8458

F.M.777

RE 1RICTED.

FLUID MOTION EANEL

AERONAUTICA:' RESEARCH COMlJITTEE

Ideal Dreg Due to e Shock Wave. Part II-By-

C. N. H. Lock, M.A.,o£ the Aerodynamics Division, N.P.L .

•

8458

F.M.777

Summery

19th February, 1945.

The method given in 5852 3 o~ calculating a £irst epproximBtinnto the ideal theoretical dreg rise due to a shock wave on en aeroioilis extended to cover the use of the Karmi~sien solution in place ofthe Glauert relation. It is shown how the drag rise can be calculatedfrom the theoretical critical Mach number and the geometrical curvatureof the eurfRce at the point of maximum suction. In pnrticular for twoBerafoila having the SBme criticnl Mnch number the drag rise isproportionnl to the rndiuB of curvnture; thus the drog rise on 017'30/0 ellipse ot zero incidence will be 3 times thot for anN.A.C.A. 0012 section hAVing the snme criticnl Mnch number.

Compnrison with experiment in the N.P.L. 20high speed tunnel sh~.s thet for 0 number of aerofoiltheoretical rise occurs consistently from M = 0".thnn the observed rise.

in. X 8 in.shapes theto 0'13 later

exceedsa "

The .rreBent °li:.e-thod is S"i.mJile pnd should giye ,"It lanEtt nbetter indicAtion of the relnti T"e merits of different aerofoil smpe'Sthan a knowledge of the theoroticol criticel epeed elone.

In 5958~ nnd 5852 2 fl method wne ndv~~ced of cnlculntingthe dreg of nn nerofoil arising from the incrense of entropy througha limited shock wove, on the basis of same rather crude assumptions.In outline these assumptions were os follows.

1. The shock wove storta from the point of mDximum velocityon the upper surface ae colculnted for incompressible flow nndfollows the course of the low speed equipotsntiol through thia point.

2. The velocity just upstrenm of the shock wove is deducedfrom the low Mach number potentinl field by menns of one of thestcndard ossumptions, (in the previous reports 1,2, by the Glnuertrelntion OB applied by Jacobs nnd nlternntively in the presentreport by the Knrmnn-Tsien relntion).

3. The shock wove occurs whenever the maximum velocitythe locel volocity of sound ond ends ot the point ot which Mas deduced from nssumption 2,

Calculations were ~~de in 5958 1 on the bosis of the nboveassumptions for two elliptic cylinders r.t zero incidence nnd for nnnerofoil shr.pe at two values of incidence. These cnlculntions involveno further simplifying ossumptions.

In!IA I REPORT

- <

In 5852" the increase of CD due to the shock wave weetreated BS an expansion in po8itive powers of (Mo - Meo )' where

Mo is the actual free stream Mach number and Meo is the free streamMach number at which the velocity of sound is first attained on thesurface. It was shown that the first term of the expansion involvesthe fourth power of (M - M ). The value of the coefficient K inthe relation 0 co

was evaluated in a general manner ae 8 function of M Bnd ~ where~ is the coefficient of ~ in the expansion co

= ••• ( 2 )

•p (= -p/ipV") is the suction coefficient in potential flow at lowMfi~h number at a point at distsnce ~ from the surfece of the bodyalong the equipotential line througn the point N of msximumvelocity, PcLN being the value of PeL at this point. Thus aknowledge of the values of Meo and K gives B first ap~roximetion

to the entropy dreg.

The numerical value of K was calculAted for the generalcBse of the elliptic cylinder nt zero incidence.

expressedcurvature

It has since beenne n function ofof the surfoce nt

noticed that the value of ~ cDn beMco(or PcLN) nnd of the geometricslthe point N, by the formuln

C£c = (2c/R)(' + PCLN). • •• (3)

Thie i8 demonstrnted in Appendix A, or it could be deduced from therelation between centrifugnl force nnd curvnture in n curved streom.It would therefore be a simple mntter to determine K for nnyshope or incidence for which the thooreticnl vnlue of M (or p )ie known, provided that the shnpe cnn be expressed es A F2nsonnb£tNsimple olgebrnicnl formuln (or r rough vnlue ann be obtnined fro~

the plotted shope). The method of Ref. 1 on the other hond, is toolaborious for exte~sive USB.

