compressible potential flow with circulation about a circular cylinder

8/7/2019 Compressible Potential Flow with Circulation about a Circular Cylinder

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REPORT No. 780COMPRESSIBLE POTENTL4L FLOW WITH CIRCULATION ABOUT A CIRCULAR CYLINDER

By MAX. A. &LASLEIT

SUMMARYhe pot en t ia l jm%On for jZow , w it h cireu l.u$im ,of a com-asibk jlu id abou i a circu lar cylin der h obt aimd in seri.am in clu din g t erm of t he ord ers of M whereM is t heMachr of thefree stream. The rem dtin.geguutium are u3ed toin premu re coejlcim i w a ju nction oj Mach number & aon thesu.qfweoj thecylinderjor di$eren$va.hm of eircw?u-The [email protected] !a derived are com pared * the Qluuer&n aW an d E&mdn-T ti apprommu tti w hich are jun c-oj th p remu re coe#ic&m%of an in comp ressible j lu id .r the cmes conmliered, the valuea of the premure coe@i-m%djrom the theoqi werefound to be somewherebetweentwo approxim a$ti, the jirst underm tim ding and thend overestimatingit.

INTRODUCIIONt he t wo-dimen sion al ir rot at ion al flow of a compr essible

id, where the expansion is assumed to be adiabat ic, theocity poten t ia l is known. to sa t isfy a nonlinear pa rt ia lerent ia l equat ion of the second order . For subsonicocit ies, a t lea st th ree m eth ods a re kn own for t he a ppr oxi-t e solut ion of this equat ion. They are usually denotedth e m et hod of sm all per tur ba tion s, t he Ra yleigh -J an zent hod, a nd t he hodogr aph m eth od.he method of small per tu rbat ions (references 1 and 2)um ea thm t velocity changea which are brought a bou t byair foil in the un iform para llel a ir st r ewn are small inpar ison with the velocity of the undisturbed str eam.der th is a ssumpt ion it is possible to in t roduce new var i-es which reduce the differen t ia l equat ion to a Laplaceuation,ad, as a con sequ en ce, th e pr oblem becomw onemnin g flow in a n in compr es sible flu id, pr ovid ed t he bodysumed dist or ted t o cor respon d t o t he ch an ge of va ria bles.

e a ssu med distor tion con sist s in expa nsior i of th e dim en-n s of t he a irfoil per pen dicu la r t o th e dir ect ion of th e ii-weam in the ra t io l/~~ where ill is the Mach numbert he u ndist ur bed st ream.Th e Ra yleigh -J a nzen m et hod (r efer en ces 3 a nd 4) a ssumeat the genera l expression for velocity potent ia l may beit t en as n ser ies in r ising powem of M a nd wit h va ria blefficien t s. These coeilicimts oan be shown to sa t isfya in P oisson dH er ent ir d equ at ion s a nd, if t he equ at ion sin tegr able, t he solu tion becom es a m at ter of det er min in g

coe fficient s .. ,Su ccwsive st eps, h owever , becom e in-

creasingly labor ious and the convergence of the serks maybe slow, even at rela t ively small Mach numbers, if the shapof the body is such that the speed of sound is approached100ally. Solu t ions, using th is m ethod of a t t ack, have beencar r ied out by C. Kaplan (references 5 and 6), S. G. Hooker(reference 7), I. Imai (reference 8), K. Ta mada a nd Y. Saito(r efer en ce 9), a nd L. P oggi (r efer en ce 10). P oggi in tr odu cedcer ta in refinements and some of the preceding refer encesemploy t h is p rocw. It is t ant am ou nt t o u sing th e so-ca lledNeum ann funct ion in solving given P oisson equat ions anwill be discu ssed in t he a ppen dix.The hodograph method is a scr ibed by writ ers on thasubject h P. Molenbrock and A. Tschaplygin. Instead oexpr ess in g t he velocit y pot en tia l a s a fu nct ion of coor din at esin the Car tesian or polar plane, the magnitude of velocityV and its inclina t ion o to an assumed axis are chosen ain depen den t va ria bles. Th e r esu lt in g differ en tia l equ at ionis linear and can be fur ther simplified by replacing the preasure-vohune rela t ionsh ip for adiabat ic expansion by thequat ion of a line tangent a t a point ecmwponding to thsta te of the flu id in the ambient st ream. This ar t ifice wasuggested by T. von Kdmm in (references 2 and 11) a nd usesu eoessfu lly by H su e-Sh en Tsien (r efer en ce 12). K. Tama da(m fer en ee 13) h as a lso a pplied Tsien s m or e gen er sl r esu lton ellipt ic cylinders to eompremible flow past a circulacylinder.One notewor thy resu lt of the hodograph method has been

