1、REPORT 1359THIN AIRFOIL THEORY BASED ON APPROXIMATE SOLUTION OF THE TRANSONIC FLOWEQUATION 1By JOHNR. SPmrrmiand ALBERTA Y. ALKSNRSUMMARYThe prtxent paper describes a methodfor the approxiqa.tesolution oj the nonlinear equattin.sof transonic small dtiturb-mmetheory. Although the solutions are nonlin
2、ear, the analys-i.sis suj%iently simple that r. =0,” or with M.= 1FUNDAMENTAL EQUATIONS AND BOUNDARY CONDITIONSConsider the steady flow of an inviscid compressible gaspast an arbitrary thin symmetriwd nordifting airfoil, andintroduce Cartesian coordinates x and z with the x axis paxal-lel to the dir
3、ection of the free-stream, as illustrated in e 1.-ik)FIGUREl.View of airfoil and coordinate system.Let the free-stream velocity and density-be U. and p., theperturbation potential be q, and the perturbation velocitycomponents parallel to the z and z axes be p., or u, and , orw, where the subscript i
4、ndicates differentiation. The bound-ary conditions require that the perturbation velocities vanishat infinity, and that the flow be tangential to the wing sur-ftice. The first condition indicates that p is constant atinfinity. The latter condition can be approximated forthin wings by(%),.0=UC4 : (1)
5、where Z represents the ordinates of the airfoil upper surface.The pressure coefficient CDis likewise approximated to firstorder by(2)These relakions are familiar from linear theory, but applyequally for transonic thin airfoil theory. The differentialequation for p is not the same as in linear theory
6、, however,but is(3)where M. is the iMach number of the undisturbed flow andy is the ratio of specific heats (1.4 for air). It is useful tonote that the coefficient of pn corresponds, in the presentapproximation, to 1M where M_is the local Mach number.Knowledge of methods for obtaining solutions of e
7、quation(3) is meager, not only because the equation is nonlinear,but because it can change type (elliptic, hyperbolic), depewi-ing on the value of Mm and p This change of type is anessential feature of transonic flow, since subsonic flows arerepresented by elliptic equations and supersonic flows byh
8、yperbolic equations. If both types of flow occur in a singleflow field, it is apparent that the differential equation mustchange type. In the present case, the type of the equation620C07-0044SOLUTION OF ITJD TRANSONIC FLOW EQUAITONis recognized by the sign of the total coefficientfollows:511of qn, a
9、s0 elliptic (subsonic)l_m2 til- - % O (7).and rewrite equation (3) in the form:(wl)s (13)whence; 1u=k (34. 1)+ (M.-lflfl 7:kU. n (14)The corresponding relation for the pressure coefficient (?, isobtained by combination of equations (2) and (14), and is2“=M.2(.y+l) (J!fm-l) (Mml)3fl-;A4.w+l)$y (mIt s
10、hould be noted that the restriction to supemonic flowimposed in the evaluation of C and in the inequality ofequation (7) requires that equation (15) is to be applied onlyto cases for which the quantity in square brackets, that is,(M.l)W (3/2)iMo2(Y+ 1) (dZ/dz), is positive.COMPARISON WITH EXfSTING H
11、IGHZR APPROXIMATIONSEquation (15) is recognized, by comparison with equation(3-15) of reference 3, page 387,2 as the precise equivalent,in the transonic small disturbance apprcmtiation, of simplewave theory for the surface pressure on an airfoil in super-sonic flow. Exact simple wave theory is lmown
12、, moreovw,to be perfectly adequate for all practical purposes up to oMach number of 3, which is considerably in excess of thopresent range of interest. Within this Mach number range,the results obtained by use of simple wave theory me almostidentical with those obtained by use of shock-exTansiontheo
13、ry. Comparisons of the variations of C, with dZ/dxin-dicated by exact simple wave theory and by equation (16)are shown in figuIe 2 for several Mach numbers from 1 to 2.As might be anticipated, the two sets of results are in C1OSOagreement for Mach numbers near 1, and ditler by an in-creasing amount
14、with increasing Mach number.Although the necessary calculations are vwy easy to ac-complish in any given case, simple wave theory is not alwaysused in actual practice. Many calculations rtre based onlinear theory or Busemanns second-order theo, Conse-quently, an additional set of graphs is shown in
15、figure 3 inwhich the curves of iigure 2 are repeated together with thecorresponding curves calculated by use of first- cud second-order theory. No comptins are shown for M.= 1 becausothe latter theories indicate infinite pressures. It can be menthat equation (15) furnishes a better approximation tlm