The opportunity hee olso been token of genernlising theformulAe slightly, so os to cover tho type of relntion between, ~

compressible and incompressible suction coefficient given by KarmPnAnd Teien. Thie generolisntion i.e ronde in Appendix B which oleo contninSthe explicit results r.riBin& both from the GIAuert-Jncobe relation,os in Ref. 2 p~d from the Kormnn-TSien relation.

In Ref. 2 the relAtion between compressible suctioncoefficient p nnd incompressible coefficient PeL wos Assumedto be of the f8rm

where M is the MOch number of the free stream. This hAS beengenernliQed into the form

which covers the

Pc- - .Kormnn...JI's~en

~ <l>(pcL'Mo )

relntion.It!

It is shown in 5852 2 that the formula for the coefficientK in terms of ~ and Meo (for B g1ven form of relation betweenDcLN end Mc ) ie of the form a: cK ~ Function of M only,Velues of a:gf heve been tebulated in Table 1b both forOths Gleuertand for the Karmen relation; the former supersedes Table 1 at 5852which is not very accurate; the corresponding relatias between P LNand Meo ere given in Table 1B. From Table 1b, values of K Ccen be ded~ced when PcLN end ~ ere known, the latter beinedsducsd from the geometr~cel curveture by uss of equation (A8) ofApDendix A,

Additional arguments in favour of the use of (1) ere:-

1. The basic BssUI!lptions become les8 end less justifiable aeMo - Mco increesss from zsro, so that it ie doubtful whsther it ieworth whils to go beyond e first order thsory.

2, Fig. 1 (re~oduced from Fig. 2 of Ref. 1) shows that fortwo elliptic oylindsrs et zero incidsnce, ths first epproximatioiof Ref. 2 egrees reeeonably well with ths celculations of Rsf. 1 •

In Teble 2 ere given veluee of Mco end K celculated bythe formulae of the Appendices using the Karman relation for anumber of typical aerofoils; values of Meo rnnge from 0.65 to 0.8.Curves of ~cD calculated for A selection of these aerofoils, byli'ormule 1 ere givsn in Fig. 2e. ,The velua of lIcn is taken Re thesum of the vAlues for the upper ~nd lower surfoce~ calculatedeeperatsly.

An interesting comparieon is thnt between N.A.C.A. 0020and the 20 per cent ellipse, both nt zero incidence. The vnlue otMco for the latter, exceeds that for the former by 0'058 but thed1££eronco o£ the vnluos of Meo £or n 60 o£ 0·02 i9 only 0·023;thus the edvantage of the ellipsa gets smnl~er ae the dreg rieee.Although this comparison does not include the effect of breakaway,it may help to explain the rather disappointing reeults of attempteto reduce shock stalling effects by improving the shape of n seotionleaving the thickness unaltered.

TobIe 2 olso illu8trr.te~ the foot thot for two nerofoilB(symmetricol and at zero incidenco) of different shape and thicknesehRving the eome criticnl speed, the ratio of the drng rises is equalto the ratio of the redii of curvatura. Thus the drag rise for a17-' per cent ellipse ie , timee that for the 0012 section whichhns the snme criticRl MAch number.

In Fig. 2b some of the calculated values of 6cn are

compered with values of profile drag measured in the N.P.L. rectangularhigh speed tunnel (detaile ere given in Appendix 0), In all caseethe theoreticel velue of lIcD has been increaeed by the opproximateamount of the observed VAlue below the shock etall.

It is remorkable that the difference between the calculatedideol drng rise ~nd the rise observed in the tunnel con be representedby B change of Moch number which lies between 0·10 end 0'13 for tourvery diverse coses. Apnrt from errors in the theoretical curvesdue to the npprox~~tions employed, this difference hoe been commonlyattributed to a brenk nWDy of the boundnrylDyer cnueed by theshock wave. Of other possible cnuses of discrepnncy, tunnelinterference ~hould be smoll, since the flexible wolle of therectangular tunnel were ehnped to ngree with free streomlines bythe method of 807",

The/

*The Glouert rel~tion is used in both enses.

- 4 -

The curve of the flight results on N.A.C.A. 2218 ('Tornado'),derived from 59908 , though of very limited extent, might suggest thatthe break Away in flight is about one half of that occurring on themodel; similar results of diving tests on the 'Mustang' and 'Spitfire'will be of the greatest interest.

