1、REPORT No. 253FLOW AND DRAGAerodynamicaI Laboratory,I?ORR!IUIASFOR SIMPLE QUADIWSBy A. J?. ZAHMBureau of Construction and Repair, U. S. Navy. 515Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networkin
2、g permitted without license from IHS-,-,-REPORT NO. 253FLOW AND DRAG FORMULAS FORBy A. F. ZPJWPREFACESIMPLE QTMIMWSk this iext are given the pressure distribution and resistance found by theory and experim-ent for simple quadrics fixed in an M.nite uniform stream of praet.icaUy incompressible fluid.
3、The experimental values pertati to air and some Iiquids, especially water; the theoretical refersometimes to perfect., again to iscid fluids. For the cases treated the concordance of theoryand measurement is so cIose as to make a r6sum6 of results desirable. IncidentaUy formulasfor the velocity at a
4、ll points of the flow field are given, some being new forms for ready usederived in a prezious paper and gi-ren in Tables I, III. A summary is given on page 536.The computations and diagrams were made by Mr. F. A. Louden. The present text is aslightly revised and extended form of Report NTO.312, pre
5、pared by the writer for the Bureau ofAeronautics in June, 1926, and by it released for publication by the ATational Advisory Com-mittee for Aeronautics. A Hs in the near fieId the resultant veloeiy g.If now the distant pressure is e-rerywhere p) md ihe pressure at any point in the disturbed flowis P
6、O+ p, the superstream pressure p is given by BernouiHis formtia,ph. = f12/!lo2, (1)where p = pgOz/2,called the .-stop1 or CCstaggation 97or CCnose pressure.At any surface element the superpressurk exerts the drag f p dy dz, whose imtegral overany zone! of the surface is the zonal pressure drag,= p d
7、y dz. (2)VaIues of p, D are here derived for various solid forms and compared with those found byexperiment. PRESSURE MEASUREMENTSThe measured pressures here plotted were obtained from some tests by Mr. R. EL Smithand myself in the Clnited States q, q,= (1+- cL3/2?J3:Q, q,= 1.5q, sin 0, (3)where 6 i
8、s the polar angle. Figure 1 shows plots of these equations. 1.4zoue is a part of the surface bounded by two pIaries normal to W. CkuaIly one pIane is assumed tangent ta the mrface at its upstreamend.517Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
9、518 REPORTNATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTo graph p/p. in Figure 1, we subtract from the line y= 1, first q.2/q$ to show the pressurealong x; then qt2/q02to portray the surface pressure. A similar procedure gives the superpressurein the equatorial pIa.ne.The little circles show the actua
10、l superpressures found with a 2-inch brass sphere in a tunnelwind at 40 mile: an hour. These agree well with the computed pressures except where ornear where the flow is naturally turbulent.By (3) and (l), on the spheres surface p/pfi= 1 2.25 sin2; hence the zonal pressure dragp.2rydy isD = xa2sin20
11、(1 sinze)p., (4)for a nose cap whose polar angle is 0. With increase of O, as in Figure 2, D/p% increases to amaximum .698 a2 for =41 50 and p = O; then decreases to zero for (?= 70 37; then to itsminimum .3927 a2 for = T)2; then continues aft of the equator symmetrical with its forepart. Thus the d
12、rag is decidedly upstream on the front half and equally downstream on theI-5 -4 -3 -2 -1Le ng+h in A the. 196.f%D. /lrno/d. Rose metal spheres mcolza oi7. d= .o/3 fo. /41 cm. Phi7.,Mug, 1911!L;ebsf er A Schifler. Steel spheres inglycerin, suqar solutions A wu+er.d=. I fo .7cm. Phy.s. Zeif., 19Z4.Air
13、 bubbles in wof er.d=. 0094 fo .061 cmAir bubbks in onifined=.007 fo .l CmPoraffih spheres in anitine Ii.S AIlend=.069 fo .316 cm Phd Mag.,1900Amber spheres in woferd= .114 fo .346 cmSfeel spheres in waferd=.318 to .792 cm 1d=.8cm)Sfeei,spheres .d=f.8 CM in ord= 62/cm1C. Wiesefsbergerd = 9.98 cm ffo