16、nlinear theory throughout the entire range of variables shownon figure 3 and a better approximation than second-orderthqmy for Mach numbers less than about 1,4. It cmi bo1 Comtin dfxfoss that the quantity .3Jmqy-!-1) tkat apmis IIIomut10n (M)isreprosentedby+.1 In ermatfon (3.15) of raferonce 3. Thod
17、ifferenceMmsooMMwithncmrc-spcmdtng dlrkonce In the eoefficfemtk of the nordlne.w term of eqnatlon (3). A1thouh thotwo morlickants are identfcal at .lf -1, and mfgbt appmrto bo orpMltYrnmhtcnt with thoother a-ptioas of tramanfo flow thmry,Ithasbeenshown tn refmmccs 8, 0,7,8, and Lwwhere that the appm
18、atlonobtafned by use of .V*(Y+l) Is rnnoh the better of tho two forMach nnrnben other than 1.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THIN AIRFOIL TKt30RY BASED ON APPROTEIL$FIGURE 2.Ccrmpftrieonof results indicated by present theory and byexa
19、ct simple wave theory.seen thnt second-order thegry furnishes a very poor approxi-mation for CPat Mach numbers applacg ity.In order to explore this behavior further, two additionalcurves labeled “third order” and “fourth order,” calctiatedusing the formulas of references 14 and 15,3are included onth
20、e graph of figure 3, even though they must be interpretedin a somewhat more restricted sense than the other curves.To be more precise, the third-order curve is restricted to air-foils for which dZ/dx is zero at the leading edge, and thefourth-order curve to airfoils for which both dZ/dz andd2Zldti a
21、re zero there. It is clear from this sketch that thenccurrtcy of second-order theory at Mach numbers nearunity is not improved by addition of lher order terms.The eqkmation resides in the fact that the larger values ofldZ/dzIshown on the graphs of figure 3 exceed the radius ofconvergence of the powe
22、r series expansion for CPfor all butthe highest Mach number shown. With the noted restric-tions on the leading edge, the higher order results of fieme 3me equivalent to the fit few terms of a power series ex-pansion, in terms of dZ/dz,of the expression for CPindicatedby exact simple wave theol. The
23、radius of convergenceof the series depends, of course, on the Mach number and isgiven by the value of ldZ/dzassociated with the occurrenceof sonic flow or, in terms of the curves shown on es 2 and3, with the termination of the left end of the exact curve.The failure of higher order theories at negat
24、ive dZ/dx is thusof purely mrrthematical origin and has no direct physioalsignitkance.$Attentfen of those who refer 10rekrark% 16 h dfed to the Idct that the first termapprfngkithefonrth.orderWIMmta4of amatfon (zWshonfd be 2/3 rather than 1/3. Thfs term Iswrltfen correctly fn the nnmmfml example ven
25、 fn -uon (IW).ISOLUTION OF THE TRANSONIC FLOW EQUATION 513ADDITIONAL PROPERTIIW, OF kPPFtOXIMATE SOLUTIONEquation (15) has some additional interesting propertiesworth noting. Of the two major components of the righbhand side, the first is recognized upon comparison withequation (5) as the expression
26、 for CPm. Since the remtigterm is zero when CP= Cfl,”it follows that the expression forthe critical value for dZc associated with the occurrence ofsonic velocity at n given hIach number M. is(16)It follows, furthermore, that a curve representing the varia-tion of CP with M. for a given dZ/dx, and he
27、nce a givenpoint on the airfoil, approaches tite slope as C, approachesc Pm.An alternative form for equation (15) that is useful forsome purposes is the following which expresses Cpin terms ofthe linear-theory solution Cprather than dZ/dx.This relation can be written in somewhat more concise formif
28、expressed in terms of the transonic similarity parameterscp=2 1, , ,4, ,r.2 1., *, i j,.4-.4 /cI .Mm=l.4 ,.-1.,.”-.2.0Cp-.2/,./FIauaE 3.Compariacm of results indicated by pent theory andin linearized subsonic airfoil theory and the solution UEatthe airfoil surface iswhere the subscriptDifferentiatio
29、n yields(22)i refers to the values for IM=O.du 1 d%= _dx . J1= 1.08$.045 .6 .7 .8 .9 1.5270-l $:.2x/cdzFIGURE 24.Chordwise variation of ordinate, Z, and slope, forthe rear half of the airfoil shown in figure 23.and slopes of the rear half of this airfoil are shown graphicallyin figure 24. No analyti
30、c e.spression is given in reference 42for the ordinates of the rear half.The pressure distribution on the front part of the nirfoilwhere the passage through sonic velocity occurs can becalculated by use of equations (45) and (46), since it doesnot depend on the shape of the rear half of the uirfoil.
