCURRED AT X" O.xxx -- gIves x locatIon where ]",mmar 

sepal"atlon occurrt!d • 

32 

--: _ .... 

, 

, 

I 

It 

1,-
/ 

:' 

/' 

,/ 

b. SHORT BUBBLE FORMED? TRANSITION TO TURBULENT FLOW 

ASSUMED, X- O.xxx 

c. LONG BUBBLE? LAMINAR STALL MAY OCCUR, X· O.xxx 

BOUNDARY LAYER CALCULATION WILL BE CONTINUED AS 

TURBULENT BUT ACCURACY OF RESULTS IS QUESTIONABLE 

d. BOUNDARY LAYER CALCULATION COMPLETED? 

NEITHER SEPARATION NOR TRANSITION WAS DETECTED 

ge. Turbulent Boundary Layer Information (Prmted &very LP cycles) 

X -- HorlZontal ordinate (-0.5 15 ludmg edge, 0.5 is tr.uling edge) 

MACH ft -- Local Mach number 

t 

DLSTR -- 6 Ie, non-dlmem51onAI boundary layer dlsplacement thickness 

DEL -- SIc, non-dlmensl0nd boundary layer thicknesses 

CUE -- Ue , transformed boundary layer edge velOCity, Ua(~/~ )u 

USTAR --Law of the wall parameter, SQRT(~II ) 

USTAR**2 -- SKm friction paramltter, t;,'1 

THETA -- 9/c, ncn-dlmensional momentum thickness 

COP -- Profile drag coefhclent u510g Squire-Young approach modified 

I.ccordmg to Reference (9) 

COW -- Wave drag coefflclent, not computed 10 thiS version of code and thus 

always zero 

CDTOT -- To\al drag coeffiCient 

9d. Turublent Boundary Layer Messages 

a. SEPARATED CP IS 0.:<::)( -- In the constant separated pressure level 

ophon <IMASS=l,lTACT=l,KSEp:zl) the pressure 10 the separated reglon 15 prmted 

33 
/ 

I 

) . 

'. 

After eAch boundary layer calcula.tIon. If solution 15 converging, 1t should 

I.pprl.och I. constant value. 

b. USTAR2 (or USTCK) LT ZERO -- Indicatn. whlre in computation separation 

was first detected. Se'! program listing for d.tall~. 

10. Final Boundary Layer Rnults 

YUORIO -- Or1gmt1l A1rfo11 upper surface ordinate 

DU -- Smoo~.lld upper surface displacement thicl<nns 

SLU -- Slope of upper du.plAcement surfAce 

YLORIG -- Original a1rfol1 lower surface ordmatu 

Dl -- Smoothed lower surface displacement thlcl<ness 

SLL -- Slope of lowtr d15.pla.cemti r ,+ surfAce 

11. Pressure Dlstrlbuhon on AlrfOll 

CPU -- Upper surface pressure coeHlc1ent 

CPL -- lower surface pressure coefficllmt 

12. Fmal DIsplacement Surface Informatlon 

YU -- Ordmate of upper dlsplacement surface 

YL -- Ordinate of lower dIsplacement surface 

SLU -- Slope of upper dIsplacement surface 

SLL -- Slope of lower dIsplacement surface 

13. Mach Chart 

The Mach numbers at the I,J coordInate points are multlphed by 100 and 

printed out in black farm. The grId pamts "mslde" the upper and low(!r 

34 

---- ~\:: 

J 

i 
~ I , 

,tj 

l~ 

, ( 

~l I \ 

! \ . 

It 

I 

, 

, 

l 

L, 

~ . r I 

!- j 

\ 

t;..1 

tI'" 

,\ tJi 

F' 

r--

I 

\ 

\ 

displacement surfacls are indicated by zeros. Velocities (UN) At the flawfilld 

grid points may 1.150 bl printed out using option IPRT2-1. 

14. Mlsclllaneous Information 

The normal forcI CDtff1Cllnt, eN, and the drag c:olfficient, WAVE CD, 

obtainild by Integration of the prlUllurct dIstribution &re printld aut. The latter 

should theor.tically bit zlro for subcritical unseparated cues. but It usually is 

non-zero due to mesh sue and grid placement. Thus values of WA Va: CD tlhauld 

only be used for comparison purposes. 

15. Printer Plot of Results 

U -- Upper surfAce prnsure coeff1clent 

L -- Lower surface pressure coefficient 

T -- Up,!r d1splacement surface 

B -- Lower d1splacement !iurface 

CPSTAR -- Pressure c09ff1clent for local Hach number of one 

CLCIR -- Lift cor:fhclent from computed Circulation 

CL -- Lift coefficient from Integl at10n of pressure du.trlbutlon 

CD -- Wava drag plus profile drag COefflCltnt (Accuracy depends upon value 

01 CDWAVE) 

CMLE -- Moment coeffiCient about the leading edge 

CDF -- Proflle drag coeffIcient using Squire-Young approach modified for 

separation and compresslblhty 

CMC4 -- Moment Coef.c~clent about quarter chord pOint 

35 

/ 

... 

.AI 
I 

, I 

, 

\, 

~ ~ , 

t 

; I 

•• 1 

· 

~~ , 

! 

, 

I 

t 

~ 

· , 

.1 

-r i 

/ I 

/ I I 

/ I 

• I 

r I 

I 

• 1 

I 

I 

~ . , , 

" I t 

, , 

P/' ,. 

" 

/ 

16. Miscellaneous Messages 

PROD .lE. ZERO -- This muuglt indicl.tu that the turbulent boundary 

calculation was un~ble to Clbtaln an appropriate sbrting solution. As a. resul1 thlt 

flowfitld calculat10ns were cont1nued wlth the dlsplacRmlnt surface ordinatn 

and wlop .. frozen &t thelr last updahd v&1UItS. ThlS ,utultl0n usu&lly occurs 

whln transltion ta~s plac. &t thl shocl< WIV. and the locil Mach number 

immediat1ly upstream of the shocl< wavrJ is greder than 1.3S. It probably 

indicatu that at a m1nimum ther. 1S local 5ep~ration In thR vicinity of the "hocl< 

wive. Since thlS phenomena is not modeled In the present code the results 

obtained m such cases should bit used carefully. Sometlmes thu. problem can b. 

"avoided" by computing an all turbulent case UlAM=O) wlth tramnUon at or 

immedl~tely upstrum of the shocK wavLt. It should be notE.'d that when this 

message appears, the subsequent computed values of CDF are in error. 

