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NATIONAL ADVISORY COMMITIEE FOR AERONAUTICS

TECHNICAL MEMORANDUM

No. 1203

THE COlvWRESSIBLE FLOW PAST VARIOUS PLANE

PROFILES NEAR SONIC VELOCITY

By B. Gothert and K. H. Kawalki

Translation of untersuchungen und Mitteilungen Nr. 1471, 1945

Washington

March 1949

AFMDC TECHN~C;.t lH!:iARY

t\FL 2811

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liATIONAL W1ISORY COMMIITEE FaR AERONAUI'ICS

TECHNICAL MEMORANlXlM 1:0. 1203

TEE CQMPRESSIBLE FLOW PPZr VARIOUS PLANE

* PROFILES NEil.R EOIITC VELOCITY

By B. GOthert and K· R. Kawalki

bUMMARY

TECH LIBRARY KAFB, NM

IllI\~ II\~ \~~ n~ H11\]II[~ t~ \I~ 0144651

In an earlier report TIM No. lll7 by GOther-lj, entitled f'Be:':'echnung kompl'essibler ebener Stralll:l,.1ngen bei hohen UnterschnJ.l­Anblasegeschwindigkeitenlf (Calculation of Compressiblo Plane Flows at High Subsonic' Airspeeds), the- single-source me-thod was applied to the compressible flow around circles, ellipses, lunes, and a~oUAd an eloDg~ted body of revolution at different Mach numbers fu~dthe result~ compared ae far as possible with the oalcul&.tions by Lamla and BtlSemEnn. Es sent:. ally i it was found that with favorable source arrangement the si~le"source method is in good 'agJ;'o9mont with the cclculations of thesF.lllle degree of approximation by' LSmla and Bueem8J.m.. NefU' son.-tc veloci tythe number of stellS must be increased considerably in order _ to sufficiently approxi­mate the adiabatic 'curve. After exceoding a cortuin Mach number vThere local supersonic fields occur already, it was no longer possible, in­spite of the substantially inc:ceased number of B't.eps, to obtain a systematic solution because the calculation divereed. This result ,was interpreted to mean that above- this point of divergence the symm~trical type of flow ceases to exist and changss into the unByJmII.etrical type­characterized by compressibi~ity shocks.

SURVEY OF THE APPROXThlATION METHOD USED

In report UM No.' 1117, it is explained how by !!leans Qf suct;:eseive approxi~tionB the veioc1ty fiel~ arounQ plane p~ofilea at high,sub~OlrlC speeds can be computed. But' instead of representing the solution in closed form as practiced in the conventionul methods up to nm', , a numerical method was developed by which a large nVlll.ber of ep:9l'oximating _ steps can be computed, such that the labor for the indJvidual ste];l :is· not increased, with progressive degree of appro:x5mation. As a conBe.quence, the restriction was rem.oved of being able to compute, in t;eneral, only

~ilDie kom}?ressib~e Sbramung um verschiedene ebe-;;P~ofile in NEllie derSchw.lgeschwindigkeit. 1I Zon-t;rale fUr wiss€nschaftlich€s Berichtsweeen der LuftfnhrtforsctlUDg des Genera1luftzel1~istere (zw.s) Bel'lin-AO.l.el'shof, Unte!'s1.1_ehungen und Mitteilur..gen Nr. 1471, January 6, 1911-5. '

:2 NACA TM No. 1203--

two individual approximations ~H1ng to the increasingcamount of p~pe~ work involved. On the other hand, howeve~J it was taken into the bargain that even at increasing d,esree of .approximetion the "mesh widthll of the n'UIllerical method is, in gezwral, me.inta1ned, so t:i:lat a small consistent residual error remains, but Judged from the model problems worked out, this error remains, on the whole, fax below the error introduced by an insuffic~.ent number ,of individual approxi!lle.tions.

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The afore-mentioned report proceeds'from the'concept that the compressible flow oan be regarded as a special incompressible flow, where the entire flow field outside of the profile in the stream is covered with elementary souroes and.. si,nks, the yield of which is dependent in a simple manner, upon the maenltude of the local Mach number. Thus, if the velocity field aro'l.md a profile were knovTn, the elementary sources theIrtSelves would be kno',m also. r.ehe a.eperture of the oompressible from the incompressible velocity distribution could then be explained simply by the fact that the ~dditional velooities produced,. by the Surtl- of the elementary sotU'ces are computed. By the method pursued in the subsequently described model problema, the el~mentary sources were determined first from the velocity field of tho incompressible flow, that is, the velooity field of the Prandtl rule, and With this first approximation formula for the source inten­sities1 the variations of the velocity field computed. With this improved velocity field, the source intensities are then computed again and from it the velooities until with progressive de6'1"99 of appro:r.i1Jl8.tlon the additional velooities due to the sources tend toward a fixed value that represents the desired velooity in the respeotivepQ1nt of the compressible-flow field. .

As the caloulation of the elemantary sources for given velocity is comparatively simple, it is important that the velocity fieJd, which is produced by the souroes distr1but~d two dimensionally over the entire floW' field, be computed in the simplest possible manner. For this purpose, the inco,mpress1ble comparative flow oovered wit;h sources is conformably transformed on the circular flatoT, where the indiv:'dual elementary sources do not vary in yield. T:Qe tiro din:ens1or.LBJ.ly tiistri­buted sources were oombined into ooncentrlcr1ngs or, for tiven roug..'ler apprQximation, into single-point sources which were dispoaed in suitable manner at such points where the :maximum yield Df' the sources is to be expected. Even this very rough method proved surprisingly satisfactory by suitably arranged single souroes for definir15 the velooities in the proximity of maximum profile thiokness as verified by a cOffiparison with the known calculations of Lamla and Busemann with the substantially more aoo~ate s,ource ring method.

The present report deals with a number of model oaloulations for profile flOW's- worked out by the sineJ.e-eouroe lIl9thod. The number of single sources in 'each flow gradient was fixed at twelve, theohoice

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Nf.CA TM No. ]203

of arrangement being sllch that the single sources disposed alone a, stream filament are of equal yield B.S much as possipl~. The .ce1cu1€~tion was further simplified by :postulating synm1etr1cal flmTs. Uusynnnetrical supersonic flows with compresSibility shocks are thus rule~ out before-hand. .

One particUlar advantage of the sin~e-eo'~ce method consjsts in being able to see at once whether the chosen number of approx1"lation steps already complies adequately with the continuity condition of the compressible flow or whether the method diverge a and so indicates that, at the p8.1-tioular Maoh number for the chosen profile, syrrimetric8.l flow free from compressibility shocks is no longer to be expected,

II. VELOCrry DISTBIBUI'ION ABour A LINE (CRillSCENr). (d/l == 0.10)

1 •. Mach Numbers with . Convergence Of Computed Flows

. The chosen eingle-aource scheme (system I, appendix) for a crescent (d/t = 0.10) at Moo = 0.77 is represented in figure 1. The 12 single sources are arranged so as to 11e particularly close together near the' point of maximum profile thickness in order. to be· able to better account for the high yield of the sources expected here. But at f3l~eater distance from the profile, the sources are spaced. farther apart because the total yield, owing to the already subsided increases of velocity and the reaction of these outer-lying sources to the "7elocities at the profile contour and on the areas of great source density, is substantially less. The additional velocities due to the single sources were cbm~uted in starting points disposed as much as possible at' e~ua1 distance from the adjacent single sources in order to minimize the error due to the 'point concentration of the elementary sources. The represented source diagram Varies with increasing Mach number so that the ind.ividual sou:rces move' farther outward into the flow. This outward travel of the sources, deaipable from the physical stand.point J offers mathematically an . alleviation or facility by reason of the fact that after application of the Prand.tl rule, the source locations in the plane of flow, conf'ormaJ.iy transformed. in a circle, are coincident for all Hach numbers so that' the additional velocities induced. by the sources .in the plane of the circle can be computed with the aid. of the same influence fac~or.

The result of the approximate calculation with the previously described system of sources and starting points at ~ = 0.77 io illustrated in figure 1; the stream density (pv) is plotted. against the Mpch number for different groupe of starting points; the correct value of the stream denSity striven for by the approximate calculation is given by the adiabatic curve which at 8Qn1c velocity M+ = 1 has its maximum a,s sign for the greai:ieat p08s1ble stream: density. At polnt

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M+ = M+oo , which corresponds to the, state of the flow velocity, the adiabatic 'curve is tangent to the stream density ourve of the Pra.ndtl rule. On the Prandtl ourve, all pOints of _ the a.pproximate oal culat.~ on by the Prandtl rule ar~ l~d; they are idontif.ied in the diagram as Oth approximation~ such as -the starting point' II at the maximum velooity, etc" for example. 'The diff~rence of the stream density in the partioular approximation points relative to the accurate adiabatic ourve indicates then the extent to whioh the etreaDldensity by Prandtl rule' is still too great and to whioh the width of the stream filament passing through the respective starting point is still too small to fulfill the condition of oont1nu:tty of the compressible flow. IJ.lhis differenoe represenl:,s~ therefore, at the same time a measure for the additionally to be applied source yield, which is to force the po.rt1cular stream file.ment apart to the oorrect width. On compuMng the' additive velocities by the souroes whioh are d1mensionaJ. aooording to the veJ,ocity field of the Prandtl rule, (oth approximation) the velooity field of the next higher is obtained or, in this case, of t;ef~rstapproxtmat10n. It is true that the points of the first approximation found this way lie already muoh oloser to the adiabatic ourve than to the Prandtl ourve; but it is seen at once that the seoond degree of approximation, espeCially at the points of maximum velocities, is still not adequate to approxink~te the adiabatio curve closely enough. Sinoe the still remairung differenoe relative to t~e adiabatio curve is immediately apparent, frolll the d:f,agl:'am even for the points of the first approximation and the additive yield is thus . lmown~ the second approx1raation and also the hiP.,ber approximations oan be caloulated at onoe. For the example in question" it is seen (fig. 1) that' about five approximative steps are neoeesal~y to approximate the adiabatio ourve aocurately enough starting from Prandtl' 6 ourve.

The number of steps required is 1 of course ,less at small Mach numbers of' flow •. Thus" figures 2 and 3 ShOl'7., for example, that at Moo = 0.75 for the same profile only about three, end at t-t" = 0.70 only about two steps are needed to attain the same degree of approxi­mation as at Moo = 0.77 with five steps. On the other hand J the number of steps required at higher M L'I1Oroases enormously. From all these approx1ma.te oalculations reproduced. 1n flgureel 1 to 3, it is apparent that in the middle section (starting pointsll to 71) the velocities in respeot to the Prandtl ourve are still considerably increased; while at the starting points 12 to 32" farther upetreaLl, this increase has alree.d.y subsid~,d appreciably. At the points 13 to 23~ farth~at upstre~., it baa practically disappeared so that~ on the whole.,>, the velocity diatribution along the oontour deoreases'6ubet~ltially with inoreasing Mach numbs'!,.'