the K6r mlm-Tsien expression for pressure coefficien t P it er ms of Ma ch nu mber M a nd l~.o, th e pr essu re coefficien tfor M= o. This exp ression maybe wr it ten

It a lwa ys gives, for n ega tive pr essu re ooeflicien ts, a r esulgrea ter in absolu te value than the Glauer&Prsndt l formulawh ich is ba sed on th e m et hod of sm all per tur ba tion s,

(2

and is current ly accepted as the m ore a ccura te of the two.From equat ions (1) and (2) it is possible to compute th

cr it ioa l Ma ch n umber M., the va lu e of M a t which th e locaspeed of sound is a t t a ined, in texms of P~.o. The rela t ions

129


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130 REPORT NO. 78 ONATIONM.I ADVISORY COMMITTED FOR ADRONAUTWSinvolving Mo and P.ar.o, coirwponding respect ively toformulas (1) and (2), a re

(3)and

The diilicult ies inherent in the two la t ter procedures arequite a s dist inct ive as their respect ive approaches to theproblem. As sta ted before, the Rayleigh-J anzen methodemploys classical mathemat ics, the required terms beingsolut ion s of P oisson equ at ions with given boun da ry condi-t ions, but the work involved is arduous. In the hodographmet hod t he pr in cipa l di.f3icu IQ is t o det ermin e p roper bou nd-ary condit ions in the V, o plane. In available ca lcula-t ions the solut ion is given with a slight distor t ion in thegiven bounda ry. It is possible to correct this distor t ion,in some cases, so tha t the iinal resdt s are not too seriouslyaffected. When the flow around the body involves circu-la tion, however , the cha nge in t he boun dar y is more ser iou s,for nonperiodic W appear and the boundary is no longera closed curve. Added circulat ion does not involve anyes sen t ia l va r ia t ion s in t he Rayle igh -J a n zen met h od , h oweve r,a nd in this r epor t t he velocity potent ia l for such compr ti-ble flow about a circular cylinder has been der ived. Sinceno theoret ical study has been prwented, as far a s is lmown,t o det ermin e t he er ror in t he Kt ir ro&n -Tsien pr essu re coef6-cient , the resu lts obtained in this repor t furnish a means ofa pp roa ch in g t h is p roblem . Th e r es ult s of s uch ca lcu la t ion s,for va riou s va h.w of cir cu la tion , a re t her efor e in clu ded.

ANALYSISConsider a gas obeying the adiabat ic law and flowing

ir rotat ionally in two dimensions. Iiw equat ion of motionmay be writ ten in polar coordinate in the form

where@VW

YcouMv

veloci ty poten ti a l

r at io of specific h ea ts of ga svelocit y of sou nd in u ndist ur bed flowvelocit y of fr ee st ream

(7Mach number of free st ream ~

loca l velocit y s qu a re d [(%)+$(%)1

By the introduct ion of the vaxiables 4 and u so that4=; ~d V=;

equat ion (5) may be writ ten in the form

where

Following the method of Rayleigh and J anzen, assume@ may be developed in a ser ies ~f ascending powersso t ha t