14、/ow coppe a and az bz= Cz,hence changing Czto C2underthe integral sign of (15), we find2 a :,)z log. y,D/pn=4bayb P+ q (16)where now C2= V az. With b fixed, the upstream pressure drg on the front half increases with6/a, becoming infinite for a thin flat plate. It is balanced by a symmetrical drag ba
15、ck of theplate.Such infinite forces imply infinite pressure change at the edges where, as is well known, thevelocity can be g = -jzP,/P = a, in a perfect liquid whose reservoir pressure is p,= co. Otherwiseviewed, the pressure is p, at the plates center (front and back) and decreases indefinitely to
16、wardthe edges, thus exerting an infinite upstream push on the back and a symmetrical downstreampush on the front. In natural fluids no such condition can exist.THE PROLATE SPHEROIDA prolate spheroid, fixed as in Table I, gives for poiuts on z, y and the solid surface, respec-tively, the flow speedsQ
17、.= (1 n)Qo, g,= (1+ )!lo, g= (1 +k.) q, sin 0, (16)1 !- !0 -8 -6 -4Wthd ,Lengih in inchesFrG. 9.Veloeity and pressure along axes and over snrface of prolate spheroid. Graphs indicate theoretirzd values; circlesindicate pressures measured at 40 miles per hour in S-footwind tunnel, Unitsd States Navy;
18、 dots give pressures foundwith an equal model in British test, R. and M. No. W British .4dvisors Committee for Aeronrmtios,_ crosses indicata pressure-drag DIPs computed from measured pressureProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FLOW AND D
19、RAG FOR.MTJLJ4S FOR SEKWX3 QUADEKkS 525where by dots fora like test in a British tunnel. (Reference 2.)By (16), for points orL the surface p/pn = I qt2/go2= I (I + k=)2 sin 6. From this, sinceSinz o = a2y2 (b thendiminishes to its minimum for y= 6. Figure 10 gives the theoretical and empirical graph
20、s ofD/p. for ajb =4.For b fixed the upstream drag on the front half decreases indefinitely with 3/u, becomingzero for infinite elongation.013LATE SPHEROIDThe flow eIocity about an oblate spheroid with its polar axis aIong stream is given byformulas in Table 1, and plotted in Figure 11, together with
21、 computed values of p/pz. NTOdeterminations of p or D ere made for an actual flow. The formula for D/pz is like (17),except that C2= ?P a2, and k= is larger for the oblate spheroid, as seen in Table 11. For 6 fixedthe upstream drag on the front half increases indefinitely with ba.Wind wFm. Il.The.me
22、tie.=); +=.-(l_n) q.cos 6, for the cylin-ders:=(lfOfzou,qn=ia4!y by ds=(ln)qo cos e, for the axialsurfaces; viz., sphere,spheroids disk.ForaJ b=a, bl TableII gives m.; whenceq,= (I+mt) q. sin 0,astheflovvelocity on afixedquadricsurface.g.:=tO for disk, sinceRemarkboth q, qncan be derived fromeitheZp
23、_or y.onl;gi; q; :. hpoint tereofqt=qt sin 8, q.=.Cos 9$= (1n) qov,b a+b= Fa+bw= (l+m) qw,b a+b=Z a+bP=(l+?n)qc%I+el2efWe -j _eIn= 2e”log. 2, See diagram B (fig. 20)Iebsin “em= easin 10bera-v- sinle= ea-sin IebSee diagram C(fig. 20)4=+(1n) oY2)II .+I=-si-:f;=-:f;-s)n-e)2 bm=(ri-u ,v, $, in elliptic
24、coordinates, can be found in textbooks; e. g., $7 ,105,10S,Lambs Hydrodynamics, 4th Ed.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FLOW AND DRAG FORXIIGAS FOE SIMZLE QUADEICSTABLE IIinertia fact ors k.* for quadric surfaces in steady translation
25、aIong axis a in Figure 20531IZUiptic cylinder, E=a/bEIonga-1 tiOQ E k=+1.00 1.0001.50.6672.00 .5002.50 .4003.00 .3334-00 , .2505.00 .2006.00 .167i. 00 .143s. 00 .1259.00 .111I 10.00 .100t .000tIProlate spheroid E=ajb om l Oblate spheroid E= b/aka= a le k.= eli7-E Sin-leIoge *2 eE sin-Ie0-500 0-.500.
