1、REPORT 1077TWO- AND THREE-DIMENSIONAL UNSTEADY LIFT PROBLEMSIN HIGH-SPEED FLIGHT By HAEVAED LOMAX, MAX. .%. HEASLET, FEtANKLYN B. FULLER, and LOMASIJJLIERSUMMARYThe problem oftransient lift on nro- and threedimemrionawing8$ging at high speed8 i8 dimmed a8 a boundary-calueproblem for the cawical ware
2、 equation. Airchhofls formulais applied so that the anaysi8 reduced, just a8 in the steadystate, to an in.reatigationof 8ource8 and dou61et8. The appi-ca/ion8 include the eraluatiw of india”ai lift and p.tching-mmnent m.trre8 for twodimenm”onal tinhl). Suppose next that there are two sensingelements
3、, or detectors, placed at the point (x,y) Iocatedsomewhere ahead of the VI axis; one of these detectors isresponsive to Iight and the other to sound. ATOW,the lightThe z-O pbne b assumedto be the “pIenc of the wtag”; tit 1s,ifrhe 6.e C4attack were zero and the wIq had m thkkness It wmdd lb entirely
4、In the z-O phne.r Quotesam + aroundtbe v7mdtime s!neethe dbmsion oft Is MueJly bmgtfs, not tkne. It b convmfmt, howerar, to refer b: as “time; I she the actual vdueoftbne fssImpIyt divided by thecozmtantUC,thh shouldmuseno confusbn.Provided by IHSNot for ResaleNo reproduction or networking permitted
5、 without license from IHS-,-,-400 REPORT 1077NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSNumbers refer h t-7._ - -A YlY4 -IXiorxFIOURE4,AcmWic wave pattern for a rnovlng I1noof sonro?5.cletwtor at any given time wiI1 show the sources lying in astraight line just as they -would appear visually at each
6、ptirticuIar in9tant. The situation is entirdy different,however, for the sound cletcctor. First it is necessary tounderstand the nature of a spherical sound wave. Such awave travels outward from its origin at a velocity G, so thatin the time t it hM t,ravelcd a distance t. Before the wavereaches a p
7、oint, th point is completely unaware (unawareis used in the sense tha an instrument will record no changein any of the physical properties of the air at the point inquestion) of its exiskmce and, further, after the wave haspassed, the point remains subsequently unaware of itso.xistcnce. Hence the on
8、ly points disturbed by the waveare those momentarily on the spherical surface itself. (Inthis connection soe reference 1, pp. 1-3.)1he sound detector, therefore, can only “hear” sourceswhich are so locrdml that their spherical sound waves arejust, at h given instant of time., reaching the detector.T
9、he locus of all th points which, at a time. f ago, emittedsound waves that are just now reaching the point P(x,w) isit,self a sphere and for convenience this sphere will be rcferreclto as an” invwse sound wave.” 8 The traces of these inversesuund waves in the z= O piano are drawn in figure 4 asconmn
10、tric circles about tho point I(x, z1, lJ the vmiablc pointsof the sources; t, Wrne” now; and tr, “time” ago. 13qufi-tion (20) is that of the inverse sound waves rtnd cquat.ion(21) represents the position of the visual plan form at a“time” 7. It is nccmsary to includo the region behind thowing covere
11、d by the vortex wake as par of tho visual planform. k case the vorbicity in the wake vanishes, as in thethickn= problem, the wake may still lw considered as partof the visual plan form, but tho strength of the source-doubletdistribution over that prt of the acoustic phm form corre-sponding to the wa
12、ke will vanish. If P 2x IW- 2s, and theIld in the mnge if,# Z Mof. Designating thesepoints by PI,PZ, and Pa (see fig. 5), it. can be shown that theirwmustic plan forms are, respect ively, a complete circle,a part circle and part eIIipse, and a compIete ellipse. Thepoints P are at the centers of the
13、circles and at focaI pointsof the ellipses. Since, moreover, the circular plan form aboutP, receives no signals from sources on the leading or traiIingWlge, conditions at F1 are consequently completely inde-pwdent of the actual (visuaI) plan form of the wing. Th and finally themixed plan form about
14、P? is in certain regions (the circdarportion”) independent of the leading edge, and in otherregions (the elliptic portion) entirely dependent upon it.Since the wing is travehng at supersonic speeds, the traihgellge and vortex wake can have no effect on the measure-ments taken on the wing and, in the
15、 same viay, a Pint aheadof the wing Ieading edge, P4 in figure 5, is undisturbed.Xext consider a Wit lies on the Ieading edge of the wing and the hyperbolic .-sides of its plan form htive its pIan form is still acombination of a hyperbcda and a circIe, but P4 is now thefocal point lying ahead of the
16、 hyperbolic branch used.Figure 6 was constructed so that the portion of the visualplan form behind the trailing edge had no effect on thepotential at the various points PI, etc. If these points hadbeen chosen at positions where the wake cmdd signal itseffect, one of two acoustic configurations would