@=@O+WV~M4A+ . . . . .Aft er su bst it ut ion of equ at ion (8) in equ at ion (7), elemeca lcu la tion s sh ow t ha t

ti=v/ +@kf+vg2M4+ . . . . .where (%)+$(%)In a sim ila r m an ner , equ at ion s (8) a nd (9) m aybe su bst iin equat ion (6) and on equat ing coefficients of thepower s of M, the following rela t ions for h, 41, da , . . .result:

&=()

If the equat ions (ha), (llb), and (llc) can be solvedcessively for #0, 41, A, . . . t he values may be subst iin equat ion (8) to get the potent ia l funct ion for the fa compr~ible fluid. A step-by-step procedure is theesta blished wher eby a ny desir ed degr ee of a ppr oximto $ maybe obta ined, provided the value of M is wit hir egion of con ver gen ce of t he r esu lt in g ser ies. E qu at ionis t he differ en tia l equ at ion sa tisfied by t he pot en tia l fu nin the case of incompressibility. Once th is potent ialt ion is known it is used to evaluate the righ&hand meof th e second equ at ion , the solut ion of -which fur nishesecond term in the development of +. The methodtaining fur ther tm follows the same general proceduConsider now the case of a right circular cylind


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coldHu?lsaIBm oTENTLm FLOW WITH CIRCULATION ADOUT A CIRCULAR CYLINDER 13in it e len gt h in a compr essible flu id , t he a xis of t he cylin dering at right anglea to the direct ion of steady flow. Inot wm in in g t he velocit y dist ribu tion a bou t t he cylin der , t heoblem may be t rea ted two dimensionally with a circle a se boundary curve and the equat ions e+blished in theayloighJ anzen method may be applied direct ly in thellowin g mann er . The radius of the circle is arbit rar ilysumed equ al t o on e, a nd a pola r coor din at e syst em is ch osenith or igin a t the center of the circle and polar& extendingwnst ream. The flow about the circle is assumed to beat result ing from the combinat ion of un iform streamloci~ and circu la t ion about the cylinder . Under thesendit ions the claasical expression for #0 is well lmown.maybe writ t en

#O=(r +;) cos O*here I is the circu la t ion around the circle, measuredsit ive in a clockwise dir ect ion. F or ea se of com put at ion itcon ven ient t o set

~=KTud , a s a con sequ en ce,

40= () r+; Cos 0$0 (12)h e bou nda ry con dit ions, in genera l, a re

$$=0 for r=l (13a)nd ~+=w~ ~ for r=m

br (13b)

r om equ at ion s (12) a nd (lOa )0.2= 1 +-#1); Cos 20+K(+++) sin 0+$ (14)

h is r esult , t oget her with equ at ion (llb), gives

v=(++~) Cos 0+; Cos 30+

( )Ksin 20 $$+$ 2$ Cos e (15)Th e mor e elem en ta ry m et hods of in tegr at ion lea d t o cer ta iniflicuhiea when an at t empt is made to solve for A, inuat ion (16). These difhcult ies result from nonper iodicrms in the pa r t icular integra l and resu ltan t t rouble inet er min in g su ch con st an ts of in tegr at ion t ha t t he n ecewwyer iodicit y, in t erm s of 0, i s main ta in ed in t he fin al expr essionr t he pot en tia l fu nct ion . Th is difficu lt y may be obvia ted,owever , by established methods. (See appendix.) Itllows t ha t t he solu tion of

sin m ev%=~ (16)t is fyin g t h e boundaxy condit ion s

(%)r.l=o($%=o

isS i nm$

{(s2) m=m (m8+2)(m+82) 7 1.,when m+2# s) (17

andn= {

Sin?m 1mrm }fi+~ log r (when m+2=s) (17

Th e ver acit y of t hese solu tion s, t oget her wit h a na logou s onexist ing when sin mOis replaced by cm m 8, may be checkeea sily by su bst it ut ion in equ at ion (16).Since equat ion (15) is a linear different ia l equat ion,