2. Maoh Numbers with Divergenoe of the Computed Flaws

The apprOximate calculation at Moo = 0.80 and Moo';' 0.82 was carried out by the same scheme and the results plotted in fig­ures 4 and 5. In spite of the muoh larger number of stops (up to 10),

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NACA TM No. 1203 5,

no satisfactory approJl'.imation, of the adiab~ti~,. cl.U'Ve was obtainable. While at ~ = 0.80, the ninth and tenth steps appear .:1,;0 ap::>roximate the theoretical curve to SOIIl9 emall extent, divergence -is plainly noticeable at MIlO = 0.82 in the fifth step even as!8 evident'in the starting points 11 end 21. This divergence eeems to be traceable to t~ starting points which extend ~to the range of supersonic speed (M+ > 1). The sigIiifioence of -the d:tvergence is discu.8sei( at the end of the report after thf9 results' of approximate celculations for sev­eral other profilos are. ~va..ilable for the d~S~USSi6n:. . , ,

3. Discussion of the Computed Ma.xllD.um Increases of Speed

Compared. with' Buserilann r s Da~a

Figure 17 reprGsents the ve1.oclties in starting point 11 at various, Mach numbers from which the increasin£ number of necessary ~eps with increasing M is ap-parent. When t!lG velocity in starting point lJ. nears the velocity of sound, this profile requires about five to six steps by the Prandtl r-~e to assure suffioiently cloSf9 approximation. So, while the approy1mete oalcuJ.ations ayailable' so far for slender profiles were, 'beoause of the enormous paper "W'.)rk involved., c,arried out only to th~ first, in a. few caSes to the second. step by the Prand.tl rule, satisfactory ccmplience with the continuity condition at those high subsonic speeas is definitely no~ to be counte~ on. This'holds'true so much more because by this appro:x::Jmation methl')d the individual steps always yield a I1ttie loss then by the e1~e-eource method, because the ir source yield was not applied in full strength , , oorreBj;l6nd~g to the momente.I7 velocity field., but merely in an approxi­mation" the quality of which is equivalent to the degree of approxima­tion of ~he whole calculation. ,

From the velocitios obtained. in che different ste.r";incr pointe, the velocity. at -the 'Point of maximum th1clmees' 'W9.S then computed by extrapolation with the aid of the tangent condition (dv = - ~ dn, , " ., '. r :t: = radius of curvature of stream. lines) , which a.t the same time represents the maximum. veloc,i:ty occurring, at the crel!lcent Cl')!~tour and pl<;>tted in fisure 19 for the ind1vidua,l,13-pproximations in com­parison with the values by Prandtl and. B~semanri.. It was found that the first approximation of the sinBle~ourca method was ip good agreem:mt with B\1.semann' s data. Thus, in this example, the arrange­IllSnt of the Single sources was obviously fortunate enough so that the errora of the rough a.~pro~imation in the zone .of maximum increase of speed are nearly obviated: , (However, it should be pointed vut here that in other cases the agreem.ent is not quite as good as .for me crescent" for example" on the eJ.;Li:pse. This d,efect 'in the qu8.1:!ty of the approximation method ie, however, already. evidenced. during the

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ITAC ^. TM Ko . 120 3

calculation in the very nonuniform distribution of the yield of thesource along the individual stranml.i.res so that a correction witsbetter (1i_stribution of the single sources would have been possibleat any time.)

After exc©eding th.e limit of the velocity of sound., fig,,,ra 19 at?a.inindicates that the number of necessary approximation steps increasesconsiderably. A substr-ntially grouter number of steps beyond the fifthapproximation were calculated and. from these the values toward whichthe velocity tends at starting point 11 for Ez unlimited nuriber ofsteps, determined by extrapolation or el9e, as in the immediate proxin!it"rof the point of divergence, by a refined rie±hod. It reouJ.ted in a. very

accuratelj defined limiting Mach rr-unber -it which the method ceased. toconverge. The thus ob aizied point of divergence lles defialtely Inthe si personi_c range at a maximum. local. Mach number of rt~max = 1.09S.

4. Effect of D'.ssimil.ar. So,lrce Arranrd meni:s

The arrangement of the sources must proceed from the point ofview of placing the sources so that the areas of high yield of theelementary sources are covered as vnifornly as possible with singlesources. Particularly, the single so»xces lying on n stream filamentshoulr', be distributed for moot unifona ^r?eld. Since et the star!, ofthe a^aproxima.tion in goner^l, only sn izlcomplete es- '̂imation of thestream dansitias is possible, ,uifavorab7 e :systems of e-)urcas mr..y beencountered. Thus, in order to gain an in3ight into t,le potentialerrors introduced. the crescent flow was computed ag^.in in the vicinityof the point of divergence with a modified source system, which infigure 16 is contrasted with the old syetam. r n the new syetem III,the single sources are i,iovod. nearer to the are& of maximum increaseof speed and disposed more closelZ ,; owing to this closer coverage nearthe maxi-mum. speed in,-reaces, the two outermost s )urce.s of syste.0 I wereomitted so as not to increase the nzaber o" swvrces much abotav tTrelve.The results of the approximate calculations with both systems of sourcesfor the starting points in the central section closest to the prof.iieare shown in figure 16; once for M. = 0.7? w .thin the convergent zor.3.of the mt:thod, then a little above the diverging point at M. = 0.30.At Moo= 0.77 both source systems manifest dietirctt convergence and.at Mm = 0.80 divergence. To be sure, it apueax•s that the nowsoilrce sr; stem TZ -:lr:.a.dy cc cE.s to c onTer^ e at a so?ne;rh:^t-•lo;rerlimiting Mach number; but this displacement 'should be extremely little.In the case of convergence, the velocities themselves, resulting forthe particular starting poi.n.to at high degree of approximation, differlittle from each other for both sour ce systems.

From this comparison, it can be concluded that, while dissimile,rchoice of source systems mr.y result in slightly different numericalvalues for the starting point velocitios, thin error remains considerably

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NACA 1M Ho. 1203 7

below that catJsecl by an insufficient number of approxjmating steps (as for example, breaking off the calculation after t.he first or second step beyond the Prendtl rule). Even the f1Jl1oamental characteristics of the computed flow, such as convergence and divergence of the solutions, remain completely vnaffected by the type of speCial source syete:n.

III. 'VELOCITY DIS'?RIBTJl'rON OVER EI,LIPSE AND ELONGATED

PROFti.:E'(d/Z = 0.).0)

As for the crescent, the compressible flow past an ellipse and. a spindle-shaped profile of the same d/"i:: 0.10 ,,7as computed end. the results for several Mach nw.~bers illustrated in figill'es 6 :\to 1;1.. The source systfJm I was used for the spindle-snaFed Frofile while for the, ellipse, a different arrangement of sources (s~stem II) was choson where the s:i.ngJ.e sources eneno. farther upstre81l1 or c.mmst:ceam so as to secure a better adaptation on the J10re complete Yelocit,)-" distr;ibu.tion of the ellipse. These profiles also evince a Fotnt of diyergence . beyond which the applied ca.lculation method ceases to con-lerge: . This point lios at differeEt Mach number of the basic flow, o.epend:i ng upcm the particular profile form. But it is always o-pse:cved when the veloci ty of sound is loc'ally eJ(ceeded. in the flow aro1Jl1d the profile. This characteristic is especially r~produced for the ellipse in figure 17, where the velocities ere compared to several Mach nVlllbers in the immediate proximity of maximum. profile thickness (starting point 11).

The maximum. increases of speed at the ccmtour 6vk/voo of the ellipse were also computed for the several deerees of approximation and plotted iJ:l figure 19 along with the values of Busemann IS s:pproiirnation. At the lfuch number of flow correspond.ing to the divergence point, a local increase of speed of 26.4 percent results, equivalent to a maximillll Jocal Mach number of M\,~x = 1.08. The compg,rison with BUSSI!18Dn t S method of approxirnatio~ indicates that ':;'he :'irst step of the single-source method. yields a little :mere than the corresponding· Busemann step. Even thoug..h fundamentally the single-source method must yield a little more for the source intensities than Buseroannts IJ1E)thod, on account of -the abserr'G series expansion, the existent dif­ference app~ars, nevertheless, to be largely attributable to the fact that the chosen system of sources was not well enough E'.daptocl to the flow aro~Dd the ellipse as evidenced by the nonuniform distribution of the ~ield8 of the source. However, since the observed difference amounts, at the most,to 0.5 percent of the parallel flow, a correction of the calculation by means of a mod.Hied sovrce system was omitted.

8 NACA TH No. 1203

IV i :, VELOCITY DISTRIBUrION ABOUT A CIRCULAR CYLINDER IN

COMPARIJ50N ,mIT tAMLAf S CALC'()T.wA.TIONS

As a further example for caicluation by the single-source method) the compressible flov1 around a circ1)~ar cylinder was chosen. The Prandtl rule certainly represents no good initial approx:i.Lns;tion for the flow around a circle, because the assumptions for the applicability of the Prand:cl rule, such as regards suffic5.ently small increases of speed, for exa.mple 3 are not complied 1-rith. For this res,son, the ordinary incompressible flow was used as the initial approximation for the, circle so that the respective sources to be applied must cover the enth:'e stream density d.ifference' Datl'leen the incompressible and tl:e compressible flow. The result of the caJ.culation in the vicinity of the divergence point is reproduced in figures 12 to 15. At Moo = 0.40, about :fOUT steps afford a sufficient approach to the eX'lct ad.iabatic curve; at Moo = 0.42, about five stellS are already neceSS8 .. CY and. at M(X) = 0.45, the calculated. oight steps indicate that at this Mach number of flow, no convergence can be any long0J:' counted upon; at Moo::: 0.50, divergence definitely prevails. The velocitios in the charactGristic starting point 11 computed at different Mach nt1lUbers clearly show (fig. 18) how the number of necessery apprOX1IDf'.tion steps from 1 to 2 at Moo = 0.25 up to the divergence point increases consjderably and ultimately after exceeding the d.ivergencepoint, no approxlmatjon satisfying the adiabatic equation can be foun.a no matter hO;'T many steps are used. ' '

The maximum increases of speed. occurring €),t the circlo circum­ference ioTere computed from the data of the single-soUTce calculation and plotted in figvre 20 for dJfferent degrees of approximation and Mach numbers. Where the point of divergence 11e£3 at around Moo= 0.4)+7 with a maximum increase of tSVk/vco = 1.70, the eQuivalent to a maximum local Nach number M+ma ::: 1.295. The values of the firs·t to third approximation by Lamla {reference 1) ;'10re also includ.ed. The conparison indicates good agreement between Lamlats data and those obtained by the single-source method.