26、305.209 1:;%.157 1.AJf3.121 L 742. 0s2 2.379.059 3.000.045 3.642.036 A 279. fM9 4915.024 5.549.0216.183.000 m1“b. this tsbIe k.=rrh of Table 1, VIZ,the value of m when a, LS=a, b. Lamb (R. and M. NTO.623,Brit. .4d.i. Corn. Aeron.) gives the numerid%YIIUCSm the third cokm.n above. For meton of effipt
27、ic cyfimler aIong b axisinertia factor is Ei=a/b.Diagram A Y Diagram B Y Diugrum C T/rzoFIG, 2(3VELOCITY AND PRESSURE IN OBLIQUE FLOW PEIX(XPLEOFWILOCITYCO%lPOSITION.4 stream qOoblique to a model can be resolyed in chosen directions into component streamseach ha-ring its inditidwd -velocity at any f
28、lo-w point, as in Figure 21. Combining the inditidualsgives their resdtant, whence p is found.FELOC!IT1 FUNCTIONLet a uniform then -ii-efind the -reloclty potential p for q. as the sum of the potentials pa, phqcfor u, V-, .In the present notation textbooks pro-ie, for any point (r, y, z) on the conf
29、oca.1 ellipsoidat b c,p= (I+m=) Ux, (26)and give as constant for t-hat surface=+”f. H.J-fmdY c * - (2T)fcothe multiplier of being constant for the model, and k= a a. .4dding to (26) analogous.A .p= (1+77?,=)Lk(l+?iiJv2J (l-iw.=)wz= (l+m)Q?7i, (28)TMsbrief trmtment of oblique flow was added by reques
30、t after the preceding taxt was olished.“ Simple formnfss for this integml and the correspmng b, c ones, published by Greene, R. S. Ed. 18SS,are :ven by Dedar Tnckerm.an inReport No. 210 of the NritimrafAdvisory Committee for Aeronadies for 1925. Some ready vsfues are lissed in TabIes LIf, IV.Provide
31、d by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-532 REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSwhere h is the distance of (x)y, z) from the plane P=O, and a,mb,mq-ma.re generalized inertiacoeffuients of a b C for the respective streams U, V, W, go. F
32、or the model itself the inertiacoefficients usually are written k., kbf k, k. The direction cosines of h are(29)as appears on dividing, (28) by (1+ m)qo, the resuhant of; (1 + m) U, (1+ m b)V, (1+ m,) .EQUIPOTENTIALS AXDSTREAMLINESOn a V c the pIane sections P= constant are equipotential ellipses pa
33、rallel to the majorsection q = O, and dwindling fore and aft to mere points, which we call stream poles, where theplane (28) is tangent to a N c. If e is the angle between any normal to a V c and the polar?/1f(l+?na) uz=-r(l-n.=)u,nn= being constant on a 6 i, as may be shown. SimiIarly, p, p,(1 n.)