AA .IT. ZERO -- This message occurs whim thl:! local !':;eed of sound is 

computed to be negatlve. It Indlcates that e1ther the Solutlon has become 

unstable or else the rotated scheme trled to use a pomt outslde of the solution 

domain due to a supersomc pOint on the JMAX-l row. Usual Solutl0n 15 to 

lncrean the supusomc dampmg. EPSS, and/or mcrease the stretchl ., and lor 

size of the Y grld. When encountered, the computatlon 15 termlr,_ced and the 

current perturba.hon flowheld solut,on 1~ prmted for dlagnoshc purposlts. 

17. Note that addltl0nal output can be obtamed for elther dlagnostlc or analysls 

purposes by removmg the CiS fr.J,ll varlOUS comm~nted prmt statements 1n the 

program. 

36 

~ 

I 

j 
-/ 

... 

I 

I 

, 

I 

! 

..... 

r, 

, 

I 

, 

I 

\. 

, .... 

.. 

~ 

, 

, 

I • 

r· 

;: 

TYPICAL RESULTS 

In the devllopment of th, prolitnt method. results havi bun obtAinld for 

NACA 4412 and N"CA 0012 .l1rfo115 for freestream Hach numbers up to 0 • .5, Angles 

of Attack up to 18.5 degren, and Reynolds numbers betweln three And nine 

mlllion. Thul conditions were selected not bec&uslt of limitations in the method 

but dut to the AvailAbility of excellllnt experiment&l prUbUre distr1but1on dAtI. 1n 

those regions.S9-21 Thus, the results pruented hero lore only meant to be 

representative • 

Figures 4 and 5 comparl rnults obtained with the present method w1th the 

low speed experimental datA of Pinkerton21 for .. UACA 4412 A1rfoll at 6.3 m1llion 

Reynolds number. In Figure 4 the experimental data ha!J been plotted USlng the 

angle of attacK correction suggested 1n Reference 21. As can be 5een, the theory 

predicts shghtly larger hft coefficients than the data &t the lower Angles of 

attacK. Whethl!r thiS difference is due to an underestimation of the a.ngle of 

atta.c\( correchon, as suggested In Reference 22. or a. problem In the theoretical 

model is unl<nown. In any event, the theory and the data are In excellent 

agreement between ten and flfteen desreesi and the present model reasonably 

predicts the loeabon of maximum lift a.t an angle of attacK of About 16 to 17 

degrees. The theoretlcal model does, however, overpredlct slightly 11.9 'Is. 1.7) 

the m&IC1mUm 11ft. coeHlclent predicted by thiS fortY-$l!ven year old data. 

Figure 5 compares pressure d1strlbutlon results obtalned wlth the present 

method wlth the experlmental data of PlnKerton at an angle of attacK shghtly 

below that correspondmg to maXlmum 11ft. In thiS case. the corrected angle of 

attacK W415 used In the computatIons, and the upper and low.r surface boundc:.ry 

37 

" i 

I 

I 

,... 

.. 
/ 

: 

lay.rs wtre assumed to be initlally lamInar follow.d by natural transtion to 

turbulent flow. For this high 11ft casR, the theory predlct.s that the lower 

surface rema10U entlroly laminar and that thlt upper surfac. transitions at on. 

percent chord followed by separation at 74.9 percent chord. As can be seen en the 

f!gurp, the predicted pres sur. dIstrIbution il 10 .xcell.nt Agreement With the 

da.ta; and the pressure coefficient in the separated zonet is slightly negative and 

constant. Experience indicates that for low spud c&Ses b.tt.r risults are 

usually obtained uSing the constant pressure optlon (KSEP=O) for the separated 

zone. For this case, the theoret1cal 11ft Co~fflclent was 1.69 Whl1t the 

experimental value was 1.68. The predlcted profIle drag ceefflcumt was 0.0200, 

Whlch 15 in rIPaso~. blIP Agreement WIth aV!1latle meas'.Jrer ..... ts.23 

As Ind1cated prev10usly, lt 15 lmportant for th1s type of me tllod to Include the 

effects of a lammar boundary layer. F1gure 6 shows lo .... .!r 5urfac~ 

lam mar-turbulent transltlon POint locat1ons predIcted by the present method fer 

a NACA 0012 a1rfe11 at Mach 0.3 and SlX m11110n Reynolds number. As can be seen, 

for these fhght condltlons the lower surface boundary layer 15 predominantly 

lammar at low angles of attack; and at angles of attacK above ten degrees lt 15 

essenbally all lammar. ObVl0usly, a method wh1ch only mcludes a turbulent 

boundary layer calculat10n andlor assumes translt10n near the leadmg edge mIght 

Y1eld mcorrect results for these fl1ght cond1tlons. 

F1gure 6 also shows for the same cond1tlons the pred1t. t£'d upper surface 

separatIon pomts determmed by the TAHSE? method. Nobee that no upper 

surface separabon IS pred1cted unhl about 12 degrees angle of attac\<. After 

tha.t, as the angle of attacl< 1ncreases, the b~qJnnlllg of separatton moves ferward 

on thl> upper surfa.ce un'lll more than half of th~ ,udell e>:perumce!> separated 

38 

-_.-- ~ 

I 

, 

I' 

i 

I 
'\ 

'1 

~ 

II 

',I 

, 

I 

lI.' , 

I-I 

1-

, 

I , i 

I 

flow at about 16 degrees. For thIs caSlt, malumum lift occurs around fuurteen 

degrees. 

Figure 7 comparn thlP predIcted lift coefficient variatIon WIth &ngl. of &ttacl< 

for & NACA 0012 a1rfCllI 4t the same nommal condlt10ns b.lt. Hach 0.3 and SIX 

mllhon Reynolds number) WIth experlmlntal data obtamed in 'the Low Turbulence 

Pressure Tunnel at NASA Langley.20 Then data were obtamed on clean aIrfOIls 

without the use of trip str1ps, and thus the theoretIcal results were obtained 

using the lamInar natura.1-tran!Sltlon turbulent model UlAH'" U. At the lower 

angles of attack the agreement 15 qUIte good. However, at tna onset of tral11ng 

edge separation, around twelve degrees, the theoretIcal 11ft curve exhlblb a 

"kmk" accompamed by it. shght decrease m 11ft coeffICient. This "l<1nK"' 15 often 

observed m the theOrlttlca.1 11ft curves when separatIon 15 flr'~t J,I:,!'dlcted and 15 

due to the fact that sepa.ratlon 15 usually +~t5~ detected on the coarse grid. On 

that grId, the flrs.t pomt forward of the trl.l11ng edge 15 about 93". chord; I,nd, 

thus, when separatIon 15 fIrst pred~ded on the coarse grId the separation POlr.t 

moves forward fr'om the tral11ng edge to at least the 93 percent pomt, wltn the 

result that the amount of separation 15 overpredlcted and the 11ft a1. that 

condItion 15 underpredlcted. Smce the model only permits the sepl.ratl0n POInt to 

move forward, thiS effect 15 mamtamed thoughout all the grids at that angle of 

attack. However, as the angle of attack IS Increased thiS effect dlsappurs. 