V.. SIGNIFICANCE OF THEPOnlT OF DIVF.;F,GENCE

In aJ.I. of the afore-ment:).onoo, calciD.ations, it Was found that the single-source method, once a certain limiting Mach number had been exceeded, was unable to prodUce satisfactory approximate solutions because the calculation diverged. Tj~ical for the divergence point was the fact that the highest local Mach number at the prof:l.le circU1Jlf'erence was only slightlJ' above the sonic velocity. To illustrate: on the

Nft.CA TM No. 1203 9

ellipse (d/l = 0.10) the highest Me,Gh n'U.rrlber relater'). to the d.ive:cGence point was M+max = 1.082, for the crescent (d/I:;: 0.10) Itj+Y'lax :;: 1.096, and for the circle, Wmax = 1.295. At, this state of affairs, the question arises whether perbaps by the rough approximation assumptions of the calculation only the l1l2,ximum local ~..ach numbers are shifted into the supersonic range while by the exa0t oalcuJ.ation the sonic velocity just represents the critical limit at the exceecling of which t11e conver­gence stops. On top of that, it is s:&-tremely important to c.ecide whethor the observed divergence merelj- represents some c.efect of the mathemaMcaJ. wEthod of calcw.ation or whether it is perhaps the evidence of a specific physical progress.

To analyze the last proble11, the single-sov.rce method TN'as lliodif:l.ed so that the calcvlation was reduced to the solution of a l2-term ~'t.iadratic e'luatlon system with 12 unknowns, (This investi~)'ation js to be published in a separate report.) After this cban,ge .. it was possible, for example J to calc-Q]ate flows P2,st profEes with ):].ig..11 local supersonic fields of Wmax :: 1. 5 End mOl~e., This therefore p't'oved that the single-source method. contains no func,amental diffic'L11t:! to push ahead into the area of high local supersonic fields. It. was also found that above a certain limHing Mach number, e'lulvalent to the d.ivergenc~ point, a soh,tion joining on the subsonic flow no longer exists, There­fore .. the reason ror th0 unsatisfactory approx5.mate solution for the compressible flow aroVIld circular .cy1lnders-, ellipseS J and crescent . above the point of divergence is clue to the fact that above the limiting Y1B.ch number, the (3ymrr'etl~ical floi>T pattern postulated by the calculating formula no longer exists, but rather is replaced by an UIlsymmetri'cal flow associated with compressibility shocks.

Concerning the maxhuurn local M8.ch numbers related. to the divergence point, it is appropriate to roexamine :figures 17 an0. 18 with the volocity variation in starting point 11. It is seen that the point 'of divergence is reached i?hen the velocity curve related to this point 11 is exaotly tangent to the adiabatic curve. But this contact pojn-c is alvays in the supersonic range as is to be inferred from the-aBcent of the velocity curve for point 11 and the adiabatic curve. The divergence point could lie then exactly at the sonic velocity only if the velocity C'LJ.L"'V'e of point 11 with horizontal tangent approached the adia'b \tic cur'Fe.

The Cl.uestion is then whether the ascent of the velocity curve of point 11 can be varied as far as the horizontal entry in the adiabatic curve by refinement of the calculation formula. The refinement of the­formula can, for example, be visualized such that the previously employed 12 single sources are consistently split up until, in the limiting case of the exact calculat50n, ir~initely many small elementary souroes are involved; but this splitting up modifies ~n no way the sum of the yields of tho source. Ev@n though this splitting may slightly modify the magnitude of the additional velocities produced by the sources,

10 NAC!. TM No. 1203

that 1s, shift the toteJ. velocity cur-fe frorq. the first approximation somewhat, it is extrem61y unlike) y that the curve up to hor:i. zontal entry would be ch~Dged as a result. It is tterefore necessary to reckon with the fact that the computed looal supersonic fields do actually occur and can be symmetrically removed without compressibility

. shock. By the worked-out model problems, the highest local sapersOlli.c Mach numbers still obt,dned in syrnLletric.s.l flow are so much greater as the Mach nV.Ilibers ofbhe airspeeds related to the di vergonce point are smaller. [.l:h1s is exemplified j.n fig1..1.re 21 where the highest local Mach numbers of the divergence point e.re plotted aga1.nst the airspeed Mach number for the cirole, crescent, ano. ellipse. At Moo::: 1 the curve of the highest local Mach numbers Las the value ~t+ max;::: 1, hence at this speed the increases of velocity eX9.ctly.disappear. It means that at sonic velocity Hoo e 1 a symmetrioal flow is possible only for the infinitely thin, flat p'late in parallel flow which is consistent with the variation of the flow clensHy curve with its maximum at sonic velocity. Figuro 21 further 8hmTS the Mach nun1bers of flow at which local sonic velocity is exactly rep,ched. A com·&erison with the Mach numbers of flow related to the d.ivergel1GG point indicates that both differ very little, the difference becames less as Mw becomes greater. At Moo::: 0.80 the difference amounts to a mere L'Mo = 0.030. SOJas far as the application of these data to the conditions jn aero­technics is concerned, it is quite irmnaterit~l whether the flovi abandons its symmetrical char(',ctel' at the sonic sp0ed lfillit or at the Mach number of the divorgence point and changes to unsymmetrical flmr with compressibility shocks. This result holds for the present only for the discussed mOdel flows arour~d the Circle, ellipse, and crescent. Howeyer, it is not very lilrely that the conditions will be mater:i.alJ.y different on other profile forms, Since on the ellipse and on the

. crescent of equal d/~) for example, profile forms with very dlssimilar curvature Variation are already involyed.

From this point of view, the. question of "whether the previously computed s;ynunet:t'ical flm'l"s with local supersonio flow fields are stable, that is, whether they return to ·the initial flo·;'/" attitude after minor disturbances, itself loses. irtrp9rtance. In the calculation itseIf., only the conditions for potential flows, such as continuity, freedom from vortices, etc." were contaJ-ned 80 that thE> problem of stabiltty would haYe to be treated additionally and t:qerefore it is not ascertainable .. for the time being.

VI • P.E8ULTS·

From the results of the calculations by the method of approxi­mation on compressible plene flows with the aid of sine~e-source arrangements, the following findin/3S 'iv-ere arrived at:

NACA 'IM No. 1203 11

1. The maximum increase of speed on the circumference of a circular cylinder computed by the single-sour0e method ane compared with the data by Iamla indicated that the calculated maxi~um speeds in all comparaple degrees of approximation (steps 1 ~o 3 beyond the incompressible flow) are in very close agreement by both methods.

In the case of the flow around a~ ellipse, a O.5-percent difference in airspeed resulted in ccmparison with~uaemaD~ls cal­culation (approximation ste-p 1 ), which obviously is attributable to the not very favorable arrangement of the single source • But the magnitude of the error inv01ved did not seem to justify t~e additional paper work involved in a calculation with an improved source distribution.

The first step for the flow armmd. a cresr;ent beyond. t1:..e Pr!Ulotl rule could be compared with BusElmanr~ 2 scalculations; here also the agreement was practically perfect.

The conclusion, therefore, is that the single-souTce methoc1. is suitable for a quick, n::imerical1y correct insight into the :tig116st increases of speed accompanying the compressible flow past profiles.

2. As the single-source method afford.s a clear view of the quality of the attained approxirnation to the adi'3-bati.c curve, it j s possible at all tilli8s to decide whether the attained degree of approxi­mation of the calc1.uation is alree.dy sufficient or whethe:c t~le mmloer of steps needs to be increased. In this respect, the. model problems worked out disclosed that a:t low airspeeds t'tro to three steps are sufficient, but that in proximity of sonic velocity, six to eig..ht steps are necessary. Thus in this range, the calcl)J.ations by Rayleigh .. Lamla, and Busemann would also r.L8.ve to be cont:inued up to sub3tantially hig,her degrees if sufficient approxL~tion to the ad.iabatic curve in the sonic speed. r~Dge is to be attained..

3. After exceea.ing a certain [1ach m.u:nber of airspeed.) it Was no longer possible to secure a satisfacto:c,f syY!llJ.6trical a.p-proxir>..ate solution with ~Ly number of steps because the calcu2ation di7erced.. Since in other cases it succeeded in cowputing OJf the S~~ method subsonic flows with large local sUJ.?8rsonic zones (Mt:m.."LX =,..., 1.5), the observed divergence cannot be attribuced to the imperZect suitability of the employed calculating method. The appearance of the divergence was, therefore, L71terpreted as tr.at the symmetrical subsonic flow can be continued only up to the Mach nUIilber related to the point of diver­gence and after that, the flO'll changes into the 'Ul."lsymmetrical type characterized by compressibility shocks.

4. The highest local velocity appearing at the point of diver­gence lies a little above sonic velocity, tne .. fuore so the sma.ller the Mach number of floy! related to the point of divergence.

l~ NACA TI'1 No. 1203

On the circle 2 for example, the maximum local Mach . 'number is M+mctx ;:: 1.29, at the crescent (d/1:::: 0.10) M*!l1'3JC "" 1.10) , and on the ellipse (d!L::: 0.10) M+~~x :::1.08. '

5. The Mach :ntmibel':s of air fJow' related to the point of di ver­gence are only a little h:!.gber than the Mach numbers of flow at which, locally at the circv.mference i the' sonic velocity is ey..actly reached for the first time: the diff'erenoe in ,Mach nu.rnber is, for e:rample) tMoo::: 0.030 for the ell:!.pse (d!l ':c 0.10), L:Mco::: 0.030 for the crescent (d!i::: 0.10), and bl1co = 0.050 on the circle. Thus for actual flight practice, it is quite irmnatorjal whether the flow already chenges into the unsymmetrical type on reaching sonic s~eed or at the Mach number of the point of divergence.

6. The flows with local supersonic fields computed by the single­source method satisfy the conditions for potential flows such as continuity, nonturbulence, etc. Ro:vever, it yet rerne.ins to be proved whether the calculated supersonic fields are stable" that is, whether, ai'ter a small disturbance, they ret"cU'n in the initis.l state.