Humidity is another possible factor tending to give aepuriously high drag coefficient in the N.P.L. tunnels which mightaccount for a pert of the discrepancy.

In e recent report, 8286, I,10naet'J.[ln and Fowler have calculatedthe dreg corresponding to the irreversible loss thro~gh a shock waveon the basis of measured static pressures near the shock wave on anaerofoil in the N.P.L. rectangular high speed tunnel. In the caaaconsidered the corresponding drag vAlues based on the assumptionsof the present peper ~ere seriously in error so that at a Mach numberelightly more than 0'1 above the critical the riee of drag coefficient8S celculsted by the formulae of the present paper occurs at e Machnumber about 0·05 higher than the same value BS besed on observedstatic pressures, and the latter again at about 0·05 higher than the68me value 88 measured in the tunnel by pitot ~raverse. The extremedifference of 0'1 agreee with the other caees illustrated in Fig. 2.

I~ it were legitimate to argue from A cross comparison oftwo different aerofoile (0 modern ehope 14'5 per cent thick of8286 6

nnd N.A.C.A. 2218 of Fig. ,) it might bo Auggeeted thot there ie noapprecinble brenk owey in flight, the drng Rgreeing with thatcalculated from the preseures nenr the shock vmve on the model.

In gener.. l the reeults eugbest thnt the preeent method maygive a better indication of the relntive merits of different Rerafoilsections thp~ 1s given by the theoreticol criticnl spoed nbove Andthat it would be worth while to undertn~ the smRll extr.. labour ofcalculating the foetor K whenever the theoretical critical speedie colculo ted.

AppendiX A

rAterminAtion of the coefficient « o~ eQuntion (2)in terms of the geonetricnl curvnture.

Toke as origin 0 the point of mnximum suction on theeurfnce of n body in two dimensions (incompressible fluid); takenxea Ox tnngentinl to the aurfnce downstrcnm nnd oy normnl intothe fluid. Then eince the totnl velocity ~ ie stotionnry nt theorigin .and pnrnllel to the Dxis of x, we cnn write,

anO so

where

u- iv ~ qo + ib:1 z + o( r·) •.• (A 1)

'"~ q y + tb.a,.(x· - y') + o( r') , •• (A2).

0

U = qo b, y + oCr") ... (A,)

If R ie the rndiue of curvnture of(which coincides with the streamline(A2) thnt

the eurfnce nt the originy , 0), it follows from

R/

- 5

and therefore by (,),

Since

(au) = _ qa!R.ay (y=o)

• .• (A 4 )

· •• (A5 )

• .• (A6)

it follows that -the value of " in equation (2) ie .e.,iven by

" = _(a::L) et 0 = -t:;L) at 0 · •. (A7)

2qo (aq) 2qo (au)= - u2 ay at 0 = -- - Rt r')

U· ay

= 2QJ'/(RU')

---;

"c = (2C/R) (1 + PcLN). • .. (Ae)

Appendix B

Slight "gone-rnlisrtion 'of' 'tho forl''lUlne. o:f Ru:f. 2 so t'El toCOV07 ~he K~rmrin-T8ien ~s Toll ~s the ~l~ue~~co~ rel~tion.

In 5852 2 and 59581, the pre9~ure P1the shock wave is deduced from the presgure atpoint in incompressible flow on the besie of aknown) of tha form

where

just in front ofthe correspondingrelation (assumed

=P - Po

tp V·a

,

are auction coefficients at corresponding points in compressible endincompre~$ible flow respectively, and Mo is the Mach number of thefree stream. This covers the use of the Glauert-Jacobs relation

whereby the

~ =more

(10) will here be replaced

which covers the use of the

Pc = ~(PCL' MO ), ,

Karmen-Ts1en relation

· .. (B1 )

· .• (B2)~ Pc *

1 + t(1 - ~)pc

If p~ is the pres~ure just in front of a shock wave(ss$umed normal to the local streamlines) and M~ the Mach numberat the same point, it follows from the ordinary adiabatic relationthet

_tl.1 + t(y - l)M~ = (1 + i-{y - 1) M6'1 (P'/Po) Y ..• (5852)(n)

while it follows from (Bl) on putting tpoV.

thet

=

· .. (B3 )

which tekes the plece of (5852) (12). At the point of IT3ximum Isuction N (et which PeL = PcLN) end at the free stream Uachnumber (Mco ) et which the velocity of Bound is first ettained, e

relation between Mea and PcL:T can be deduced by putting M1 = 1in (5852) (n) end eliminating' P,/Po between (5852) (3) and (B3).