Thus, at low and medium freestream Hach r,umbers, the 11ft coeffiCients predicted 

a.t angles of attack J:,Jst above the onset of trall1ng edge separa.tlon are u~ually 

shghtly low. 

However', as can be $oeen on Flgul'e 7, the 11ft coeffiCients predIcted at the 

higher angles of attack are, at least for thiS case, In reasonabl!> agreemenl; \ 'lth 

39 

-----'-= 
I 

• ~ t 

. 

• 

t'-

-. 

L 

j; 

.. .. 

t". 

t • .. 

~ . 

. : 

'. 

. ---

thlt Itxpltrimlntd data. Noticl that for this C4U thl Ixplrlmlntal data 1ndica tItS 

an apparent maximum lift around fourtltltn dlgr ... follow.d by a d.cNlUQ and 

thin an mcr.lIl. The thlor.tical 11ft bUld upon the c&1culatld Circulation 

prltdicta a maximum lift cOIffiCllnt of about t .40 at J4 to 16 d.oren. On thl 

other hand, thl th.orltical lUt bUld upon 1ntlgrat1on af thl pre.llr9 

di5tNbutiona indiCAt.. a maximum lift at fourtltltn doorn., which nominally 

aOr .. s with expltrlmlntal data. This sUght divlrglncl bltw .. n tho values 

prltdlct.d by circulation and thou by prnlur. tntloration ia, baud upon 

oxp.ri.nc., an excellent indication of the mllClmum lift location at low &nd 

mldium ape'ds. Thl Nilan for this 5t&t.mlnt will bt tvident whln th~ Pl't"'ut~ 

distributIons for these CUts art dlllcusud • 

Tht pt'edlcted varIatIon of drag With lift f:::- -:'12' cnl 11 compared to 

angle of attul< plot and lS du'" t:1 tnt Qver::- Jlctlon of thlt Slllt of thlt imtial 

slparated zon.. Nrverthlllu, thlt a91'1lm.nt btt .... nn thlt thooretical pr,.dlctJOnS 

and experimental values. partlcularly ne.r maximum 11ft. is good and should bIP 

acc.phbllP for apph.d englneerlng calculations. 

Flgures 9(a-c) comp.re pressure dlstrlbut10ns obtained wlth th. present 

Langley20 at thru d1HUf'nt "n9)ir~ of attolcl<. The 42rst corresponds to an 

unsepara'ted flow sltuabon, th~ second 15 near maXlmum hft. and the third lIS for 

An 4ngl. of attacK above the mllXImum hft condlt1on. Smce at m~dlUm frnfli.r."m 

Mach number!> and hIgher eVldence9,19-20 mdIca.1es that ihe preuure variation In 

01 !5eparated :one 15 not constant. these and !Subsequent CABeG were .U run UIln(J 

40 

! . I 
-----_._--_.--..- ----- ------~-

thl vuilbl. pt'l!!Slure option (KSEP-U. As cln bl Ilin on Figure 9(a) th. 

th.oreUcal pt'tlIUI'e distribution for th. unuparat.d can is in .lCceUttnt 

agr.lmlnt with Ixplrimental d&t&. For thig case, the thloretical 11ft and drAg 

coefficient, wtr. 1.27 &nd 0.0116 whillt the corr .. ponc:hng dun airfoil 

Ixptrimontal v&lutl wIre 1.23 a.nd 0.0123. 

For tht C&llt. nur maKimum lift, Figurlt 9(b), it 5hould bl nDt.d that lome 

Gupttrsanic flow exilts over the upper lurf&C. of thl &irfoil in a very small regie.., 

nur the It&ding edge linel the critical Cp for this cau il -6.09. In addition, the 

theoretical method predicts upper surface npU&tlon at 74.9 percent chord &nd 

boundary l&yer irlstablhty en the lower suNAc. &t about eo parcent chord. 

However, due to thv favorable pressure gradient, the lower Gurface boundArY 

layer n.v.r transItIons. F or thIS CUI th. theoretIcAl bft CO.'ficllnt of S .39 

COIncides WIth the experImentally meASured VAlulh And the two pres lure 

distributIons exhIbit rusonable agrnm.nt. 

At an angle of attAcl< greater thin that correspondlng to maximum lift, the 

flow about an airfoil IS tYPically chlracterlZed by a large regIon of unstt'ldy 

separated now; and a study state soluUon method such IS tho prl!!Sent one II not 

reAlly apphcable. Thus, the 'ppuent IACI< of agrnment botwun the pruont 

study theory and the experImental measurements shown on Figure 9(c) is not 

aurpr1!ung. Nev~rth~lnr., th. generAl pattern of thrl prnsurt dlstrlbutlon 

includmg the eXlstlmce of a large separAtion zone III predIcted; and the predicted 

hft Co~H1Clent of 1.44 bued upon clrculahon IS In surprlsmg a.greement With the 

experImentally menured value of 1.437. However, CAreful el:&mlnahon of the 

solutlon lnd1cat~!5 thAt It 15. not complttt~ly convt'rged, thAt the thtorL'hcAl 11ft 

may be o~cllh.hng mhghtly, and that the hft bAUd upon prenurfP lntegrolUon 

41 

_-_---0 \! 
I 

'/ 

---------_ .. , .... ~ -- ... --- ------- - { 

compubd At the wnd of & run u.ing default paramoters is only 1.27. Interestingly, 

result. ulif19 th. thin lay~r Havitr Stokes tqu&tion.n,18 for a sunillr cu. 

(NACA 0012, Huh No, • 0.3, Reynolds • 1 million. angll of attuK. 18 dlgrlll) 

in:Ucatl A 11ft coefficitnt vArying with timl from 0.65 to 1.6 with a Strouh&1 

numbtr of (I.li and both the thwo,..Ucal and Ixplrimlntal pr .. surtt profiles sh:lwn 

on FigLA'1 9(C) ara rtpres.ntative of thou comput.d &t various timn in tho cyell 

with the th·.n l&yer Navier Sto~s model. Thus. thl pres.nt theoretical result is 

reprnenta1:ivt of the! type of prnsUNt dIstribution and lift which might .,ust for 

this condition. 