31、 Thismeans that the. pressure distribution on the front half ofthe airfoil described above is given specifically by equation(53) for x/c befnveen O and )4. It is clear that the pressuredistribution on the entire rear half of the airfoil cannot bedetermined by use of equations (45) and (46) because t
32、heresults so calcuhted indicate a point of zero pressuregradientiu the vicinity of the inflection point. Although this detail,in itself, is not incorrect, it signals the breakdom of theparabolic method that occurs when A is zero. Positiveevidenm of the breakdown is provided by the fact that thecalcu
33、lated pressures decrease downstream of the point ofzero pressure gradient rather than increase as indicated bythe experimental data shown on sketch (w) or by simpleconsiderations of supersonic flow. These results, further-more, cannot be joined to those obtained by use of equations(75) and (76) for
34、the part of the airfoil downstream of thepoint of zero pressure gradient because the two sets of equa-tions do not indicate the same location for this point. Thissituation should not be too surprising since the proceduresshould not be expected to fail abruptly w-henx is preciselyzero, but gradmdly a
35、s A approaches zero.There exists another possibility for the determination ofthe preswmedistribution on the rear half of the present airfoilby joining together solutions. It is to use the formulasdeveloped for supersonic flow, but with the fial constantof integration adjusted so that the pressure is
36、 equal, at thepoint, of connection, to that given by the solution for the for-ward part of the airfoil. This procedure corresponds to theuse of simple wave theory for the calculation of the differencein pressure between an arbitrary point on the rear of the air-foil and the point of connection. In t
37、his way, the followingequation results for the pressures on the rear of the airfoil atMach numbers near unity:(7s)where Z refers to dZc, and UP(X) is the value of P atx=x.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . . . . . _ . .CA52S REPORT 1
38、35)=%-: (l+ln4)? z (;)=0 (79)and the following expression result: for the pressures on therear half of the airfoil at Mach numbers near unity:(80)Figure 25 shows a comptin of the experimental pressuredistribution for Mach number 1 given by Michel, Marchaud,and Le Gallo in reference 42 and the corres
39、ponding theo-retical values calculated using equations (53) and (80)together with the valuea for dZ/dx given in figure 24. Thetheoretical and experimental results bear about the samerelationship to each other as those shown previously forbiconvex circular-arc airfoils although effects of boundaxy-la
40、yer shock-wave interaction extend over a larger fractionof the chord of the cusped airfoil. This di .iVm = 1.-4 Eq (78)/-2qo 2 / M* =1o f .2 .4 .6 .8 1.0WcFxmmm 26.4mparison of pressure distributions for the rear pnrt ofa oirmdar-aro airfoil as indica M.= 1.L further example involving accelerating a
41、nd decelerat-ing flow at Mach number 1 is furnished by examining thecase of the symmetrical double+vedge airfoil of arbitrarythickness ratio for hich a solution has been given byGuderley and Yoshihwa in reference 32. Figure 27 shoa plot of their result together With the corresponding resultcalculate
42、d by the procedures described above. The resultfor Ocj2:Cpf=-:rl+(-wal 83)The results illustrated in tlgure 29 display a remarkable prop-erty that the subsonic part of the pressure distribution atMach number 1 diffem from the pressure distribution at thelower critical Mach number by nearly a constan
43、t, and thatthe supemonic part of the pressure distribution differs fromthe prwsure distribution at the upper critiwd lMach numberby nearly the same constant, although of opposite sibm.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . . . . . . . .
44、.530GREPORT 135NATIONAL ADVISORY COE FOR AERONAUTICSx/cFmuaE 29.-Summary of presure distributions for upper and lowercritical Maoh numbers and for Mach number 1 for half-oiroular-aroairfoils.In order to investigate e this d.Merence further, the pres.iredistributions at the upper and lomer critical M
45、ach numbershave been calculated for the complete biconvex circular-arcairfoil rmd each of the four related airfoils having mnxhnumthickness forward and aft of the midchord station for whichthe results for Jlach number 1 are shovim in fres 16through 20. The results are shorn in ure 30. It can beseen
46、that the three pressure distributions for each airfoil .-berm the same general rdationship to each other as notedabove, although the difference between the pressure disti-butions is not almays quite so constant as is observed forthe wedge and circukr-arc profles.PRRE DRAGOnce the pressure distributi
47、on is knoxvn for a given airfoil,the pressure drag can be obtained directly by integration ofequation (48). The corresponding expression in terms ?f ;aand ?v (seeref. 5, 6, or 9 for additional information on this point),together with the results computed by use of tlm presenttheory. The new results
48、are indicated by tho solid lines, thoformer by dashed lines and by data points. The short verti-cal lines on the data points indicate Brysons estimate of i,heexperimental accuracy of the data. As can be seen, the onlypoimt of difference between the present results and tho pre-viously existing result
49、s is at Mach numbers slightly in excessof the ,upper critical, and results from the error incurrecl inapproximating the pressure jump through the bo;v shoclcwave by simple wave theory (i. e., by eq. (16) rather thtineq. (6). The positive slope of the drag curve at .= O, orMach nic flow is that the drag is not the same in forward andreverse flow, as is indicated by line