· .. (B4)1 )M~o

The formulae of (5852) .Appendix 1 are then modified eefollows. Writing es before ~1M end ~1P for the result of

differentieting ths vslue of M1 dsduc~ble from (5852) (13) end (B3)with'rsspect to Mo end PeL rcspecti,ely end then futting M1 = 1,M' 't:; M ,p L = P LII' after differentiation, we findo co c c

~1M = t(y + 1)Mco {1 + t(y1

where

• .. (B5)

end

=1

4(y + ... (B6)

where ~M and ~P

and PeL = PcIJIThe formule for K

stand for the result of

in a~/aMo end a~/apcL

then becomes

putting Mo = MeAfter differentiction.

1)(P /p )(3Y+1)/2Y r r2(y + 1 Po1 0 •«cK = + - . ~f;

3Mco ~p l1 + t(y - 1)M" P1co

... (B7)°nl

*This formula is usually quoted L~ ter~s of pressure coef~icients inplace of suction coefficients; t~e pl~s sign in the denomir.ator thenbecomes Po minus.

(Bl) we

, ,On substituting the Karmen--Tsien relation (Ji}) for

have

4'M = ~--. {1 + ~l 4'11co

4'6 = 1t""24'{ 1 - !M~o(' - ~ll

~p = ~(4'!PCLN)2.

For the G~acob8 relation,

4'M = ~-. M ~c

4'6 = (1 -!M2 ) ~-.~co

4'p = ~-'.

Appendix C

Notes on Fig, 2b. experimental curves, Drag c~efficipnt

by bitot traverse method.

Muetang Section: 5 in. chord model L~ 20 in. X 8 in. high speedtunnel at N.P.L. A.R.C.8135".

N.A.C.A. 2218 (ITornAdo') 5 in. chord model in 20 i!l. X 8 in. ·~unnel.A.R.C. 666,7.

N.A.C.A. 22'8 ('Tornado') Flight experiment. A.R.C.5990B •

E.C. 1250 5 in. chord model. in 20 in. aIlS 8 in. tunr..cl. :ro"." yet issued g.

(Compared with theoretical curve for '2 par cent ellipoe).

N.A. C.A. 0020 2 in. chord20 in. X 8 in. tunnel. Not yet iB~uedo.Circular h4It speed tunnel. 4709'0.5 in. X 2 in. tunnel. 7153".

All experimental curves ere subject to an unkno".;n correction(in the direotion of increased Mach number for given Cn ), for theeffect of condensation of moisture. This will be less Tor the5 in. x 2 in. tunnel and mey in part account for the diRcrepencybetween this tunnel and the other two for the N.A.C.A. 0020 scction.

Critical Mach numbers ere indicated by errow8~ values for'Mustang' are derived from pressure observation; the re~ind~~ nrccalculated.

Refercnces!

Owen, P. R.Waves.

- 8 -

References

and Young, A I D.: Note onR.A.E. Report Aero 1749.

the Drag Effect of Shock5958.

2. Lock, C. N. H.: The Ideal Dreg due to a Shock WAve. 5852., ,

,. Karman, Th.von,: Compressibility Effects ~ Aerodynamics.From I'Journal of the Aeronautical Sciences". July, 1941.5" 4.

4. Lock, C. N. H. and Beavan, J. A.: Tunnel Interference atCompressibility Speeds using the Flexible Walle of theRectangular High Speed Tunnel. 8073.

5 • Monaghan, R. J. and Fowler, R.Experimental Ieta, of theR.A.E. Report Aero 1987.

G.: An Estimate, Based onIdeai Drag of a Shock Wave.8286.

6. Thomp8on, J. S., Markowicz, Mo, Beavan, J. A. and F~wler, R. G.:Pressure Distribution and Wake Traverses on Models of'Mustang' Wing Section in the R.A.E. and N.P.L. High SpeedTunnele. R.A.E. Report Aero 1948. 8'35.