Aneth,pr lnt.rntmg futur~ assoclat,d With thlt resulb $hown on Figura 9(c) 

is thAt th r drag co.fficitnt predlctetj us:ng the methrd of Rlf~rence 6 b.e. CDI') 

was 0.060' whU. thAt pr.dicted uSIng thl methcJ of R,ferttnc. 9 ( M!. COP) wn 

0.1190. tI'ormAl1y. these two values arlt In good I.grelmant WIth nch other. 

Apparently, in the present m!tthod when the ma)"mum hft condition IS Ixceeded 

the solutlOfl becomes OSCillatory and not compllttely converged. the 11ft computed 

by prl.surt mtegratl0n dlvergu. from and 15 lower thAn that from CIrculatIon, and 

the CDI' And COP valullS differ signifIcAntly. It 15 blJhev.d that these threll 

1tems can be uud to determm. for medium Mach numbers the .. ngle of attack 

correspondmg to maXImum hft. 

Figure 10 shows for thlt urnl NACA 0012, Mach 0.3, RlPynolds number 6 

mllhon case, the dIsplacement surfaces pl'edlctt!d by the present method for the 

upper surface region b.twnn 70 perclnt chord and th" tralhng edge at varIous 

angles of attacl<. Below 11.13 degrees, ''''here the How 15 unsep&rated, the 

dlsplAcem~nt thlcknIPS!le5 in the truhng edge 20ne arE.' relatlvi'ly small. However, 

With the onset of sl,Opuatlon at 12.09 degrl!eu th!? thlckm''S!\t'S bl!9tn to mcrease 

42 

I· 

-
. • / 

I 

" . I 

'I I 

rapidly, and the dlsplacement lurfac!:n; t4i<o en shapes ch&rl.ctlrhtic of thl flow 

oVlr a .tal1ad a1rfeil. It should bct noticld that at an angll of Attack of 16.12 

dlgr ... , thl displacement lurfAce st&rtl to curve bad( toward a thl fre .. trltolm 

angll pr10r to thl tralling crdgl. It 15 blUlvld that thl dllpl&c.mlnt lurfacn 

shown en this figure have the correct b.havior and arl an adlquatl .ngin~ltrlng 

rlprt&lntation of thl real flow. 

Obviously. it would bl desirabtl for thcp prtacmt ""thod to accuratltly modftl 

transOnlc flows wlth and without significant traihng edgl slparation. 

Consequently, pr.dlctionl obtained with TAHSEP have bun comparld with dat" 

obtained in thl NASA a Ft. Transomc Pressure Tunnel by H&rris.!9 This data 

probably repruents th. but high lift transomc experimental A1rfoil data 

avallable today. and it has been used b)' many lnvestigators. How.vctr, in mo~t 

cue. preViOUS studi.s have compared results using the tunnel geometric ang11 of 

attacK values and havlt ignorld the i<nown corrltctions auoclated wlth t.hll dAta. 

Since such comparlsons could leAd to erroneous conclusions, thl present results 

wire obhlned by matching the tunnel Mach and Rltynolds numb.r. and uSing the 

corrected angles of AttACI< suggluited by HarriS. 

Flgur! 11 shows a trAnsomc 5tparatL'd flow result compAred With data 

obtained by HArl'1!.!? Whlle the pressure dlltrlbutlon shown was obtained uSing 

the lam mar-turbulent boundary layer option, lndlstlngulshable results were 

obtamed assummg tramutlon at SlX percent chord. In the actual experIment, 

boundary layer trlp strIp! were lo::&ted at flve percent chord. For th15 case, the 

present method predlcts upper surface separabon at 87 percent chord And low!:'r 

surface trans! han very near the traIling edge. \.Jhlle the experImental shocl< 

locatIon 15 sbghtly forward of the 1htorehcal value and whIle the e)'perlmenhl 

43 

I 

.~ 

• 
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.. ~-, 

I 

I 

I 

'I 

.I 

1 

I 

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.. 

t 

:j 

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! 

, 

I 

I 

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, 

preslurn in thl trailing edge r.gien ar~ llightly low,r than ~holl pr.dietld by 

the theory. the overall agreement i. probably Acclptabl. fer IInginllrlng studin • 

For this CU" the meuUt'ld normal forel cOlffitiont WAS 0.994 while the 

prodictltd lift coefficient wn 0.991 • 

A lift v.rsus &ngl. of AttAck curv. typi.:&1 of thOat predicted by thl pr .. ent 

method is compared WIth IICperimlnt&1 data on Figure 12. For this fr .. strlam 

Hach numb.r of 0.5, significAnt tranlonic flow accompaniod by " stroni! ahoci< 

wave is prctllnt on thl upper surfacil At angles of atbd< groater than sbe 

dlgren. As CAn btt sun, thl thlerlttical prediction, which wn obtain.d assuming 

transition At SIX perCint chord, agrc .. well WIth thlt .xp.rimflntal data up to 

about 7.35 d.gl·~I5. Abovi that 4ngl. of attacl<, tht: or.nnt mlPthod prldicta a 

maxImum 11ft coeffiCIent of 1.09 At 8.24 degrees With no tra111ng edge !Sepilr4t~on. 

At 9.21 dtgrects thl theol'Y prediCts A decrull 1n lift CO.fflCltnt to C.9S2 ~ .. ith 

upper surface sep.a.r&uon at 87 percent chord. The elCparil'lent&l data, hcwever, 

indIcates that maxImum lift 15 1.02 At 9.21 degrees and that traihng edge 

separ&tlon probably started At about 7.S degrees. Examinatlon of the theoretical 

res~llts &t 7.35 degren re-vealli that thl local Hach number immed14tvly upstream 

of the upp~r surfac~ shocK wave is 1.42. Such" strong shocK wave, who&& 

strength increas~s wlth .. ngl~ of attacK, should lnduce sigr.lfi::ant shocl< boundary 

layer interAchon wh1ch, unfort:.mately, is not modeled 10 the pregent ~hl?r:ry And 

coda. Th~refore, in thls case tht' dIfferences between the theorehcAl fc- _ ... Us and 

the experImental data near maxImum 11ft are probably due to shocK boundary layer 

1Oteraction and lts subsequent effect on boundary layer growth and trall1ng edge 

separatlon. Nevt'rtheleS5, the present model and code does glve a.. reasonable 

1ndlcAtlon of the 10c4hon a.nd m.lgnltude of the mAXlmum 11ft coefflclent. 

44 . 

,,. 

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1 

I I 

, ,-
I 

, 

I 

, ' 

I I 

~ \ 

. I 

: I 

I 

\ 

. 

• 

-, 

. , . 

A similar 11ft curv. for I. frentrum Mach numb.r of 0.55 as shown on Figur. 