Translated by J. Vanier National Advisory Committee ,for Aero:p~utics

•

NACA TM No. 1203 13

P.PPENDIX

:t.ffiIT'HOD OF CALCULJI..TION

1· Choioe of Starting Points and Source Systems

Since the sources to be applied "3 '" 0 on the crescent and especially at the

. ., '\ 2 ,* 1'01 ~O spindle-shaped body are predomin£ltely Star-c:mg pOlnts ----;-.1 \~0-+ r placed "lithin a cOlll:paratively narrow

·3 3 ,j ~ 2~ r---- \ !' 2 range near the :profile center, u, Source-line "", /<., \ \ \ ss-stem of sources was chosen at 'I.hich {j* :::; Const ;" /-+" \. \ ....-C'r--t' 1'1 the three-s01.n'ce lines {}*:::; ,Constant

'l~~~'- "'t>/+'\ \ J - lie close :'or::ether (s~stem of poin~s I). -r "'--.' \.,) ~\ l 1'0 For the circular proflle and espec~ally

Source circle I /' ~.~ \\ l for the ellipse on the other hand, the r = Const [ . I "'-.''0\\ \ area of the additional sources is more

....... I f i 13 "-:~ extended toward the stagnation :point. 1'/1'0 1. 0 Owing to the stee:p increase of s:peed

at the elli:pse aft of the Jlrofile nose, the maximu,m of the additional sources lies for this profile in the forward half of the :profile even at high Mach numbers (system of :points II). For the study of the ini'luence of the chaI'.ge of the source arrangemEnt, a second system. was includ.ed for the case of the crescent, where, instead of the 12 so'urces extending far into the outer s:pace, 13 closely spaced sources were chosen (system III).

Starting (f.Lv)

Starting Point and Source System I (Polar Coordi~ates in the Plane of the Circle)

[j3tarting point (rf.L' 'l'lV), Source point (rk' 'l'l A *), 1"0 = Radius of image circle]

points Starting :point and radii of oircle of source rf.L,k/ro

Source points 1.075 1.242 1.436 1.659 2.21~ 2·959 3·952 (kA)

Starting point angles 13v = 90 f.Lv (1 1) (2 1) (3 1) (4 1) (5 1) (6 1) (7 1)

= 74 (1 2) ~2 2) (3 2) ::0 60 (1 3) 2 3)

Source-line angle -o*A = 78.75 kA (1 1) (2 1) (3 1)

= 67·5 (1 2) (2 2) (3 2) (1+ 2) (5 2) (6 2) (7 2) = 52·5 (1 3) (f') ,,) c :)

14 N/DA TM No. 1203

Starting Point and Source System II

Starting points Starting point and :radii of cir01e o.f S01ll"ce ()l. V )

Source points 1.075 1.242 1.436 1.659 2.061 2.572 3.952 (k;,,)

Startin8~point tly :::: 90.0

angles /J,V (1 1) (2 1) (3 1) (h 1) (5 1) (6 1) (7 1)

= 67·5 (1 2) (2 2) (3 2) :::: 45·0 (1 3) (2 3)

--'--~'

Source-line angles (2 1) tl*?\ ::; 71·75 kA, (1 1) (3 1)

:::: 67·5 ,

(4 2) (5 2) (6 2) (7 2) ::; 56.25 (1 22 (2 2) (3 2) ::: 38.0 (1 3) (2 3)

StartIng Point and SOl.1r()8 System III

-Starting points I Starting point and radii of circle of source

(p,V ) -

Source points 1.075 1.2!.~2 1.1~36 1. 6~9 2.061 (kiv)

I Starting-paint I ang1es\

-8v -= 90.0 ' I flY (1 1) (2 1) (3 1) <4 1) (5 1) ;: 78·75 I (1 2) (2 2) (3 2)

(5 3) = 67·5 (1 3) (2 3) (3 3) (4 3) !

Source-line a~gles I k/\ (1 1) {) *;,. ;: 84. 35 (2 1) (3 1) == 73·12 I (1 2) (2 2) (3 2) (4 2) (5 2) ;: 61.88 (2 3) (3 3) (4 3) (5 3) : (1 3) - -.

2. Influence Factors in the Cyclic (circular) Plane

With ~~* ::: Reitl * denoting the local vector of a source and

;s = re itl the starting point., the velocity j,nduced in the starting point is

lThroughOUG this report ;f and y are substituted for tl1e German script x and v which were used in the original German Version.

l5

y '" R2 x - x* E ro -;- ... == -f2

-13 2rr R H 00 r

when represents the source strength and

the spacing-, If the system of the coo:cdir.ca-ses is rotated through 'J, radial and tangentie.l components in the sta2.~ting points change into the x a.'rld y' components of tLe new syste:w.- Th1.l.S, vTith r/:'{ ,:= T,

-I3*~a==)'

Yo - i1Q vr + i"'"v-8 -e ., == Voo Veo

'" ro (T "'" .r'.i == 2rr R

hence for the ~ components

with

E ro == --f

2n R v ~

- sin r

- c.os )') - i sln r 1 + ,2 - 2'1" cos r

16 NIGA TM 1.0 " 1203 '\

The contour conditions are satisfied by adding to each principal SOlJrCe (Hqu) the sou.r.co (SWu) reflected at the contour circle. If the source arrangement is in addition doubly symmetrical, that is, sources of equal strength in ±-0* sources and in ±(rr - -a*) sources, four principal sources each can be comMned. For I' = (v*. - ij) Wo.d 1';: -(-a* + -a), f VfJ is to be assumed positive, for "I = rr - (13* --8) and I' = -rr + ({l* - 13), negat1ve, so that

Mn (-3 -l(. + v) (~*+~)]2} + sin2(-a* + tl) + LT + cos

-[ Sin2(~* sin (-0* - 'a)

- v) + [! - cos (v*--:-{) D 2

-"821 sin (v*-iJ) (1) + sin2(a * - v) + G + cos (-a *

)

In order to better assess the influence of a continuous source distribution by a system of 12 single source12, the latter are visualized as being replaced by a source-line system with two-dimensional source strength distributed along rays {J* = Constant. Combining then the influence of one radius at the source points by averaging and taking the source strength along the interval as constant, the influence of a step-lilee sovrce distrib-u.tion along the ray {l*:;: Constant that sub­stitutes the line superposition is closely epproximai:.ed although in the practical calculatioI~only single sources are involved. Unless the influence quantity t-fVtl plotted for fixed -a and fixed {)-)(-

against T varies excessively, the mean influence is equal to the influence in the center, or within the, framework of the distribution by powers of 1.075 in the intermediate partial points. In the vicinity of the extremes of the influence curVes at T = 1, however, and at intervfl...l increases at the fourth source ring the average value of the interval must be computed more accurately. Indicating the average formation by oVers coring and including the factor 1/2rr in the influence factor, the final 19 component of the induced velocity follows as

ivith

UACA 'IM No. 1203 17

3. Conformal Transformation. and the EDsuing Velocit,y pistox:tion

The conformal transi'ormation Z:i.k'Y = f(1:) gives the flow around

the profile in the ik* plane from, the flow around the image circle I s I = roo The complex potential )( = cp + i-r is

dX = dZ *::

'ik"·

'1Z * o .. , , anCL the CListortion fw...ction

J.,K is

The

anCL

anCL

F := (v /V(fJ) circle

(v /To:J ik. * - CL circle

flow arounCL the circle I s I := ro ro

X := S + -

as contoUl' is ''lith S

S

CLX ro2 [: (r" 2 ] J.'Crr:)2 - - - 1 - - - _1 - \ 'rC2 i. cos 2-Q .J-, -d1:-- S2-, ,,/ IJ

------c,~ 1 + er olr) 2

{) = 2 --.--:;.) 2 sin {J

r = ::Co

sin 2'13

To CLeterrnine then the velocity of a specific profile in the ik*-pl~.:r:£ , only the CListortion neeCLs to be computed.

a. Crescent.- If 5 is the edge aD~le of the crescent and

18 TM No~ 12q3

the thickness ratio of the given comparative prof5.le, the geometry of the crescent gives "

or

with

tan Q . ,4

The conformal transfor.mation is attained by

~ '/'J. X'o - 1. c' ll-' 06 :::: __ c_

o !

fa + 1

,The quanti ties S cUld J3 can be represented as functions of 'the circular coordinates

/ r \2 2.1:. ;'1 ::;: :re/2 / . ~ I + 1 cos ~ ':':'-...y 1 82 \rQ) J~Q = ,'')'" 2

+ 1. + 2.1:· cos -S r 1:'0 ii"l~O ) fr r 0

nrc tan 2-~ sin ~ ro 13 = ----.-..:;:.. __ _

" 2 I r\ _ 1. \reV

1"1 ::: r~o

-~'"7 ~r/2

tan2 ( 8/2)

arc 2r/ro

ton ~'/'-'-.2:----f X'o) - 1

N.~.cp. TM JiJo. 1203 19

The profile parameter n is given by the thickness ratio. As exponent it indicates the multiplication of the angle rr in the stagnation point of the circle. It is

rrn ::: 2rr - 5

or

n __ 2 - 4 a:rc tB.Yl (~)~ rr v J.k

The velocity distortion follovTs from.

as

:8' ::: n2

::: n (' if3'\~ \.2e )

(-:13' -~ - \2e ")

(1'1

s+ 8) - 2 cos f3

sn + (t~ - 2 cos nj3

€ = 2 arc tan (s + 1 tan .§\ - 2 arc tan. (~:... + 1 tan !1§\ \2 - 1 2) ~n - 1 V

(4)

The formula of the conformal tran.sformati-:.lD ,:"as so chosen that parallel flow ¥.revails at infinity. Actually it is

S ::: 1, f3 - 0> -71

which cond:i.tions that = n.

20 NI.CA TM No. 1203

b. Elongated body of revolution ...... The confo:rmal transformabion dZ *'

is Supplied by the formula satisfying ik 'co '. 1

rtr K". + j L ~o - :;/r

o) SIn 13

1 ., K

3 (1~o\3 J '\...r) sin 3J

~/2 = 22 __ + K Sj,nce and ro 3 (gji)' * = 2~(}... - K~

,(-ro ik 3 is obtained) the profile parameter K is

and

The velocity distorti on folloi'TS from the ik* - veloci ty at

21

With

N = (F",4 + 2(1)2 (1 -K) cos 2-a + (1 - K)2 '£0) \:0'··

we get

(Vx - iV?\* = IfF. )2~(+,2 + \: v 00 -) • • N\t 0 I \:- 0)

. lK L (1 + K) cos - K)

and

hence, with (3)

= A + B cos 2{l + C cos 4-a

"There

rr" 1.:_

C = -2(1 - K) ~i)

22 NIle!. TM No. 1203

c. E1~ipse.-A, corresponding formula supplies the confor.mal trans fOrTuat i on:

Zik* !: a2 1'0 = -+ 1'02 1'0 1'0 !:

I

2\ + i~ -;~)sin ~ J = ~ [~+ ~ cos tJ

10 ' 1'2)

with

as profile parameter. The velocity distort':'on follow's from

=~-,"D = ~ - ;: cos 2V + \r~ sin 2V

as

, 4 2 Ij' = VI + (;) - ~{;) cos 213

/,,;,\2 C~.J sin 20

= -----.-

1 - (~)2 sin 213

tan € ( 6)

hence; with (3) the velocity becomes

* ~k~ voo

(a\4. (a)2 1 + r) - 2 r cos 211

~ ::: n/2~ 1 + (ro/r)2 = 1 + (a/r)2 1 +

2 sin 11

sin .i]

4. Detorm..i.nat:).on of Source Strength for Assessing the Difference Between the Exact Stream

Density and the Em:P.1oyed A:p:P.roximation

is the stream denflity of the com:pressi ble fJ.ow) we get Hi th

(a ::: local sonic velocity, ffL::.: crj:tical velocity of sound) as a

result of the adiabatic :phase change

1

K - 1

wi th It := ~J2 ::: 1. )+05. Considering only the linear terms in the interference velocity Cv

conformably to Prandtl and :putting

1--" ~'I o . F0 o

CA)

. M};:')!- = 1 + ~k* Moo* Moo*

gives with

with B = M 2

CO

yielding

as linearized Prand.tl-s'bream density.