34、W; hence the whole normaI component isalong the surface normal n(31)By (26) the inward normal(32)contribute m(l n) V, nq=l(lna) um(lnb) vn(lnc)w=. Cos e, (33)where n= (1 n)z P+ (1 nD)zV+ (1 n,)2TP”5 =m the normal speed (Q. cos e) equals the polar speed times the cosine of the obliquity.This theorem
35、applies to all the confocak, een at the model where gn= O.*Incidentally the normal tlux through a V c is fn cos e- dS=7n J d+, There S6 is the pro-jection of E on the plane of = const. and equals the cross secion of the tangent cylinder.The whole flUY through a b c is therefore zero, as shouId be.PO
36、LAR STREAMLINESome of the foregoing relations are portrayed in Figure 22 for a case of plane flow. Note-worthy is the poIar streamline or hyperbola. Starting at intlnity paralIel to qO,the poar fla-ment runs with waning speed normally through the front poles of the successive confocal sur-faces; abu
37、ts on the model at its front pole, or stop point; spreads round to the rear pole; thenaccelerates downstream symmetric with its upstream part. Its equation qt= O=ZWfh can bewritten from (28)1 + fibV*gt=(l+rn=) Usti8-(l+rnh)Tcos 0=0, ortanO= (34)aThis asymptotes the stream axis y/x = V/U; for at infi
38、nity m=, m,= O, and tan O= V/ U. P1ane-flow values of m., ra, are given in Tables I, 111.AU the confocal poles are given by (34); those of the model are at the stops whereano=wa v (#y=.li-ku 62X (37)Phua on an elliptic cylinder they are w-here y/z= 63/a3. V/U; on a thin la.mina they are atz = + c co
39、s CY,as given in the footnote. Tables 11, IT give values of ?G, . Uis the sIoPe of g. or the aaymptote to (34). Thus (34) becomes a/b= (I+md/(I+?n.), which with the tabtiated dues ofm., m. reduces toz -1 , (36)# zmza hyperbola whose aemiazes are c ma a, c sti CC,c each itspolar streamlinegiven by (3
40、4).A c,lose-graded family of confocal ellipses and hyperbolas therefore portrays all the poles andpolar streamlines intheplane abforall angles of attack. The family can rewrittenx=a cos a, y = V sin cf. (38)Thus, giving a, b a set of fixed values, then a a set, we have the confocal familiesthe first
41、 being ellipses, the second hyperbolas like (36) below.Similarly, the locus qn= O, or q= ij, is written from (33). With F= O,(39)(40)Its discussion is of minor interest.DRAG AND MOMENTFormulas for the pressure p all over the simple quadrics here trea.tied are well known, forobIique as well as axial
42、flow, and serve to find the drag and moment. For uniform flow theresultant drag is zero; its zonal parts can be found as heretofore. The moment about z is thesurface integral of p (g dy dz x ,ji . =za+vwhich determines A, B, 0, and thence /3 in terms of a V. Thus, for an endless elliptic cylinderof
43、semiaxes a=4, b = 1, yawed 10 to the stream, i. e., V/U= tan 10 = .1763, the graph of (42)has the form shown full line in Figure 23. This graph takes the dotted form when V= O, g, = U,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FLOW AND DRAG FORM
44、ULAS FOR HMI?IIE QUADEICS 535For a proIate spberiod of semiaxes a =4, b =1, yawed 10, the graph of (42) is shown inFigure IT.FIG. 23.-Lines of steady flow, lin of consknt, speed end pressure, for irrlWte frictionkm IiquIdstreaming across endIeS elliptic cylinder. Dotted curve refers to stresm parslk
45、l to z : full-linecurve q=q. refers to stream imled W to zThe two values of tan 13in (42) aretan (45)hence the asymptotes confiue rect.angar, as Fre 23, Me with Varying angle of attackthey rotate through (pl + i%). Or more generally one may show that - for the spheroids it is 2tan-2= 109 28 in the a
46、b plane.Figure 17 is an example. If the flow past the spheroids is paraIIel to the h pkne the inter-asymptote angle for the cwes q = qa ha$ plane is obviously affecked by stream direction. “It k 90 for infinitely elongated spheroids; 109” .28 for m others. Excluded from the gen-erakations of this pa
47、ragraph are the for cylinders it is90, for spheroids it is 2tan-l = 109028.6. The velocity and pressure distribution are closely the same as for air of the samedensity, except in or near the region of disturbed flow.7. The zonal drag is upstream on the fore half; downstream on the rear half; zero on
48、 thewhole. These zones may be bounded by the isobars, e const.For the same stream, but with kinematic viscosity v, if the dynamic scale is R= qd/v,d being the models diameter:8. The drag coefficient of a sphere is 24/R for R.2; 28R-”85+ .48 for 0.2 R200,000;and 0.5 for 10RlOs.9.The drag coefficient of an endless round cyh.nder fixed across stream is 8r/R(2.002 Iog,R) for