13. As bllore, the! theorobcal r.sults were obtained using an all turbulent 

boundary layer; and an this CASe no upper surface trailing edge separation was 

detected until An angle of &ttacl< of 8.266 d.g,.. ... Tho maximum lift co.fficient 

of 0.983 and 8.266 degru.. For this c .... significant transonIc flow existad at 

all Anglu of attack Above 4.5 d.grltts and by 7.29 d.gr .. s thlt Mach numb.r at the 

uppal' surface shock way. had incrustd to 1.50. At 9.33 degre .. a complete 

solution could not be obtaintd with the preunt method. On th. medium grid the 

upper surface now separated at 97 percent chord, and on the fine grad the 

separatIon paint movtd forward to tho shocl< wav, at about Ie p.rcent chord and 

the solution fa1led. QU1t. cbviouuly 51gn14icant shocl< boundary layer 1ntor4ctloo 

.IUSts at thn. high angln of Attack and thl decru!i' in lift or Ita11 is probably 

due morlt to shocl< induced !ll!paration than to the on9lt of sngnHicant trall1ng 

edge .eparation. 

Nevl?rthelns, the present method can be used to IIst1matlt rusonably 

accurately thlt occurence of this situatlon. As can be seen on Figure 13, the 

method pred1cts reuonably well the magnitude of the m&XlmUm hft coefhc11!nt 

and 15 connrv&t1ve as to the correspondang angllt of &tt&cl< location. In addItIon, 

by notlng th. mechamsm of code "fallure-, an thIS case sudden separation at the 

shocl< wave, .. user can probably determIne the type of stall phcmumena present. 

Tht'oretical pressure distrlbuhon results Are compared With expt'rimlnh.l data 

for a freestream Mach number 0.6 case 10 Flgure 14. Th1S casCl At a. corrected 

angle of attacK of 5.59 dllgrees and three mllhon Reynclds number 15 Glgmfico1nt 

for" vArlety of reasons. First, It IS an example of A transomc cage With a strong 

45 

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, 

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, 

I 

1 

,:1 

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I , ,'t , , 

, " t , , I 

upper surface shock wa.ve. Second. the flow is unseparAtCld And the data should 

serve as & good tnt of the prnent method for & situaUon without IIparation. 

trin&1ly. this cue ha. &150 beetn tsolvCld by Anderson It aU7 using both An Bulltl' 

boundI.ry l&y.,r method and & thin layer Navier Stokes method. 

It should be noticed that thl prultlt results. 11~ tho.e of Anderson, &gree 

very weU with the IlCperiment&l pressure dlGtribution with respect to shock 

lOCAtion and p"nure levels. Also, fDr' this cue the IlCperimentall1ft coefficient 

was 0.781. The present TAKSBP method predlcted 0.809, the Euler boundary layer 

method of Anderson yielded 0.(104, while h15 thin layer Nav.er StoKu result was 

0.793. ObvioUGly, the present method is capable of Yleld' 10 excellent result. that 

&re in agreemQnt wlth uperimClntal data and other analy1ical methods. How&ver, 

.incI it 1. a. non-'Conservative full potent1al method, lt ~hould ob't&m such result. 

with accurAte shocl< wave locations mol'. easl1y and futer than other more 

comp11cated methods. 

Amother 11ft versus angle of attack cpmparlson lS shown on Flgure S 5 for the 

N.\CA 0012 &t & freestrum Hach number of 0.6 and a Reynolds number of mne 

mUllon. A. can be nen thlP hlghest theoretlcal point plotted is at a hft 

coeffICIent vdue of 0.945 and an angle of attack of 6.392 degrees. At 7.348 

degrees the boundAry layer solutlon became -frozen h.e. PROD.LE.ZERo) on the 

medlum grid due to ~hocl< bounda.ry layer 1Oter~ctlon, and on the -hne grld the flow 

separated a.t the shocK WAve and a converged Solutl0n was not obtamed. At E:.371 

degrees the flow nparAted At the upper surface shocK WAve on the coarse grld, 

And &galn a converg~d soluhon WAS not obtalned. For these cases, comput!'d local 

H~ch numbers 10 the viclmty of the shocK wave were as hlgh as 1.56. Thus, the 

theory mdlca1u thAt above 6.392 degrees slgmflcant shocK boundary layer 

46 

, 

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, . 

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, , 

, 

.; 

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11 .. 

t-

/~ n -- .. I 

:. \ 

, / 

. . , , 

r' 

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interaction probably accompAnied by nparatlon exist~ and that the m&ICimum bft 

coefficient occurs At that Anglll of Att&C\(. As can be sllan on the flour., the 

magnitude of the prltdict.d maximum lift coefficlltnt is in good agrlt.ment with the 

experimental datA; Although the Anglll of attad< location is Aoain conservative. 

Based upon the results pNtuenttd in this slction, it is believed that thlt 

pre.ent method and code CAn be used At low and medium MAch numbers to 

accurately predict lift and prnaurR distributions At anglea of AttacK up to that 

""sociAted with maximum 11ft. At transonic sp.edB, tho mQthod should give good 

results for un"paratlJd flows and for flows having tr&111ng edge set)ar4tlcn 

without significant shock boundary layer interact10n. Thus, &t trAnsoniC 

condlt1ona, the method is probably currently lilnlted, for Accurate results, to 

Reynolds numbers of three million and higher And to 10C41 upper 5urf&cCt Mach 

numbers less than 1.4 to 1.-15. In addition, 1t should yield rftl.sonabllt utimates 

for the maximum bft coefflcumt at transonic speeds wlll1e bling conservativi as 

to the corrnpondmg value of angle of at1&cl<; and the method ~Duld indicatt the 

onset of slgnif1cant shocl< boundary layer mtcractl0n • 

47 

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,-

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"-~I '/ i! 

,../" :: 

I' II 

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P.'" - ••• . .... _ ..... _ .... __ - -

CONCLUSION 

A dir.ct-inv.rn t.chniqu, baud upon & nonconslrv&tin full potential 

invistid mothod, I. Thw.utes laminar boundary layor techniquo, and the Barnwlll 

tUI'bullnt momentum integral m.thod has bien dlvllop.d. This mlthod i5 !luitabl. 

for predicting the subsonic And trAnsonic: flow field .. bout airfoils having trailing 

edg. "eparatld flow. EKtensiv~ comparl.ons with olCplrimlnt&l data tndicatl that 

the method should bo I. useful tool for applied aerodynl.mic: englneerlng an&1ytslS. 

In addition, it is b.li.vld that thlt rang It of applicabllity of the mlthod could bit 

lI(tended !SlgnHIC.1ntly by the Addition of I. shoc\( boundary layer tntlracticn modi. 