Ivfk orr A-- + B

M * 00

( 8)

Differer:.tiation of (7) gives as slope- of, the s~ream density C'LU"ve at point M of:

1. !J.2

The Prandtl-st:ceam density epT' therefore, reprEJS0'f.t" the tangent to the stream densi ty CD.rve in the approacll flow point, whj le the incom­pressible stream density at speed Mk*

can be regarded as secant through the points Mk* = 0 and 1%:* """ Moo*.

Now ..:in order to be able to secure an impr0"TE;;d solution for the compressib10 flow with the aid of the approximated stream density eN = 8fr or 0ib respectively, the compressible flow in a flo'" tube

is visuali zed as being replaced by a flO''VT of apPl'oximated stream density eN with identical veloei ty fleld, in \vhich the additional displacellient is produced by an addi tio.i.1al source distribu.tion. (Compare 'OM 1117, p. 3)· If d..n.k is tbe vlidth of the flow tube at a given point, ~

NAC;>., 1M No ~ 1203

1<lill be valid for the flow along the fl,ow tube. The approximation flow with additional sources is ·to show the saw.e -,telocitJes as the compressible flow. Due to the source qistribution, the flow tLen differs from point to point:

t'V

' .. d'fN = d'W'7\J

rc7 00

= ~n~ eN.

ro Boo

25

Both ek and eN are functions of tbe Yelocity M-i{*/i':h,*. A.-to infinity, the flow is the same in both Gases, due to eNoo = Boo' The sum of all vclume sources to be pla0ed upstream then is (ccmpare uN 1117, p. 28)

d~ ':N~

If one introduces tho constant flow d1jr,_, whj cll is calcuJ.ated, say" .r;.

in the center section of the doubly-syY~etrical profile,

dEN = CN* -...!: w'i th Conk e.~~ - ro e center section

(10)

is valid. CN* denotes, therefore, the relative error of the approxi­

mated stream density in contrast to the exact stream density for the, compres sible veloc ity 1'k *.

a. IncOlllJ?ressible basic flow (flow a:r·o1..rr~d circular profile). - The curve of the source strength to be disllosed about the circular profile in the incompressible flow is, therefore

'" dEik with (10 t)

The width of the flow tubes at the circle in t~1.e center Bectlon is

Qu.nk~ r ,', . o cen~er seC~lon

26 JeIWA 'HI No. 1203

E.:2randtl basic flow (crescent, spind.le~sha'ped body, elHpse).-For their profiles the Prandtl approximation is considered as comparative flow and the profile analyzed in the affinely d:tstor~ed ik*-plane instead of in the compressible plane. Witlf Mik*o denoting the velOCity at the profile of the ik*-pl,une,. as_it iso.ttained 'fx'om the cyclic flow free from sources by cop~ormal transformation, the velocity of the Prandtl approximation M,.\: *.) OIl the profile of the k-plano obtained by the transforrrLation

corres1?onding: to the linearization of , the stream density in (8). The related stream density, therefore, is

Bik* M'k* with _...b:...- denoting the inco:m1?ressiblE;l stream density related

Boo .Moo to Hk * ° in the ik*'-pl£lnc, the transf,'ormution of the stream density follows as

(12)

NOIv, if the Prandtl flow 1v\c *0 in the compressible plane corresponds

to the flow Mik* 0 in the -ik*-plane) the Prandtl :i:low Hk'j(-

provided with additional sq,urces replacing the compressible flOl'1' corresponds to a flow in th6 :i.k*-plnne f1 tterd ',vi th sources, the velocity of which is equal to the velocity Mik* transformed by (J,l)

The throughflow of the Mik * - flow wi tIl SOi..U'ces at a point of flo'll, tube width dnik,)i-, is then, since by (12) Bik* - 8Pr Prandtl is the related stream density,

d'" * 1.VUc :=

Owing to the source <listribution this throughflmr varies from CI'..e

place to fu'1other. An irlfini te <listance

the sum of the source strengths in the

dljrik*co = ik*-plf'rJ.6

hence}

27

dnik~ o.n,~ For their profiles) ho-;-rever, _ * = dn" L.DO and in t;he correspond.ir<...g

C!...."'1ik k

compressible floi'T dn~k = d.hko-}~~ DO by reaDon of the continuity. Therefore, the SOUIce strength to be al'plied in the H:*I)l~ is

while. by (10) the le-p.lcne follows at

~ * - C * .--:::.-!: - ..:lUI * '" (dnk- e1~ rv

ctEt: -. PI' ,ro G~ - f.L u..i,ik

The source streI'~th per width of flow tube remains, therefore, unchanged by the transformation. On comparing (lOti) ~Tith (lCJ I) H is seen that, owing to (12), the same source strength dEik* had to be placed in the ik*-pleDL, if a ccro.:pressible fl,)w '\'TaS to 1:e cv.mPUI;€d oJ' illc..ans. of addi tive souj~ces, which is to flo" throueh- the flow tube 0'::' the ik*-~lnne and there cttuin the velocitJ ~L'r* of the incam-

- eJ .-Lh~ pressible flow with sources.

5· Iteration Process

Since in the fo:rro.ula for CQ'np1.1tip..g the SOlITCe strer..gth the un..i{nown compressible velocity £.1::;:* enters on the righthaneL sieLe of (10), the source strength must be calculated step by step. With Q-1t*o)f.L V as the velocity in the startinc point f.L V of the ?cX!rparati ve flow used as basis, that is, of the incompressible ci:rcular flow or P~~anrctl flow around one of the thin profiles the relative error of tile related approximation stream density BN(Mk:o). in respect to the actual

stream density e (M,,~o) is computed at the -pvints (M. * \" :..-fter -- '. K O. Il\<

which (10) affords a first approy..irD.ation for the lJoked for source

28

sum I of the mati on

with

N!CA TM No. 1203

-L. == """ 1ro ...

-. . . Oeneraliy .. if ( M *(n-1)') . are the velocIties , \. k , flY

rovco' Ix-(n,..~)th app1:'o.xiInatior~,: the, source strength sum. of nth approx.i­is

tV (n) _ [I J ;." Bk: A. • ~tV -

where ek(MJr*(n~1)) is preferably taken i'rom a diagranl,l and A/.l \ / '-

represents the 1.,ridth of the stretmJ. 'Cube in mid.section. It is

.AtJ. = r- and -r- i3::o 11 respecti ve1y. Since the outermost source 61: ~ik) . o 0 2

ring must inc1u e all the sources existing i~ the outer space, we get, R( R7

for example} for the residual source ring ~. = 3.952 , if ro = 3. 298 :1."0

ring (~:r) related \.:t 0 6 represents the limiting ring of the influence

r6 to r-'

o

where 0·38 indicates a reduction factor for the .~nflu) ence of the outer space. The individual source strengths EXA n follow then

from the difference of two sou:l;'ce surns in adjacent points on the same ring, since they relate to the same flow tubos. The additive

veloci ties .6.M1\:-)(·(n) of nth approximation induced by the sources E~)" (n)

in the starting points flY are obtained by means of the influence factors . fe'X' at the cirole, by neglecting tl;le r-component and taking the -6 -Qomponent equal to the total velocity. (Co.mpare UM W·'Y,p· 3·)

lFor thin profiles for i'lhich the velocity differs IHtle from the approved flO'l", the exact stream density curve (7) can, in vicinity of sonic velocity, be closely app.coximated by a parabola, (compare Oswati tsch) . On expanding the bracket in povler of DMk .)(·k ~ ~k~' - 1 an~ discounting all but the squared terms, I-Te eet ell: ~ 1 - -2 ~ - Mk*) .

I&-.Cf .. 'l'1vi lio .• 1203

The influence factors for the velocity

the circular flow are, because of 0

'6v\. <

--) :!co/'tJ-V

(2) simply

1"V = f.o* KA, 1J

29

of the source of

The total effect of all sO,urces in the starting point 'tJ-V is o-Dtained then by summation over all sm.1I'ces ~A,

-,---\.

=L- (14) KA

For the profile f10'l1' the influence factors (2) m1)st be, in correspondence with the conformal transfo::cmation, dh"io.sd by the distortion g,uont::.ties (4, 5, 6):

and the additive velocities according to Prandtl distorted corresponding to (11)

with this velocity connection the totnl velocities cf the nth approxi­mation are:

M *(n) - M * .I- P-,U ·:fen) L-k -"k 0 . ,,-,,-'Jk

If to the ir~tial velocity ~*o there corresponds the stream density

of the basic comparative f10w- 81l: (0) = en: Q\: *~ OJ:' 8Pr ~1r*~ ,­respectively, then to the improved velocity Mk*{n) there pertains already a stream density nearer to the COYTect 8~curve; ek(n) is calculated from (13) by solution of tlle eg,u.ation "Yrith respect to SkI

and by substituting for all magr~tudes on the right side the values that correspond to the nev! approximation

30

with

Oonnecting the points Mlr*(n), a Cn) k

NtCf, TM No. 1203

(16)

for each starting point giveo a set of curves which at convergence of 'r,he method approach the a-cur've consistently. The points of eg;ual approxi:matil)n eive a picture of the already attained approximation of the O~ourve.