7. Peercey, H. H.: Dreg Measurements on N.A.C.A. 2218 Section atCompressibility Speede for Comparison with Flight Testa andTheory. 666'.

8. Metr, W. A.: Flight Measurements or Profile Drag at Hi.gh Speede.R.A.E. Report No. Aero '767. 5990.

9. Beavan, J. A.: Note on rise of Dreg above the Critical MachNumber. (Unpublished).

'0. Lock, C. N. H., Hilton, W. F. And Goldstein, S.: Determinationof Profile Drag at High Speeds by a Pitot TrAverse Method.4709.

11, Fage, A. end SArgent, R. F.: Effect on Aerofoil Drng ofBoundnry Lnyer Suction behind n Shock Wove. 7153.

Table "'/

- 9 -

Table 1 a

Values of Moo and pcUl

, ,Karman

•

Glauert------ij----------------------lie, I PcLN PcLN0.50 I 1 .6170 1 .84760.55 1.2184 1.38490.60 0.91680 1.035460.625 0.79300 0.892440.65 0.68365 0.766410.675 0.58675 0.655040.70 0.50062 0.55636~.725 0.42391 0.468810·75 0.35546 0.391030.775 0.29440 0·322000.80 0.23993 0.260780.85 0.~44B7 0.15908------~-----------------------

Tabl.e 1b

Values a.cK for given MKarman

--------------------------------------------------------------------------------

- 10 -

Table 1b Continued

Glauert__________~_______________ G'- tM ' ---------------.-- ,u1uerI log "oK I "oK 1 Diff.' ----T-----------------------

_________________~_________ 1M, log "oK '''oJ( +Diff

0.50 ~:~~~~~ ! ~~:~~ -~;~~-r~~675-'--~:68;;;;--~-48:~;;-- -----:2.75865 573.65 52.l,.9 I 1.65l,.17 l,.5.1oo 3·0322.72101 526.03 l,.7.62 1.62612 l,.2.279 2.8212.6837l,. l,.82.77 l,.3.26 1,.59826 39.652 2.627

0.525 2.6l,.68J,. l,.l,.3.l,.5 39.32 1.57059 37.20l,. 2.l,.l,.82.61030 l,.O7.66 35.79 0.700 1.5l,.309 3l,.·921 2.2832.57l,.13 375.09 32.57 1.51578 32.793 2.1282.53830 3l,.5.38 29.71 : 1.l,.8865 30.807 1.9862.50282 318.29 27.09 I 1.l,.6170 28·953 1.85l,.

0.550 2.l,.6767 293.5l,. 2l,..75 1 .l,.3l,.92 27·222 1 ·7312.l,.3285 270.,3 22.61 0.725 I' 1o,l,.0830 25.60l,. 1.6182.39836 250.2l,. 20.69 1.3818J,. 2l,..090 1 ·51l,.2.36l,.18 231.30 18.9l,. 1·35553 22.67l,. 1.l,.162.33033 213.96 17.3l,. 1.32938 21 ·3l,.9 1 ·325

0.575 2.29677 198.05 15·91 1.30337 20.108 1.2l,.12.26352 183.l,.5 1l,..60 0·750 1.277l,.9 18.9l,.5 1.1632.23056 170.0l,. 13.l,.1 1 .25176 17 .855 1 .0902.19790 157.73 12.31 1.22615 16.833 1.0222.1655l,. 1l,.6.l,.0 11.33 1.20066 15·873 0.960

0.600 2.133l,.5 135.97 10.l,.3 0.775 1.17528 1l,.·972 0·9012.10166 126.37 9.60 1.15001 1l,..126 0.8l,.62.07011 117.52 8.85 1.12l,.85 130331 0.7952.03879 109.3l,. 8.18 1.09977 12·583 0.7482.00777 ,101.81 7.53 1.07l,.78 11.879 0.70l,.