48 
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rl 

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REFERENCES 

S. Carlson, L. A., "Transonic Air;oil Flowfield Analysis Using Cartul"n 

CoordInates," NASA CR-2577, August 1975. 

2. Carlson, L. A., "Transonic Airfoil Dnign Using Cartnian Coordinatn," NASA 

CR-2579, April 197(,. 

3. BaLWr, F., GarabedIan, P., Korn, D., and J~mncn, A., Supcrcrltlct.l WIng 

S.ction, n .• New York., Springer-Verlag, 1975. 

4. Carlsen, L. A., "TRANDES: A Fortran Program for Transonic AirfoIl AnAlysis 

and Design," NASA CR-2821, June 1917. 

5. Bamwell, R. W., "Two Invlsc1d Comput&tional CJmulltions of Separated Flow 

About AIrfoils", AIAA PA~tr 76-379, July 19··6. 

6. Carlson, L. A., "TRANSEP:A Program -for High Lift Siparated Flow about 

Airfoils," NASA CR-3376, December 1980. 

7. Carlson, L. A., "A Dlrect-Invers. TechmquI for Low Spited Hlgh LIft Au'foll 

Flowfleld AnalysIs," AGARD CP-291, pp. 26-1 -- 26-10, February 1981. 

Et. BlucOV1Ch, J. D., "ChAracterIstics of S,parated Flow AIrfoIl AnalysIs 

Methods," AIAA Paper 84-0048, January 1984. 

9. Barnwell, R. W" "A Potttnbal-Flow/Boundary Layer Method for Calculatmg 

Subsomc and Transomc AirfoIl Flow wIth TrAilmg P'dge Separation," NASA 

TM-818S0, Jun2 1981. 

10. Dvorak, F. A. and Chei, D. H.,"Sepa:-ation Hoc'l"l for Two-Dlmemnonal Alrfolls 

In Transomc Flow," AIAA JOl/rn .. l, Vol. 22,1-'0.8, August 1984, pp. 1064-1070. 

11. Taverna, F., " Prediction o~ SubsonIc/Transonic Stpar&ted Flow About 

Alrfolls," AIAA Paper 85-020:;, January 1985. 

12. RIZZi, A. And Viviand, H., editors, NUm!rlCal Hethods for' the ComputatIon of 

lnvlsc:id TranSOniC Flows with Shock WAves, Vleweg, 1981. 

13. Salas, M. D. and Bumbrr, C. R" "BreaKdown of thll' Conservat1ve Potential 

EquatIon," AIAA Paper 85-0367, January 1985. 

14. Schlchtmg, H., Boundary Layer Theory, 6th ed., New York, McGraw-HIll, 1968, 

p.479. 

15. HasKew, B. and Dvorak, F. A., "Invest1gahon of Separahon Models for tht' 

Pred1ctJon of CLmax, AHS Paper 77-33-01, May 1977 • 

16. Hendor:lon, M. lo,"A Solution to the 2-D Separated Wake Modellng Problem and 

Its Use to PredIct CLma:< of Arb1trary A1rfoll Sec:tions, " AIAA Paper No. 7E:-1S6, 

,January 1978. 

49 

--~ 
!/ 

I" 

( 

r I 

,,' 

/' I 

'I' 

J! 

I I 

Ii 

1 

I 

I 

I 

, , 

, . 

I' 

, ~ 

. 

,,-

. . 

, 

17. Andersen, W. K., Thomu, J.L., and Rum"ey, C. L., "Apphcation of Thm-LAyer 

Navilr-Stol<es Equl.H:3nl Near Maximum lift," AIAA 84-0049, January 19&4. 

18. Rumsey, C. L, "Timlf-Depcmdent NAvlwr-StoKas Compub.tions of Slparated 

Flows OVlr AirfoIls", AIAA-8S-i684, July 1985. 

19. Harris, C. D., "Two-DimenSIonal AerodynamIc CharacterIstIcs of the NACA 

0012 Airfoll in the langley a-Foot TranSOnic Pressure Tunnel." NASA TH-81927, 

April 1981. 

20. Ladson, C.L., PrivAti Communication, NASA Langley. 

21. PinkC!rton, R. H., "Thlt Variation wIth Reynolds Number of Pressure 

Distribut·on over an AirfOil SectIon," NACA Report 6HI, 1938. 

22. Coles, D. and Wad cock, A.t II A FlYlng-Hot-Wlre Study of Two-Dimlfnsional 

Hun Flow Put an NACA 4412 Alrfml at Ml.)(lmUm Lift," AIAA 78-1196, July 

1978. 

23. Abbot, I.H. and Von DCI(!n~oH, A. E., Theory of WinO Sections, Dover, New 

York, 1959, p. 489. 

50 
It 

I 

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l'-- I ' .. ! r. I 

Direct Solution with 

Boundary Layer Interaction 

(Airfoil Specified) 

---

solution) 

Inverse Solution 

----} c const<lnt 

~-____________ -:_::=: - - - - -- ..... .... bound~ry conditIo,' 

- -- Z:irec~s:u::' vith Boundary Layer Int~raCtion (Airfoil Specified) 

Flgure 1 -- Problem I'ormula'hon 

51 
~; 

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I 

, 

.. 

( 

I 

L. 

~' ... 

, ,. I 

: 

~ 

" , ------ --- ------ --- -- - ---. 

Input & Initialization I 

I ----------TI ___ ----------- _II Gr I a s. tup 11--II-R.=-fP"'!l~n~.-Gr=-I"""a"'l _______ I--IYn 

I I_ ~50,,-,J-=n~u I ~~ ":"'0;: '!":, a:r-roC~yc:":"'Ir-::.:-::s~ I Gr I a 1 I -------------_1 ? I --~_r_--I No I 

I --------------------_1 I--~vr:",-=sc'=o~u s=---

I-I Int'~actlon Us I '---....,-1----- I Nol 

I I 

I I I----T---~- No 1-1 ... 0.....,1 n""u""I""'s'""c""I""'a-1 

I Cycln WI th I 

I -N.w· Shall' I 

1 and Cps.p(x)1 I I 

CarnlnarTurbulent 

B. L. ? 

C'lunar B. lol 

Ca 1 cu 1a t I on I V.s I 

I ---.:r.:---- -t""u""r""'b""'u 1 ""'.":"'n"'"t -.8 ....... C-. I 

No Calculation I 

I ~O"'p~p:-::.-=r-r5u~r~f~a~c~e-1 --~I----

S~puatlon? I I Compute Ora,nahs '--I and Slopes of , V.s I I DI~plac.ment Surfaces' 

I I I 1 __________________ Xlt , X < Xte or Xsep I 1 

I --,:RnS~EBp ....-r '-.,... __ _ --..".,r---I 1 01 

I 

I -C,...,p"..,_5,."e-=p-... ·~C,...,o".."n:-::s:-rt.".,an~t.---

Xs.p < X < Xt" 

I 

I 

Compute Cpsep(x) Xs.p < X < Xh 

-C'f"'cm=p::":'u:":lt:"":.:-'''U'''r''''a-I ""n a""'t""'e""s'-"'a"""n""'a""SP<T"':1 o:":p""'e""s'- ________ of Displacem.nt Surfaces 

Xs.p < X < Xt. 