1.03

1.02

1.0f

1.00

0.99

0.98

0.97

0.96

0.95

Stream ·density l{ p v I p'~ I V " 1 Prandtl's rule \r- Starting p , , I V. '1 I

2f P:'4 .... ~6 "'-0, ~~ I

oint 11

l' 1 1 1

\ I \ 1 \ 1 \ 161

+ \ 1

~ 1 \ I

I e I I

~ I I I

V ~-' V<

~V V 1--= Adiabatic c

\ 1 \ ~51 I Starting pOints + ~ 1 + urve

5~ V

71 ~ Parallel flow

~v ~ I- Mach number 11 I I 1

II Vx =(lif

\ : I Source system I \ 141 I

, i31 32? ~t·ti'r' 11~ 121~i22I 21

~

sources III

sinks e

, 0.&1 0.82 0.84 .0

0.86 0.00 ago O.g2 0.94 096 0.98 1.00 2. lA( I I

1.02

vi \ 1.0

Stream density t p v ~ V- Starting

p' \/l(

~ v point 12-

t I ' 1 q:2

Prandtl's rule-? ~~ " V- 5 ./

v: ~ ,,,,-I-Adiabatic curve

~ V

~ ---' ~ Mach number M! v:

1.00

0.99

0.96

0.97

0.96

Stream density f v Iv Vo

~ I-Starting

! \ point 13 I I V ~ Prandtl's rule- ~1 ..... V

l/i ~ ~2 W V"-I-Adiabatic curve

V x V

? Mach number M-:: :

f.01

1.00

0.99

0.98

Q97

0.96

V I I I I 0 V I I I I I a o.~ a~

().lJ() 0.82 0.8<4- 0.86 0.88 0.90 0.92 0.94 0.91; O.9B 1.00 0.80 0.82 O.8-f M6 D.c!18 0.90 O.~2 0.94 0.96 0.98 1.00

Figure 1. - Line (crescent) (d/~ := 0.10), Moo:= 0.77. Velocity and stream density for various approximations to the adiabatic curve (source system I).

LV 1-'

/ .- ar mg pOl Stream density v~r /V 8~ I '. J

P" vi. 11 1/'

V 7 !

I 1 V/ Prandtl's rU~17Y , "-Adiabatic curve

St t' 'nt 11

0.96

0.92

FI VI

~ 7 I /, Parallel flow MQ)(

Jf ./ I YK

Mach number - ~,Af)(~ ax:

O.~

o·t

0.7/ o.r2 (J.11{ 0,76 0.78 o.ao t:J2. au 0.86 !J.se ().'lO 0.92.

0.95

r'St;eaml

den~it;-r-" (0/ ar mg.....---

point 12 pv I ,,-

il ~ 7' ,pJ( v' i X i

. I V !~-Adiabatic curve Prandtl's rule--

/; ~

;; (/

St t' 0.99 -Stream density V

/' pv

I-~:arting p" v'A 7

Prandtl's rule A ~ point 13 6 I L

i. 7 - Adiabatic curve

b ~13

:

0.'18 o,'fIJ

0/17

0.9,

V / / )( VK Mach number 11 : -x-a 7 )<~

~ Mach numbet' M;: x I I I CJ ~ ga

C.72 0.7- .7. .80 em· 8 (; 'f 0.72 Q,f,f iJ,'/,j 1,78 d,SO 0-12. (t8f Mff o.~ tJ.90 Ml. ..,. Q 6 1-78 () 0. f 0.8 fJ.8lJ (). Q 1).92-

Figure 2. - Line (crescent) (d/~ = 0.10), Moo = 0.70. Velocity and stream density for various approximations to the adiabatic curve (source system I).

w f\)

I-' f\) o w

1.0l

r I I I Stream density A /.01 f v - Starting point 1

f"v~ V \\\ I I

1.00 -t / [;<, 21

k31;:ccurJ o.~ / '!... ./

Prandtl's rule)L~ /v I O~

V-ff/

0.97 // :V 0.96 -r 71 .

0.95

k~arallel flow I~C: ~ ~ vI<. - Mach number H::.y:

0.94 I I I I a

1

e

o.V8 ,/ , 0.80 a 81 a 84 0.86 0.00 a9I 0.9' a94 tJ.g~ O9IJ 1.1){}

I,or JJt. Stream density

/ / I I Stream density /i

V t,O '~ ~I

Starting p v

j / point 12 , I V \1 --------Prandtl's rule V ,

~ 12 /'

/ /" K-- Adiabatic curve

7J 32/,

7. //1

~~

0.99

0.96

0.97

p v fJiV'I.

{-Starting j I ~ point 13 1 I

~ ~~ -Prandtl's rule~

7 ~V ~ Adiabatic curve /. ~

c. /,,2 // I

~y

1.0/

liJI

0.98

/ " Y K -"t Mach number 141'" a)<.

I I I O.I}!; / /-1 .... Mach number M~v',(

I I I I aX 0.95

0.80 0.82. 0.84 aBO IM8 oso 0.92 OrH 0 '18 1.98 t1lJ 0.- a8t /184 0.66 a86 G!JO fl.9t O!H 0.96 0.98 1.9Q

Figure 3. - Line (crescent )(d/L := 0.10), Moo == 0.75. Velocity and stream density for various approximations to the adiabatic curve (source system I).

w w

/.03

1.02

1,01

/.00

0.99

Q98

0.97

0,96

Stream density pv

1--p< v~

~ I V_

/ Prandtl's ruley L'--r

/ r,/ / 50 6V= r.,/

71// I-parallel flow /op/ j M~ I I

/ vo(

C1 ,

.31 - ..... -..0-. I

~--r\ '-

I!-+-

/ 0 I I I I \

\

\-;- Starting pOint 11 " -,

" 1 0 ,

, 2 , '" 0,....,; <I

........ ~oo. "0 .5, 6

--~~;;-r'-

i-Adiabatic curve

Mach number ~ 2JL M::: x-I I L let

~ I

I I

+71 , , I , I

" I I I

I I I I , I

Source system 1

I ),61 I e I I ,

\ 1 I , I \ I I \ '..SI 1

~ , ~ I I

\ : / Starting points + I -t 41 :

t ! , -1-0 +31 ,/21

I I " , 2' j, +0 +.21_$. 2~ o. I I' T"'.

Sources 6)

sinks e

OBC' 0.84 0.8(5 0.88 0·90 0.92 0.91 0.96 0·98 /.00 /.02 /.04 /.06 /.08 hi ~+o Ht ~f2~!~ ~T ~~~~e.

1.03 ./

I Stream density V( I I -~ ,.,..- Starting

tl' v' / point 12 -+ I , \

" _.:1. .J , , ! V . - _ 22 , " ' 2 Prandtl's ruley " o,~

..J!? -0"

V -'-~"( - iJOS6

-

/ / ~ - Adiabatic curve

I I~VI /,

1/ /'

/.02

/ .0/

1.00

0.99

0,98

0,97

!.O3

/,0

. Stream density V 2!- P V

pXv'< ~

I I'Ll/-Starting

V'II point 13

0 Prandtl's rule") \

1

V'Z3 ~f- --, Q 2

9 / ,;.. B' 3

~ V v ll-Adiabatic curve 8

,/ /"

I,D

I,D

0.9

0.9

0.97

0.9 I v

~ Mach number fvI ~-4-I I I ,a 0,96

.; v ~ Mach number M ~--}

6 J I I I a 0.84 0.86 0.88 0.90 0.92 0.94 0. 96 0.98 lOO ;.02 ;.04- 0.84 0.86 0.88 0.90 0.92 0.8., 0.96 0.98 /.00 1.02 1.04

Figure 4. - Line (crescent)(d/l, = 0.10), M <Xl;: 0.80. Velocity and stream density for various approximations to the adiabatic curve (source system I).

.04- I I I Stream density :/0

.03 I' v I

p~ v"" I V ',.....- Starting pain t 11 .02 1 PrandU's rule H'

, , V .........

(.I ...... 'c/ / ....

....... 0 ..... "-.00 v-- 3/ ....

--0 '0_ ..... 'J L-f)!~

--0_ 0--0 "'0 __

f--o :0... .... ............ 3 -... -. 0 .99 --

4 ~ ----- ....

~ ~

-.............. ....... 04

Adiabatic curve ....... ......... 0S'

98 61 r-..........

~/ .......

~ .97 -71 c& t........

%parallel flow M )( Mach number fvI~ \I~ ...........

.~ I I J (1) I , , I a

.14 0.8& O.S~ P.'JO D.9/! aM 0.96 1290 1.00 /.02 . '6 .8 /.Z I .

o

0.

IJ

0. 1.04 I.(J. /.1/ /.lfJ I"

1.04 1.04

Stream density p v I / /.03 1-1'" '11- I V 0

/.03

1 I , './ Starting L02 '\ point 12 f--- I.oe

I I ~ ", PrandU's rule y ............ 22

" 1 /.0/ - '" /.01

/ V 3Z 0 ......

1'0 "" J 0 __ 0 ....... 0...

/.01 _ 0'0-0, ,~

/.00

Stream density pv I

pI< v'l. I /V W- I I II k(/ Starting Prandtl's rule '/ point 13

V \

/\?3 \ 0 I

S'

/~ v---~ Ie -~ Adiabatic curve 1---, ~ ~ ~ fJ7z

0- • 3 5 1- \ --r----. !---

0.99 0.99

~ V

/

~ v- I- Adiabatic curve -

fJ.9t 0.98 I I I I Mach number M~!;J Mach number

/) "1 I I I I .> k-__ ~ __ ~~~~ __ ~ __ ~ __ ~ __ ~~~ __ ~ IJ.y I i I a88 /).90 1J.9Z 0.94 1296 ags f.fJO I.Ot 1.04 I.f)6 !.dB O.EM 0.'» o.9Z O.9~ Q.g6 0.98 {DO

Figure 5.- Line (crescent) (d/L ::: 0.10), Moo::: 0.82. Velocity and stream density for various approximations to the adiabatic curve (source system I).

1.0Z 1.04

1 v M=-'1 ,a 1.06 1.08

LV \Jl

------------------------............

1.03 Stream density

! ~L pYvx ;< ~ Starting poi .(

I /fS, ' / T 21 'oz ~ ~I " 'd& 0,. ~

- Prandtl's rule" V41 ~ ~f-

L -Qojl I

6~~ '/ oJ/ V \. ~ Adiabatic curve ~

7/~;'<Y ~ Parallel flow .Af~

v/ /' -+ Mach number M ~ ~

I aX

'.0 I

1.00

0.93

0.98

0,'27

nt 11

~ ? \ I \ I \ I \ I \ I I \ +61 / ~ I r

, I I

I I I Starting points + , I , +5/ I

$ e • I ,

Source system IT. : I I

~ +.f.' ~ , -?~, ~11f -+.