17、 result.First, if the wing is symmetric about the =0 plane, no liftis deveIoped and the vorticity in the wake is zero so that- thev-iwd pIan form need not include the wake, but effectivelyends at the trailing edge. In this case, the leading edge ofthe acoustic plan form is then determined as before,
18、 whileits modified trailing edge may be made up, in part, of circulararcs formed by the prima wave and, in part, by an arc of “-the hyperboIa formed by the (acoustic) intersection of thestraight visual trailing edge with the prima wave (such anarc bethe acoustic plan form has a traiIing edge made up
19、 entirelyof an arc of the prima inverse saund wave. The spacebetween this arc and the acoustic trace of the visual trmilingedge is covered by a sheet of doublets, the strength of whichis determined by the vorticity distribution of the vortexwake.It is interesting to notice the conversion of terminol
20、ogywhich one to find the partial differentialequation for , and the other to find the physical problemand consequent boundary vaIucs leading to a homogeneousflow field, The Iatter path will be first expIored.First, consider an exampIe of a homogeneous boundary-value problcmo Suppose that a rectangul
21、ar flat pIate startssuddenly from rest and moves forward at an angIe of attackat a supersonic Mach munber MO. At time” tl the initiaspherical wave generated by the forward righhhand cornerlMS traveled outward to a radius tl and, at “time” 2tl, to aradius 2tl. Fe 7 indlcatw the traces of these sphere
22、s inthe z= o plane together with the original and present positionof tle wing leading edge. Let the points .PI and Pz belocatid on the same rays through the origin of the circIesand the wing corners. The probIem is to find the pressuresat PI and Pa.Itis apparent that, if every dimension in the figur
23、e involv-ing Pa is divided by 2L and every dhcnsion in the figureinvoIving PI is divided by tl,the two” figures will be similarin every respect and point PI d coincide with point Pa.T2M0ti1“TM. t,-L “$PiT.-. - - - 3/t_ - -Wing e9of time zero -wing ge af tj ond $?tiFIGUUR7.-OeometrIc reIatIonshlpfor
24、homogoncouaflow.Since the vertical -reIocity wOis constant over the plan form,a simple change in scale hfis made the boundary conditionsfor both problems idmtictd. But this means that tho solu-tions at PI and P2 are identical since the wave equation isinvariant to change in scale. Hence, in regions
25、of u rec-tangular wing unaffoctcd by the waves from tho trailingedge, the pressure can be written(22)and the prw.sure is a homogeneous function of degrco zero,A generalization of this example is contained in tho folIowingstatement:(1) The pressure in any region affected by onIy twointersecting edges
26、 of a straighided flat plate travelingat a uniform subsonic or supersonic speed is homogene-ous and of degree zero (i. e., satisfies equation (22).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TWO- AND THREE-DIMENSIONAL UNSTEADYConsider as another
27、example the case of a flat- rectangularwing traveling forward at. a subsonic or supersonic speed androlling about one edge, taken to be coincident with the x W s d, d: (CO Z)C=*(CO z, t)iiydr (29).whee q is the maximum chord (see fig. 11). In the develop-ment. of equation (29) each element is assume
28、d to have thesame variation of motion with time.I+XFnNM 11.EsaIIIpie of useof iiftlm?demerit.Xotice that when the wiug is a flat plate fIying at a steadyspeed so that. all transient effects have disappeared, CI=*isindependent. of t and of the chord length, being, in fact,vqmd to 4/fl. Then equation
29、(29) becomes4 s“=.s ct(x,y)dxdy=where Z is the average angle ,.of attack of the wing. Thisresuit has already beeu obtained in reference 12. Whencdr, y is independent- of z (as for a flat wing sinking orrolling), equation (29) becomes(30)vhtw c is the local rhord which is, in general, a function of y
30、.Equation (30) simply indicates that longitudinal strip theoryis (xact for calculating the total lift on such wings.Finally, notice tlmt the calculation of the unsteady lift onthree-dimensional wings with supersonic leading edges andstraight trailing edges perpendictiar to the fretream(1irwt ion has
31、 been reduced to an integration involving thervlat ively simple results for a twodirnensional wing under-going the same unsteady motion. For example, the lifto a fiat unya-wed triangular wing with supersonic leadingLIFT PROBLEMS IN HIGH-SPEED FLIGHT 405edges rising and sinking with a harmonic motion
32、 can becomputed from a single integration of the results presentedin reference 5.There is another simplitled method for obtaining the totaIlift and moment on a wing with all supersonic edges and astraight. trailing edge. In this presentation it wilI be assumedthat the traihg edge is normrd to the fr
33、ee stream. Hovr-ever, since the wave equation is invariant to a rotation, itwill be apparent that the solution can be generalized to in-clude n straight, supersonic trailing ede- yawed with respectto the free stream.Consider equation (9) and integrate each term with respectto y between the limits mi