--.,.."..-_I--cDiiVergea or 

No 'Exceeded Max 

I lteratlons ? 

I---~-v'~- I Yes 

I -R"'e""'f""",'=n e-Gr=-,""a,-o-

? Yes 

final Output 

No 

FIgure 2 -- Iteration and InteractIon Procedure Used 10 TAMSEP 

52 

---\i 

r. 
1& 

, 

,-, VALUE 

I I 

, 

I 

I , 

, , 1-' COORD 

, 

I 

, , I , 

, , 1-' FOIL ARC , , 

,--, SPLINE 

, 

, 

, , 

, , FLrx.ll , ,--, ,--, , , 

I , , 

I , , , ,-I SOLVE , MAIN , , , FLOW2 , , 

, ,-,--, ,--, , 

, , , 

, , , 

, , , 1 1 1 

1 1 ,--, FLrx.l3 , 1 ,--, , I -, TRIO 

, , 

I , 

1 , 

, 

1 1 

1 WAKE 1 1 ,--, ,- , 

, 

, 1 , 1 , 1 VISACT Ttl.'AJT 1 1 FJT2 ,-, I ,-, , , 

, , , 

, , ,- I 

, I SHAPE I PRESS , ,--, I ,---, , 

I 1 ARC , , SPLINE 

, I ,-, , , 

, , PLOT ,--, 

, 

, 

, 

, , HALVE --, 

Flgul'e 3 -- Subrouhne Tree for TAHSEP 

53 
• 

• 'I"'1'C .,,-.,,~; •• , .......... 

til ..: ~ •• ~, ." '",.. "-

.. if 

_ .. --

, 

, . 

l 

, 1 

. • . I 

I 

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. 

, • 

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, 

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'j t: 

" 

t 

, 

C 

L 

1 

1 

o 

.5 

c 

.0 

.5 SYMBOLS ARE EXPT. 

LINE IS THEORY 

5 10 15 20 25 30 

ANGLE OF ATTACK (OEG) 

FIgure 4 -- ComparIson of Prcdlct!d lift COffflclent wl1h Experlml!nta.l Data 

NACA 441~ Ca!;l' 

54 

J 

, 
H 

E 

G 

A 

T 

I 

U 

E 

C 

P 

--.. -

, \ 

8 ~ 

6 

4 

2 ~ SEPARATION 

0 

-2 

-4 LINES=THEORY 

-6 S 'y' ~1 B 0 L S = E X P T 

-8 

0 .2 .4 .6 .8 1 .0 

X/C 

Fioure:; -- Comparl\:lJn of Theor@hc.al Pres~ ... re Distribution with Experimental 

Data 

NACA 4412 Case 

5S 

I/": 
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i . . __ i 

.,/ 

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X 

/ 

C 

N CA0012 M=0.3 RN=6E6 1.0~----~~~~~~~~====~-. 

o 

.8 

.6 

)(/C SEP--) 

.4 \ 

.2 

2 4 6 8 10 12 14 16 18 

ANGLE OF ATTACK (OEG) 

Figure 6 -- Lower Surface Transihon and Upper Surface Separa.hon Loc:ahons 

NACA 00l~ Case 

56 

; 

. 

v::. 
< 

I I . I 

~ I 

I 

Flgurt" 7 -- ComparIson oi PredIcted Ltit CoeHlclent wIth uxperlmental Data 

NACA 0012 Mach No ... 0.3 Case 

57 
I 

1" 

. , 

• 

\ 

, I 

.\ 

, 

. 

t 

0 

R 

A 

G 

C 

" 0 r.~ 

~. 

,----- .. I U , N , 

... 

; 

-.a. 

• . . 

i 

- -- --

800 

LINE=THEORY 

680 SYMBOLS=LTPT 

400 

[] 

," ~ 0iJ . 

[] 

[] 

0 I 

.8 1 .0 1 .2 1 .4 

CL 

rlgure 8 -- Companson of Theoret1C:al Llft and Drag wlth E>:penmental Data 

NACA 0012 Mach No. II 0.3 Case 

59 

--.~ 

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; , 
1 

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. --

, 

I 

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.' ---- '. ' , -- / ---1"'_ .. --.1 

10 

8 

6 

4 

2 

0 

-2 

-4 

-6 L I NES=THEOR'y' 

-8 SYMBOLS=LTPT 

-10 

(3 .2 .4 .6 .8 1 . e 

X/C 

Flgure 9(a) -- Theoretlcal and Experlmenta.l Pressure Dlstrlbuhon Compa.rlsoras 

NACA 0017 at 11.13 Degrees Angle of AttacK 

59 
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.. ~,. ~ .. ~ 

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10 

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0 

-2 

-4 

-6 

-8 

-10 

e .2 .4 .6 

I ,// 

I 

LINES=THEORY 

SYMBOLS=LTPT 

.8 1 .0 

FIgure 9(b) -- Theoretical and experImental Pressure Dlstrlbutlol1 Compi'lr150nS 

NACA 0012 at 14.3 Degrees Angle of AttacK 

60 

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-8 SYMBOLS=LTPT 

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X/C 

FIgure 9(c) -- Theoretlcal and ':>:perlmental Pressure Dlstrlbuhon Comparlslons 

NACA 00l::! at 16.1 Degrees Angle of Attack 

61 

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Flgure 10 -- Predlcted Dlsplacement Surfaces fClr NACA 0012 at a Ma.ch No. of 0.3 

62 

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Flgure 11 -- Theoretlcal and E)'perlmental Pressure Dlstrlbutlon Comparison 

NACA 0012 at Mach No. 0.5 

63 

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Flgure 12 -- Comparlscn of Pred1C:ted lift Coefflclent wlth Experlmental Data 

NACA 0012 Mach No. = 0.5 Case 

64 

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ANGLE OF ATTACK (OEG) 

Figure 13 -- Comparison of Predicted lift CoeffiCient with EXperlmp.ntal Data 

NACA 0012 Mach No. = 0.55 Cas!? 