solu tion is det er mined by con sider in g ea ch ter m of t he righhand member a nds umm in g t he in dividu al in tegr als obt ain eby means of equat ions (17a) and (17b). The fina l result

c0s3(=++)+~s~2(:+(1

In the evaluat ion of & the calcula t ion follows the sampa tt er n of developm en t. From equat ion (lOb), togethewit h equa tion s (12) a nd (18)

c0s4e(3+Khe(x+*-i%( )log rKsin30 ~~~ 1 4r 6? 29+%7 9 +

log r log r) (

1 @_r_ +F sin 8 +@+ @2P + ) (1This result , imgether with equat ions (12), (14), and (1su bst it ut ed in equ at ion (1 lc), gives{(7%$g=y 1)Cos e -++=+$)+WS3

() ( )os i50 $ +K sin 20 ~; &-$+& +fi4(~+=J+mcOsG-Hcos(al+lmse(=+=+


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132

.

RE PORT NO. 78 *NATIONAL ADVISORY

To in tqgr at e, for mu la s (17a ) a nd (17b) a re a ga in r esor tedto. The method of in tegra t ion given in the appendix alsoprovides in tegra ls cor r esponding to the new type of termappea ring in the r ighkha nd m em ber of equat ion (20). Thus,t he solu tion of ~%=log r sin mOv (21)s a tis fying bounda ry cond it ion s(al=o(am=ois tinme$2= { 1 1 1m.P (m+82)a+(m8+2)2 sin me I 2m log r2mF2 (m+s2) (m s+2)+-@bP-(-:+a (22a)when m#-B2. When m=.s -2,

P roceed in g d ir ect ly wit h t he in tegr at ion r esu lt s in t he followi-n g expnxs ion :{(2=(?-1) Cos e ~L+A )0r 8? 12+&7+


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COMPRESS IE fLE POTENTIAL FLOW WITH CIRCULATION ABOUT A CIRCULAR CYLINDER 13Resu lts der ived from th is equat ion will be presen ted inrms of pr essu re coefficien t wh ich is defin ed a s

2?2? !-l/2pJJ2 (26)ere the zero subscr ipt s refer to free-st ream condit ions.m erms of Ber nou llis equ at ion it follows t ha t

r)Af.~=1 ~ , (27)PH.O den ot es t he p ress ur e coefficien t for in compr es-

le fluids. If P d en ot ea p ress ur e coefficien t for a compr es -le flu id obeyin g t he a dia ba tic la w, t hen

P= [[ ( 71+1-11 28)? l+~&lw lmere ill is the Mach number of the free str eam and y ise r at io of specific h ea ts (1.4o for a ir ).As an approximation for P the Gla uer t-P ra ndt l r es ultfer en ce 2) is given by equ at ion (2) a nd t he K&rmAn -Tsiensult (r efer ence 2) is given by equa tion (1).

Claubf -PkcW farmida-w ------ K&r-m&n- Tsien fwmula- Che term appmxio)aflon-- Two term approximation

-.37IIII

cIrculotion-0 I-CM I1

Cfjf~calrI Moth n,wber. 1 ....-a5 I/ {I

1 /-34 !/ ,

I /-a3 If /

/ / /ti // /-3= / / ,./ / //f , ,,

/ / ./ /,/1 //

-tiuo ,./ .2 .3 .4Mach numbefi-jdl.voriouao of InfnImum plWsOro ccdklent wf th Mad numbr whan circdatbnfsaS48110-5&10