0.625 1.97702 9l,..8J,.7 6.963 1 ·0l,.987 11.217 0.662'.9l,.651 88.l,.12 6.l,.35 0.800 1.02503 10.593 0.62l,.h9162l,. 82.l,.60 5.952 1.00025 10.006 0.5871.88621 76.951 5.509 0.97553 9·l,.523 0·55371.856l,.2 71.8J,.9 5.102 0·95085 8·9300 0·5223

0.650 1.82688 67.125 l,.·72l,. I 0.92620 8·l,.373 0·l,.9271.79755 62.7l,.1 l,..38l,. 0.825 0.90156 7.9720 0.l,.6531.768l,.6 58.676 l,..065 I 0.87693 7 ·532l,. 0.l,.3961.73957 5l,..9OO 3.776 I 0.85231 7.1173 I 0.l,.1511.71090 51.393 3.507 0.82770 6.7252 I 0·3921

0.675 I 1.682l,.3 48.132 3.261 I 0.850 I 0.80309 6.35l,.7 003705_____J c ~~~~~~__~ ~~~~_~ __:~~~::_.

TabJa 2/

- 11 -

Ta.ble 2

Ideal Drag Rise SUJUI:18l"Y

K

7·95

25·85

11 .88

2.83

8.25

5.56

1 .93

2q! Ellir-se

29.0il Ellipse

N.A.C.A. 0020

N.A.C.A. 0012

1;:lb Ellipse

-------------------------------------------------------------,------------------, at.· I

. Inci- 1 PeLN I xc'denoe = -Cpe - = Mbc

~:-~ I ra~5~;- -~~~--~-;~~-- ~.697(1) -;~~~-- -~;.2;-o· LO.256 0.l,.6 O.l,.- 0.792(1) 1.005 12.18

Or. [0.763 0.161 2.3l,. 0.632(2)

0.1 LO.620 0.0361 741+5 0.666(2) 2l,..12

O· 0.25l,.l,. 0.5 0.2l,. 0.793(2) 0.602

O· 0.l,.l,.0 0.5 0.l,.0 0.720(2)

O· 0.663 0.151 1.67 0.655(2)

O· 0.663 0.5 0.58 0.655(2)

O· 00375 0.151 1.030 0.7l,.2(2)

17." Ellipse i O· I 0.3751 0.5 • 0.3l,.5 0.7l,.2(2) 0·9l,.9 23·7

--------------------------------------------------------------------------------,

WApproximate values by interpolation.(1)From observed pressures.(2)By calculation.

All.

.

•

40% thick 12% !:.hick•",mp6e. cdlipoe.

.

-

Mf~(4C%V Mf(l?%U

0·07

0·01

oo·!> o-~ 0·7 M 08

Maeh number_0-9 J-O

Tnc<o~c ..J VOIri ..!:ion of Qr"2'9 incrClmene d".. 1:0 ..node

waves wieh Mach nurnbu- For I',. !:hie-I. ana 40% !:.hick e1li~.

Calc"l",eions of 5'358' .~ Cl e Fir-.,c appr-oK;rTlacion of 5852~

f.E.M.

FIG.7bFiC1. '1a

_1- . I 1 ;; IN.".C.".2218 CL 0·/ -<>-1 c

~ IN.".C.A.OO20 O· --+--, C I:20% EIBp&e o· -+-/ ~ c c.t;' 'il I

Mu.~ 5ccQon O!..-*-, .d. I12% Ellipse: O'

~'0 l ,t

!/ I x .'"I c ~. / , .£ ~ ~I c , II .il I 'Ii / ~I U~ ,, I.£ I .£ , I2 '0

+ X I '" I I/ I ~ I 'l""'C ... I f'.C.1250 I N.A.cAJI , :g I 221S ;;h (12". Elli~1 0020I I

,T.".,,,d~ r;Iv~UOUnq I N /

I x 2 &czee;~, : ,, I I l 'i / 1/

~/ /( I'Ii / i / I

I~~I I // .S!' I I

t : .~. I Ii::' I -_/i !<l I I f-

1 x/ I _/I f I I

I I / is t.s. I, ./ ,- II --- t I• I l

,S. l.S. _/

!1/V)~/~ 0·' 07 O·S.. ..7 & 0·' 0·' 0" 0·7 O'S 0·' 0·7 0·8 0", 0·7 O·S 0·'o

O~,

0-0

0·02

lie

~

~

~s. LiS. Me.

Ideal '*''''3 ,...115e (F,;"Sc. apP",oJCjrn~bOn)EJlCperjrnet")cal (pil:::cC Cravqrse).