34、nus and PIUS infinity.11 Thereresults the equationIf y=y,(r, z, t and y=y,(z,z,t) are the equations of the 31achwaves streaming back from the leading edges on the left andright sides of the wing, respectively (see fig. 12), then, sinceq is continuous across these waves but w., PH, q., and ware not,w
35、here Ur andthe right andYaIues of theal are the values of u on the interior faces ofleft lach waves, respec.t-ively, andsYr zpy,dy=nrv,11 aterms involving 9, and w are similar to thoseinvolving p= so that finalIy, ii -(31)tzPh7e-._- _ _Plane Wing Jeacfinqe7eg=-m(x+.WOtj -Truce of s far fing .erim wo
36、ve:in x= O planeFIGCEr t2.-Forvmrd portionof Mach ww systemfora supmonfe+dge.i, trfangdac wfng.u TM Mh Iin fact, the problem of a wing tip of specified camber in a freestream at hlach number. Fc 13 shows a lifting sur-face in the z,t phme, The solution for the potential in thesteady-state problem ca
37、n be writtenwhere a is the area on the wing plan form in figure 13 that isincluded in the forecone (ttl)2= (z X1)2,INow from equation (12) we haveI tF1auKE13.Reglonof m plane III whloh bxmdmy condltbmsfor # uro known.whore s is tho local semispan of the triangular wing. Thisequation shows how the so
38、lution to the lifting surface prob-lem in will a-id in the solution of the unsteady problcm, forthe unsteady Iift on the triangular wing is given bywhere S is the area of the triangular wing. It is t.hercforseen tobybe convenient to evaluate the quantity o,s-Mrmc M-CMSS of tw-odimemionel IMw WIds.So
39、lutions to the twodimensional unsteady problems mesometimes especially easy to find because of the anrdogy theyhave With threedimensional, steady %ate, lift.ing+urfticeproblems. For emmple, consider an infinitely long unya-wedving which starts from rest and trayds forward at a velocity1“0which may o
40、r may not be a function of time. The traceof this wing in the x, t plane is like that shown in figure 15.(In the re shown, the wing velocity is varying and isalways Iess than the speed of sound.) The boundary condi-tions are that q, is spccifiel over the shaded area and theloading Apt is zero everyw
41、here except within the shadedarea. But if z is replaced bj- y and t by r, these boundaryconditions are exartly the same as those for a plate of knowncamber and angle of atttick, with a plan form as indicatedby the shaded area, placed in a free stream directed alongthe positive z axis at a lIach numb
42、er equal to , Thesolution for the one problem may be used, therefore, as asolution to the other with only a change in notation.- - T-S of CLVrocft%-kfic cones-. .“:-Tree of Ikooing edgeitFtG,.!?I Iil.-WemtIng wing In ai pIsne.BOUNDARY CONDITIONS FOR WRY SL?WDER WI?+GSiThen the wing plan form is slen
43、der in the sense that itslength in the streamwise direction is Iarge compared to itsspan, an estimation of the loading on it can be obtained neglecting in the partial differential equation the gradientof the induced wIocity component in the stream direction.Thus, if the wing is moving in the negativ
44、e z direction,equation (9 reduc= to%+%=% (36)which is again the wave equation but in two space dimen-sions. Since equation (36) is independent of z, study can bemade independently in each plane x= constant. This is anextension of steadytate sIender wing theory, see, e. g.,reference 13. Figure 16 (a)
45、 shows a typical section in the ytplane. If the Iving is a flat plate at a constant angle of attack,AI x IatI istation/ n occupiedingAI II xl II ISection AA -(b)(a) M fixedreIStlTe to SW alr at InnI!Jty.(-b)AxeSExed OlllFfnlrFIGE M.-Hemler Kfng in unsteady Sow.Provided by IHSNot for ResaleNo reprodu
46、ction or networking permitted without license from IHS-,-,-40s REPORT 1077NATIONAL ADIISORY COMMITPEE FOR AERONAUTICSthe value of w, over the prirt of the plane occupied by thowing is a constant and the jump in p across the vortex wakemust be consistent with its value at the wing traiIing edge.The a
47、nalogy with three-dimensional, steady-stat,e, super-sonic, lifting surface theory is apparent.There is another way by means of which the effects ofunsteady motion on slender wing pressures can be estimated.If instead of using the stationary wst coordinate system, thereference axes are fixed on the w
48、ing by the simple set oftransformationsrl=z+A40ttl=tYl=ll21=2equation (9) becomes(1 MO? $%l.l 2340 .l:l + %jul +Pzlr,+w,tlAgain, if the induced velocity components in the streamdirection are neglcctcd, the simplified equation%lul+%,.l=%,rlresults. The latter equation is identical in form to equation
49、(36). However, now the axes are Mo-O.between this problem and the more general caso of Lwo-dimcnsional compressible flow lies in the fact that in thiscase the traces of the characteristic canes are normal to thet axis, The boundary conditions are tberefom satisfiedaIong Iatw-al strips and, in Iifting-surfaco termiqology, theanalysis corresponds to sl