The velocity a t the topmost pa r t of the cylindar may bfound by set t ing 0= 90 in equat ion (25) and the rewdtanexpression is a funct ion of K and M. In figure 1 , p r es su rcoefficien t a t this poin t is plot ted against M for K=As a testfor rapidity of convergence the expressions fovelocity, using only I@ (one-term approximation) as weas W and M4 (two-term approximat ion), a re used. Itto be noted that the curves diverge grea t ly near the cr it icaMach number , but tha t for smaller va lues of M , th e cu rveder ived from equat ion (25) are together and definit ely lbetween the resu lt s der ived from the Glauer t -l?randt l anKti&n-Tsien rela t ions. Figures 2 and 3 show the samequat ions applied for K= 1/4 and 1/2, respect ively. It thuappea rs from these calcula t ions tha t the true value of P lisomewhere between the approximations applied. On thoth er ha nd exper im en ta l da ta , a s det er mined fr om a ir foilshave shown a bet t er agreement with the lUrm&n-Tsienequa tion t ha n h ave t he t heor et ica l r cm dt s obta in ed h er e fothe cyl inder .In figures 4 and 5 the same point on the cylinder is unde

considera tion, but circula t ion is made negat ive by set t inK equal b X and ~ in the two cmwa. As the pressur

I I G/abet-f- Wadf formula-43 ----- KZrmbn - ?%ien formula- he term approximation-- Two term qaproximafion

i-42 1

1ICicfflofim - + 1-4J I

; CPlfjcalMach I nunber% ./-g%--4.0 I$ 1.. i ,~ 1 )t / /b -3.9 ,*5 1 /,~ If /-38 # // !i/ :,

I /.-3.7 / / .i/ ,, ./ /

/1 / .//{/

-36 //

-35 Jo ./ 2 .3 .4 .Mod numbeq MFmmus2.-Vaii8tfmfxnfnfmum ruemure maftlciont wfth Nnch number whan akcalatfob*. u.


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. . .. ---...-, -----

134 REPORT NO- 78 ONZWION~ ADVISORY COMMITTEE FOR ~RONA~lCSI IGlouerIPrano?l formula

-4.8 -- h&m6n - Tsien formula- tie fcrm opproximoiion .- Tw o t erm opproximm%n~ .1

-4.7 I/I

Circulo fim w ;X7 [-46 I

i CP1/icalMach Q ndcr1 --- ./ /


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COMPRESS IBLE POTENTIAL FLOW WITH fZE tCULiTION &f OUT A CIRCULAR C!YLINDR2 135r equ ir ed p ot en tia l ~, s at is fyin g equ at ion (29), is t her efor eSs l#(r,O=+T log [P+lP-2Rr cos (ou)]+ .

[ 1log $+B: cm (00) +

ore the in tegra t ion extends over the region of the planen g ext er nal t o t he u nit cir cle.In tho equa t ions under considera t ion in th is repor t the(R , u) is r est rict ed t o on e of t he for ms

an example of t he in tegra t ion process, take the fir st caseted. Then, set r= l/ r , wh ich r esu lt s in

In tegra t ing It fir st with respect to u shows immediatdya t ii% va lu e is zer o.F or pu rposes soon eviden t L is wr it ten in t he for m

=y%fiRa (35)n ce t he log E t er m va nish es, aft er in tegr at ion wit h r espectu , the expression for 11 may be simplified fur ther by the

1g p+(;)2; co (ea) = 2 ~ ; (;) co n(ea l)(36)r


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. ..- .-A . ..=

136 REPORT N O. 78 ONATIONAL ADVISORY COMELITI?EE FOR AERONAUWCSI

.8 Tw o f eem qc .prox imof lon 0 + Circulation- ~ n UGlourt -Prandff formula7 K&r&n - Tsien formulo Much number - ~

+ \ !o i

\ [} + +

~. -.8~ IA! I +;oz~Q?&

L-2.4 \ {\

-t

-32

+-40-0 60 /20 180 240 300~ dcg 3

~GUBE 6.-Frssme m&kl@nt an smfecaof aylhder for mmpremible flowwItb drmdat[on.