4f 1. I I Q 23Jt.. ,pfII +~ e + l +"1" I I I I V+

sources $

Sinks e

PrOfile D.t?J6 0.82 0.84 0.86 0.88 O.SO 0.92 0.94 0.96 0.98 /.00 1.02 +~ +,2~ +11 ~ ~ f -4-~

, 7"

1.03 Stream density Stream density

0.96

P v L ./ pl(v)< ~ iStarting --4+ ,-~- ,/

point 12 -~<22

\ , PrandU's rule - '-V

02 ... \Q~

/ /;~ ~-

v/ "'J Adiabatic curve ~,

.~ /' ,

_____ Parallel flow Mr: / ;/ ,

Mach number M ~ 1* ~ I I

1.01

",00

O,e9

v.eo

1.02 p v

fv'l. ~[, Starting

// ',I...;, point 13 -

• / c!>

Prandtl's rule\,/ .~ --- ,.--1>'-

~ :/ ./ ~ '- Adiabatic curve

~ /

/~ V rparallel flow MQ')x

,/ "/ -f Mach number M~ VK J f£X

1.00

O~8

0.97

o.8Z ~B4 0.86 o.g8 0.90 0.9Z 0.94 0.96 a.9S 1.00 /.02. 0.82 D.94- 0.86 0.88 0.90 0,92 0.94- 0.%' 0.98 r.oo /.02

Figure 6.- Ellipse Cd/]' = 0.10), Moo = 0.80. Velocity and stream density for various approximations to the adiabatic curve (source system IrJ

.............. -------------------------

I.Ot

/.01

1.00

0.99

0.911

0.97

Stream density I I I c-I'V I ~/Starting r V

X Prandtl's rule - V \

~ /' ,

" 21 ,

~1,(~31 " ",I

~ '~ "'41 .... ..... "-. ~~'7

~ ..... -~

~ -K-~

~ v Adiabatic curve-

b

L-~/ Parallel flow M{() )( ~

-AJ _ Mach number ~w~ "~ I I I CI

I.O~

1.02

1.0/

1.00

0.99

0.98

0.97

point 11

0.98 0.84 0.86 aB6 O.!)(J 0.92 09f a96 0.98 1.00 !.IlZ 1.04

Stream density TI -f v I v -Starting-

I /'V point 12 pv; v'" Prandtl's rule:> J " ,

./ " 2Z "

/.~2 ",- ~J.

/~-<: -~ ';5 -

. Adiabatic curve-~/

h-

Stream density !.D2 P V j--j--l----If--.-I--l-A-F­

p'" v~

a~I--~~~--74~~---4~MI Adiabatic curve

/ ~ ,.,- Parallel flow MtXi x

I v -I -Jr- Mach number M'f."" ~

,

a~~~--~--~---4---+--~--~---+~-+--~ f)!J6 0.93 /.00 1.02 I.fJ4 0.8'1 0.86 M8 ago 0.92 O.9.f ag6 0.98

I I , I ()

0.88 0,90 (/.92 0.94

Figure 7.- Ellipse (d/L :::= 0.10), Moo:::= 0.82. Velocity and stream density for various approximations to the adiabatic curve (source system n).

/.00 1.02 /.04

0.99 --::MI:IIIII--=-,f-----t--f---t-->o.-Adiabatic curve

J.J---+--Parallel flow MO')JI.

o.97~~---4---+---t---~--+---+---r-~~-+~v~ )( K Mach number 11,. x a

~- t. a66 0.88 a90 aJe a.9JI ~ 0.98 1.()() 1.02 1.Q.j. .116 f.(J8

'.

i03r-~--~--~---r--'---.---.-~.--'r--. Stream density

1. OJ. I I Stream density f'="Starting

1.02 p v ---+--+--+--+--I-:r''-t~:-1 :tey7r

vI I J ~. 2 ~, V"'~ pOlnt 13

p ! v Prandtl's rule V 1",

1,0.

O.991----!--:>I'~"'-+-_+_-~--=f Adiabatic curve

.J.3--+- Parallel flow M~ Q971----!~-4---+---+---~--;---+---r_--~~

Mach number M~v~ Q

to 1

1.0 0

,J

0.9 7

Cl~ 0.9 t

, V /

~ V ~ V io'"

'" (!' Parallel flow Me:

---;.

! ........ -f

~J " , 1:

I' 'Q. J,. b.

~. I -~ Adiabatic curve

X Y K t- Mach number M::(iYC I I I

~

a8G O,e8' 0.90 0.$2 o.9If 0.96 o.Slt t()() (.02 to,," f.06 0.86 0.6(3 0.90 0.92 0.94- 0.96 0.96 1.00 f.fJ2. 1.011 1.0&

Figure8.- Ellipse(d/l, =0.10), Moo=0.838. Velocity and stream density for various approximations to the adiabatic curve (source system IT).

f-' I\) o w

3 / ,

V IV I I Stream density ',/ I-Starting p v /

p" vI( Prandtl's rule-V , ,

! / 11 01 , -,

I V~ if '::( 3

/ ~1). r~s

V31 ~ 'I- r-

k' - - ~ • -o1l/J:I'

Af V V '\: r Adiabatic curve

_O"~

~ V 5f~

6f", ~ i ~.parallel flow tV! x )

V I I I ~ --'I I- Mach number x.2Js. I I I ~·ax

to

1.02

1.0

1.00

0.99

0.98

0.97

0.96

0.95

point 11

0.940.78 aeo 0.82 O.BIf 0.84 0.88 0.90 0.92 0,94 0.06 o.~ f.{)O '.02

1.0c'c----,,-----,r----r--.---r---,.-....-,,L.,---,.---.-- 1.f12,,.......-~-~---,..-----,----,----r---r---r---,

Stream density Stream density 1.01 p v --t----t----t---I----r4---4-"4--

p'L. vI(

"OOI-'f--I---I----t--t-r'-i--::---l---I-"'t

M~ vK

L........~'---L..........--.--l.--.-.l-_L_.........L_.........L-.-!...1 -.~~ o.9:J 0,94 0,96 0,98 1.00 0.00 0,88 0.90

1.01 .E...:!.---I---I----I-----I---/---I--+---+-----l----1 pI( vI(

1. (}()1-----.!iF----'I---I--t--l----?4---t---I----\-::;:;--.-+---!

Q97~~~7Y~~--+----+----I--+---+--~-; ~ Mach number /1: v K

aX a96L-~~~---L-J--L--L-~-~---L~

a D.Be 0.90 0.92 0.94 0 . .96 0.96 f.OO

Figure 9. - Spindle-shaped body(d/l, == 0.10), M ()()::: 0.76 Velocity and stream density for various approximations to the adiabatic curve (source system I).

w \0

I.OJ

/.QZ pv Tv~

1.0/

/.1M

1J.99

0.98

1.0/

-~Malch number M":~ I I I q

1J.8Z Q.# 0.16 D.RB D.9tJ D.9C 0.94 0.96 9.98 1.00 1.02 I.IJ4 I.IJG lU8

I.OJ Stream density

/ /

I I I _I' v /''' v" Prandtl's ruiey

/

-t /.02

1.0/

/ / ~

~-/' ,...Adiabatlc curve

23ZV Starting point 13

~i~1 -=t M~Ch number ~# 1 J I

1.00

0.99

0.96

O.lIB 0.90 agz a9t 0.98 0.98 1.00 tot tI.1T

a .~84 0.88 P.fM a~ 0.12 0.91- arM 0.49 1./1 /.fJ2

. Figure 10. - Spindle-shaped bOdy.(d/L = 0.10), Moo:= 0.78.

Velocity and stream density for various approximations to the adiabatic curve (source system I) .

.................. --------------------------

1.05 Stream density

f.Ot fl v fl" \f"

i I

V Prandtl's rule

/ ....

/- - - -o-3J_ -<3-

102

1.01

/ "!-o-V-- ~ -~ / ..".

~~ A 71/ V

-Parallel flow

~ I M!, I I

1.00

0.99

0.98

0.97

0.96

). /

V \

v Starting point 11-,

~~ 'q

" ....... .... "" .... - ~- - p ~ -<>- -..( "

' 3

\ -I"--' ....

.... '0 ..

Adiabatic cu~~ r---... '" ~ ...... .........

)(.Y K --:'J - Mach number M::-x

j I I I Gt

\ \ \

~ \ \

~

_?1 I I I I ~61 I

I tf, I I I I

+51 I

I +

I I

~ ,

Source system I \ ~:4-1 I' Starting points +

~ ~ ~ EDt' ,t31 ~~ \ ~J,. 121 I ~

'fIt;n' t ~~t~ ~

sources EB

sinks e

. 0,82 0.84 Q8G 0,88 O,g() 0,92 0,9-1- 0,96 0.98 (.00 1.02 t04 t.OG 1,08 1,10 1,(2 ,.,4-1mr-~--~--~---r--~--~--~--r-~~~

Stream density fr Starting 1.02 P v I_--'-__ --I-__ --I-__ --I--_?"'-_+----;.,'.- point 12

p)( v~ \

Stream density 1.02 p v I--+---t---t----+--~-+--+---+____I

P'vx

1.011-i'-+---+---I--+--;r'~-I -t---I-Q.1 -+--4 1.0( f---!l'-Il----I---j-----1---:A

0.991---1-

Mach number M: v~ a

0.88 0,90 0,92 0.94 0.96 0.913 (,(10 f.02 1.04

Figure 11. - Spindle-shaped body (d/L ;:;: 0.10), Moo;:;: 0.80. Velocity and stream density for various approximations to the adiabatic curve (source system I).

Stream density

t2 p v---r~-r---+'F--r---r--~--~--1-~ p" vI(

1.0

0.9

0.5

0.7 Parallel flow Me:

0.6 Mach number 11~ v~

a 0.4- 0-5 0.6 0.7 0.8 U9 (,0 f.f

1.3 S d't tream enSl y

1.2 L.':!.- --r--+-----+t'---+---+---J--+---I----l p< V

X

0.9 \---i-----1ff

Adiabatic curve

Mach number M~ :~ Q6~~--~--~---4--~--~--~--~--~--~

OJ.. 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Figure 12.- Circle. Moo = 0.40.

~ \ \

\ \ \

\

<it \

\ \

Source system IT ~

Stream density

+ 171 , I I I 1 161 + I I I ~S1

I ~41

I

I I

e I

I

Starting pOints + sources ffi

sinks e

12 P V b . Fi?- /' Incompressi Ie flow

0.19 Adiabatic c1urve

Velocity and stream density for various approximations to the adiabatic curve (source system IT).

...... f\) o w

1.3

1.2

1. f

to

0.9

0.8

0.7

1. 3,r--~~-.---r-----r---:;---.-----'----r--.---.....-,

Stream density t2 p v.-~---+--r.~-r-~-+---r---~~

pI( vI(

().5 0.6 0.7 0.8

Stream density I I

j-I-Incompressible flow Stream density pv I

pI( VX V I

~ 1.0 ;--Starting I I' II point 12

\1

~ ,22 ~\ ~~ 'yS

1.2 P v:---------r--t---+-:f--t---+--+---t----II--l F' V

x

~ V \. I-Adiabatic curve Adiabatic curve

~ Parallel flow MIXl'l..