65 
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FIgurE? 14 -- Theoretlcal and E>:perlmental Pressure DIstrIbution ComparIson 

NACA 0012 at Mach No. 0.6 

66 

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Flgure 15 -- Comparlson oi PrE'dlCied llit CoeHlcleni wlih Experlmenial Da ta 

NACA 0012 Mach No. = 0.6 Case 

67 

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APPENDIX A 

Sampltt Ca. .. Input And Output 

68 . -
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, 

OR;'::~::".:' r'''~ , .. _- IS 

OF POOR QUALITY 

T[ST CASE. NACA 0012. MACH 0 5. RN 6 MILLION. AlrHA 9.25 DEQ 

&FINP M-O 50l.ALP-9 150.RN-6 00-06. -" (PSS-I .XPC- 5.&ENO 

&IWP IKAS[-S09.NIIAU-2.&ENO 

96 

00 00 o 002500 o 008717 o 005000 o 012213 o 007500 o 014849 

o 010000 o 017C37 0012500 o 018939 o 015000 o 020637 o 017500 o 022179 

o 020000 o 023598 o 025000 o 026147 o 037500 o 031374 o 050000 0.03!>547 

o 062500 o 039027 o 075000 o 041999 o 087500 o 044575 o 100000 o 046828 

O. 112500 o 048808 o 125000 o (l50'S54 o 137500 o 052094 0.150000 o 053452 

0 162500 o 054644 o 175000 o 055608 0.187500 o 056594 o 2()()()(x) o 057:175 

o 212500 o 058040 o 225000 o 058596 o 237500 o 059052 0.250000 o 059412 

o 262500 0.059684 o 275000 o 059873 o 287500 o 059982 0.300000 o 060017 o 31~500 o 0':9982 o 325000 o 059879 o 337500 o 059713 o 350000 0.059486 

o 362500 o 059202 o 375000 o 058863 o 387500 o 058471 o 400000 0.058030 

I - o 412500 o 057541 o 425000 o 057006 o 437500 o 056428 o 450000 o 1)55607 

o 462500 o 055147 o 475000 o 054448 o 487500 o 053712 o 500000 o 052940 

o 512500 o 052134 o 525000 o 051296 o 537500 o 050425 o 550000 o 049524 

o 562500 o 048594 0.·575000 o 047634 o 587500 o 046647 0600000 o 045634 

o 61:1500 o 044594 o 625000 o ,)43529 o 637500 0.042439 o 650000 o 041325 

o 662500 o 040188 o 675000 o ,)39027 o 687500 0.037844 o 700000 o 036639 

o 712500 o 035412 0.725000 o 034104 o 737500 o 032894 o 750000 o 031603 

o 762500 o 030291 o 775000 o 028959 o 787500 o 027605 o 800000 o 026'31 

o 812500 C 024836 o 825000 o 023420 o 837500 o 021984 o 850000 o 0205:16 

o 862!>00 o 019016 o 875000 o 017545 o 887500 o 016022 o 900000 o 014477 

o 912500 J 012909 o 925000 0.011318 o 937500 o 009704 o 950000 o 008066 

o 962500 o 006402 0 975000 o 004714 0 977500 o 004374 o 980000 o 004032 

o 982500 o 003689 0 985000 o 0033-15 0 987500 o 00300:) o 990000 o 002654 

o 992500 o 002307 0 995000 o 001959 0 997500 o 001610 I 00000o o 001260 

00 I 0 00 00 

96 

00 00 o 00:1500 -0 008717 o 005000 -0 012213 o 007500 -0 01.1819 

o 016000 -0 017037 o 01:1500 -0 018939 o 015000 -0 020637 o 017500 -0 On179 

o 020000 -0 023598 o 025000 -0 026147 o 037500 -0 031374 o 050000 -0 035547 

o 062500 -0 039027 o 075000 -0 0-11999 o 087500 -0 04-1575 o 100000 -0 046828 

0 11:1500 -0 0-18808 0 1:25000 -0 050554 o 137500 -0 052094 0 150000 -0 ()5345:1 

0 16:1500 -0 05-1644 0 175000 -0 055688 0 187500 -0 056594 o 200000 -0 057375 

o 212500 -0 058040 0 225000 -0 058596 o 237!>00 -0 059052 o 250000 -0 059-11J 

-. o 262500 -0 059684 0 275000 -0 0591173 0 287500 -0 059902 o :'100000 -0 060017 

o 312500 -0 059902 0 325000 -0 059879 0 337500 -0 059713 o 350000 -0 059486 

o 362500 -0 059202 o 375000 -0 058863 0 387'.;00 -0 058471 o -100000 -0 050030 

IJ .112500 -0 057541 o 425000 -0 057006 0 437500 -0 056-128 0 450000 -0 055807 

o 4"2500 -0 055147 o 475000 -0 05-1448 o 487500 -0 053712 0 500000 -0 0529-10 

o !il25('10 -0 oc;2134 o 525000 -0 051296 o 537500 -0 050-125 o 550000 -0 04'l52.a 

o 56"'~OO -0 0.aS594 o 57Sooo -0 047634 o 5117500 -0 0466-17 o 600000 -0 0456:'1.a 

o 612!J00 -0 01459-1 o 625000 -0 04352!) o 637500 -0 0-12439 o 650000 -0 0413:>5 

o 6132500 -0 040188 o 67!:>000 -0 039027 o 687500 -0 037814 (I 700000 -0 036<039 

0 712500 -0 035412 o 725000 -0 0341"1 o 737500 -0 03:!1l'l4 o 750000 -0 03H;03 

0 762500 -0 030291 o 775000 -0 028959 o 787500 -0 027605 o 1100000 -0 026231 

0 812500 -0 024836 o 825000 -0 023·120 o 837500 -0 021984 0 050000 -0 020'.l;!6 

0 862500 -0 019046 o 875000 -0 017!> 15 0 867500 -0 01G02:1 o ~ooooo -0 014-177 

o 912500 -0 012909 o 925000 -0 011318 0 937500 -0 009704 o 950000 -0 OOR066 

o 962500 -0 006·102 o 975000 -0 00471-1 0 977500 -0 004374 o !)Boooo -0 001032 

0 982500 -0 003689 0 965000 -0 0033~5 0 987500 -0 OC.OOt' o 990000 "0 00~G5-1 

0 9<1:1500 -0 002107 0 995000 ·0 COl959 0 9'l75CO -0 0016\0 I 000000 -0 00"60 o 0 -, 0 00 00 

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OF. POOR QUAUTY 

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