h the same mannerJa=& J

r r~~ ~R~ sin mt ldl?P Sinme. 2m (m+82)@~-

F rom equ at ions (37), (38), (39), a ndequa tion (29), for th e ca se in which

f(r , e) =~~ ,

ia consequentlySillme

((82) m

4=m(7?a-8+2) (m+82) P 1fx when m+2#a

when m+2=8F or t he ot her ca ses t he in tegr at ion pr ocess follows

(40), t he solu tion of he -e mcedme&s AERONAUTICAL LABORATORY,

NATIONAL AwnsoRY C!omwrrrm FOR AERONAUTICSMOFFETT FIELD,CALIF.


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COMPRESS IBLE POTENTIAL FLOW WITH CIRCU IJATION ABOUT A CIRCULAR G _YLINDER 13REFERENCES

, Ghmer t , H.: The Effect of Compressibility on . the Ltit of anAer ofoiL R. & M. No. 1135, Br it ish A. R. C., 1927.

. von K6r m6n , Th.: Com pr ewibilit y IMects in Aer odyn am ics.J ou r . Aer o. SCL, vol. 8, n o. 9, J u ly 1941, p p. 337-366.

. J an zen , O.: Beit ra g zu ein er Th eor ie der st at ion am n St rom un g Kompr ea eibler F lu ea igk eit en . P hys. Zeit ich r., 14 (1913), pp.630-643.. Lord R.ayleigh : On the Flow of Compressible Flu ids Past anObstaole. Phil. Msg. 32, J uly 1916, pp. 44; or Soi. Papers,VOL ~, p. 402.

. Ka pla n, Ca rl: Com pr edble F low a bou t Symmet rica l J ou kowsk iP roffles. NACA Rep. N o. 621, 1938.Kap la n , Car l: Two-D imen s ion a l Subwmio Compre ss ib le F low Pa stE llip tio Cylin dem . NACA Rep . N o. 624, 1938.

, H ook er , S. G.: Two-Dim en sion al F low of Com pr wible F lu ids a tSub-Sonio Speedsaat Ellipt io Cylin der s. R. & M. No. 1684,Br it ish A. IL C. 1936.

. Imrd, I.: On the Flow of a Com prmsible F lu id P xt a Cir cu la rCylin de r. P r oo. P h ys . Ma th . S oo. J a pa n, 20 (1938) p p. 636-646.

9. Tamada, K. and Sa@ Y.: Note on the Flow of a Comp re sa fbF lu id P ast a Cir ou la r Cylin der . P roo. P hys. Ma th . Sot . J apa21 (1939), pp. 402-409.10. P oggi, L.: Campo d i velooit a in u na COr rer h pia na di flu ido oop reedble, LAer ot eon iq VOL 12, 1932, p. 1579. P ar te II, Ca

dei p roi ili ot t ehu t i con r ep r es en t a zi on e con forme dal ce r ch io edp ar t icola r e d ei p rofili J ou k ows ki. LAer ot eon ica , I bid . vol.1 934, p . 532.11. von K&rm6n , Th .: Th e E ngin eer Gr app lex wit h N on lin ea r P rolem s. Bu ll. Am . Ma th . Soo, vol. 46, n o. 8, Au g. 1940, pp. 61683. .12. Tsien , Hsue-Sh en: Two-Dh nen eior ud Su bwnio F low of Compressible Fluids.J ou r. Aer o. Sci., vol. 6, no. 10, Aug. 193pp. 399-407.

13. Tam ada K.: Applica t ion of t he H odogr aph Met hod t o t he F loof a Compr essible F lu id P ast a Cir ou la r Cylin der . P roo. P hyMa th . S ot . J a pa n, 22, 1940, p p. 207219.

14. Taylor , G. I. and Sharman, C. F .: A Mechanical Method fSolving Problems of F low in Compressible F luids. R. &N o. 1195, Br it ish A. R. C. 1928.

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compressible potential flow with circulation about a circular cylinder

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