(J.13

/ v

0.5

)(~ -I+- Mach number Ms ct)( 1 I I J

0.6 0.7 0.8 0.9

Figure 13.-1.0 1.1 1.2 1.3 "If 0.5 0.6 0.7 0.8

Circle. Moo = 0.42. Velocity and stream density for various approximations to the adiabatic curve (source system TI).

1.3 r--~-~-~---r----ft'r-""---r-----'r-----r-~ Stream density p v

t2 pit. v.--~-+--+7L-~~--+--+-~~

0.9 f----1---+-/.

x a7~~~-'--~--~--~~~~--+--r-~~

ltV Mach number Jt1'" #

0.~~~-a~5--a6~-aL7-~-~~9~~1~.D--~~71-~~2~~1~~~~t.·

Stream density Incompressible flow , ,

t 2 P v -'--------1f----I--+-.f---+= Starting --j----t----j l V It. point 12

Stream density 1.2 P v --+--+--+-~-h P Incompressible flow

tf~~--+--+-'~~~-~-+_-r---1r-~ 1. f ~H--t-----+-,f--t---t---+---t------1~-+--I

Adiabatic curve Q.81----I------,~--+-_t_---+---+-----+_-t_-1____I

Mach number M~ ;: a~~~-~--L_~_-L_~_~_+-_~~

<44- 0.5 0.6 0.7 O.S ().9 to 1.1 1.2 1.3 1 .•

Figure 14. - Circle. Moo = 0.45. Velocity and stream density for various approximations to the adiabatic curve (source system II).

.... l\) o w

Stream densitYI 1.3 I' Y rvx lZ t

0.8 1---/-/,1--7""-

Stream density l3 p V~--~--~--~~~-4--~ __ -+ __ -+ __ -'

rv~

47n I.I,S O.G 0.7 as 1.1 /./ [3 [5

Stream density t3-fv+---+---+---~--r-~~~~~---1--~

f~ y'l. Incompressible flow lz-t~--~--~-r~--~--+---+---+---~--I

/./ t------It-----t----rhA

6.9 t---t---"f

O.R aT 06 I.! /.0 1I

Figure 15.- Circle. Moo == 0.50. Velocity and stream density for various approximations to the adiabatic curve (source system I) .

~ o

I-' f\) o w

-I=" \Jl

------------------------.......... .....

Source system I Moo: 0.8(}

\ \

\

~ I

I I ~ I

I

~

I

~ I

I

"";0.80

Source system I ....... Source system J: .0--0-

Starting points + Sources $

sinks e

1.03 -f.:-V~ -+--+--i----I-~t"____t 11«/0.17,-P v I

1.()2

1.0f~--+---~--~--~--4-~Mr---+---T--__t----1

Source system I ~----t--- Source system III _-1-__ -/-_---'-_-1

Mach number M: y: ,0

0.92 0.9'1- O.9G 0.98 1.00 1.02 1.0'1- a9~.J)J!> 90 0 0 098 V.I.)(] a. .92 0. 94 .96 .

[Title almost wholly illegible.] Various ........ source systems . . . . . . . .. .

f.OO 1.02 1.0,," f.06 1.08

................. on the line (crescent) (d/z, = 0.10) .

•

2: o

I-' (\) o w

NACA TM No. l203 47

Stream density I I 1.04 pv MIX) =0.82

pXv~ 0.80 -",-- 0

0.96

------/ ,

", 0.77 ____ " ~ 1st approximation , , ,/

, '-

o.~.-v/ " , .-1- ", 1 " , ,

_'0 r--o.." " -... 2

Prandtl - --- ~o..:-/ " "'..,-- ,..-- r-- 1-":::" 1..--- - -~ I ',- -'0-';;:::: ~-= f--:: t--r-_ :::., .... 3

J v/ /'n ~ -~ t:.oo- ::::::-2 ;::~

v~ ~ ,...

:'-~ I"-.. ~

Q70/ .......

~ v '/' i'---. /V'\ /~ Adiabatic curve -

r ~ V/ / /' "'

Line (crescent) dll=o.lO / /

V I I

/ Source system I

1/

1.02

/.00

0.98

v ~ Mach number M'= -+

0.94-0.80 0.84 0.88 0.92 0.96

r I ~ a 1.00 1.04 1.08 1,12

1.02

/ .04-Stream density I !

I' V I1IX) = 0.84

~)( v" 0.82 -'" 0.80 ~ ~[ 1st approximation

~ t ,

I 0.78ff",/ --'\ , , ',I ,

I

----, , --0,..1- f--'-'

... 2 0. 7S/ V , .-4-

=-~ , -- ~,j 4-.,- ~ - >=

V'\ --~ ....... - --~N Prandtl-r--/ \\ V; V r-.-.:::r--..,

p V ~ / // Adiabatic curve-

/.00

0.98

0.96

0.70;( / /' /' ~

/ P / V Ellipse djl:::o./fL ,

V / Source system II

/ V V . MX~

~ Mach number = aX I _I ~

I

0.94 0.76 0.80 Q84 0.88 0.92 0. 96 /. 00 /.04- /.08

Figure 17. - Velocity at starting point 11 for various degrees of apprOximation for the line (crescent) (d/~ = 0.10) and the ellipse Cd/? = 0.10).

•

/.4 Stream density . I

-I v Mer/aliO

fX V X

/\ / ,

// 0.45 \ Circle \

/) \ SourCe system II

\ I n.42 \

/ \ \ II 1st approximaiion /~ \

aJ~/ \ \ \ , \

Incompressible flow I~ /\ \ \ /\ I

1.3

/.2

I \ \ \ '.-1- -~ \

, , 1'1-- " O.J.F // , " ..- "- '" \ 0( '. .,

'"... "... , ~o ,. 2

1./

/1 \ /~, ~ .2.'b -~, \

// -,

fs;::: _'0.:.1 ~- ~ \ ./ l==:::: - r- " -= " oJ (

\~~ /~ ~ I-" -..

~ ~ ~~ ~), /' " ......... P, V ........... , r-....' V/ /~

~ " r,~ 4-

~ " ,~,

/ Y Adiabatic curve --~ ~,

[ ~ " s

1.0

/.9

/ ~ '" ~

0.2S # ~

<. j /' " "" / " "'-.7

-~ Mach number 11~ ~/f I I I

aX

0.4 0.5" 0.6 0.7 0:8 0.9 lO II 1.2 1.3 1.1- 1.5 .,. Figure 18. - Velocity at the starting point 11 for various degrees of

approximation for the circular cylinder.

I

" '" 1.6 I--' I\) o w

!2,

f;

0.# /; Approx.5 ;,::0

D.4/)

0.36

0.32

0.28

0.21

0.20

0./6

(:x _I) I .. 1II4,l' I

0.26

System of 12 sources Approx. 5 ;;:-

1 I IV'" O.U

Lim:tillg Ma~h n~bet-i 022 4 Point of divergence"\ i I

, I 1 # I-Line (crescent)d/!:o.lO

/1 VII Source system I

1\ J'A VI Sonic limit- IJ ~ 1: 1'\ "

I~ V). ~

Busemann Ll V ~\! ~ Approx.1 .....

~ ~ ~ ~ I' Approx.O ..... ....." I\.

~ ~ ~ ~ Kprandtl ! '\

.-1111

~ t: i.--- --- I I--

Incompressible flow '\ I I I I I , ,

Approx. L _---'

Approx. 1 I

3 0.20

v I r Approx. 2

I I " I I

0.18

Approx.

0.16 It'

0.1-1-

1\

0.12

~VH -1 1

v(J> ma):

Ellipse d/l ~().I0 Source system II

~ ~ ~ V

~ V k V --- v---~ I-' \ Busemann

r Approx. 1

J J Incompressible flow

Point of divergence VI A pprox.4

pprox. 3

Sonic limit -

/

~ ~ V

~ ~ V

L'

V I'-

J ~A 1\ 1/'.1 V\\ ~) Ap . 1/

prox.2

\ II II J<fAP

'f! V;': I prox.1

/; 0' Syste m of 12 sources Approx.O ,

~ V 1\ U

V' / V\ ~

<prandtl+ f\ 1 1+ I--Busemann

Approx. 0 t I--

I

Limiting Mach number-i I I I I I

I

0./2 0.60

I 0. 7 Mach number At,. as

0./0 0.6

I I I I [\ I 0.7 a8

Figure 19.- Maximum excessive velocity A,,\(. ~-1 on the contour according to various v. v.

approximation calculations for line lcrescent) (d/l. '" 0.10) and ellipse (d/l. '" 0.10).

2.0

J.9

1.8

1.7

/.6

1.5

1.4

/3

/.2

J. J

09

50 -NACA TM No. 1203

I ! I I I II r--k -1 (- v ~

v~ ma.:J. System of 12 sources-

Approx.5 '1 /

Circle / Source system n Limiting Mach number 17 / ) I ! I

Point of divergence -~ 1/

/ / N

I- / I) Approx

:i fl / J ,I / '/ /

\ I~/ II r- Approx. I /

Sonic liInit - ~\ I i'fl V 1/ /1 /

1\ ijj V / / /' ~ Approx.

\ /Ij ~ 1/ / \

) ~ v/ n LamIa /~ ~ ~r \

Approx.3 -//7 k"l \ .....-to- ,../''- Approx.

~ V r;-

Approx.2 -? k::---' I

Approx.1 -~ -......... r-.~

;;0-~

~- \ r---.... ;...--..... ~

--~ "'/ ~

~-- \ ...- ----~ I -- - ..-::;;:-:; ~. -- :.-:;:::-::- '1 ---~ ..-~ Incompressible flow :---'\ - --- \ 1

I 1

,

Mach number Moo 0.20 0.24 0.28 0..36 a40 0.#

Figure 20. - Maximum excessive velocity on the circle contour according to various approximation calculatiollS.

. 4

3

2

1

•

NACA TM No l203 5l

!.f )( 1 . L. . JC -----'-- . J. L. .1 .1,

Nmax~ Maximum local Mach number!

.~ '" ~ 1.3

l~ r-.... I ..........

N"mox. For point of divergence I /.2 , ... Circle "" 1 ~

I ~ ! i".... Ellipse d/l=~ /.I I Line (crescent) dJl~O.JO , ~ I ~ I I I

I SOnic limitf\ I I ~ I I

~ I J '-...

I I

I " / /

1.0

I

f / I Mach number M., I --L -0.9

I 0.4 05 QtJ 0.7 / 0.8 09 1.0

Figure 21. - Maximum local Mach number at the point of divergence as a function of the Mach number of the air velocity. i

, r